Figure 5.11 Examples A5.1 and A5.2: geometry for cubic enclosure
Table 5.42 Example A5.1: surface data
Surface Area / m2 U-value Emissivity Convective Inside surface Temp. on number / W.m–2.K–1 of surface, εn heat transfer resistance, Rsi outer side of
coefficient, hc / m2.K.W–1 surface / °C
1 100 1.00 1.00 3.00 0.12 –1.0
2 100 1.00 1.00 3.00 0.12 –1.0
3 100 1.00 0.8 3.00 0.12 –1.0
4 100 1.00 0.6 3.00 0.12 –1.0
5 100 1.00 0.4 3.00 0.12 –1.0
6 100 1.00 0.2 3.00 0.12 –1.0
10 m
10 m 3 2 5
1 1
4 3
4 6
10 m
Example A5.1: Cubic enclosure; uniform U-values, varying emissivities
See Figure 5.11 and Tables 5.42 to 5.45. This example demonstrates a weakness of the Simple Model when emissivities differ from 0.9. It offers a means of checking that a computer program can account for variable emis- sivity.
Operative temperature: 21 °C Outside air temperature: –1 °C Infiltration rate: 0.5 h–1
Table 5.43 Example A5.1: heat loss
Emitter Component Heat loss using Percentage difference using stated model characteristics of heat loss Reference Model
(% convective) / W Basic Model Simple Model
100 Fabric 12912 0 –6.4
Ventilation 4258 0 –0.6
Total 17170 0 –4.8
70 Fabric 13209 0 –4.6
Ventilation 3886 0 0.7
Total 17095 0 –3.2
30 Fabric 13600 0 –2.4
Ventilation 3396 0 2.1
Total 16996 0 –1
Table 5.44 Example A5.1: surface temperatures
Emitter Surface Surface temperature Difference in calculated surface characteristics number using Reference temperature using stated model / K
(% convective) Model / °C
Basic Model Simple Model
100 1 17.52 0 –0.2
2 17.52 0 –0.2
3 17.49 0 0.11
4 17.43 0 0.52
5 17.29 0 1.09
6 16.97 0 1.9
70 1 18.6 0 0.28
2 18.6 0 0.28
3 18.29 0 0.32
4 17.88 0 0.4
5 17.28 0 0.52
6 16.33 0 0.75
30 1 20.01 0 0.89
2 20.01 0 0.89
3 19.34 0 0.59
4 18.47 0 0.22
5 17.26 0 –0.24
6 15.48 0 –0.79
Table 5.45 Example A5.1 radiation view factors
Surface View factor for stated surface as viewed from surface indicated in
number first column
1 2 3 4 5 6
1 0 0.2000 0.2000 0.2000 0.2000 0.2000
2 0.2000 0 0.2000 0.2000 0.2000 0.2000
3 0.2000 0.2000 0 0.2000 0.2000 0.2000
4 0.2000 0.2000 0.2000 0 0.2000 0.2000
5 0.2000 0.2000 0.2000 0.2000 0 0.2000
6 0.2000 0.2000 0.2000 0.2000 0.2000 0
5ã0 m
1
4
6 3
7 2
4ã0 m
2ã8 m 0ã8 m
3ã0 m
1ã0 m 1ã2 m
2ã7 m
0ã8 m 2ã0 m
5 1 8
8 5
7 2
Figure 5.12 Example A5.2: geometry for typical room
Table 5.46 Example A5.2: surface data
Surface Area / m2 U-value Emissivity Convective Inside surface Temperature on number / W.m–2.K–1 of surface, εn heat transfer resistance, Rsi outer side of
coefficient, hc / m2.K.W–1 surface / °C
1 20.0 0.4 0.9 4.3 0.1 –1.0
2 10.4 0.6 0.9 3.0 0.12 –1.0
3 11.2 2.5 0.9 3.0 0.12 18.0
4 14.0 2.5 0.9 3.0 0.12 18.0
5 9.6 0.6 0.9 3.0 0.12 –1.0
6 20.0 0.2 0.9 1.5 0.14 –1.0
7 3.6 3.0 0.9 3.0 0.12 –1.0
8 1.6 2.0 0.9 3.0 0.12 –1.0
Table 5.47 Example A5.2: heat loss
Emitter Component Heat loss using Percentage difference using stated model characteristics of heat loss Reference Model
(% convective) / W Basic Model Simple Model
100 Fabric 959 0.1 3.9
Ventilation 444 0 0.2
Total 1403 0.1 2.7
50 Fabric 1018 –0.2 0.8
Ventilation 408 0 0.3
Total 1426 –0.1 0.7
Example A5.2: Typical room; varying U-values, uniform emissivities
See Figure 5.12 and Tables 5.46 to 5.49.This example indicates the uncertainties likely to occur in a typical design situation.
Operative temperature: 21 °C Outside air temperature: –1 °C Infiltration rate: 1.0 h–1
Table 5.48 Example A5.2: surface temperatures
Emitter Surface Surface temperature Difference in calculated surface characteristics number using Reference temperature using stated model / K
(% convective) Model / °C
Basic Model Simple Model
100 1 19.85 –0.01 0.28
2 19.12 0.12 0.14
3 19.77 –0.01 –0.35
4 19.72 –0.05 –0.4
5 19.01 0.01 0.02
6 19.37 –0.02 –0.45
7 12.5 0.04 –0.31
8 15.15 0.05 0.19
50 1 21.15 –0.06 –0.04
2 19.7 0.18 0.11
3 20.21 0.03 0.03
4 20.16 –0.04 –0.02
5 19.58 0.07 –0.01
6 20.44 –0.1 –0.02
7 12.82 0.11 –0.41
8 15.53 0.15 –0.3
Table 5.49 Example A5.2: radiation view factors
Surface View factor for stated surface as viewed from surface indicated in first column number
1 2 3 4 5 6 7 8
1 0 0.1358 0.1458 0.1849 0.1289 0.3387 0.0491 0.0169
2 0.2611 0 0.1544 0.1690 0.1449 0.2611 0 0.0095
3 0.2604 0.1434 0 0.1845 0.0943 0.2604 0.0411 0.0159
4 0.2641 0.1255 0.1476 0 0.1158 0.2641 0.0511 0.0318
5 0.2686 0.1570 0.1101 0.1689 0 0.2525 0.0429 0
6 0.3387 0.1358 0.1458 0.1849 0.1212 0 0.0491 0.0246
7 0.2727 0 0.1279 0.1986 0.1143 0.2727 0 0.0136
8 0.2107 0.0616 0.1116 0.2782 0 0.3073 0.0306 0
Figure 5.13 Example A5.3: geometry for atrium (9)
(7) 5
(9)
3 Hidden surfaces
indicated by parentheses
8
(10)
(2)
4 10 m
30 m
20 m 10 m 1
Table 5.50 Example A5.3: surface data
Surface Area / m2 U-value Emissivity Convective Inside surface Temperature on number / W.m–2.K–1 of surface, εn heat transfer resistance, Rsi outer side of
coefficient, hc / m2.K.W–1 surface / °C
1 200.0 3.0 0.8 4.3 0.1 –4.0
2 200.0 0.7 0.9 1.5 0.14 –1.0
3 600.0 2.0 0.9 3.0 0.12 21.0
4 300.0 2.0 0.9 3.0 0.12 21.0
5 200.0 2.0 0.8 3.0 0.12 21.0
6 100.0 3.0 0.8 3.0 0.12 –1.0
7 200.0 3.0 0.8 3.0 0.12 –1.0
8 100.0 3.0 0.8 3.0 0.12 –1.0
9 300.0 0.45 0.9 3.0 0.12 –1.0
10 600.0 0.45 0.9 3.0 0.12 –1.0
Table 5.51 Example A5.3: heat loss
Emitter Component Heat loss using Percentage difference using stated model characteristics of heat loss Reference Model
(% convective) / W Basic Model Simple Model
100 Fabric 46923 4.7 8.6
Ventilation 6976 0.8 1.7
Total 53489 4.2 7.7
50 Fabric 49903 4.5 6.1
Ventilation 5928 0.4 1.3
Total 55831 4.1 5.6
Example A5.3: Atrium; varying U-values, varying emissivities
See Figure 5.13 and Tables 5.50 to 5.53. This example illustrates a non-typical application, in this case an atrium.
It is intended mainly to assist in comparing computer programs.
Operative temperature: 21 °C Outside air temperature: –1 °C Infiltration rate: 0.1 h–1
Table 5.52 Example A5.3: surface temperatures
Emitter Surface Surface temperature Difference in calculated surface characteristics number using Reference temperature using stated model / K
(% convective) Model / °C
Basic Model Simple Model
100 1 11.84 –1.02 –0.69
2 18.44 0.99 0.37
3 20.83 0.56 –0.17
4 20.93 0.51 –0.07
5 18.71 –1.88 –2.29
6 11.22 –1.14 –1.34
7 11.35 –1.06 –1.2
8 11.19 –1.17 –1.36
9 19.79 0.65 0.83
10 19.96 0.9 1.01
50 1 11.97 –0.99 –1.22
2 19.78 1.05 0.95
3 21.46 0.47 0.46
4 21.55 0.54 0.55
5 19.14 –1.88 –1.86
6 11.58 –1.12 –1.51
7 11.71 –1.06 –1.38
8 11.56 –1.14 –1.54
9 20.55 0.69 0.8
10 20.72 0.78 0.97
Table 5.53 Example A5.3: radiation view factors
Surface View factor for stated surface as viewed from surface indicated in first column
1 2 3 4 5 6 7 8 9 10
1 0 0.0362 0.0740 0.0509 0.2406 0.1164 0.2406 0.1164 0.0509 0.0740
2 0.0362 0 0.3081 0.1617 0.0065 0.0056 0.0065 0.0056 0.1617 0.3081
3 0.0247 0.1027 0 0.1595 0 0.0121 0.0539 0.0121 0.1595 0.4756
4 0.0339 0.1078 0.3190 0 0.0242 0.0255 0.0242 0 0.1464 0.3190
5 0.2406 0.0065 0 0.0362 0 0.1164 0.2859 0.1164 0.0362 0.1617
6 0.2329 0.0112 0.0725 0.0766 0.2329 0 0.2329 0.0686 0 0.0725
7 0.2406 0.0065 0.1617 0.0362 0.2859 0.1164 0 0.1164 0.0362 0
8 0.2329 0.0112 0.0725 0 0.2329 0.0686 0.2329 0 0.0766 0.0725
9 0.0339 0.1078 0.3190 0.1464 0.0242 0 0.0242 0.0255 0 0.3190
10 0.0247 0.1027 0.4756 0.1595 0.0539 0.0121 0 0.0121 0.1595 0
Example A5.4: Enclosure with multiple surfaces; varying U-values, uniform emissivities
See Figure 5.14 and Tables 5.54 to 5.57. In this example, the calculation methods are applied to multiple surfaces. It is intended to assist in checking computer programs to determine the accuracy with which surface temperatures are calculated.
Operative temperature: 21 °C Outside air temperature: –1 °C Infiltration rate: 1.0 h–1
Figure 5.14 Example A5.4: geometry for enclosure with multiple surfaces
Table 5.54 Example A5.4: surface data
Surface Area / m2 U-value Emissivity Convective Inside surface Temperature on number / W.m–2.K–1 of surface, εn heat transfer resistance, Rsi outer side of
coefficient, hc / m2.K.W–1 surface / ºC
1 50.0 1.0 0.8 1.5 0.14 –1.0
2 30.0 5.6 0.8 3.0 0.12 –1.0
3 3.0 1.0 0.8 3.0 0.12 –1.0
4 3.0 1.0 0.8 3.0 0.12 –1.0
5 3.0 1.0 0.8 3.0 0.12 –1.0
6 3.0 1.0 0.8 3.0 0.12 –1.0
7 3.0 1.0 0.8 3.0 0.12 –1.0
8 30.0 1.0 0.8 3.0 0.12 –1.0
9 15.0 1.0 0.8 3.0 0.12 –1.0
10 10.0 1.0 0.8 4.3 0.10 –1.0
11 10.0 1.0 0.8 4.3 0.10 –1.0
12 10.0 1.0 0.8 4.3 0.10 –1.0
13 10.0 1.0 0.8 4.3 0.10 –1.0
14 10.0 1.0 0.8 4.3 0.10 –1.0
Table 5.55 Example A5.4: heat loss
Emitter Component Heat loss using Percentage difference using stated model characteristics of heat loss Reference Model
(% convective) / W Basic Model Simple Model
100 Fabric 6560 –0.2 0.4
Ventilation 1402 0.1 –1.8
Total 7962 –0.2 0.1
50 Fabric 7116 0.1 0.2
Ventilation 1141 –0.1 0.0
Total 8256 0.0 0.1
10 m
5 m (9)
(1) Hidden surfaces indicated
by parentheses
(8) 3 m
4 5 3
2 6 7
10 11 12 13 14
Table 5.56 Example A5.4: surface temperatures
Emitter Surface Surface temperature Difference in calculated surface characteristics number using Reference temperature using stated model / K
(% convective) Model / °C
Basic Model Simple Model
100 1 15.07 0.25 –1.13
2 5.74 0.02 –0.06
3 16.06 –0.91 –0.64
4 16.71 –0.26 0.01
5 17.07 0.1 0.37
6 17.3 0.33 0.6
7 17.46 0.48 0.75
8 16.94 0.09 0.23
9 16.92 –0.01 0.22
10 17.26 –1.03 0.24
11 17.81 –0.47 0.79
12 18.18 –0.11 1.16
13 18.42 0.14 1.4
14 18.61 0.32 1.59
50 1 17.66 –0.19 0.05
2 6.28 0 –0.08
3 17.42 –0.77 –0.75
4 18.14 –0.06 –0.03
5 18.53 0.34 0.36
6 18.77 0.58 0.61
7 18.92 0.73 0.76
8 18.37 0.04 0.21
9 18.35 0.11 0.19
10 17.87 –0.85 –0.64
11 18.51 –0.21 0
12 18.91 0.2 0.41
13 19.17 0.45 0.66
14 19.33 0.61 0.82
Table 5.57 Example A5.4: radiation view factors
Surface View factor for stated surface as viewed from surface indicated in first column number
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 0 0.1867 0.0155 0.0188 0.0196 0.0188 0.0155 0.1867 0.0083 0.0783 0.0957 0.1019 0.0957 0.0783 2 0.3112 0 0.0359 0.0220 0.0151 0.0109 0.0082 0.1934 0.0921 0.1311 0.0803 0.0490 0.0307 0.0211
3 0.2585 0.3593 0 0 0 0 0 0.0821 0.0416 0.1361 0.0652 0.0304 0.0168 0.0101
4 0.3136 0.2198 0 0 0 0 0 0.1092 0.0438 0.0652 0.1361 0.0652 0.0304 0.0168
5 0.3272 0.1505 0 0 0 0 0 0.1505 0.0445 0.0304 0.0652 0.1361 0.0652 0.0304
6 0.3136 0.1092 0 0 0 0 0 0.2198 0.0438 0.0168 0.0304 0.0652 0.1361 0.0652
7 0.2585 0.0821 0 0 0 0 0 0.3593 0.0416 0.0101 0.0168 0.0304 0.0652 0.1361
8 0.3112 0.1934 0.0082 0.0109 0.0151 0.0220 0.0359 0 0.0921 0.0200 0.0307 0.0409 0.0803 0.1311 9 0.2943 0.1842 0.0083 0.0088 0.0089 0.0088 0.0083 0.1842 0 0.0517 0.0627 0.0654 0.0627 0.0517
10 0.3916 0.3934 0.0408 0.0196 0.0091 0.0050 0.0030 0.0600 0.0776 0 0 0 0 0
11 0.4786 0.2409 0.0196 0.0408 0.0196 0.0091 0.0050 0.0921 0.0941 0 0 0 0 0
12 0.5094 0.1471 0.0091 0.0196 0.0408 0.0196 0.0091 0.1471 0.0982 0 0 0 0 0
13 0.4786 0.0921 0.0050 0.0091 0.0196 0.0408 0.0196 0.2409 0.0941 0 0 0 0 0
14 0.3916 0.0600 0.0030 0.0050 0.0091 0.0196 0.0408 0.3934 0.0776 0 0 0 0 0
5.A5.1 Introduction
The Simple (dynamic) Model is based on the assumption that all loads and heat flows comprise a daily mean (or steady state) value and an alternating component (i.e.
swing, or deviation from the mean). The total load (or temperature) is the sum of these two components. Thus the calculation of the space load requires the evaluation of each.
The space load comprises the following elements:
— solar gain through glazing
— internally generated heat
— conduction through walls and glazing
— air infiltration.
The first of these is treated separately, see Appendix 5.A7.
Internally generated heat gains are calculated from design conditions. For the purposes of the Simple Model a single design level is assumed constant during the occupied hours.
Thus the mean level is the sum of the gain over the occupied hours divided by 24, the number of hours in a day.
The swing is the difference between the design load during the occupied hours and the daily mean value. These two elements must be divided into convective and radiant components. The model also assumes that heat is added (or removed) at two nodes, the air node and the environmental node. In the case of radiant gains 150% of that gain is realised at the environmental node (a hypothetical heat transfer node) with the excess 50% removed at the air node, see Appendix 5.A3, section 5.A3.5. The whole of the convective gain is realised at the air node.
Conduction and infiltration gains are discussed in the following sections where the mean and alternating compo- nents are handled separately. Furthermore, it is recognised that the design intent may be to cool to either a specific operative or air temperature therefore calculations for both cases are given.
For the purposes of this appendix, emitter output is taken as positive for heating and negative for cooling. When apply- ing the equations, they are presented such that both heating and cooling loads are expressed as positive values. This also necessitates the use of a single symbol (Φp) for the emitter output. When applied, cooling is distinguished from heating by replacing Φpby Φkin the case of cooling load.
5.A5.2 Notation
Symbols used in this appendix are as follows. Some of the quantities occur in three forms: the instantaneous value, which is denoted by the appropriate letter (e.g. X). The 24- hour mean or steady state value, denoted by X–
; and the instantaneous variation about the mean value, denoted by X~. Where appropriate, the variation symbol is given a subscript to indicate the time at which it occurs, e.g. X~tis the value of X~at time t.
A Surface area (m2)
Cv Ventilation conductance (W.K–1) cp Specific heat capacity of air (Jã kg–1.K–1) f Decrement factor
Fau Room conduction factor with respect to air node Fay Room admittance factor with respect to air node Fcu Room conduction factor with respect to operative
temperature
Fcy Room admittance factor with respect to operative temperature
F1au, F2au Conduction factors related to characteristics of heat source with respect to air temperature
F1cu, F2cu Conduction factors related to characteristics of heat source with respect to operative temperature F1ay, F2ay Admittance factors related to characteristics of
heat source with respect to air temperature F1cy, F2cy Admittance factors related to characteristics of
heat source with respect to operative temperature ha Heat transfer coefficient between air and environ-
mental nodes (W.m–2.K–1)
hc Convective heat transfer coefficient (W.m–2.K–1) hr Radiative heat transfer coefficient of a black body
(W.m–2.K–1)
qv Total ventilation (mechanical plus infiltration) rate (m3.s–1)
R Radiant fraction of the heat source t Time (h)
U Thermal transmittance (W.m–2.K–1) Y Thermal admittance (W.m–2.K–1) θai Inside air temperature (°C) θao Outside air temperature (°C)
θc Operative temperature at centre of room (°C) θei Environmental temperature (°C)
θeo Sol-air temperature (°C) θm Surface temperature (°C) θr Mean radiant temperature (°C) Φa Heat flow to the air node (W) Φk Cooling load (W)
Φe Heat flow to the environmental node (W) Φp Emitter output (W)
Φpa Emitter output supplied to the air node (W) Φpe Emitter output supplied to the environmental node Φsg (W)Solar gain (W)
ρ Density of air (kgãm–3)
ΣA Sum of room surface areas, unless otherwise indicated (m2)
Σ(AU) Sum of the products of surface area and correspon- ding thermal transmittance over surfaces through which heat flow occurs (W.K–1)
Σ(A Y) Sum of the products of surface area and correspon- ding thermal admittance over all surfaces (W.K–1)
Σ Φ–con Sum of daily mean convective heat gains (W)
Σ Φ–rad Sum of daily mean radiant heat gains (W)
5.A5.3 Calculation of the steady state load
The Simple (steady state) Model assumes that heat enters the space at two points: the air node and a hypothetical node called the environmental temperature node. For a