Repository navigation
Expand file tree
/
Copy pathheat-transfer.html
More file actions
384 lines (327 loc) · 174 KB
/
Copy pathheat-transfer.html
File metadata and controls
384 lines (327 loc) · 174 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
<!doctype html>
<!-- Generated from content/heat-transfer/ by the local content build. Edit the Markdown source. -->
<html lang="en">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1">
<meta name="description" content="ES1 course reader, Chapter 07: steady-state heat transfer through the building envelope. Heat and temperature, conduction, convection and radiation, surface resistances, U-values and temperature profiles, air layers, thermal bridges, insulation materials and when transient effects matter. Metric, with US and German conventions.">
<meta name="author" content="Timur Dogan">
<meta name="theme-color" content="#ffffff">
<title>Heat transfer · ES1 Course Reader</title>
<link rel="icon" href="./assets/course-logo.png" type="image/png">
<script src="./src/theme.js"></script>
<link rel="stylesheet" href="./styles.css">
<link rel="stylesheet" href="./theme.css">
<link rel="stylesheet" href="./solar.css">
<link rel="stylesheet" href="./heat.css">
<script defer src="./assets/heat.bundle.js"></script>
</head>
<body>
<a class="skip-link" href="#main">Skip to chapter</a>
<div class="mobile-bar"><a href="#top" class="wordmark"><img class="course-logo" src="./assets/course-logo.png" alt="ES1" width="372" height="123"><span>Course reader</span></a><div class="mobile-actions"><button type="button" class="theme-toggle" data-theme-toggle aria-label="Switch to dark mode" aria-pressed="false"><svg viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="1.6" aria-hidden="true"><circle cx="12" cy="12" r="8"/><path d="M12 4a8 8 0 0 1 0 16Z" fill="currentColor" stroke="none"/></svg></button><button id="menu-toggle" aria-expanded="false" aria-controls="sidebar">Contents <span aria-hidden="true">☰</span></button></div></div>
<aside class="sidebar" id="sidebar" aria-label="Chapter navigation">
<a class="brand" href="#top"><img class="course-logo" src="./assets/course-logo.png" alt="ES1" width="372" height="123"><span class="brand-sub">ES1 / COURSE READER</span></a>
<p class="sidebar-intro">Environmental systems<br>as a design material.</p>
<div class="chapter-label">THE CHAPTERS</div><div class="course-chapters"><a href="./site-climate.html"><span>01</span>Site + climate</a><a href="./index.html"><span>02</span>Solar geometry</a><a href="./shadow-drawing.html"><span>03</span>Drawing shadows</a><a href="./sun-paths-shading.html"><span>04</span>Sun paths and shading</a><a href="./solar-radiation.html"><span>05</span>Solar radiation</a><a href="./hdr-photography.html"><span>06</span>HDR photography</a><a href="./heat-transfer.html" aria-current="page"><span>07</span>Heat transfer</a><a href="./heat-balance.html"><span>08</span>Heat balance</a><a href="./psychrometrics.html"><span>09</span>Psychrometrics</a></div>
<nav aria-label="On this page">
<a href="#symbols" class="active"><span>00</span>Quantities and symbols</a>
<a href="#heat"><span>01</span>Heat and temperature</a>
<a href="#modes"><span>02</span>Three modes</a>
<a href="#conduction"><span>03</span>Conduction</a>
<a href="#surfaces"><span>04</span>Surfaces</a>
<a href="#uvalue"><span>05</span>U-value, profile</a>
<a href="#airlayers"><span>06</span>Air layers</a>
<a href="#bridges"><span>07</span>Thermal bridges</a>
<a href="#insulation"><span>08</span>Insulation materials</a>
<a href="#transient"><span>09</span>Steady vs transient</a>
<a href="#quiz"><span>10</span>Self-test</a>
</nav>
<div class="sidebar-bottom"><button type="button" class="theme-toggle" data-theme-toggle aria-label="Switch to dark mode" aria-pressed="false"><span data-theme-label>Dark mode</span><svg viewBox="0 0 24 24" fill="none" stroke="currentColor" stroke-width="1.6" aria-hidden="true"><circle cx="12" cy="12" r="8"/><path d="M12 4a8 8 0 0 1 0 16Z" fill="currentColor" stroke="none"/></svg></button><div class="reading-label"><span>Reading position</span><span id="progress-number">0%</span></div><div class="progress-track"><div id="reading-progress"></div></div><a href="./index.html#author">About the instructor <span aria-hidden="true">↗</span></a><a href="#figures">Figure files (SVG) <span aria-hidden="true">↗</span></a><a href="#references">Sources and model notes <span aria-hidden="true">↗</span></a><p>Environmental Systems I<br>Cornell AAP</p></div>
</aside>
<main id="main">
<header class="hero" id="top">
<div class="topline"><span>ENVIRONMENTAL SYSTEMS I</span><span>CONDUCTION · CONVECTION · RADIATION · U-VALUE · STEADY STATE</span></div>
<div class="hero-heading"><div><p class="eyebrow">CHAPTER 07</p><h1>Heat transfer through the envelope<span class="title-dot">.</span></h1></div></div>
<p class="hero-lead">How heat moves through walls, roofs and windows, and how a few layers of material decide how much energy a building needs and how warm its surfaces feel. The chapter stays with steady state: temperatures that do not change with time. Section 09 explains when that is not good enough.</p>
<div class="hero-bottom"><p>All units are metric (SI). Symbols follow ISO 7345 and ISO 6946, as used in DIN 4108 and the <em>Lehrbuch der Bauphysik</em>; the US terms of ASHRAE and Incropera are listed beside them. Equations are numbered 7.1–7.13. <a href="./heat-balance.html">Chapter 08</a> adds the flows up to a building heat balance.</p>
<a href="#symbols" class="text-link">Start <span aria-hidden="true">↓</span></a>
</div>
</header>
<noscript><p class="notice">The text, equations, tables and figures work without JavaScript. Enable JavaScript for the interactive drawings and the self-test.</p></noscript>
<section id="symbols" class="reader-section">
<div class="section-heading"><span class="section-number">00</span><div><p class="eyebrow">ISO 7345 · ENGLISH AND GERMAN TERMS</p><h2>Quantities and symbols</h2></div></div>
<p class="section-lead">Building physics in Germany and in the US uses the same physics with different words and letters. This reader uses the ISO symbols. Keep three quantities apart: energy Q, a flow of energy per time Φ, and that flow per area q.</p>
<div class="table-wrap"><table class="nomenclature heat-table"><caption>Symbols used in Chapters 07 and 08</caption><thead><tr><th scope="col">Symbol</th><th scope="col">Quantity</th><th scope="col">German (DIN)</th><th scope="col">Unit</th><th scope="col">Also written</th></tr></thead><tbody>
<tr><th scope="row">Q</th><td>Heat, energy</td><td>Wärmemenge</td><td>J, kWh</td><td>E (lecture)</td></tr>
<tr><th scope="row">Φ</th><td>Heat flow rate</td><td>Wärmestrom</td><td>W = J/s</td><td>Q (lecture, Szokolay), q (Incropera), Q̇</td></tr>
<tr><th scope="row">q</th><td>Heat flux, heat flow density</td><td>Wärmestromdichte</td><td>W/m²</td><td>q″ (Incropera)</td></tr>
<tr><th scope="row">θ, T</th><td>Temperature (Celsius, kelvin)</td><td>Temperatur</td><td>°C, K</td><td>T in °C (lecture), t (ASHRAE)</td></tr>
<tr><th scope="row">λ</th><td>Thermal conductivity</td><td>Wärmeleitfähigkeit</td><td>W/(m·K)</td><td>k (Incropera, ASHRAE)</td></tr>
<tr><th scope="row">d</th><td>Thickness</td><td>Dicke</td><td>m</td><td>b (Szokolay), L (Incropera)</td></tr>
<tr><th scope="row">R</th><td>Thermal resistance of a layer</td><td>Wärmedurchlasswiderstand</td><td>m²K/W</td><td>RSI; US labels use ft²·°F·h/Btu</td></tr>
<tr><th scope="row">h<sub>c</sub>, h<sub>r</sub></th><td>Convective, radiative heat transfer coefficient</td><td>Wärmeübergangskoeffizient</td><td>W/(m²K)</td><td>α (older German texts)</td></tr>
<tr><th scope="row">R<sub>si</sub>, R<sub>se</sub></th><td>Inside, outside surface resistance</td><td>Wärmeübergangswiderstand</td><td>m²K/W</td><td>surface film (ASHRAE)</td></tr>
<tr><th scope="row">R<sub>T</sub></th><td>Total thermal resistance, air to air</td><td>Wärmedurchgangswiderstand</td><td>m²K/W</td><td>R-value of the assembly</td></tr>
<tr><th scope="row">U</th><td>Thermal transmittance</td><td>Wärmedurchgangskoeffizient</td><td>W/(m²K)</td><td>k (older German texts), U-factor</td></tr>
<tr><th scope="row">c, ρ</th><td>Specific heat capacity, density</td><td>spezifische Wärmekapazität, Rohdichte</td><td>J/(kg·K), kg/m³</td><td>c<sub>p</sub></td></tr>
<tr><th scope="row">ε</th><td>Emissivity (long-wave)</td><td>Emissionsgrad</td><td>–</td><td></td></tr>
<tr><th scope="row">ψ, χ</th><td>Linear, point thermal transmittance (thermal bridges)</td><td>längen-, punktbezogener Wärmedurchgangskoeffizient</td><td>W/(m·K), W/K</td><td></td></tr>
<tr><th scope="row">f<sub>Rsi</sub></th><td>Temperature factor of the inside surface</td><td>Temperaturfaktor</td><td>–</td><td></td></tr>
<tr><th scope="row">α (a), b</th><td>Thermal diffusivity, thermal effusivity</td><td>Temperaturleitfähigkeit a, Wärmeeindringkoeffizient b</td><td>m²/s, J/(m²K·s<sup>½</sup>)</td><td>α (Incropera)</td></tr>
</tbody></table>
</div>
<div class="callout"><span>US LABELS</span>
<p class="measure">Product labels in the US give R-values in imperial units. Convert once and work in SI: <strong>R<sub>SI</sub> = R<sub>IP</sub> / 5.678</strong> and <strong>U<sub>SI</sub> = U<sub>IP</sub> × 5.678</strong>. An R-13 batt is 2.29 m²K/W; R-30 is 5.28 m²K/W. In Germany λ and U are given directly, for example λ = 0.035 W/(m·K) (formerly "WLG 035").</p>
</div>
</section>
<section id="heat" class="reader-section">
<div class="section-heading"><span class="section-number">01</span><div><p class="eyebrow">ENERGY · TEMPERATURE · EQ. 7.1</p><h2>Heat, temperature and specific heat</h2></div></div>
<p class="section-lead">Heat is energy in transit: molecular motion in matter, or electromagnetic radiation in space. Temperature is its symptom. Heat flows by itself only from higher to lower temperature, as water flows downhill.</p>
<dl class="defs">
<div><dt>Energy Q · J, kWh</dt><dd>1 J = 1 W·s. 1 kWh = 1000 W × 3600 s = 3.6 MJ. Heating fuel and electricity are billed in kWh (Btu in the US: 1 Btu = 1055 J = 0.293 Wh).</dd></div>
<div><dt>Power, heat flow rate Φ · W</dt><dd>Energy per time, 1 W = 1 J/s. A heater, a person (≈ 100 W) and a wall losing heat are all described by Φ.</dd></div>
<div><dt>Temperature θ · °C, T · K</dt><dd>Celsius: water freezes at 0 °C and boils at 100 °C at sea level. Kelvin starts at absolute zero: T = θ + 273.15. A temperature <em>difference</em> is written in K: 40 °C − 10 °C = 30 K. Radiation laws need T in K.</dd></div>
<div><dt>Specific heat c · J/(kg·K)</dt><dd>The heat needed to warm 1 kg by 1 K. Metals 400–900, masonry and concrete 800–1000, timber 1300–2100, air 1000, water 4180 J/(kg·K), the highest of all common substances.</dd></div>
</dl>
<div class="eq" id="eq-7-1"><span class="eq-math"><span class="eq-line"><span class="eq-label">sensible heat to change the temperature of a mass</span><math><mi>Q</mi><mo>=</mo><mi>m</mi><mo>·</mo><mi>c</mi><mo>·</mo><mi>Δ</mi><mi>θ</mi></math><span class="eq-note"><span class="eq-unit">J</span></span></span></span><span class="eq-n">(7.1)</span></div>
<div class="calc calc-wide"><div class="calc-row"><span class="calc-k">Example</span><span class="calc-e">0.5 kg of water at 20 °C in an 800 W kettle (efficiency 1.0): how long to boil?</span><span class="calc-v"></span></div><div class="calc-row"><span class="calc-k">Q</span><span class="calc-e">m · c · Δθ = 0.5 × 4176 × (100 − 20)</span><span class="calc-v">167,040 J</span></div><div class="calc-row calc-key"><span class="calc-k">t</span><span class="calc-e">Q / Φ = 167,040 J / 800 W</span><span class="calc-v">209 s ≈ 3.5 min</span></div></div>
<ul class="facts">
<li>Heating 1 m³ of air (1.2 kg) by 1 K takes 1.2 kJ; 1 m³ of water takes 4180 kJ, about 3500 times more. Air carries little heat; water and heavy materials store a lot. This returns in ventilation losses (Chapter 08) and thermal mass (section 09).</li>
<li>Latent heat, the energy of evaporation and condensation (about 2450 kJ per kg of water), belongs to psychrometrics and comfort; this chapter treats sensible heat only.</li>
</ul>
</section>
<section id="modes" class="reader-section">
<div class="section-heading"><span class="section-number">02</span><div><p class="eyebrow">CONDUCTION · CONVECTION · RADIATION</p><h2>Three modes of heat transfer</h2></div></div>
<p class="section-lead">Heat crosses a wall in three ways. Conduction carries it through solids and still fluids. Convection carries it between a surface and moving air. Radiation exchanges it between surfaces that see each other, with no medium needed.</p>
<!-- FIG:heat-modes_START -->
<figure class="fig" id="fig-heat-modes"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 760 400" class="diagram" role="img" aria-label="Conduction, convection and radiation in a wall section"><title>Conduction, convection and radiation in a wall section</title><defs><marker id="hm-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="hm-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="hm-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5"/></marker><marker id="hm-arrow-third" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5"/></marker><marker id="hm-arrow-muted" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="1.5"/></marker><marker id="hm-arrow-red" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5"/></marker><pattern id="hm-hatch-shadow" patternUnits="userSpaceOnUse" width="3.4" height="3.4" patternTransform="rotate(45)"><rect width="3.4" height="3.4" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="3.4" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="hm-hatch-shade" patternUnits="userSpaceOnUse" width="7.5" height="7.5" patternTransform="rotate(45)"><rect width="7.5" height="7.5" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="7.5" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="hm-hatch-screen" patternUnits="userSpaceOnUse" width="4.2" height="4.2" patternTransform="rotate(135)"><line x1="0" y1="0" x2="0" y2="4.2" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern></defs><g font-family="Helvetica Neue, Helvetica, Arial, sans-serif"><g id="hm-title"><text x="20" y="22" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">Winter · inside θi = 20 °C (left) · outside θe = 0 °C (right)</text></g><g id="hm-wall"><rect x="360" y="70" width="12" height="300" fill="light-dark(#ffffff, #202124)" stroke="none"/><circle cx="362" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="89" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="89" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="96" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="96" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="103" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="103" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="110" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="110" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="117" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="117" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="124" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="124" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="131" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="131" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="138" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="138" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="145" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="145" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="152" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="152" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="159" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="159" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="166" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="166" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="173" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="173" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="180" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="180" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="187" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="187" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="194" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="194" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="201" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="201" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="208" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="208" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="215" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="215" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="222" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="222" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="229" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="229" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="236" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="236" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="243" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="243" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="250" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="250" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="257" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="257" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="264" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="264" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="271" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="271" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="278" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="278" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="285" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="285" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="292" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="292" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="299" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="299" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="306" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="306" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="313" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="313" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="320" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="320" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="327" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="327" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="334" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="334" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="341" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="341" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="362" cy="348" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="367" cy="348" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="363.6" cy="355" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="368.6" cy="355" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="365.2" cy="362" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="370.2" cy="362" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><rect x="360" y="70" width="12" height="300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="372" y="70" width="60" height="300" fill="url(#hm-hatch-shade)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="372" y="70" width="60" height="300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="432" y="70" width="60" height="300" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M434.4,72 C427.5,74.6 496.5,95.4 489.6,98 C496.5,100.6 427.5,121.4 434.4,124 C427.5,126.6 496.5,147.4 489.6,150 C496.5,152.6 427.5,173.4 434.4,176 C427.5,178.6 496.5,199.4 489.6,202 C496.5,204.6 427.5,225.4 434.4,228 C427.5,230.6 496.5,251.4 489.6,254 C496.5,256.6 427.5,277.4 434.4,280 C427.5,282.6 496.5,303.4 489.6,306 C496.5,308.6 427.5,329.4 434.4,332 C427.5,334.6 496.5,355.4 489.6,358 C496.5,359 427.5,367 434.4,368" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="432" y="70" width="60" height="300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><polyline points="352,70 420,70 424,63 428,77 432,70 500,70" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><polyline points="352,370 420,370 424,363 428,377 432,370 500,370" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="426" y="400" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">plaster · masonry · insulation</text></g><g id="hm-conduction"><line x1="356" y1="180" x2="498" y2="180" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-alt)"/><line x1="356" y1="220" x2="498" y2="220" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-alt)"/><line x1="356" y1="260" x2="498" y2="260" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-alt)"/><text x="426" y="46" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="700">CONDUCTION</text><text x="426" y="62" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle">q = λ · Δθ / d</text></g><g id="hm-inside"><path d="M342,110 L342,330 Q342,350 320,350 L240,350 Q220,350 220,330 L220,130 Q220,110 240,110 L320,110" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-dasharray="6 4" stroke-linecap="round" stroke-linejoin="round"/><line x1="342" y1="200" x2="342" y2="240" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-alt)"/><line x1="220" y1="260" x2="220" y2="220" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-alt)"/><text x="58" y="206" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">FREE CONVECTION</text><text x="58" y="224" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">Air cools at the cold surface</text><text x="58" y="240" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">and sinks (downdraught).</text><text x="58" y="256" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">h_c ≈ 2.5 W/(m²K)</text><path d="M50.00,150.00 L51.87,152.56 L53.80,153.66 L55.81,152.72 L57.89,150.33 L59.97,147.94 L61.98,147.00 L63.91,148.11 L65.78,150.67 L67.65,153.22 L69.57,154.33 L71.59,153.39 L73.67,151.00 L75.74,148.61 L77.76,147.67 L79.69,148.78 L81.56,151.33 L83.42,153.89 L85.35,155.00 L87.37,154.06 L89.44,151.67 L91.52,149.28 L93.54,148.34 L95.47,149.44 L97.33,152.00 L99.20,154.56 L101.13,155.66 L103.15,154.72 L105.22,152.33 L107.30,149.94 L109.31,149.00 L111.24,150.11 L113.11,152.67 L114.98,155.22 L116.91,156.33 L118.92,155.39 L121.00,153.00 L123.08,150.61 L125.09,149.67 L127.02,150.78 L128.89,153.33 L130.76,155.89 L132.69,157.00 L134.70,156.06 L136.78,153.67 L138.85,151.28 L140.87,150.34 L142.80,151.44 L144.67,154.00 L146.53,156.56 L148.46,157.66 L150.48,156.72 L152.56,154.33 L154.63,151.94 L156.65,151.00 L158.58,152.11 L160.44,154.67 L162.31,157.22 L164.24,158.33 L166.26,157.39 L168.33,155.00 L170.41,152.61 L172.43,151.67 L174.35,152.78 L176.22,155.33 L178.09,157.89 L180.02,159.00 L182.03,158.06 L184.11,155.67 L186.19,153.28 L188.20,152.34 L190.13,153.44 L192.00,156.00 L193.87,158.56 L195.80,159.66 L197.81,158.72 L199.89,156.33 L201.97,153.94 L203.98,153.00 L205.91,154.11 L207.78,156.67 L209.65,159.22 L211.57,160.33 L213.59,159.39 L215.67,157.00 L217.74,154.61 L219.76,153.67 L221.69,154.78 L223.56,157.33 L225.42,159.89 L227.35,161.00 L229.37,160.06 L231.44,157.67 L233.52,155.28 L235.54,154.34 L237.47,155.44 L239.33,158.00 L241.20,160.56 L243.13,161.66 L245.15,160.72 L247.22,158.33 L249.30,155.94 L251.31,155.00 L253.24,156.11 L255.11,158.67 L256.98,161.22 L258.91,162.33 L260.92,161.39 L263.00,159.00 L265.08,156.61 L267.09,155.67 L269.02,156.78 L270.89,159.33 L272.76,161.89 L274.69,163.00 L276.70,162.06 L278.78,159.67 L280.85,157.28 L282.87,156.34 L284.80,157.44 L286.67,160.00 L288.53,162.56 L290.46,163.66 L292.48,162.65 L294.56,160.33 L296.62,158.24 L298.62,157.59 L300.55,158.64 L302.44,160.67 L304.34,162.47 L306.29,163.10 L308.30,162.40 L310.33,161.00 L312.36,159.82 L314.35,159.55 L316.29,160.22 L318.22,161.33 L320.16,162.22 L322.13,162.47 L324.11,162.16 L326.11,161.67 L328.10,161.41 L330.07,161.51 L332.03,161.80 L334.00,162.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-third)"/><path d="M50.00,310.00 L51.87,312.56 L53.80,313.66 L55.81,312.72 L57.89,310.33 L59.97,307.94 L61.98,307.00 L63.91,308.11 L65.78,310.67 L67.65,313.22 L69.57,314.33 L71.59,313.39 L73.67,311.00 L75.74,308.61 L77.76,307.67 L79.69,308.78 L81.56,311.33 L83.42,313.89 L85.35,315.00 L87.37,314.06 L89.44,311.67 L91.52,309.28 L93.54,308.34 L95.47,309.44 L97.33,312.00 L99.20,314.56 L101.13,315.66 L103.15,314.72 L105.22,312.33 L107.30,309.94 L109.31,309.00 L111.24,310.11 L113.11,312.67 L114.98,315.22 L116.91,316.33 L118.92,315.39 L121.00,313.00 L123.08,310.61 L125.09,309.67 L127.02,310.78 L128.89,313.33 L130.76,315.89 L132.69,317.00 L134.70,316.06 L136.78,313.67 L138.85,311.28 L140.87,310.34 L142.80,311.44 L144.67,314.00 L146.53,316.56 L148.46,317.66 L150.48,316.72 L152.56,314.33 L154.63,311.94 L156.65,311.00 L158.58,312.11 L160.44,314.67 L162.31,317.22 L164.24,318.33 L166.26,317.39 L168.33,315.00 L170.41,312.61 L172.43,311.67 L174.35,312.78 L176.22,315.33 L178.09,317.89 L180.02,319.00 L182.03,318.06 L184.11,315.67 L186.19,313.28 L188.20,312.34 L190.13,313.44 L192.00,316.00 L193.87,318.56 L195.80,319.66 L197.81,318.72 L199.89,316.33 L201.97,313.94 L203.98,313.00 L205.91,314.11 L207.78,316.67 L209.65,319.22 L211.57,320.33 L213.59,319.39 L215.67,317.00 L217.74,314.61 L219.76,313.67 L221.69,314.78 L223.56,317.33 L225.42,319.89 L227.35,321.00 L229.37,320.06 L231.44,317.67 L233.52,315.28 L235.54,314.34 L237.47,315.44 L239.33,318.00 L241.20,320.56 L243.13,321.66 L245.15,320.72 L247.22,318.33 L249.30,315.94 L251.31,315.00 L253.24,316.11 L255.11,318.67 L256.98,321.22 L258.91,322.33 L260.92,321.39 L263.00,319.00 L265.08,316.61 L267.09,315.67 L269.02,316.78 L270.89,319.33 L272.76,321.89 L274.69,323.00 L276.70,322.06 L278.78,319.67 L280.85,317.28 L282.87,316.34 L284.80,317.44 L286.67,320.00 L288.53,322.56 L290.46,323.66 L292.48,322.65 L294.56,320.33 L296.62,318.24 L298.62,317.59 L300.55,318.64 L302.44,320.67 L304.34,322.47 L306.29,323.10 L308.30,322.40 L310.33,321.00 L312.36,319.82 L314.35,319.55 L316.29,320.22 L318.22,321.33 L320.16,322.22 L322.13,322.47 L324.11,322.16 L326.11,321.67 L328.10,321.41 L330.07,321.51 L332.03,321.80 L334.00,322.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-third)"/><text x="30" y="386" font-size="10.5" style="--fs:10.5" fill="light-dark(#6a4fb3, #c3b2ff)" font-weight="600">LONG-WAVE RADIATION between room surfaces · h_r ≈ 5 W/(m²K)</text><rect x="20" y="70" width="24" height="300" fill="url(#hm-hatch-shade)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><text x="20" y="62" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)">room surfaces</text></g><g id="hm-outside"><path d="M562,96 Q510,146 518,196" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-azi)"/><path d="M562,196 Q510,246 518,296" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-azi)"/><path d="M562,276 Q510,316 518,356" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-azi)"/><text x="582" y="180" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">FORCED CONVECTION</text><text x="582" y="198" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">Wind along the facade</text><text x="582" y="214" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">h_c ≈ 4 + 4·v W/(m²K)</text><text x="582" y="230" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">≈ 20 W/(m²K) at 4 m/s</text><path d="M502.00,120.00 L504.91,121.10 L507.06,120.97 L508.13,119.10 L508.43,116.00 L508.73,112.90 L509.79,111.03 L511.94,110.90 L514.86,112.00 L517.77,113.10 L519.92,112.97 L520.99,111.10 L521.29,108.00 L521.59,104.90 L522.65,103.03 L524.80,102.90 L527.71,104.00 L530.63,105.10 L532.78,104.97 L533.84,103.10 L534.14,100.00 L534.44,96.90 L535.51,95.03 L537.66,94.90 L540.57,96.00 L543.49,97.10 L545.63,96.97 L546.70,95.10 L547.00,92.00 L547.30,88.90 L548.37,87.03 L550.51,86.90 L553.43,88.00 L556.34,89.10 L558.49,88.97 L559.56,87.10 L559.86,84.00 L560.16,80.90 L561.22,79.03 L563.37,78.90 L566.29,80.00 L569.20,81.10 L571.35,80.97 L572.41,79.10 L572.71,76.00 L573.01,72.90 L574.08,71.03 L576.23,70.90 L579.14,72.00 L581.84,72.75 L583.68,72.12 L584.74,70.25 L585.57,68.00 L586.71,66.25 L588.35,65.29 L590.24,64.75 L592.00,64.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-third)"/><text x="602" y="60" font-size="10.5" style="--fs:10.5" fill="light-dark(#6a4fb3, #c3b2ff)" font-weight="600">RADIATION to the sky</text><text x="602" y="74" font-size="10.5" style="--fs:10.5" fill="light-dark(#6a4fb3, #c3b2ff)" font-weight="600">and the surroundings</text><circle cx="720" cy="300" r="12" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><path d="M734.5,300 L738,300 M730.25,310.25 L732.73,312.73 M720,314.5 L720,318 M709.75,310.25 L707.27,312.73 M705.5,300 L702,300 M709.75,289.75 L707.27,287.27 M720,285.5 L720,282 M730.25,289.75 L732.73,287.27" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M704.00,306.00 L701.85,304.48 L699.82,303.98 L697.96,304.94 L696.23,306.92 L694.50,308.91 L692.64,309.87 L690.61,309.37 L688.46,307.85 L686.31,306.32 L684.28,305.83 L682.43,306.78 L680.69,308.77 L678.96,310.76 L677.10,311.71 L675.07,311.22 L672.92,309.69 L670.77,308.17 L668.74,307.67 L666.89,308.63 L665.15,310.62 L663.42,312.60 L661.56,313.56 L659.54,313.06 L657.38,311.54 L655.23,310.01 L653.21,309.52 L651.35,310.48 L649.62,312.46 L647.88,314.45 L646.03,315.41 L644.00,314.91 L641.85,313.38 L639.70,311.86 L637.67,311.36 L635.81,312.32 L634.08,314.31 L632.34,316.29 L630.49,317.25 L628.46,316.76 L626.31,315.23 L624.16,313.71 L622.13,313.21 L620.27,314.17 L618.54,316.15 L616.80,318.14 L614.95,319.10 L612.92,318.60 L610.77,317.08 L608.62,315.55 L606.59,315.06 L604.73,316.01 L603.00,318.00 L601.27,319.99 L599.41,320.94 L597.38,320.45 L595.23,318.92 L593.08,317.40 L591.05,316.90 L589.20,317.86 L587.46,319.85 L585.73,321.83 L583.87,322.79 L581.84,322.29 L579.69,320.77 L577.54,319.24 L575.51,318.75 L573.66,319.71 L571.92,321.69 L570.19,323.68 L568.33,324.64 L566.30,324.14 L564.15,322.62 L562.00,321.09 L559.97,320.59 L558.12,321.55 L556.38,323.54 L554.65,325.52 L552.79,326.48 L550.77,325.99 L548.62,324.46 L546.46,322.94 L544.44,322.44 L542.58,323.40 L540.85,325.38 L539.11,327.37 L537.26,328.33 L535.23,327.83 L533.08,326.31 L530.93,324.85 L528.93,324.54 L527.08,325.54 L525.31,327.23 L523.51,328.70 L521.61,329.28 L519.60,328.94 L517.54,328.15 L515.50,327.60 L513.54,327.66 L511.64,328.28 L509.77,329.08 L507.87,329.65 L505.92,329.86 L503.96,329.88 L502.00,330.00" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#hm-arrow-alt)"/><text x="532" y="362" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="600">SOLAR (short-wave): α · G absorbed</text></g></g></svg></div><figcaption><span><strong>Figure 1.</strong> Conduction, convection and radiation in a wall section</span><a href="./assets/diagrams/heat/heat-modes.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:heat-modes_END -->
<div class="table-wrap"><table class="nomenclature compact heat-table"><caption>What controls each mode in a building (lecture frame 590)</caption><thead><tr><th scope="col">Mode</th><th scope="col">Driven by</th><th scope="col">Design levers</th><th scope="col">Rate equation</th></tr></thead><tbody>
<tr><th scope="row">Conduction</th><td>Temperature gradient in a material</td><td>Insulation thickness, conductivity λ, continuity (thermal bridges)</td><td>Eq. 7.2–7.3</td></tr>
<tr><th scope="row">Convection</th><td>Surface-to-air temperature difference, air speed</td><td>Exposure to wind, air gaps and their width, airtightness</td><td>Eq. 7.4</td></tr>
<tr><th scope="row">Radiation</th><td>Difference of T⁴ between surfaces</td><td>Emissivity ε (low-e coatings, foils), solar absorptance (color), view to the sky</td><td>Eq. 7.5</td></tr>
</tbody></table>
</div>
<div class="callout"><span>STEADY STATE</span>
<p class="measure">In this chapter temperatures on both sides are held constant until nothing changes any more. Then the heat flowing into each layer equals the heat flowing out of it, and the same q passes every layer. Only the conductivity λ and thickness d matter; density and heat capacity drop out. This is the basis of U-values, design heat loads and seasonal energy estimates.</p>
</div>
</section>
<section id="conduction" class="reader-section">
<div class="section-heading"><span class="section-number">03</span><div><p class="eyebrow">FOURIER · CONDUCTIVITY · RESISTANCE · EQ. 7.2–7.3</p><h2>Conduction</h2></div></div>
<p class="section-lead">In a solid, energy spreads from more to less agitated molecules (and, in metals, by free electrons). Through a flat layer in steady state the temperature falls linearly, and the flux is proportional to the temperature difference and inversely proportional to the thickness.</p>
<div class="eq" id="eq-7-2"><span class="eq-math"><span class="eq-line"><span class="eq-label">Fourier's law for a plane layer between surface temperatures θ<sub>1</sub> > θ<sub>2</sub></span><math><mi>q</mi><mo>=</mo><mi>λ</mi><mo>·</mo><mfrac><mrow><msub><mi>θ</mi><mn>1</mn></msub><mo>−</mo><msub><mi>θ</mi><mn>2</mn></msub></mrow><mi>d</mi></mfrac></math><span class="eq-note">general form q = −λ · dθ/dx <span class="eq-unit">W/m²</span></span></span></span><span class="eq-n">(7.2)</span></div>
<p class="measure">The conductivity λ is the flux through a layer 1 m thick with 1 K across it, W/m² × m / K = W/(m·K). Its inverse, scaled by the thickness, is the layer's thermal resistance:</p>
<div class="eq" id="eq-7-3"><span class="eq-math"><span class="eq-line"><span class="eq-label">thermal resistance of a layer (Wärmedurchlasswiderstand)</span><math><mi>R</mi><mo>=</mo><mfrac><mi>d</mi><mi>λ</mi></mfrac></math><span class="eq-note"><span class="eq-unit">m²K/W</span></span></span><span class="eq-line"><math><mi>q</mi><mo>=</mo><mfrac><mrow><mi>Δ</mi><mi>θ</mi></mrow><mi>R</mi></mfrac><mo>,</mo><mspace width="0.8em"/><mi>Φ</mi><mo>=</mo><mi>A</mi><mo>·</mo><mfrac><mrow><mi>Δ</mi><mi>θ</mi></mrow><mi>R</mi></mfrac></math><span class="eq-note">resistances in series add: R = R<sub>1</sub> + R<sub>2</sub> + …</span></span></span><span class="eq-n">(7.3)</span></div>
<div class="calc calc-wide"><div class="calc-row"><span class="calc-k">Example</span><span class="calc-e">200 mm of mineral wool (λ = 0.032) or of basalt (λ = 3.5), 1 m², Δθ = 10 K between the surfaces (lecture frames 597–600)</span><span class="calc-v"></span></div><div class="calc-row"><span class="calc-k">R<sub>wool</sub></span><span class="calc-e">0.2 / 0.032</span><span class="calc-v">6.25 m²K/W</span></div><div class="calc-row calc-key"><span class="calc-k">Φ<sub>wool</sub></span><span class="calc-e">1 m² × 10 K / 6.25</span><span class="calc-v">1.6 W</span></div><div class="calc-row"><span class="calc-k">R<sub>basalt</sub></span><span class="calc-e">0.2 / 3.5</span><span class="calc-v">0.057 m²K/W</span></div><div class="calc-row calc-key"><span class="calc-k">Φ<sub>basalt</sub></span><span class="calc-e">1 m² × 10 K / 0.057</span><span class="calc-v">175 W</span></div><div class="calc-row"><span class="calc-k">Both</span><span class="calc-e">50 mm wool + 200 mm basalt: R = 1.56 + 0.057 = 1.62 → Φ = 10 / 1.62</span><span class="calc-v">6.2 W</span></div></div>
<div class="table-wrap"><table class="nomenclature compact heat-table"><caption>Typical properties of building materials at about 20 °C</caption><thead><tr><th scope="col">Material</th><th scope="col">λ W/(m·K)</th><th scope="col">ρ kg/m³</th><th scope="col">c J/(kg·K)</th><th scope="col">R of 100 mm, m²K/W</th></tr></thead><tbody>
<tr><th scope="row">Steel</th><td>50–60</td><td>7850</td><td>450</td><td>0.002</td></tr>
<tr><th scope="row">Basalt, granite</th><td>2.8–3.5</td><td>2650–2850</td><td>775–1000</td><td>0.03</td></tr>
<tr><th scope="row">Concrete, normal weight</th><td>1.4–2.0</td><td>2300–2400</td><td>880–1000</td><td>0.06</td></tr>
<tr><th scope="row">Glass</th><td>1.0</td><td>2500</td><td>750</td><td>0.10</td></tr>
<tr><th scope="row">Brick, common</th><td>0.7–0.8</td><td>1800–1920</td><td>835–1000</td><td>0.14</td></tr>
<tr><th scope="row">Aerated concrete (AAC) masonry</th><td>0.08–0.24</td><td>300–700</td><td>1000</td><td>0.4–1.25</td></tr>
<tr><th scope="row">Gypsum board</th><td>0.17–0.25</td><td>700–900</td><td>1000–1090</td><td>0.5</td></tr>
<tr><th scope="row">Hardwood / softwood, across the grain</th><td>0.16 / 0.12</td><td>720 / 510</td><td>1255–2100</td><td>0.6 / 0.8</td></tr>
<tr><th scope="row">Insulation (section 08)</th><td>0.02–0.05</td><td>15–200</td><td>800–2100</td><td>2–5</td></tr>
<tr><th scope="row">Still air</th><td>0.026</td><td>1.2</td><td>1000</td><td>(convection, section 06)</td></tr>
</tbody></table>
</div>
<ul class="facts">
<li><strong>Density.</strong> Pores of still air make materials light and poor conductors: for masonry and concrete λ rises roughly exponentially with density (Cammerer). Very light fiber insulation is the exception: below about 20–30 kg/m³ convection and radiation inside the pores raise λ again.</li>
<li><strong>Moisture</strong> raises λ, since water conducts about 25 times better than air. Design values (Bemessungswert, DIN 4108-4; ASHRAE design values) include an allowance for moisture and aging; use them, not the lab values on a data sheet.</li>
<li><strong>Direction.</strong> Timber conducts about twice as well along the grain as across it (pine 0.35 vs 0.14 W/(m·K)).</li>
<li>Tables: Incropera Table A.3; DIN 4108-4 and EN ISO 10456; ASHRAE Handbook – Fundamentals, Chapter 26.</li>
</ul>
</section>
<section id="surfaces" class="reader-section">
<div class="section-heading"><span class="section-number">04</span><div><p class="eyebrow">NEWTON · STEFAN–BOLTZMANN · ISO 6946 · EQ. 7.4–7.6</p><h2>Convection and radiation at surfaces</h2></div></div>
<p class="section-lead">The air next to a wall does not take the surface temperature at once: a thin boundary layer separates them. Convection and long-wave radiation act in parallel across it, and together they form the surface resistance.</p>
<div class="eq" id="eq-7-4"><span class="eq-math"><span class="eq-line"><span class="eq-label">convection (Newton's law of cooling)</span><math><mi>q</mi><mo>=</mo><msub><mi>h</mi><mtext>c</mtext></msub><mo>·</mo><mrow><mo>(</mo><msub><mi>θ</mi><mtext>s</mtext></msub><mo>−</mo><msub><mi>θ</mi><mtext>air</mtext></msub><mo>)</mo></mrow></math><span class="eq-note">h<sub>c</sub> is not a material property: it depends on air speed, surface direction and roughness</span></span></span><span class="eq-n">(7.4)</span></div>
<div class="table-wrap"><table class="nomenclature compact"><caption>Convection coefficient h<sub>c</sub> in air, W/(m²K)</caption><thead><tr><th scope="col">Situation</th><th scope="col">h<sub>c</sub></th><th scope="col">Source</th></tr></thead><tbody>
<tr><th scope="row">Free convection (buoyancy)</th><td>2–25 (Incropera) · 3–10 (Bauphysik)</td><td>Incropera Table 1.1; Bauphysik Tafel 1.5</td></tr>
<tr><th scope="row">Forced convection (wind, fans)</th><td>25–250 · 10–100</td><td>as above</td></tr>
<tr><th scope="row">Room surface: up / horizontal / down</th><td>5.0 / 2.5 / 0.7</td><td>ISO 6946 Annex C</td></tr>
<tr><th scope="row">Outside surface, wind speed v in m/s</th><td>4 + 4·v (≈ 20 at 4 m/s)</td><td>ISO 6946 Annex C</td></tr>
</tbody></table>
</div>
<p class="measure">Every surface emits long-wave radiation in proportion to the fourth power of its absolute temperature. Between a surface and large surroundings the net exchange is</p>
<div class="eq" id="eq-7-5"><span class="eq-math"><span class="eq-line"><span class="eq-label">long-wave radiation, gray surface in large surroundings</span><math><mi>q</mi><mo>=</mo><mi>ε</mi><mo>·</mo><mi>σ</mi><mo>·</mo><mrow><mo>(</mo><msubsup><mi>T</mi><mtext>s</mtext><mn>4</mn></msubsup><mo>−</mo><msubsup><mi>T</mi><mtext>sur</mtext><mn>4</mn></msubsup><mo>)</mo></mrow><mo>≈</mo><msub><mi>h</mi><mtext>r</mtext></msub><mo>·</mo><mrow><mo>(</mo><msub><mi>θ</mi><mtext>s</mtext></msub><mo>−</mo><msub><mi>θ</mi><mtext>sur</mtext></msub><mo>)</mo></mrow></math><span class="eq-note">σ = 5.67 × 10⁻⁸ W/(m²K⁴), T in K</span></span><span class="eq-line"><math><msub><mi>h</mi><mtext>r</mtext></msub><mo>=</mo><mi>ε</mi><mo>·</mo><mn>4</mn><mi>σ</mi><msubsup><mi>T</mi><mtext>m</mtext><mn>3</mn></msubsup></math><span class="eq-note">≈ 5.7 ε at 20 °C, ≈ 5.1 ε at 10 °C <span class="eq-unit">W/(m²K)</span></span></span></span><span class="eq-n">(7.5)</span></div>
<ul class="facts">
<li><strong>Emissivity is not color.</strong> In the long-wave range almost all building surfaces, white or black, have ε ≈ 0.9; bare metals have 0.02–0.3 (Incropera Table A.11; Bauphysik Tafel 1.8). Color matters for sunlight: white acrylic paint absorbs α<sub>s</sub> = 0.26 of the sun but emits with ε = 0.90 (Incropera Table A.12). That is why a white roof stays cool.</li>
<li><strong>Kirchhoff:</strong> at the same wavelength a surface absorbs as well as it emits (α = ε). A low-e coating reflects long-wave radiation back and emits little itself.</li>
</ul>
<div class="eq" id="eq-7-6"><span class="eq-math"><span class="eq-line"><span class="eq-label">surface resistance: convection and radiation in parallel</span><math><msub><mi>R</mi><mtext>s</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>h</mi><mtext>c</mtext></msub><mo>+</mo><msub><mi>h</mi><mtext>r</mtext></msub></mrow></mfrac></math><span class="eq-note">inside: 1 / (2.5 + 0.9 × 5.7) = 0.13 · outside: 1 / (20 + 0.9 × 5.1) = 0.04 <span class="eq-unit">m²K/W</span></span></span></span><span class="eq-n">(7.6)</span></div>
<div class="lab">
<div class="lab-canvas"><div class="canvas-heading"><span class="small-label">SURFACE RESISTANCE · EQ. 7.4–7.6</span><button type="button" class="quiet-button" data-export="surface">Save SVG ↓</button></div><div class="diagram-host" id="d-surface"></div><div class="diagram-inputs" id="in-surface"></div></div>
<div class="lab-controls"><p class="control-kicker">EQ. 7.6 WITH CURRENT VALUES</p><div class="calc" id="calc-surface" aria-live="polite"></div></div>
</div>
<div class="table-wrap"><table class="nomenclature compact"><caption>Conventional surface resistances, m²K/W (ISO 6946, DIN EN ISO 6946)</caption><thead><tr><th scope="col">Heat flow</th><th scope="col">Element</th><th scope="col">R<sub>si</sub></th><th scope="col">R<sub>se</sub></th></tr></thead><tbody>
<tr><th scope="row">Horizontal (±30°)</th><td>Walls, windows</td><td>0.13</td><td>0.04</td></tr>
<tr><th scope="row">Upward</th><td>Roofs, ceilings in winter</td><td>0.10</td><td>0.04</td></tr>
<tr><th scope="row">Downward</th><td>Floors over outside air</td><td>0.17</td><td>0.04</td></tr>
</tbody></table>
</div>
<ul class="facts">
<li>US practice (ASHRAE Handbook – Fundamentals) tabulates the same films with nearly the same values: still air, horizontal heat flow 0.12; outside, 24 km/h wind, 0.03 m²K/W.</li>
<li>Moisture checks use a larger R<sub>si</sub> = 0.25 m²K/W (behind furniture, in corners, DIN 4108-2 and ISO 13788): less convection, a colder surface.</li>
</ul>
</section>
<section id="uvalue" class="reader-section">
<div class="section-heading"><span class="section-number">05</span><div><p class="eyebrow">U-VALUE · TEMPERATURE PROFILE · MOLD · EQ. 7.7–7.9</p><h2>U-value and the temperature profile</h2></div></div>
<p class="section-lead">From room air to outside air the heat passes the inside surface, each layer and the outside surface in series. Their resistances add. The U-value is the inverse of the total: the heat flow through 1 m² for 1 K between the air on both sides.</p>
<!-- FIG:resistor-network_START -->
<figure class="fig" id="fig-resistor-network"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 790 430" class="diagram" role="img" aria-label="Wall section, temperature profile and the equivalent series resistor network"><title>Wall section, temperature profile and the equivalent series resistor network</title><defs><marker id="rn-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="rn-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="rn-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5"/></marker><marker id="rn-arrow-third" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5"/></marker><marker id="rn-arrow-muted" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="1.5"/></marker><marker id="rn-arrow-red" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5"/></marker><pattern id="rn-hatch-shadow" patternUnits="userSpaceOnUse" width="3.4" height="3.4" patternTransform="rotate(45)"><rect width="3.4" height="3.4" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="3.4" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="rn-hatch-shade" patternUnits="userSpaceOnUse" width="7.5" height="7.5" patternTransform="rotate(45)"><rect width="7.5" height="7.5" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="7.5" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="rn-hatch-screen" patternUnits="userSpaceOnUse" width="4.2" height="4.2" patternTransform="rotate(135)"><line x1="0" y1="0" x2="0" y2="4.2" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern></defs><g font-family="Helvetica Neue, Helvetica, Arial, sans-serif"><g id="rn-title"><text x="20" y="26" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">Lecture example · Δθ = 10 K · the temperature drops in proportion to each resistance</text></g><g id="rn-section"><rect x="220" y="50" width="10" height="160" fill="light-dark(#ffffff, #202124)" stroke="none"/><circle cx="222" cy="55" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="55" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="223.6" cy="62" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="228.6" cy="62" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="225.2" cy="69" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="222" cy="76" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="76" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="223.6" cy="83" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="228.6" cy="83" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="225.2" cy="90" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="222" cy="97" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="97" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="223.6" cy="104" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="228.6" cy="104" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="225.2" cy="111" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="222" cy="118" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="118" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="223.6" cy="125" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="228.6" cy="125" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="225.2" cy="132" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="222" cy="139" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="139" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="223.6" cy="146" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="228.6" cy="146" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="225.2" cy="153" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="222" cy="160" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="160" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="223.6" cy="167" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="228.6" cy="167" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="225.2" cy="174" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="222" cy="181" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="181" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="223.6" cy="188" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="228.6" cy="188" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="225.2" cy="195" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="222" cy="202" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="202" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><rect x="220" y="50" width="10" height="160" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="230" y="50" width="100" height="160" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M234,52 C222.5,54.6 337.5,75.4 326,78 C337.5,80.6 222.5,101.4 234,104 C222.5,106.6 337.5,127.4 326,130 C337.5,132.6 222.5,153.4 234,156 C222.5,158.6 337.5,179.4 326,182 C337.5,184.6 222.5,205.4 234,208" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="230" y="50" width="100" height="160" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="330" y="50" width="250" height="160" fill="url(#rn-hatch-shadow)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="330" y="50" width="250" height="160" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="225" y="226" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">10</text><text x="280" y="226" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">100 mm · λ 0.032</text><text x="455" y="226" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">250 mm · λ 3.5</text></g><g id="rn-profile"><path d="M110,71.66666666666669 L190,71.66666666666669 Q216,71.66666666666669 220,76.15859248289965" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><polyline points="220,76.16 230,76.5 330,184.48 580,186.95" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M580,186.95120231295397 Q584,188.33333333333334 610,188.33333333333334 L690,188.33333333333334" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="112" y="63.67" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">θi = 20 °C</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="688" y="180.33" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" text-anchor="end" font-weight="700">θe = 10 °C</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="234" y="94.16" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">θsi = 19.61 °C</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="338" y="174.48" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)">10.33 °C</text><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="574" y="204.95" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" text-anchor="end" font-weight="700">θse = 10.12 °C</text></g><g id="rn-network"><path d="M74.2,300 Q82.96,300.05 91.72,300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><path d="M179.73,300 Q188.37,299.98 197.01,300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><path d="M88,280.56 Q87.99,299.42 88,318.27" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><path d="M182,280.28 Q181.88,299.35 182,318.42" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><polyline points="88,284 111.5,284 115.42,278 123.25,290 131.08,278 138.92,290 146.75,278 154.58,290 158.5,284 182,284" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><polyline points="88,316 111.5,316 115.42,310 123.25,322 131.08,310 138.92,322 146.75,310 154.58,322 158.5,316 182,316" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text x="135" y="272" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">1/h_c</text><text x="135" y="338" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">1/h_r</text><text x="135" y="356" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="600">R_si</text><text x="135" y="370" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">0.130 m²K/W</text><polyline points="206,300 235.5,300 240.42,294 250.25,306 260.08,294 269.92,306 279.75,294 289.58,306 294.5,300 324,300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text x="265" y="330" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="600">R_1 plaster</text><text x="265" y="344" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">0.010 m²K/W</text><polyline points="336,300 365.5,300 370.42,294 380.25,306 390.08,294 399.92,306 409.75,294 419.58,306 424.5,300 454,300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text x="395" y="330" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="600">R_2 wool</text><text x="395" y="344" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">3.125 m²K/W</text><polyline points="466,300 495.5,300 500.42,294 510.25,306 520.08,294 529.92,306 539.75,294 549.58,306 554.5,300 584,300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text x="525" y="330" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="600">R_3 basalt</text><text x="525" y="344" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">0.071 m²K/W</text><path d="M593.23,300 Q602.64,300.01 612.04,300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><path d="M698.39,300 Q708.01,300.02 717.63,300" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><path d="M608,280.8 Q608,300.33 608,319.87" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><path d="M702,280.21 Q701.9,299.5 702,318.79" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><polyline points="608,284 631.5,284 635.42,278 643.25,290 651.08,278 658.92,290 666.75,278 674.58,290 678.5,284 702,284" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><polyline points="608,316 631.5,316 635.42,310 643.25,322 651.08,310 658.92,322 666.75,310 674.58,322 678.5,316 702,316" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text x="655" y="272" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">1/h_c</text><text x="655" y="338" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">1/h_r</text><text x="655" y="356" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="600">R_se</text><text x="655" y="370" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">0.040 m²K/W</text><circle cx="70" cy="300" r="5" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="70" y="260" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="700">θi 20.00</text><circle cx="200" cy="300" r="5" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="200" y="260" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="700">θsi 19.61</text><circle cx="330" cy="300" r="5" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="330" y="260" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="700">θ1 19.59</text><circle cx="460" cy="300" r="5" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="460" y="260" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="700">θ2 10.33</text><circle cx="590" cy="300" r="5" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="590" y="260" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="700">θse 10.12</text><circle cx="720" cy="300" r="5" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="720" y="260" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="700">θe 10.00</text><line x1="110" y1="234" x2="70" y2="246" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-dasharray="2 4" stroke-linecap="round" stroke-linejoin="round"/><line x1="220" y1="234" x2="200" y2="246" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-dasharray="2 4" stroke-linecap="round" stroke-linejoin="round"/><line x1="230" y1="234" x2="330" y2="246" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-dasharray="2 4" stroke-linecap="round" stroke-linejoin="round"/><line x1="330" y1="234" x2="460" y2="246" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-dasharray="2 4" stroke-linecap="round" stroke-linejoin="round"/><line x1="580" y1="234" x2="590" y2="246" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-dasharray="2 4" stroke-linecap="round" stroke-linejoin="round"/><line x1="690" y1="234" x2="720" y2="246" stroke="light-dark(#a6bdce, #6b879b)" stroke-width="0.6" stroke-dasharray="2 4" stroke-linecap="round" stroke-linejoin="round"/><line x1="300" y1="392" x2="490" y2="392" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#rn-arrow-alt)"/><text x="395" y="410" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="700">q = Δθ / R_T = 10 / 3.38 = 2.96 W/m² · U = 1 / R_T = 0.296 W/(m²K)</text></g></g></svg></div><figcaption><span><strong>Figure 2.</strong> Temperature profile and the equivalent resistor network (lecture example)</span><a href="./assets/diagrams/heat/resistor-network.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:resistor-network_END -->
<div class="eq" id="eq-7-7"><span class="eq-math"><span class="eq-line"><span class="eq-label">total resistance and thermal transmittance</span><math><msub><mi>R</mi><mtext>T</mtext></msub><mo>=</mo><msub><mi>R</mi><mtext>si</mtext></msub><mo>+</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><msub><mi>d</mi><mi>j</mi></msub><msub><mi>λ</mi><mi>j</mi></msub></mfrac><mo>+</mo><msub><mi>R</mi><mtext>se</mtext></msub><mo>,</mo><mspace width="0.8em"/><mi>U</mi><mo>=</mo><mfrac><mn>1</mn><msub><mi>R</mi><mtext>T</mtext></msub></mfrac></math><span class="eq-note"><span class="eq-unit">m²K/W, W/(m²K)</span></span></span></span><span class="eq-n">(7.7)</span></div>
<div class="eq" id="eq-7-8"><span class="eq-math"><span class="eq-line"><span class="eq-label">heat flow through an element of area A, and the energy over a time t</span><math><mi>Φ</mi><mo>=</mo><mi>U</mi><mo>·</mo><mi>A</mi><mo>·</mo><mrow><mo>(</mo><msub><mi>θ</mi><mtext>i</mtext></msub><mo>−</mo><msub><mi>θ</mi><mtext>e</mtext></msub><mo>)</mo></mrow><mo>,</mo><mspace width="0.8em"/><mi>Q</mi><mo>=</mo><mi>Φ</mi><mo>·</mo><mi>t</mi></math><span class="eq-note"><span class="eq-unit">W, J</span></span></span></span><span class="eq-n">(7.8)</span></div>
<p class="measure">Because the same q passes every layer, each layer takes a share of the temperature difference equal to its share of the resistance. Starting from the inside air:</p>
<div class="eq" id="eq-7-9"><span class="eq-math"><span class="eq-line"><span class="eq-label">temperature at the inside surface and at each interface</span><math><msub><mi>θ</mi><mtext>si</mtext></msub><mo>=</mo><msub><mi>θ</mi><mtext>i</mtext></msub><mo>−</mo><msub><mi>R</mi><mtext>si</mtext></msub><mo>·</mo><mi>q</mi><mo>,</mo><mspace width="0.8em"/><msub><mi>θ</mi><mi>j</mi></msub><mo>=</mo><msub><mi>θ</mi><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>R</mi><mi>j</mi></msub><mo>·</mo><mi>q</mi></math><span class="eq-note">q = (θi − θe) / R<sub>T</sub></span></span></span><span class="eq-n">(7.9)</span></div>
<div class="calc calc-wide"><div class="calc-row"><span class="calc-k">Example</span><span class="calc-e">Lecture wall (frames 609–610): 10 mm plaster λ 1.0, 100 mm mineral wool λ 0.032, 250 mm basalt λ 3.5; 1 m², θi = 20 °C, θe = 10 °C, for 1 hour</span><span class="calc-v"></span></div><div class="calc-row"><span class="calc-k">R<sub>T</sub></span><span class="calc-e">0.13 + 0.01/1.0 + 0.1/0.032 + 0.25/3.5 + 0.04 = 0.13 + 0.010 + 3.125 + 0.071 + 0.04</span><span class="calc-v">3.38 m²K/W</span></div><div class="calc-row calc-key"><span class="calc-k">U</span><span class="calc-e">1 / 3.38</span><span class="calc-v">0.296 W/(m²K)</span></div><div class="calc-row"><span class="calc-k">Φ</span><span class="calc-e">0.296 × 1 × 10</span><span class="calc-v">2.96 W</span></div><div class="calc-row calc-key"><span class="calc-k">Q</span><span class="calc-e">2.96 W × 3600 s</span><span class="calc-v">10.7 kJ</span></div><div class="calc-row"><span class="calc-k">θ<sub>si</sub></span><span class="calc-e">20 − 0.13 × 2.96: the wool takes 9.3 K of the 10 K</span><span class="calc-v">19.6 °C</span></div></div>
<div class="lab">
<div class="lab-canvas"><div class="canvas-heading"><span class="small-label">LAYERED CONSTRUCTION · EQ. 7.7–7.9</span><button type="button" class="quiet-button" data-export="wall">Save SVG ↓</button></div><div class="diagram-host" id="d-wall"></div><div class="diagram-inputs" id="in-wall"></div></div>
<div class="lab-controls"><p class="control-kicker">EQ. 7.7–7.9 WITH CURRENT VALUES</p><div class="calc" id="calc-wall" aria-live="polite"></div><p class="control-note">Plotted against thermal resistance (choose the axis below the drawing) the profile becomes a single straight line: the graphical method of the Lehrbuch der Bauphysik, Bild 2.11.</p></div>
</div>
<h3>Why the inside surface temperature matters</h3>
<ul class="facts">
<li><strong>Comfort.</strong> People exchange radiation with the room surfaces. A cold wall lowers the mean radiant temperature and causes radiant asymmetry: EN ISO 7730 and ASHRAE 55 limit a cold wall to 10 K below the opposite surfaces. Insulation keeps θ<sub>si</sub> close to the air temperature.</li>
<li><strong>Mold and condensation.</strong> Room air at 20 °C and 50 % RH has a dew point of 9.3 °C. Mold grows when the surface stays above about 80 % RH, which happens below 12.6 °C. DIN 4108-2 therefore requires a temperature factor <strong>f<sub>Rsi</sub> = (θ<sub>si</sub> − θ<sub>e</sub>) / (θ<sub>i</sub> − θ<sub>e</sub>) ≥ 0.70</strong> (with R<sub>si</sub> = 0.25), at every point including corners and junctions: at θe = −5 °C the surface stays above 12.5 °C.</li>
<li><strong>Where to put the insulation.</strong> Outside insulation keeps the masonry warm and covers slab edges. Inside insulation leaves the wall cold, needs moisture control and leaves the junctions with floors and partitions as thermal bridges (section 07), but heats up quickly in rooms used occasionally.</li>
</ul>
</section>
<section id="airlayers" class="reader-section">
<div class="section-heading"><span class="section-number">06</span><div><p class="eyebrow">CAVITIES · LOW-E FOIL · EQ. 7.10</p><h2>Air layers</h2></div></div>
<p class="section-lead">Still air is an excellent insulator (λ = 0.026 W/(m·K)), but an air layer is not 'still air'. Beyond about 15–20 mm a convection cell forms, and radiation crosses the gap regardless of its width. The resistance of a cavity therefore stops growing at about 0.18 m²K/W.</p>
<!-- FIG:air-cavity_START -->
<figure class="fig" id="fig-air-cavity"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 760 412" class="diagram" role="img" aria-label="Heat transfer across a vertical air layer: conduction, convection and radiation"><title>Heat transfer across a vertical air layer: conduction, convection and radiation</title><defs><marker id="ac-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="ac-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="ac-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5"/></marker><marker id="ac-arrow-third" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5"/></marker><marker id="ac-arrow-muted" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="1.5"/></marker><marker id="ac-arrow-red" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5"/></marker><pattern id="ac-hatch-shadow" patternUnits="userSpaceOnUse" width="3.4" height="3.4" patternTransform="rotate(45)"><rect width="3.4" height="3.4" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="3.4" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="ac-hatch-shade" patternUnits="userSpaceOnUse" width="7.5" height="7.5" patternTransform="rotate(45)"><rect width="7.5" height="7.5" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="7.5" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="ac-hatch-screen" patternUnits="userSpaceOnUse" width="4.2" height="4.2" patternTransform="rotate(135)"><line x1="0" y1="0" x2="0" y2="4.2" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern></defs><g font-family="Helvetica Neue, Helvetica, Arial, sans-serif"><g id="ac-panels"><rect x="40" y="60" width="34" height="270" fill="url(#ac-hatch-shade)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="40" y="60" width="34" height="270" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="86" y="60" width="34" height="270" fill="url(#ac-hatch-shade)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="86" y="60" width="34" height="270" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><line x1="75" y1="120" x2="85" y2="120" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-alt)"/><line x1="75" y1="195" x2="85" y2="195" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-alt)"/><line x1="75" y1="270" x2="85" y2="270" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-alt)"/><path d="M77.00,140.00 L76.21,141.63 L76.06,142.45 L76.83,142.13 L78.25,141.00 L79.67,139.87 L80.44,139.55 L80.29,140.37 L79.50,142.00 L78.71,143.63 L78.56,144.45 L79.33,144.13 L80.75,143.00 L82.17,141.87 L82.68,141.87 L82.15,143.17 L82.00,144.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-third)"/><path d="M77.00,240.00 L76.21,241.63 L76.06,242.45 L76.83,242.13 L78.25,241.00 L79.67,239.87 L80.44,239.55 L80.29,240.37 L79.50,242.00 L78.71,243.63 L78.56,244.45 L79.33,244.13 L80.75,243.00 L82.17,241.87 L82.68,241.87 L82.15,243.17 L82.00,244.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-third)"/><text x="40" y="34" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">a · 10 mm</text><text x="80" y="352" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="700">R_g = 0.15 m²K/W</text><text x="57" y="370" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">warm</text><text x="103" y="370" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">cold</text><rect x="270" y="60" width="34" height="270" fill="url(#ac-hatch-shade)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="270" y="60" width="34" height="270" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="364" y="60" width="34" height="270" fill="url(#ac-hatch-shade)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="364" y="60" width="34" height="270" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M311,300 L311,90 Q311,72 334,72 Q357,72 357,90 L357,300 Q357,318 334,318 Q311,318 311,300" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-dasharray="5 4" stroke-linecap="round" stroke-linejoin="round"/><line x1="311" y1="210" x2="311" y2="170" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-alt)"/><line x1="357" y1="170" x2="357" y2="210" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-alt)"/><path d="M307.00,140.00 L309.08,141.93 L311.23,142.83 L313.49,142.26 L315.83,140.67 L318.17,139.07 L320.44,138.51 L322.59,139.40 L324.67,141.33 L326.74,143.26 L328.90,144.16 L331.16,143.60 L333.50,142.00 L335.84,140.40 L338.10,139.84 L340.26,140.74 L342.33,142.67 L344.41,144.60 L346.56,145.49 L348.83,144.93 L351.17,143.33 L353.49,142.03 L355.69,142.28 L357.83,143.34 L360.00,144.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-third)"/><path d="M307.00,240.00 L309.08,241.93 L311.23,242.83 L313.49,242.26 L315.83,240.67 L318.17,239.07 L320.44,238.51 L322.59,239.40 L324.67,241.33 L326.74,243.26 L328.90,244.16 L331.16,243.60 L333.50,242.00 L335.84,240.40 L338.10,239.84 L340.26,240.74 L342.33,242.67 L344.41,244.60 L346.56,245.49 L348.83,244.93 L351.17,243.33 L353.49,242.03 L355.69,242.28 L357.83,243.34 L360.00,244.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-third)"/><text x="270" y="34" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">b · 50 mm</text><text x="334" y="352" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="700">R_g = 0.18 m²K/W</text><text x="287" y="370" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">warm</text><text x="381" y="370" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">cold</text><rect x="530" y="60" width="34" height="270" fill="url(#ac-hatch-shade)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="530" y="60" width="34" height="270" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="624" y="60" width="34" height="270" fill="url(#ac-hatch-shade)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="624" y="60" width="34" height="270" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><path d="M622.5,57.38 Q622.7,195.49 622.5,333.61" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><path d="M571,300 L571,90 Q571,72 594,72 Q617,72 617,90 L617,300 Q617,318 594,318 Q571,318 571,300" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-dasharray="5 4" stroke-linecap="round" stroke-linejoin="round"/><line x1="571" y1="210" x2="571" y2="170" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-alt)"/><line x1="617" y1="170" x2="617" y2="210" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-alt)"/><path d="M567.00,140.00 L568.84,141.58 L570.75,142.33 L572.76,141.91 L574.83,140.67 L576.91,139.42 L578.92,139.01 L580.83,139.76 L582.67,141.33 L584.51,142.91 L586.41,143.66 L588.42,143.24 L590.50,142.00 L592.58,140.76 L594.59,140.34 L596.49,141.09 L598.33,142.67 L600.17,144.24 L602.08,144.99 L604.09,144.58 L606.17,143.33 L608.22,142.33 L610.18,142.56 L612.07,143.44 L614.00,144.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-third)"/><path d="M567.00,240.00 L568.84,241.58 L570.75,242.33 L572.76,241.91 L574.83,240.67 L576.91,239.42 L578.92,239.01 L580.83,239.76 L582.67,241.33 L584.51,242.91 L586.41,243.66 L588.42,243.24 L590.50,242.00 L592.58,240.76 L594.59,240.34 L596.49,241.09 L598.33,242.67 L600.17,244.24 L602.08,244.99 L604.09,244.58 L606.17,243.33 L608.22,242.33 L610.18,242.56 L612.07,243.44 L614.00,244.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-third)"/><text x="530" y="34" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">c · 50 mm, foil ε = 0.05</text><text x="594" y="352" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="700">R_g = 0.66 m²K/W</text><text x="547" y="370" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">warm</text><text x="641" y="370" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">cold</text></g><g id="ac-key"><line x1="40" y1="392" x2="70" y2="392" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-alt)"/><text x="78" y="396" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">conduction through still air</text><line x1="270" y1="392" x2="300" y2="392" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-dasharray="5 4" stroke-linecap="round" stroke-linejoin="round"/><text x="308" y="396" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">convection cell (rises warm, falls cold)</text><path d="M560.00,392.00 L561.88,393.41 L563.75,394.00 L565.63,393.41 L567.50,392.00 L569.38,390.59 L571.25,390.00 L573.13,390.59 L575.00,392.00 L576.88,393.41 L578.75,394.00 L580.63,393.41 L582.50,392.00 L584.38,390.59 L586.25,390.33 L588.13,391.41 L590.00,392.00" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#ac-arrow-third)"/><text x="598" y="396" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">long-wave radiation</text></g></g></svg></div><figcaption><span><strong>Figure 3.</strong> Air layers: conduction, convection cell and radiation</span><a href="./assets/diagrams/heat/air-cavity.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:air-cavity_END -->
<div class="eq" id="eq-7-10"><span class="eq-math"><span class="eq-line"><span class="eq-label">unventilated air layer between two large parallel surfaces (ISO 6946 Annex D)</span><math><msub><mi>R</mi><mtext>g</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>h</mi><mtext>a</mtext></msub><mo>+</mo><mi>E</mi><mo>·</mo><msub><mi>h</mi><mtext>r0</mtext></msub></mrow></mfrac><mo>,</mo><mspace width="0.8em"/><mi>E</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>/</mo><msub><mi>ε</mi><mn>1</mn></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mi>ε</mi><mn>2</mn></msub><mo>−</mo><mn>1</mn></mrow></mfrac></math><span class="eq-note">h<sub>a</sub>: conduction (0.025/d) or convection, whichever is larger; E: exchange factor (Incropera Eq. 13.19; Bauphysik Gl. 1.21)</span></span></span><span class="eq-n">(7.10)</span></div>
<div class="lab">
<div class="lab-canvas"><div class="canvas-heading"><span class="small-label">AIR LAYER · EQ. 7.10</span><button type="button" class="quiet-button" data-export="cavity">Save SVG ↓</button></div><div class="diagram-host" id="d-cavity"></div><div class="diagram-inputs" id="in-cavity"></div></div>
<div class="lab-controls"><p class="control-kicker">EQ. 7.10 WITH CURRENT VALUES</p><div class="calc" id="calc-cavity" aria-live="polite"></div></div>
</div>
<div class="table-wrap"><table class="nomenclature compact"><caption>Unventilated air layers between ordinary surfaces (ε = 0.9), R<sub>g</sub> in m²K/W (ISO 6946 Table 8; Bauphysik Tafel 2.2)</caption><thead><tr><th scope="col">Thickness</th><th scope="col">Heat flow up</th><th scope="col">Horizontal</th><th scope="col">Down</th></tr></thead><tbody>
<tr><th scope="row">5 mm</th><td>0.11</td><td>0.11</td><td>0.11</td></tr>
<tr><th scope="row">10 mm</th><td>0.15</td><td>0.15</td><td>0.15</td></tr>
<tr><th scope="row">15 mm</th><td>0.16</td><td>0.17</td><td>0.17</td></tr>
<tr><th scope="row">25 mm</th><td>0.16</td><td>0.18</td><td>0.19</td></tr>
<tr><th scope="row">50 mm</th><td>0.16</td><td>0.18</td><td>0.21</td></tr>
<tr><th scope="row">100 mm</th><td>0.16</td><td>0.18</td><td>0.22</td></tr>
<tr><th scope="row">300 mm</th><td>0.16</td><td>0.18</td><td>0.23</td></tr>
</tbody></table>
</div>
<ul class="facts">
<li>Heat flowing <em>down</em> (a floor over a cold cavity) meets stable layering: warm air stays on top, so convection is weak and R keeps growing.</li>
<li>A low-emissivity face (aluminum foil, ε ≈ 0.05) cuts the radiation by a factor of about 16: a 20–40 mm cavity reaches 0.55–0.65 m²K/W (Bauphysik Tafel 2.3). This is how insulating glass works: Chapter 08, section 05.</li>
<li><strong>Ventilated layers</strong> (rain screens, ventilated roofs): ISO 6946 ignores the air layer and everything outside it, and uses R<sub>se</sub> = R<sub>si</sub>. Slightly ventilated layers are interpolated between the two cases.</li>
</ul>
</section>
<section id="bridges" class="reader-section">
<div class="section-heading"><span class="section-number">07</span><div><p class="eyebrow">PARALLEL PATHS · ISO 6946 LIMITS · ψ, χ · EQ. 7.11</p><h2>Parallel paths and thermal bridges</h2></div></div>
<p class="section-lead">Real walls are not homogeneous layers. Studs cross the insulation, slabs and balconies run through the facade, corners have more outside than inside surface. Heat takes the easy path, and that path is also where the inside surface is coldest.</p>
<div class="eq" id="eq-7-11"><span class="eq-math"><span class="eq-line"><span class="eq-label">parallel paths with area fractions f<sub>j</sub> (Σ f<sub>j</sub> = 1): conductances add, not resistances</span><math><mi>U</mi><mo>=</mo><munder><mo>∑</mo><mi>j</mi></munder><msub><mi>f</mi><mi>j</mi></msub><mo>·</mo><msub><mi>U</mi><mi>j</mi></msub></math><span class="eq-note">upper limit of R; ISO 6946 averages it with the lower limit of Figure 4</span></span></span><span class="eq-n">(7.11)</span></div>
<!-- FIG:iso6946-bounds_START -->
<figure class="fig" id="fig-iso6946-bounds"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 780 390" class="diagram" role="img" aria-label="Upper and lower limits of the thermal resistance of an inhomogeneous construction"><title>Upper and lower limits of the thermal resistance of an inhomogeneous construction</title><defs><marker id="bd-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="bd-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="bd-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5"/></marker><marker id="bd-arrow-third" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5"/></marker><marker id="bd-arrow-muted" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="1.5"/></marker><marker id="bd-arrow-red" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5"/></marker><pattern id="bd-hatch-shadow" patternUnits="userSpaceOnUse" width="3.4" height="3.4" patternTransform="rotate(45)"><rect width="3.4" height="3.4" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="3.4" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="bd-hatch-shade" patternUnits="userSpaceOnUse" width="7.5" height="7.5" patternTransform="rotate(45)"><rect width="7.5" height="7.5" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="7.5" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="bd-hatch-screen" patternUnits="userSpaceOnUse" width="4.2" height="4.2" patternTransform="rotate(135)"><line x1="0" y1="0" x2="0" y2="4.2" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern></defs><g font-family="Helvetica Neue, Helvetica, Arial, sans-serif"><g id="bd-upper"><rect x="40" y="70" width="300" height="16" fill="light-dark(#ffffff, #202124)" stroke="none"/><circle cx="42" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="47" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="52" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="57" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="62" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="67" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="72" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="77" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="82" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="87" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="92" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="97" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="102" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="107" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="112" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="117" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="122" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="127" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="132" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="137" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="142" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="147" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="152" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="157" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="162" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="167" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="172" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="177" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="182" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="187" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="192" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="197" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="202" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="207" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="212" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="217" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="222" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="232" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="237" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="242" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="247" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="252" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="257" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="262" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="267" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="272" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="277" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="282" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="287" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="292" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="297" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="302" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="307" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="312" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="317" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="322" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="327" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="332" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="337" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="43.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="48.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="53.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="58.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="63.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="68.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="73.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="78.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="83.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="88.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="93.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="98.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="103.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="108.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="113.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="118.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="123.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="128.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="133.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="138.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="143.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="148.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="153.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="158.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="163.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="168.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="173.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="178.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="183.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="188.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="193.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="198.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="203.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="208.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="213.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="218.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="223.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="228.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="233.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="238.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="243.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="248.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="253.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="258.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="263.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="268.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="273.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="278.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="283.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="288.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="293.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="298.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="303.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="308.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="313.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="318.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="323.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="328.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="333.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="338.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><rect x="40" y="70" width="300" height="16" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="40" y="86" width="300" height="168" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M42,92.72 C44.6,73.4 65.4,266.6 68,247.28 C70.6,266.6 91.4,73.4 94,92.72 C96.6,73.4 117.4,266.6 120,247.28 C122.6,266.6 143.4,73.4 146,92.72 C148.6,73.4 169.4,266.6 172,247.28 C174.6,266.6 195.4,73.4 198,92.72 C200.6,73.4 221.4,266.6 224,247.28 C226.6,266.6 247.4,73.4 250,92.72 C252.6,73.4 273.4,266.6 276,247.28 C278.6,266.6 299.4,73.4 302,92.72 C304.6,73.4 325.4,266.6 328,247.28 C329,266.6 337,73.4 338,92.72" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="40" y="86" width="300" height="168" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="165" y="86" width="50" height="168" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M177.5,86 Q181.5,170 177.5,254" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M190,86 Q194,170 190,254" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M202.5,86 Q206.5,170 202.5,254" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="165" y="86" width="50" height="168" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="40" y="254" width="300" height="16" fill="light-dark(#ffffff, #202124)" stroke="none"/><circle cx="42" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="47" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="52" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="57" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="62" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="67" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="72" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="77" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="82" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="87" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="92" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="97" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="102" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="107" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="112" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="117" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="122" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="127" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="132" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="137" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="142" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="147" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="152" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="157" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="162" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="167" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="172" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="177" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="182" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="187" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="192" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="197" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="202" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="207" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="212" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="217" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="222" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="227" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="232" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="237" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="242" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="247" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="252" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="257" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="262" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="267" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="272" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="277" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="282" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="287" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="292" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="297" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="302" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="307" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="312" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="317" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="322" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="327" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="332" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="337" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="43.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="48.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="53.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="58.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="63.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="68.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="73.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="78.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="83.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="88.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="93.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="98.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="103.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="108.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="113.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="118.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="123.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="128.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="133.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="138.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="143.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="148.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="153.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="158.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="163.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="168.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="173.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="178.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="183.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="188.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="193.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="198.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="203.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="208.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="213.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="218.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="223.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="228.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="233.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="238.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="243.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="248.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="253.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="258.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="263.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="268.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="273.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="278.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="283.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="288.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="293.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="298.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="303.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="308.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="313.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="318.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="323.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="328.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="333.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="338.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><rect x="40" y="254" width="300" height="16" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="40" y="40" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">a · upper limit R′T: adiabatic planes</text><text x="32" y="80" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="end" font-weight="700">in</text><text x="32" y="268" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="end" font-weight="700">out</text><line x1="165" y1="62" x2="165" y2="278" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5" stroke-dasharray="6 3" stroke-linecap="round" stroke-linejoin="round"/><line x1="215" y1="62" x2="215" y2="278" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5" stroke-dasharray="6 3" stroke-linecap="round" stroke-linejoin="round"/><line x1="80" y1="74" x2="80" y2="266" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#bd-arrow-alt)"/><line x1="120" y1="74" x2="120" y2="266" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#bd-arrow-alt)"/><line x1="190" y1="74" x2="190" y2="266" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="3" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#bd-arrow-alt)"/><line x1="260" y1="74" x2="260" y2="266" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#bd-arrow-alt)"/><line x1="300" y1="74" x2="300" y2="266" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#bd-arrow-alt)"/><text x="40" y="300" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">Each section is its own wall; no heat crosses</text><text x="40" y="316" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">the section boundaries. Sections add in parallel:</text><text x="40" y="332" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">1 / R′T = Σ f_j / R_T,j (overestimates R)</text></g><g id="bd-lower"><rect x="420" y="70" width="300" height="16" fill="light-dark(#ffffff, #202124)" stroke="none"/><circle cx="422" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="427" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="432" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="437" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="442" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="447" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="452" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="457" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="462" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="467" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="472" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="477" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="482" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="487" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="492" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="497" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="502" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="507" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="512" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="517" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="522" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="527" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="532" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="537" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="542" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="547" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="552" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="557" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="562" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="567" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="572" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="577" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="582" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="587" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="592" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="597" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="602" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="607" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="612" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="617" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="622" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="627" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="632" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="637" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="642" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="647" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="652" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="657" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="662" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="667" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="672" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="677" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="682" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="687" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="692" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="697" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="702" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="707" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="712" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="717" cy="75" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="423.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="428.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="433.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="438.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="443.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="448.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="453.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="458.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="463.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="468.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="473.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="478.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="483.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="488.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="493.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="498.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="503.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="508.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="513.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="518.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="523.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="528.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="533.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="538.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="543.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="548.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="553.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="558.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="563.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="568.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="573.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="578.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="583.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="588.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="593.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="598.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="603.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="608.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="613.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="618.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="623.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="628.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="633.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="638.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="643.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="648.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="653.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="658.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="663.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="668.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="673.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="678.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="683.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="688.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="693.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="698.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="703.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="708.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="713.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="718.6" cy="82" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><rect x="420" y="70" width="300" height="16" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="420" y="86" width="300" height="168" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M422,92.72 C424.6,73.4 445.4,266.6 448,247.28 C450.6,266.6 471.4,73.4 474,92.72 C476.6,73.4 497.4,266.6 500,247.28 C502.6,266.6 523.4,73.4 526,92.72 C528.6,73.4 549.4,266.6 552,247.28 C554.6,266.6 575.4,73.4 578,92.72 C580.6,73.4 601.4,266.6 604,247.28 C606.6,266.6 627.4,73.4 630,92.72 C632.6,73.4 653.4,266.6 656,247.28 C658.6,266.6 679.4,73.4 682,92.72 C684.6,73.4 705.4,266.6 708,247.28 C709,266.6 717,73.4 718,92.72" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="420" y="86" width="300" height="168" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="545" y="86" width="50" height="168" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M557.5,86 Q561.5,170 557.5,254" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M570,86 Q574,170 570,254" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><path d="M582.5,86 Q586.5,170 582.5,254" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="545" y="86" width="50" height="168" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><rect x="420" y="254" width="300" height="16" fill="light-dark(#ffffff, #202124)" stroke="none"/><circle cx="422" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="427" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="432" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="437" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="442" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="447" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="452" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="457" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="462" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="467" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="472" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="477" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="482" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="487" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="492" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="497" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="502" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="507" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="512" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="517" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="522" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="527" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="532" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="537" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="542" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="547" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="552" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="557" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="562" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="567" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="572" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="577" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="582" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="587" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="592" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="597" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="602" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="607" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="612" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="617" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="622" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="627" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="632" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="637" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="642" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="647" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="652" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="657" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="662" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="667" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="672" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="677" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="682" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="687" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="692" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="697" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="702" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="707" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="712" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="717" cy="259" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="423.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="428.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="433.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="438.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="443.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="448.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="453.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="458.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="463.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="468.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="473.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="478.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="483.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="488.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="493.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="498.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="503.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="508.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="513.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="518.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="523.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="528.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="533.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="538.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="543.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="548.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="553.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="558.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="563.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="568.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="573.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="578.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="583.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="588.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="593.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="598.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="603.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="608.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="613.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="618.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="623.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="628.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="633.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="638.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="643.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="648.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="653.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="658.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="663.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="668.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="673.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="678.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="683.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="688.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="693.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="698.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="703.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="708.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="713.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><circle cx="718.6" cy="266" r="0.55" fill="light-dark(#26333e, #edf4fa)" stroke="none"/><rect x="420" y="254" width="300" height="16" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="420" y="40" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">b · lower limit R″T: isothermal planes</text><text x="412" y="80" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="end" font-weight="700">in</text><text x="412" y="268" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="end" font-weight="700">out</text><line x1="410" y1="86" x2="730" y2="86" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5" stroke-dasharray="6 3" stroke-linecap="round" stroke-linejoin="round"/><line x1="410" y1="254" x2="730" y2="254" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5" stroke-dasharray="6 3" stroke-linecap="round" stroke-linejoin="round"/><path d="M470,90 Q560,140 570,250" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#bd-arrow-alt)"/><path d="M680,90 Q580,140 580,250" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#bd-arrow-alt)"/><line x1="575" y1="90" x2="575" y2="250" stroke="light-dark(#1f6fa3, #8bcfff)" stroke-width="3" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#bd-arrow-alt)"/><text x="420" y="300" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">Each layer is averaged first; heat may move</text><text x="420" y="316" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">sideways into the stud. Layers add in series:</text><text x="420" y="332" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">R″T = R_si + Σ d_i / λ_mean,i + R_se (underestimates R)</text></g><g id="bd-note"><text x="390" y="366" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="600">ISO 6946 uses R_T = (R′T + R″T) / 2, valid while R′T / R″T ≤ 1.5.</text><text x="390" y="382" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" text-anchor="middle" font-weight="600">Beyond that (steel studs, metal fixings) a numerical 2D or 3D calculation is required (ISO 10211).</text></g></g></svg></div><figcaption><span><strong>Figure 4.</strong> Upper and lower limits for inhomogeneous layers (ISO 6946)</span><a href="./assets/diagrams/heat/iso6946-bounds.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:iso6946-bounds_END -->
<div class="lab">
<div class="lab-canvas"><div class="canvas-heading"><span class="small-label">STUD WALL · EQ. 7.11 · ISO 6946 §6.7</span><button type="button" class="quiet-button" data-export="studs">Save SVG ↓</button></div><div class="diagram-host" id="d-studs"></div><div class="diagram-inputs" id="in-studs"></div></div>
<div class="lab-controls"><p class="control-kicker">PARALLEL PATHS WITH CURRENT VALUES</p><div class="calc" id="calc-studs" aria-live="polite"></div><p class="control-note">Between 12.5 mm gypsum board and 12 mm OSB. Real frames add plates, noggings and headers: a framing fraction of 15–25 % is common.</p></div>
</div>
<dl class="defs">
<div><dt>Thermal bridge · Wärmebrücke</dt><dd>A local area with a higher heat flux than its surroundings, and therefore a lower inside surface temperature. "Cold bridge" (Kältebrücke) is a misnomer: heat flows out, cold does not flow in.</dd></div>
<div><dt>Material bridge</dt><dd>A better conductor crosses the insulation: steel and concrete through the facade, a balcony slab, a lintel, fixings, window frames.</dd></div>
<div><dt>Geometric bridge</dt><dd>The outer surface is larger than the inner one: corners, especially three-dimensional corners at the roof edge. Heat lines converge and the corner is the coldest point of a room.</dd></div>
<div><dt>ψ · W/(m·K) and χ · W/K</dt><dd>The extra heat flow per meter of a linear bridge (slab edge, window reveal) and per point bridge (a column, a bracket), calculated numerically (ISO 10211) or taken from catalogues (DIN 4108 Beiblatt 2). They add to the transmission heat loss: H<sub>T</sub> = Σ U·A + Σ ψ·l + Σ χ (Chapter 08, Eq. 8.2).</dd></div>
</dl>
<ul class="facts">
<li>A continuous concrete balcony slab can lower the corner temperature below the mold limit and add ψ ≈ 0.5–0.9 W/(m·K); a thermally broken connector (insulated, with stainless steel bars) brings ψ to about 0.1–0.2.</li>
<li>In well-insulated buildings bridges can add 10–30 % to the transmission loss. Draw the insulation line around the building section in one stroke without lifting the pen: every interruption is a bridge.</li>
</ul>
</section>
<section id="insulation" class="reader-section">
<div class="section-heading"><span class="section-number">08</span><div><p class="eyebrow">MATERIALS · EN PRODUCT STANDARDS · CARBON</p><h2>Insulation materials</h2></div></div>
<p class="section-lead">Insulation (λ < 0.1 W/(m·K)) traps gas in fibers or closed cells. They differ less in λ than in what else they do: carry load, tolerate water, burn or not, open or block vapor, store or emit carbon, and how they can be installed.</p>
<!-- INSULATIONTABLE_START -->
<div class="table-wrap"><table class="nomenclature compact heat-table"><caption>Insulation materials · typical design values · d for U = 0.20 W/(m²K) with R_si + R_se = 0.17 m²K/W and no other layers</caption><thead><tr><th scope="col">Material</th><th scope="col">Product standard</th><th scope="col">λ W/(m·K)</th><th scope="col">ρ kg/m³</th><th scope="col">c J/(kg·K)</th><th scope="col">d for U 0.20</th><th scope="col">Typical use</th></tr></thead><tbody><tr><th scope="row">Mineral wool (rock, glass)</th><td>MW · EN 13162</td><td>0.035</td><td>30</td><td>830</td><td>169 mm</td><td>Cavities, roofs, ETICS, ventilated facades; non-combustible</td></tr><tr><th scope="row">Expanded polystyrene (EPS)</th><td>EPS · EN 13163</td><td>0.035</td><td>20</td><td>1210</td><td>169 mm</td><td>ETICS, under screeds; cheap, combustible</td></tr><tr><th scope="row">Extruded polystyrene (XPS)</th><td>XPS · EN 13164</td><td>0.030</td><td>35</td><td>1210</td><td>145 mm</td><td>Perimeter and inverted roofs: wet ground, loads</td></tr><tr><th scope="row">Polyisocyanurate (PIR) board</th><td>PUR/PIR · EN 13165</td><td>0.023</td><td>32</td><td>1400</td><td>111 mm</td><td>Thin roof and wall build-ups; foil-faced boards</td></tr><tr><th scope="row">Wood-fiber board</th><td>WF · EN 13171</td><td>0.042</td><td>160</td><td>2100</td><td>203 mm</td><td>Timber frame, roofs; high c damps summer heat</td></tr><tr><th scope="row">Cellulose, blown</th><td>Loose fill, ETA</td><td>0.040</td><td>50</td><td>2000</td><td>193 mm</td><td>Blown into frame cavities; recycled paper</td></tr><tr><th scope="row">Straw (compressed panel)</th><td>ETA (e.g. EcoCocon)</td><td>0.052</td><td>110</td><td>2000</td><td>251 mm</td><td>Prefabricated panels; stores biogenic carbon</td></tr><tr><th scope="row">Cork board</th><td>ICB · EN 13170</td><td>0.039</td><td>120</td><td>1800</td><td>188 mm</td><td>ETICS, floors; moisture tolerant</td></tr><tr><th scope="row">Cellular (foam) glass</th><td>CG · EN 13167</td><td>0.045</td><td>120</td><td>1000</td><td>217 mm</td><td>Under slabs and perimeter; load-bearing, vapor-tight</td></tr><tr><th scope="row">Mineral insulation board (AAC)</th><td>Mineral board, ETA</td><td>0.045</td><td>115</td><td>1300</td><td>217 mm</td><td>Interior insulation (e.g. Multipor); vapor-open</td></tr><tr><th scope="row">Aerogel blanket</th><td>Blanket, ETA</td><td>0.015</td><td>150</td><td>1000</td><td>72 mm</td><td>Renovation where space is tight</td></tr><tr><th scope="row">Vacuum insulation panel (aged)</th><td>ETA</td><td>0.007</td><td>190</td><td>800</td><td>34 mm</td><td>Very thin build-ups; cannot be cut or pierced</td></tr></tbody></table></div>
<!-- INSULATIONTABLE_END -->
<ul class="facts">
<li>The thickness d for a target U follows from Eq. 7.7: d = λ · (1/U − R<sub>si</sub> − R<sub>se</sub> − R<sub>other layers</sub>). Halving λ halves the thickness, which matters most in renovations and at junctions.</li>
<li>Diminishing returns: on a 365 mm solid brick wall (U ≈ 1.3) the first 50 mm of insulation lower U by about 0.85 W/(m²K), two thirds of the loss; going from 200 to 250 mm lowers it by only 0.03 W/(m²K).</li>
<li>Foams with a blowing gas age as the gas diffuses out; foil facings slow this. Vacuum panels lose their vacuum when pierced.</li>
<li>Bio-based materials (wood fiber, cellulose, straw, cork) store carbon taken up by plants; the EcoCocon straw panel of the lecture stores about 98 kg CO<sub>2</sub> per m² of wall at U = 0.12 W/(m²K).</li>
<li>Selection by application in Germany follows the codes of DIN 4108-10 (e.g. WAB: external wall behind cladding; DAD: roof under covering; PW: perimeter); the lecture's DETAIL matrix maps them to materials.</li>
</ul>
</section>
<section id="transient" class="reader-section">
<div class="section-heading"><span class="section-number">09</span><div><p class="eyebrow">HEAT STORAGE · TIME LAG · EQ. 7.12–7.13</p><h2>Steady state and transient heat transfer</h2></div></div>
<p class="section-lead">Outdoor temperature and sunshine change every hour, and walls store and release heat. Steady state ignores storage. That is exact when nothing changes and a good approximation over long periods. When it fails, density and specific heat enter the problem beside conductivity.</p>
<!-- FIG:steady-transient_START -->
<figure class="fig" id="fig-steady-transient"><div class="fig-plate"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 800 384" class="diagram" role="img" aria-label="Temperature profiles in a concrete wall: steady state, after a step change and under a daily cycle"><title>Temperature profiles in a concrete wall: steady state, after a step change and under a daily cycle</title><defs><marker id="st-arrow-ink" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1.5"/></marker><marker id="st-arrow-alt" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5"/></marker><marker id="st-arrow-azi" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5"/></marker><marker id="st-arrow-third" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#6a4fb3, #c3b2ff)" stroke-width="1.5"/></marker><marker id="st-arrow-muted" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="1.5"/></marker><marker id="st-arrow-red" viewBox="0 0 10 10" refX="9" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse" markerUnits="userSpaceOnUse"><path d="M0,1 L9,5 L0,9" fill="none" stroke="light-dark(#b13d4c, #ff9daa)" stroke-width="1.5"/></marker><pattern id="st-hatch-shadow" patternUnits="userSpaceOnUse" width="3.4" height="3.4" patternTransform="rotate(45)"><rect width="3.4" height="3.4" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="3.4" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="st-hatch-shade" patternUnits="userSpaceOnUse" width="7.5" height="7.5" patternTransform="rotate(45)"><rect width="7.5" height="7.5" fill="light-dark(#ffffff, #202124)"/><line x1="0" y1="0" x2="0" y2="7.5" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern><pattern id="st-hatch-screen" patternUnits="userSpaceOnUse" width="4.2" height="4.2" patternTransform="rotate(135)"><line x1="0" y1="0" x2="0" y2="4.2" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6"/></pattern></defs><g font-family="Helvetica Neue, Helvetica, Arial, sans-serif"><g id="st-steady"><rect x="40" y="70" width="200" height="230" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M45.0,76.5l2.5,4.5h-5zM79.0,76.5l2.5,4.5h-5zM113.0,76.5l2.5,4.5h-5zM147.0,76.5l2.5,4.5h-5zM181.0,76.5l2.5,4.5h-5zM215.0,76.5l2.5,4.5h-5zM62.0,106.5l2.5,4.5h-5zM96.0,106.5l2.5,4.5h-5zM130.0,106.5l2.5,4.5h-5zM164.0,106.5l2.5,4.5h-5zM198.0,106.5l2.5,4.5h-5zM232.0,106.5l2.5,4.5h-5zM45.0,136.5l2.5,4.5h-5zM79.0,136.5l2.5,4.5h-5zM113.0,136.5l2.5,4.5h-5zM147.0,136.5l2.5,4.5h-5zM181.0,136.5l2.5,4.5h-5zM215.0,136.5l2.5,4.5h-5zM62.0,166.5l2.5,4.5h-5zM96.0,166.5l2.5,4.5h-5zM130.0,166.5l2.5,4.5h-5zM164.0,166.5l2.5,4.5h-5zM198.0,166.5l2.5,4.5h-5zM232.0,166.5l2.5,4.5h-5zM45.0,196.5l2.5,4.5h-5zM79.0,196.5l2.5,4.5h-5zM113.0,196.5l2.5,4.5h-5zM147.0,196.5l2.5,4.5h-5zM181.0,196.5l2.5,4.5h-5zM215.0,196.5l2.5,4.5h-5zM62.0,226.5l2.5,4.5h-5zM96.0,226.5l2.5,4.5h-5zM130.0,226.5l2.5,4.5h-5zM164.0,226.5l2.5,4.5h-5zM198.0,226.5l2.5,4.5h-5zM232.0,226.5l2.5,4.5h-5zM45.0,256.5l2.5,4.5h-5zM79.0,256.5l2.5,4.5h-5zM113.0,256.5l2.5,4.5h-5zM147.0,256.5l2.5,4.5h-5zM181.0,256.5l2.5,4.5h-5zM215.0,256.5l2.5,4.5h-5zM62.0,286.5l2.5,4.5h-5zM96.0,286.5l2.5,4.5h-5zM130.0,286.5l2.5,4.5h-5zM164.0,286.5l2.5,4.5h-5zM198.0,286.5l2.5,4.5h-5zM232.0,286.5l2.5,4.5h-5z" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="40" y="70" width="200" height="230" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="30" y="40" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">a · steady state</text><text x="36" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="end" font-weight="700">in</text><text x="244" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700">out</text><text x="140" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">300 mm concrete</text><polyline points="40,122 240,248" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="140" y="171" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" text-anchor="middle" font-weight="700">θ(x) fixed in time</text><text x="30" y="344" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">q in = q out at every instant.</text><text x="30" y="360" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">Only λ and d matter (R, U).</text></g><g id="st-step"><rect x="300" y="70" width="200" height="230" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M305.0,76.5l2.5,4.5h-5zM339.0,76.5l2.5,4.5h-5zM373.0,76.5l2.5,4.5h-5zM407.0,76.5l2.5,4.5h-5zM441.0,76.5l2.5,4.5h-5zM475.0,76.5l2.5,4.5h-5zM322.0,106.5l2.5,4.5h-5zM356.0,106.5l2.5,4.5h-5zM390.0,106.5l2.5,4.5h-5zM424.0,106.5l2.5,4.5h-5zM458.0,106.5l2.5,4.5h-5zM492.0,106.5l2.5,4.5h-5zM305.0,136.5l2.5,4.5h-5zM339.0,136.5l2.5,4.5h-5zM373.0,136.5l2.5,4.5h-5zM407.0,136.5l2.5,4.5h-5zM441.0,136.5l2.5,4.5h-5zM475.0,136.5l2.5,4.5h-5zM322.0,166.5l2.5,4.5h-5zM356.0,166.5l2.5,4.5h-5zM390.0,166.5l2.5,4.5h-5zM424.0,166.5l2.5,4.5h-5zM458.0,166.5l2.5,4.5h-5zM492.0,166.5l2.5,4.5h-5zM305.0,196.5l2.5,4.5h-5zM339.0,196.5l2.5,4.5h-5zM373.0,196.5l2.5,4.5h-5zM407.0,196.5l2.5,4.5h-5zM441.0,196.5l2.5,4.5h-5zM475.0,196.5l2.5,4.5h-5zM322.0,226.5l2.5,4.5h-5zM356.0,226.5l2.5,4.5h-5zM390.0,226.5l2.5,4.5h-5zM424.0,226.5l2.5,4.5h-5zM458.0,226.5l2.5,4.5h-5zM492.0,226.5l2.5,4.5h-5zM305.0,256.5l2.5,4.5h-5zM339.0,256.5l2.5,4.5h-5zM373.0,256.5l2.5,4.5h-5zM407.0,256.5l2.5,4.5h-5zM441.0,256.5l2.5,4.5h-5zM475.0,256.5l2.5,4.5h-5zM322.0,286.5l2.5,4.5h-5zM356.0,286.5l2.5,4.5h-5zM390.0,286.5l2.5,4.5h-5zM424.0,286.5l2.5,4.5h-5zM458.0,286.5l2.5,4.5h-5zM492.0,286.5l2.5,4.5h-5z" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="300" y="70" width="200" height="230" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="290" y="40" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">b · step: heating switched on</text><text x="296" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="end" font-weight="700">in</text><text x="504" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700">out</text><text x="400" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">300 mm concrete</text><polyline points="300,122 306.67,132.01 313.33,141.92 320,151.64 326.67,161.07 333.33,170.14 340,178.77 346.67,186.9 353.33,194.48 360,201.49 366.67,207.9 373.33,213.7 380,218.9 386.67,223.52 393.33,227.57 400,231.1 406.67,234.14 413.33,236.73 420,238.92 426.67,240.75 433.33,242.27 440,243.51 446.67,244.52 453.33,245.33 460,245.98 466.67,246.5 473.33,246.92 480,247.25 486.67,247.53 493.33,247.78 500,248" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-dasharray="5 3" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="397.33" y="223" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">2 h</text><polyline points="300,122 306.67,126.51 313.33,131.02 320,135.53 326.67,140.02 333.33,144.5 340,148.96 346.67,153.41 353.33,157.83 360,162.23 366.67,166.6 373.33,170.94 380,175.25 386.67,179.53 393.33,183.78 400,188 406.67,192.18 413.33,196.33 420,200.45 426.67,204.54 433.33,208.6 440,212.63 446.67,216.63 453.33,220.61 460,224.56 466.67,228.5 473.33,232.42 480,236.33 486.67,240.22 493.33,244.11 500,248" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-dasharray="5 3" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="470.67" y="209" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">12 h</text><polyline points="300,122 306.67,126.2 313.33,130.4 320,134.6 326.67,138.8 333.33,143 340,147.2 346.67,151.4 353.33,155.6 360,159.8 366.67,164 373.33,168.2 380,172.4 386.67,176.6 393.33,180.8 400,185 406.67,189.2 413.33,193.4 420,197.6 426.67,201.8 433.33,206 440,210.2 446.67,214.4 453.33,218.6 460,222.8 466.67,227 473.33,231.2 480,235.4 486.67,239.6 493.33,243.8 500,248" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1.5" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="470.67" y="195" font-size="10.5" style="--fs:10.5" fill="light-dark(#2479ad, #8bcfff)" font-weight="700">48 h</text><polyline points="300,122 500,248" fill="none" stroke="light-dark(#607181, #aabccb)" stroke-width="1" stroke-dasharray="2 3" stroke-linecap="round" stroke-linejoin="round"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="360" y="151.8" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)">steady</text><text x="290" y="344" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">Heat goes into storage first; the wall needs</text><text x="290" y="360" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">days to settle (ρ, c and λ all matter).</text></g><g id="st-periodic"><rect x="560" y="70" width="200" height="230" fill="light-dark(#ffffff, #202124)" stroke="none"/><path d="M565.0,76.5l2.5,4.5h-5zM599.0,76.5l2.5,4.5h-5zM633.0,76.5l2.5,4.5h-5zM667.0,76.5l2.5,4.5h-5zM701.0,76.5l2.5,4.5h-5zM735.0,76.5l2.5,4.5h-5zM582.0,106.5l2.5,4.5h-5zM616.0,106.5l2.5,4.5h-5zM650.0,106.5l2.5,4.5h-5zM684.0,106.5l2.5,4.5h-5zM718.0,106.5l2.5,4.5h-5zM752.0,106.5l2.5,4.5h-5zM565.0,136.5l2.5,4.5h-5zM599.0,136.5l2.5,4.5h-5zM633.0,136.5l2.5,4.5h-5zM667.0,136.5l2.5,4.5h-5zM701.0,136.5l2.5,4.5h-5zM735.0,136.5l2.5,4.5h-5zM582.0,166.5l2.5,4.5h-5zM616.0,166.5l2.5,4.5h-5zM650.0,166.5l2.5,4.5h-5zM684.0,166.5l2.5,4.5h-5zM718.0,166.5l2.5,4.5h-5zM752.0,166.5l2.5,4.5h-5zM565.0,196.5l2.5,4.5h-5zM599.0,196.5l2.5,4.5h-5zM633.0,196.5l2.5,4.5h-5zM667.0,196.5l2.5,4.5h-5zM701.0,196.5l2.5,4.5h-5zM735.0,196.5l2.5,4.5h-5zM582.0,226.5l2.5,4.5h-5zM616.0,226.5l2.5,4.5h-5zM650.0,226.5l2.5,4.5h-5zM684.0,226.5l2.5,4.5h-5zM718.0,226.5l2.5,4.5h-5zM752.0,226.5l2.5,4.5h-5zM565.0,256.5l2.5,4.5h-5zM599.0,256.5l2.5,4.5h-5zM633.0,256.5l2.5,4.5h-5zM667.0,256.5l2.5,4.5h-5zM701.0,256.5l2.5,4.5h-5zM735.0,256.5l2.5,4.5h-5zM582.0,286.5l2.5,4.5h-5zM616.0,286.5l2.5,4.5h-5zM650.0,286.5l2.5,4.5h-5zM684.0,286.5l2.5,4.5h-5zM718.0,286.5l2.5,4.5h-5zM752.0,286.5l2.5,4.5h-5z" fill="light-dark(#ffffff, #202124)" stroke="light-dark(#26333e, #edf4fa)" stroke-width="0.6" stroke-linecap="round" stroke-linejoin="round"/><rect x="560" y="70" width="200" height="230" fill="none" stroke="light-dark(#26333e, #edf4fa)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round"/><text x="550" y="40" font-size="12" style="--fs:12" fill="light-dark(#26333e, #edf4fa)" font-weight="700">c · daily cycle outside ±10 K</text><text x="556" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" text-anchor="end" font-weight="700">in</text><text x="764" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)" font-weight="700">out</text><text x="660" y="318" font-size="10.5" style="--fs:10.5" fill="light-dark(#607181, #aabccb)" text-anchor="middle">300 mm concrete</text><polyline points="560,154.46 566.67,153.74 573.33,152.97 580,152.14 586.67,151.25 593.33,150.29 600,149.26 606.67,148.15 613.33,146.96 620,145.68 626.67,144.3 633.33,142.82 640,141.22 646.67,139.51 653.33,137.67 660,135.69 666.67,133.56 673.33,131.27 680,128.81 686.67,126.16 693.33,123.32 700,120.26 706.67,116.97 713.33,113.43 720,109.63 726.67,105.54 733.33,101.15 740,96.42 746.67,91.34 753.33,85.87 760,80" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><polyline points="560,173.54 566.67,174.26 573.33,175.03 580,175.86 586.67,176.75 593.33,177.71 600,178.74 606.67,179.85 613.33,181.04 620,182.32 626.67,183.7 633.33,185.18 640,186.78 646.67,188.49 653.33,190.33 660,192.31 666.67,194.44 673.33,196.73 680,199.19 686.67,201.84 693.33,204.68 700,207.74 706.67,211.03 713.33,214.57 720,218.37 726.67,222.46 733.33,226.85 740,231.58 746.67,236.66 753.33,242.13 760,248" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="2.25" stroke-linecap="round" stroke-linejoin="round"/><polyline points="560,169.42 566.67,169.2 573.33,168.89 580,168.47 586.67,167.94 593.33,167.28 600,166.48 606.67,165.53 613.33,164.41 620,163.12 626.67,161.63 633.33,159.93 640,158.02 646.67,155.87 653.33,153.49 660,150.85 666.67,147.94 673.33,144.76 680,141.31 686.67,137.57 693.33,133.55 700,129.25 706.67,124.66 713.33,119.81 720,114.69 726.67,109.34 733.33,103.77 740,98.01 746.67,92.1 753.33,86.08 760,80" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-dasharray="3 3" stroke-linecap="round" stroke-linejoin="round"/><polyline points="560,155.99 566.67,155.01 573.33,153.97 580,152.89 586.67,151.76 593.33,150.6 600,149.4 606.67,148.19 613.33,146.96 620,145.73 626.67,144.52 633.33,143.34 640,142.21 646.67,141.14 653.33,140.18 660,139.33 666.67,138.62 673.33,138.09 680,137.78 686.67,137.71 693.33,137.92 700,138.47 706.67,139.38 713.33,140.72 720,142.54 726.67,144.87 733.33,147.79 740,151.35 746.67,155.6 753.33,160.62 760,166.45" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-dasharray="3 3" stroke-linecap="round" stroke-linejoin="round"/><polyline points="560,159.05 566.67,159.32 573.33,159.7 580,160.17 586.67,160.77 593.33,161.5 600,162.37 606.67,163.39 613.33,164.58 620,165.95 626.67,167.51 633.33,169.27 640,171.25 646.67,173.46 653.33,175.91 660,178.6 666.67,181.54 673.33,184.75 680,188.22 686.67,191.96 693.33,195.97 700,200.25 706.67,204.78 713.33,209.55 720,214.56 726.67,219.78 733.33,225.17 740,230.73 746.67,236.39 753.33,242.12 760,247.86" fill="none" stroke="light-dark(#2479ad, #8bcfff)" stroke-width="1" stroke-dasharray="3 3" stroke-linecap="round" stroke-linejoin="round"/><line x1="668.05" y1="76" x2="668.05" y2="294" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1.5" stroke-dasharray="6 3" stroke-linecap="round" stroke-linejoin="round"/><line x1="668.05" y1="86" x2="760" y2="86" stroke="light-dark(#007c9a, #6bd8f0)" stroke-width="1" stroke-linecap="round" stroke-linejoin="round" marker-end="url(#st-arrow-azi)"/><text paint-order="stroke" stroke="light-dark(#ffffff, #202124)" stroke-width="4" stroke-linejoin="round" x="662.05" y="90" font-size="10.5" style="--fs:10.5" fill="light-dark(#007c9a, #6bd8f0)" text-anchor="end" font-weight="700">δ = 138 mm</text><text x="550" y="344" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">Envelope of the daily swing; at depth δ the</text><text x="550" y="360" font-size="10.5" style="--fs:10.5" fill="light-dark(#26333e, #edf4fa)">amplitude is 37 %, at 3δ only 5 %.</text></g></g></svg></div><figcaption><span><strong>Figure 5.</strong> Steady state, a step change and a daily cycle in a concrete wall</span><a href="./assets/diagrams/heat/steady-transient.svg" download>Editable SVG ↓</a></figcaption></figure>
<!-- FIG:steady-transient_END -->
<dl class="defs">
<div><dt>Steady state · stationär</dt><dd>Temperatures do not change with time; q is the same in every layer; λ and d decide everything. Over periods longer than about two weeks heat stored and released by the walls cancels out, so steady-state U-values give the seasonal heating demand reliably (Lehrbuch der Bauphysik II 2).</dd></div>
<div><dt>Transient · instationär</dt><dd>Temperatures change; part of the heat entering a layer is stored in it. How fast a temperature change travels into a material is set by its diffusivity α = λ/(ρc); how much heat a surface absorbs from a change by its effusivity b = √(λρc).</dd></div>
</dl>
<div class="eq" id="eq-7-12"><span class="eq-math"><span class="eq-line"><span class="eq-label">diffusivity and the penetration depth of a periodic temperature (period P)</span><math><mi>α</mi><mo>=</mo><mfrac><mi>λ</mi><mrow><mi>ρ</mi><mo>·</mo><mi>c</mi></mrow></mfrac><mo>,</mo><mspace width="0.8em"/><mi>δ</mi><mo>=</mo><msqrt><mfrac><mrow><mi>α</mi><mo>·</mo><mi>P</mi></mrow><mi>π</mi></mfrac></msqrt></math><span class="eq-note">amplitude at depth x: e<sup>−x/δ</sup>, time lag: (x/δ) · P/2π (Incropera Eq. 5.69) <span class="eq-unit">m²/s, m</span></span></span></span><span class="eq-n">(7.12)</span></div>
<!-- PERIODICTABLE_START -->
<div class="table-wrap"><table class="nomenclature compact heat-table"><caption>Daily cycle (P = 24 h): diffusivity, penetration depth, effusivity and heat capacity</caption><thead><tr><th scope="col">Material</th><th scope="col">α = λ/(ρc) · 10⁻⁷ m²/s</th><th scope="col">δ (24 h)</th><th scope="col">b = √(λρc) J/(m²K·s^½)</th><th scope="col">ρ·c kJ/(m³K)</th><th scope="col">Stored per K in the first δ</th></tr></thead><tbody><tr><th scope="row">Concrete (stone mix)</th><td>6.9</td><td>138 mm</td><td>1683</td><td>2024</td><td>279 kJ/(m²K)</td></tr><tr><th scope="row">Brick, common</th><td>4.5</td><td>111 mm</td><td>1074</td><td>1603</td><td>178 kJ/(m²K)</td></tr><tr><th scope="row">Aerated concrete masonry (AAC)</th><td>4.0</td><td>105 mm</td><td>379</td><td>600</td><td>63 kJ/(m²K)</td></tr><tr><th scope="row">Softwood (fir, pine)</th><td>1.7</td><td>68 mm</td><td>291</td><td>704</td><td>48 kJ/(m²K)</td></tr><tr><th scope="row">Wood-fiber board</th><td>1.3</td><td>59 mm</td><td>119</td><td>336</td><td>20 kJ/(m²K)</td></tr><tr><th scope="row">Mineral wool (rock, glass)</th><td>14.1</td><td>197 mm</td><td>30</td><td>25</td><td>5 kJ/(m²K)</td></tr><tr><th scope="row">Expanded polystyrene (EPS)</th><td>14.5</td><td>199 mm</td><td>29</td><td>24</td><td>5 kJ/(m²K)</td></tr><tr><th scope="row">Water (still)</th><td>1.5</td><td>63 mm</td><td>1597</td><td>4180</td><td>265 kJ/(m²K)</td></tr></tbody></table></div>
<!-- PERIODICTABLE_END -->
<p class="measure">A whole room responds with a time constant: its effective heat capacity C (J/K) divided by its heat loss coefficient H (W/K, Chapter 08). After a sudden change the room covers 63 % of the way to its new steady state in one τ.</p>
<div class="eq" id="eq-7-13"><span class="eq-math"><span class="eq-line"><span class="eq-label">time constant of a building or a zone</span><math><mi>τ</mi><mo>=</mo><mfrac><mi>C</mi><mi>H</mi></mfrac></math><span class="eq-note">the lecture house (H = 258 W/K, 70 m² floor): light construction C ≈ 110 kJ/(m²K) × 70 m² → τ ≈ 8 h; heavy, 260 kJ/(m²K) → τ ≈ 20 h (ISO 13790 classes) <span class="eq-unit">s, h</span></span></span></span><span class="eq-n">(7.13)</span></div>
<div class="compare-grid"><div><p class="control-kicker">STEADY STATE IS ENOUGH FOR</p>
<ul class="facts">
<li>U-values, code compliance (GEG in Germany, IECC / ASHRAE 90.1 in the US) and comparing constructions.</li>
<li>The design heat load on a cold, overcast winter night (Chapter 08).</li>
<li>Seasonal and annual heating demand in heating-dominated climates (degree-day and monthly methods).</li>
<li>Inside surface temperatures and the mold check at a design condition.</li>
</ul>
</div>
<div><p class="control-kicker">TRANSIENT IS CRITICAL FOR</p>
<ul class="facts">
<li><strong>Summer overheating and peak cooling loads</strong>: sun and internal gains peak for hours; mass shifts and damps the peak (time lag, decrement; TAV in German texts).</li>
<li><strong>Night ventilation</strong> and passive cooling, which only work with accessible mass (lecture frames 572–574).</li>
<li><strong>Intermittent heating</strong>: churches, schools, holiday homes. Light inner surfaces (small b) warm up fast; with a night setback heavy and light buildings save about the same (Bauphysik II 3.2).</li>
<li><strong>Passive solar storage</strong> (Trombe walls, slabs in the sun) and how much solar gain can be used at all (Chapter 08, section 07).</li>
<li><strong>Touch and floors</strong>: a concrete floor feels cold, cork warm, at the same temperature. Skin (b ≈ 1000) at 28 °C on concrete (b ≈ 2200) at 15 °C meets at 19 °C; on cork (b ≈ 100) at 27 °C.</li>
</ul>
</div>
</div>
<ul class="facts">
<li>Only the first 10–15 cm of concrete or brick take part in the daily cycle (δ ≈ 110–140 mm): thicker mass adds little to daily storage. Mass must be exposed to the room: a suspended ceiling, carpet or interior insulation disconnects it.</li>
<li>Insulation has a high diffusivity but stores almost nothing (ρc ≈ 25–50 kJ/(m³K)): it is not thermal mass. Wood fiber, with c ≈ 2100 J/(kg·K) and ρ ≈ 160 kg/m³, stores more than ten times as much heat per volume as mineral wool at a similar λ.</li>
<li>Hourly building simulation (EnergyPlus, ClimateStudio, ISO 52016) solves the transient problem; Chapter 08 introduces the steady balance it builds on.</li>
</ul>
</section>
<section id="quiz" class="reader-section quiz-section">
<div class="section-heading"><span class="section-number">10</span><div><p class="eyebrow">9 QUESTIONS · CALCULATION AND CONCEPT</p><h2>Self-test</h2></div></div>
<p class="section-lead">"New values" generates a new variant of a calculation. Each answer shows a worked solution.</p>
<form id="quiz-form" data-quiz="heat" novalidate><div id="quiz-questions"></div><div class="quiz-actions"><button class="button" type="submit">Check all answers</button><button id="reset-quiz" class="quiet-button" type="button">Clear</button><p id="quiz-score" role="status"></p></div></form>
</section>
<section id="figures" class="reader-section references-section">
<p class="eyebrow">EDITABLE VECTOR FILES</p><h2>Figure files</h2>
<p class="measure">Editable SVG files of every figure. Interactive drawings save their current state with "Save SVG".</p>
<!-- FIGLIST_START -->
<ol class="fig-list"><li><a href="./assets/diagrams/heat/heat-modes.svg" download>heat-modes.svg</a> <span>· Fig. 1 · Conduction, convection and radiation in a wall section</span></li><li><a href="./assets/diagrams/heat/resistor-network.svg" download>resistor-network.svg</a> <span>· Fig. 2 · Temperature profile and the equivalent resistor network (lecture example)</span></li><li><a href="./assets/diagrams/heat/air-cavity.svg" download>air-cavity.svg</a> <span>· Fig. 3 · Air layers: conduction, convection cell and radiation</span></li><li><a href="./assets/diagrams/heat/iso6946-bounds.svg" download>iso6946-bounds.svg</a> <span>· Fig. 4 · Upper and lower limits for inhomogeneous layers (ISO 6946)</span></li><li><a href="./assets/diagrams/heat/steady-transient.svg" download>steady-transient.svg</a> <span>· Fig. 5 · Steady state, a step change and a daily cycle in a concrete wall</span></li></ol>
<!-- FIGLIST_END -->
</section>
<section id="references" class="reader-section references-section">
<p class="eyebrow">SOURCES</p><h2>Sources and model notes</h2>
<ol class="source-list">
<li>T. L. Bergman, A. S. Lavine, F. P. Incropera, D. P. DeWitt, <em>Fundamentals of Heat and Mass Transfer</em>, 6th ed., Wiley, 2007. §1.2–1.3, §3.1–3.2, §5.1–5.2, §5.9, §9.8, §12.4–12.7, §13.2; Tables 1.1, A.3, A.11, A.12.<span>Rate equations (Eq. 7.2–7.5), composite walls, Biot number, periodic heating (Eq. 7.12), enclosures, emissivity and solar absorptivity.</span></li>
<li>H.-M. Fischer, R. Jenisch, M. Stohrer et al., <em>Lehrbuch der Bauphysik</em>, 6th ed., Vieweg+Teubner, 2008. Part II Wärme: §1.4, §2.1–2.4, §3, §6–7, §8.3–8.5.<span>ISO 7345 terms, surface and air-layer resistances, temperature profile and graphical method, effusivity and contact temperature, TAV, thermal bridges, f<sub>Rsi</sub>.</span></li>
<li>EN ISO 6946:2017, <em>Building components and building elements — Thermal resistance and thermal transmittance — Calculation methods</em> (DIN EN ISO 6946).<span>R<sub>si</sub>, R<sub>se</sub>, Annex C surface coefficients, Annex D air layers (Eq. 7.10), upper and lower limits.</span></li>
<li>DIN 4108-2:2013, <em>Wärmeschutz und Energie-Einsparung in Gebäuden — Mindestanforderungen an den Wärmeschutz</em>; DIN 4108-4 (design values); DIN 4108 Beiblatt 2 (thermal bridges).<span>f<sub>Rsi</sub> ≥ 0.70, design values of λ.</span></li>
<li>ASHRAE, <em>Handbook — Fundamentals</em>, SI edition, Chapters 4 (heat transfer), 25–27 (heat, air and moisture; material properties; assembly U-factors).<span>US surface films and design values; R-value conversion.</span></li>
<li>EN ISO 10211 (thermal bridges, numerical), EN ISO 13788 (surface temperature to avoid mold), EN ISO 10456 (design values), EN ISO 13790 / EN ISO 52016-1 (heat capacity classes, hourly method).<span>Section 07–09.</span></li>
<li>S. V. Szokolay, <em>Introduction to Architectural Science</em>, 3rd ed., Routledge, 2014; N. Lechner, <em>Heating, Cooling, Lighting</em>, 4th ed., Wiley, 2015.<span>Units, specific heat, the water analogy and the lecture examples.</span></li>
<li>T. Dogan, ES1 lecture "Heat transfer, insulation materials", Cornell AAP, 2026, frames 516–636.<span>Order of topics and the worked examples.</span></li>
</ol>
<details class="math-details"><summary>Model assumptions and precision</summary><p>One-dimensional, steady conduction through plane, homogeneous layers with constant properties. Surface coefficients follow ISO 6946 Annex C with the inside at 20 °C and the outside at 10 °C; the radiation coefficient is linearized around that mean temperature. Air layers follow Annex D for large, unventilated cavities. The stud lab applies the ISO 6946 combined method; it does not replace a numerical calculation where R′<sub>T</sub>/R″<sub>T</sub> > 1.5. The dew point uses the Magnus formula (Sonntag constants); the mold limit is the surface temperature at which the room air reaches 80 % RH. Material values are typical design values; use product data and national tables for a real project. Section 09 uses a finite-difference solution for Figure 5 and the semi-infinite solution of Incropera Eq. 5.69 for the penetration depth.</p>
</details>
<p><a class="text-link" href="./heat-balance.html">Continue to Chapter 08: building heat balance ↗</a></p>
<div class="reader-end"><span>ES1 / ENVIRONMENTAL SYSTEMS I</span>
<a href="#top">Back to top ↑</a>
</div>
</section>
</main>
</body>
</html>