SE-99-1-1
© 1999, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions 1999, Vol 105, Part 2. For personal use only. Additional distribution in either paper or digital form is not permitted without ASHRAE’s permission.
Development of Periodic Response Factors for Use with the Radiant Time Series Method Jeffrey D. Spitler, Ph.D., P.E.
Daniel E. Fisher, Ph.D.
Member ASHRAE
Member ASHRAE
ABSTRACT Harris and McQuiston (1988) developed conduction transfer function (CTF) coefficients corresponding to 41 representative wall assemblies and 42 representative roof assemblies for use with the transfer function method (TFM). They also developed a grouping procedure that allows design engineers to determine the correct representative wall or roof assembly that most closely matches a specific wall or roof assembly. The CTF coefficients and the grouping procedure have been summarized in the ASHRAE Handbook —Fundamentals (1989, 1993, 1997) and the ASHRAE Cooling and Heating Load Calculation Manual, second edition (McQuiston and Spitler 1992). More recently, a new, simplified design cooling load calculation procedure, procedure, the radiant time series method (RTSM), has been developed (Spitler et al. 1997). The RTSM uses periodic response factors to model transient conductive heat trans fer. fer. While not a true manual load calculation procedure, it is quite feasible to implement the RTSM in a spreadsheet. To To be useful in such an environment, it would be desirable to have a pre-calculated pre-calculated set of periodic response factors. Accordingly, Accordingly, a set of periodic response factors has been calculated and is presented in this paper.
INTRODUCTION The transfer function method (TFM) for design cooling load calculations has been in use for a number of years. This method uses conduction transfer functions (CTFs) to calculate the transient, one-dimensional conduction through the building wall and roof elements. Conduction transfer functions are a closed form representation of a conduction response factor series. Conduction response factors, as derived by Mitalas and
Stephenson (1967) and Hittle (1981), are an exact solution to the transient conductive heat transfer problem for a multilayer wall or roof with boundary conditions that can be represented by a piecewise linear profile. The response factor series is infinite, so in practice it must be truncated, resulting in some minor, but controllable, loss of accuracy. Stephenson and Mitalas (1967) compared a response factor method to an analog computer simulation and showed that for the one-hour time steps commonly used in building energy and thermal load calculations, the expected error due to truncation of the infinite response factor series is small. Procedures for developing conduction transfer functions from response factors are described by Peavy (1978) and Hittle (1981). The methods are necessarily inexact but have been compared to both analytical and numerical solutions with excellent results. Maloney (1985) showed that for a one-dimensional, transient slab with linearly varying surface temperatures, the differences between a CTF solution and an analytical solution based on Duhamel’s method are negligible. ASHRAE research project RP-472 provided a set of conduction transfer function coefficients (Harris and McQuiston 1988) corresponding to 41 representative roof types and 42 representative wall types. In addition, a grouping procedure was developed so that, theoretically, any wall or roof could be mapped into one of the representative wall types. The walls are mapped using four parameters: primary wall material (the most thermally massive element), secondary wall material, R-value, and the thermal mass location (in, out, or integral). Roofs are also mapped using four parameters: primary roof material, thermal mass location, R-value, and presence or absence of a suspended ceiling. Once the representative wall or roof type has been identified, the CTF coefficients may be “unnormalized” so that the U-factor of the
Jeffrey D. Spitler is associate professor profes sor at Oklahoma State University, Stillwater. Daniel E. Fisher is senior research engineer at the University of Illinois at Urbana-Champaign.
ASHRAE Transactions: Symposia
49 1
TABLE 1 Nomenclature
actual wall or roof is preserved in the CTF coefficients. coefficients. Once the coefficients have been determined, they are applied using the conduction transfer function: 6 ″
qθ =
6 ″
∑ b n T e, θ – n δ – ∑ d n qθ – n δ – T rc ∑ c n n=0
Subscripts
6
n=1
q´´
heat flux, (Btu/h)/ft 2 (W/m2)
P
periodic
T
temperature, ºF (°C)
j
index
b, d, c
conduction transfer function coefficients
n
index
Y
periodic response factors
δ
time interval, h
e
sol-air
(1)
n=0
Since originally appearing in the 1989 ASHRAE Handbook—Fundamentals, the tabulated CTF coefficients have proved to be useful to designers. The radiant time series method (RTSM) uses periodic response factors to model transient conductive heat transfer: 23
Matrices
23
″
j = 0
q´´
column vector containing conductive heat fluxes
rc
constant room air
T e
column vector containing sol-air temperatures temperatures
θ
current hour
(2)
qθ = ∑ Y Pj T e, θ – j δ – T rc ∑ Y Pj j = 0
Since the RTSM is amenable to spreadsheet-level implementation, a tabulated set of periodic response factors is a useful addition to the literature. The objective of this paper is to present a set of periodic response factors that correspond exactly to the CTFs previously published. If desired, they can be used with exactly the same grouping procedure and a slightly different “unnormalization” procedure.
B, D
b d CTF coefficient and matrices
Y
periodic response factor matrix
T rc
column vector containing T rc in every row
METHODOLOGY Periodic response factors can be derived directly from a set of CTF coefficients. coefficients. The procedure is discussed in greater detail in another paper (Spitler and Fisher 1999), but is presented here to document the method to generate the periodic response factors directly from the tabulated CTF coefficients. The periodic response factors developed with this procedure yield exactly the same results as the conduction transfer functions, if the boundary conditions (sol-air temperatures) are steady periodic. Table 1 defines the terms used in the discussion. Noting that a fundamental property of conduction transfer functions is that Σbn = Σcn, the general CTF equation (1) can be written out for each hour to form a set of 24 hourly equations, as follows: 6 ″ q1
=
″
q2 =
∑ b n T e, 1 – n δ – ∑
– T rc ∑ bn
n=1
n=0
6
6
6 ″
∑ bn T e, 2 – n δ – ∑ d n q 2 – n δ – T rc ∑ bn 6
=
6 ″ d n q1 – n δ
n=0
n=0
″ q24
6
n=1
n=0
n=1
d 1 1 0 0 …
q″ 2
(3b)
d 2
d 2 d 1 1 0 0 …
•
…
q ″ 1
… d 3 d 2 d 1 1
b0 0 0 …
b2 b1
T e, 1
b2
T e, 2
b1 b0 0 0 …
=
q″ 3 …
b 2 b 1 b0 0 0 … …
6 ″ d n q 24 – n δ
– T rc ∑ bn
(3c)
b4 b3 b2 b1 b0
•
T rc Σ bn
T e, 3
–
(4) …
…
T e, 24
T rc Σ bn
Equation 4 can be rearranged and represented more simply as:
n=0
The 24 hourly equations can be rearranged and written in a matrix form, as shown in Equation 4.
49 2
q″ 1
n=0
6
∑ b n T e, 24 – n δ – ∑
(3a)
1 0 0 … d 3 d 2 d 1
q´´ = D-1 B T e − D-1 B T rc = D-1 B (T e − T rc)
(5)
ASHRAE Transactions: Symposia
where D
= the left-hand side coefficient matrix,
q´´
= the column vector containing the conductive heat fluxes,
B
= the right-hand side coefficient matrix,
T e
= the column vector containing the sol-air temperatures,
T rc
= a column vector containing T rc in every row.
Similarly, the response factor equations can be written as a matrix formulation, as in Equation 6.
″ q1
Y P 0 Y P 23 Y P 22 …
″
″
q3
Y P 2
Y P 2 Y P 1 Y P 0 Y P 23 Y P 22 …
=
…
… ″
RESULTS AND DISCUSSION Periodic response factors were calculated for the 41 representative wall types summarized in Table 2 and the 42 representative roof types summarized in Table 3. The layers are described using the code numbers detailed in Table 11, chapter 28, 1997 ASHRAE Handbook—Fundamentals. The periodic response factors are given in Tables 4 through 9. They are given to accuracy of six decimal places, which should be more than sufficient for any practical application. SI versions of the periodic response factors are tabulated in the appendix in Tables A-1 through A-6.
Y P 2 Y P 1
Y P 1 Y P 0 Y P 23 Y P 22 …
q2
In order to generate the periodic response factors, the following algorithm was implemented in a FORTRAN program: • Read wall and roof CTF coefficients from the ASHRAE RP-626 database. (Falconer et al. 1993). • Fill D and B matrices. • Calculate D-1 and D-1 B. • Extract periodic response factors from D-1 B. • Output results in tabular form.
Y P 2 Y P 1 Y P 0
q24
APPLICATION
T rc Σ Y Pn
T e, 1 T e, 2 •
T e, 3
–
…
(6)
…
T e, 24
T rc Σ Y Pn
Equation 6 can be represented as: q´´ = Y T e − Y T rc = Y(T e − T rc)
(7)
where q´´
= the column vector containing the conductive heat fluxes,
Y
= the periodic response factor matrix,
T e
= the column vector containing the sol-air temperatures,
T rc
= a column vector containing T rc in every row.
Together, Equations 5 and 7 yield the following relationship between the conduction transfer functions and the periodic response factors: Y = D-1 B.
(8)
Consequently, the periodic response factors can be determined from the matrix, D-1 B. In fact, the first column of D-1 B is a column vector containing Y P0,Y P1,Y P2, …, Y P23.
ASHRAE Transactions: Symposia
The representative wall types and roof types may be selected using exactly the same procedure as originally given by Harris and McQuiston (1988) and later described in ASHRAE Fundamentals (ASHRAE 1989, 1993, 1997) and Cooling and Heating Load Calculation Manual , second edition (McQuiston and Spitler 1992). However, if this procedure is used, a slightly different “unnormalization” procedure is necessary. (The “unnormalization” procedure modifies the CTF coefficients so that they reflect the U-factor of the actual wall rather than the U-factor of the typical wall.) The CTF coefficients are “unnormalized” by multiplying the b coefficients by the ratio of the actual U-factor to the typical wall or roof’s U-factor. To “unnormalize” the periodic response factor, each Y coefficient is multiplied by the ratio of the actual U-factor to the typical wall or roof ’s U-factor. Finally, the error associated with using the tabulated periodic response factors is the same as that associated with using the tabulated CTF coefficients. The grouping procedure developed by Harris and McQuiston (1988) was designed with two criteria: 1.
The typical wall or roof has a peak heat gain within ± one hour of the actual wall or roof.
2.
The typical wall or roof has a peak heat gain at least as high as, but no more than 20% higher than, the actual wall or roof.
If users of the RTSM desire a higher degree of accuracy, periodic response factors may be calculated for the actual wall or roof type using software developed as part of ASHRAE 875-RP. (Pedersen et al. 1998).
493
TABLE 2 Wall Types Layers (Inside to Outside)
Description
1
E0 A3 B1 B13 A3 A0
2
U (Btu/h ⋅ft2 °F)
U (W/m2 K)
Steel siding with 4 in. (100 mm) insulation
0.066
0.375
E0 E1 B14 A1 A0
Frame wall with 5 in. (125 mm) insulation
0.055
0.312
3
E0 C3 B5 A6 A0
4 in. (100 mm) h.w. concrete block with 1 in. (25 mm) insulation
0.191
1.084
4
E0 E1 B6 C12 A0
2 in. (50 mm) insulation with 2 in. (50 mm) h.w. concrete
0.047
0.267
5
E0 A6 B21 C7 A0
1.36 in. (35 mm) insulation with 8 in. (200 mm) l.w. concrete block
0.129
0.732
6
E0 E1 B2 C5 A1 A0
1 in. (25 mm) insulation with 4 in. (100 mm) h.w. concrete
0.199
1.130
7
E0 A6 C5 B3 A3 A0
4 in. (100 mm) h.w. concrete with 2 in. (50 mm) insulation
0.122
0.693
8
E0 A2 C12 B5 A6 A0
Face brick and 2 in. (50 mm) h.w. concrete with 1 in. (25 mm) insul.
0.195
1.107
9
E0 A6 B15 B10 A0
6 in. (150 mm) insulation with 2 in. (50 mm) wood
0.042
0.238
10 E0 E1 C2 B5 A2 A0
4 in. (100 mm) l.w. conc. block with 1 in. (25 mm) insul. and face brick
0.155
0.880
11
8 in. (200 mm) h.w. concrete block with 2 in. (50 mm) insulation
0.109
0.619
12 E0 E1 B1 C10 A1 A0
8 in. (200 mm) h.w. concrete
0.339
1.925
13 E0 A2 C5 B19 A6 A0
Face brick and 4 in. (100 mm) h.w. concrete with 0.61 in. (16 mm) ins.
0.251
1.425
14 E0 A2 A2 B6 A6 A0
Face brick and face brick with 2 in. (50 mm) insulation
0.114
0.647
15 E0 A6 C17 B1 A7 A0
8 in. (200 mm) l.w. concrete block (filled) and face brick
0.092
0.522
16 E0 A6 C18 B1 A7 A0
8 in. (200 mm) h.w. concrete block (filled) and face brick
0.222
1.261
17 E0 A2 C2 B15 A0
Face brick and 4 in. (100 mm) l.w. conc. block with 6 in. (150 mm) ins.
0.043
0.244
18 E0 A6 B25 C9 A0
3.33 in. (85 mm) insulation with 8 in. (200 mm) common brick
0.072
0.409
19 E0 C9 B6 A6 A0
8 in. (200 mm) common brick with 2 in. (50 mm) insulation
0.106
0.602
20 E0 C11 B19 A6 A0
12 in. (300 mm) h.w. concrete with 0.61 in. (15 mm) insulation
0.237
1.346
21 E0 C11 B6 A1 A0
12 in. (300 mm) h.w. concrete with 2 in. (50 mm) insulation
0.112
0.636
22 E0 C14 B15 A2 A0
4 in. (100 mm) l.w. concrete with 6 in. (150 mm) insul. and face brick
0.040
0.227
23 E0 E1 B15 C7 A2 A0
6 in. (150 mm) insulation with 8 in. (200 mm) l.w. concrete block
0.042
0.238
24 E0 A6 C20 B1 A7 A0
12 in. (300 mm) h.w. concrete block (filled) and face brick
0.196
1.113
25 E0 A2 C15 B12 A6 A0 Face brick and 6 in. (150 mm) l.w. conc. blk. with 3 in. (75 mm) insul.
0.060
0.341
26 E0 A2 C6 B6 A6 A0
0.097
0.551
27 E0 E1 B14 C11 A1 A0 5 in. (125 mm) insulation with 12 in. (300 mm) h.w. concrete
0.052
0.295
28 E0 E1 C11 B13 A1 A0 12 in. (300 mm) h.w. concrete with 4 in. (100 mm) insulation
0.064
0.363
29 E0 A2 C11 B5 A6 A0
0.168
0.954
30 E0 El B19 C19 A2 A0 0.61 in. (15 mm) ins. with 12 in. (300 mm) l.w. blk. (fld.) and face brick
0.062
0.352
31 E0 E1 BI5 C15 A2 A0 6 in. (150 mm) insul. with 6 in. (150 mm) l.w. conc. and face brick
0.038
0.216
32 E0 E1 B23 B9 A2 A0
2.42 in. (60 mm) insulation with face brick
0.069
0.392
33 E0 A2 C6 BI5 A6 A0
Face brick and 8 in. (200 mm) clay tile with 6 in. (150 mm) insulation
0.042
0.238
34 E0 C11 B21 A2 A0
12 in. (300 mm) h.w. conc. with 1.36 in. (35 mm) insul. and face brick
0.143
0.812
35 E0 E1 B14 C11 A2 A0 5 in. (125 mm) insul. with 12 in. (300 mm) h.w. conc. and face brick
0.052
0.295
36 E0 A2 C11 B25 A6 A0 Face brick and 12 in. (300 mm) h.w. conc. with 3.33 in. (85 mm) insul.
0.073
0.414
37 E0 E1 B25 C19 A2 A0 3.33 in. (85 mm) ins. with 12 in. (300 mm) l.w. blk. (fld.) and face brick
0.040
0.227
38 E0 E1 B15 C20 A2 A0 6 in. (150 mm) ins. with 12 in. (300 mm) h.w. block (fld.) and face brick
0.041
0.233
39 E0 A2 C16 B14 A6 A0 Face brick and 8 in. (200 mm) l.w. concrete with 5 in. (125 mm) insul.
0.040
0.227
40 E0 A2 C20 B15 A6 A0 Face brick, 12 in. (300 mm) h.w. block (fld.), 6 in. (150 mm) insul.
0.041
0.233
41 E0 E1 C11 B14 A2 A0 12 in. (300 mm) h.w. conc. with 5 in. (125 mm) insul. and face brick
0.052
0.295
494
E0 E1 C8 B6 A1 A0
Face brick and 8 in. (200 mm) clay tile with 2 in. (50 mm) insulation
Face brick and 12 in. (300 mm) h.w. concrete with 1 in. (25 mm) insul.
ASHRAE Transactions: Symposia
TABLE 3 Roof Types Layers (Inside to Outside)
Description
U U (Btu/h⋅ft2 °F) (W/m2 K)
1
E0 A3 B25 E3 E2 A0
Steel deck with 3.33 in. (85 mm) insulation
0.080
0.454
2
E0 A3 BI4 E3 E2 A0
Steel deck with 5 in. (125 mm) insulation
0.055
0.312
3
E0 E5 E4 C12 E3 E2 A0
2 in. (50 mm) h.w. concrete deck with suspended ceiling
0.232
1.317
4
E0 E1 BI5 E4 B7 A0
Attic roof with 6 in. (150 mm) insulation
0.043
0.244
5
E0 BI4 C12 E3 E2 A0
5 in. (125 mm) insulation with 2 in. (50 mm) h.w. concrete deck
0.055
0.312
6
E0 C5 BI7 E3 E2 A0
4 in. (100 mm) h.w. concrete deck with 0.3 in. (8 mm) insulation
0.371
2.107
7
E0 B22 C12 E3 E2 C12 A0
1.67 in. (40 mm) insulation with 2 in. (50 mm) h.w. concrete RTS
0.138
0.784
8
E0 B16 C13 E3 E2 A0
0.15 in. (4 mm) insul. with 6 in. (150 mm) h.w. concrete deck
0.424
2.407
9
E0 E5 E4 B12 C14 E3 E2 A0
3 in. (75 mm) insul. with 4 in. (100 mm) l.w. conc. deck and susp. clg.
0.057
0.324
10
E0 E5 E4 C15 B16 E3 E2 A0
6 in. (150 mm) l.w. conc. dk with 0.15 in. (4 mm) ins. and susp. clg.
0.104
0.591
11
E0 C5 BI5 E3 E2 A0
4 in. (100 mm) h.w. concrete deck with 6 in. (150 mm) insulation
0.046
0.261
12
E0 C13 B16 E3 E2 CI2 A0
6 in. (150 mm) h.w. deck with 0.15 in. (4 mm) ins. and 2 in. (50 mm) h.w. RTS
0.396
2.248
13
E0 C13 B6 E3 E2 A0
6 in. (150 mm) h.w. concrete deck with 2 in. (50 mm) insulation
0.117
0.664
14
E0 E5 E4 C12 B13 E3 E2 A0
2 in. (50 mm) 1 w. conc. deck with 4 in. (100 mm) ins. and susp. clg.
0.057
0.324
15
E0 E5 E4 C5 B6 E3 E2 A0
1 in. (25 mm) insul. with 4 in. (100 mm) h.w. conc. deck and susp. clg.
0.090
0.511
16
E0 E5 E4 CI3 B2O E3 E2 A0
6 in. (150 mm) h.w. deck with 0.76 in. (20 mm) insul. and susp. clg.
0.140
0.795
17
E0 E5 E4 B15 C14 E3 E2 A0
6 in. (150 mm) insul. with 4 in. (100 mm) l.w. conc. deck and susp. clg.
0.036
0.204
18
E0 CI2 B15 E3 E2 C5 A0
2 in. (50 mm) h.w. conc. dk with 6 in. (150 mm) ins. and 2 in. (50 mm) h.w. RTS
0.046
0.261
19
E0 C5 B27 E3 E2 C12 A0
4 in. (100 mm) h.w. deck with 4.54 in. (115 mm) ins. and 2 in. (50 mm) h.w. RTS
0.059
0.335
20
E0 B21 C16 E3 E2 A0
1.36 in. (35 mm) insulation with 8 in. (200 mm) l.w. concrete deck
0.080
0.454
21
E0 CI3 B12 E3 E2 C12 A0
6 in. (150 mm) h.w. deck with 3 in. (75 mm) insul. and 2 in. (50 mm) h.w. RTS
0.083
0.471
22
E0 B22 C5 E3 E2 C13 A0
1.67 in. (40 mm) ins. with 4 in. (100 mm) h.w. deck and 6 in. (150 mm) h.w. RTS
0.129
0.732
23
E0 E5 E4 C12 B14 E3 E2 C12 A0 Susp. clg, 2 in. (50 mm) h.w. dk, 5 in. (125 mm) ins, 2 in. (50 mm) h.w. RTS
0.047
0.267
24
E0 E5 E4 C5 E3 E2 B6 B1 C12 A0 Susp. clg, 4 in. (100 mm) h.w. dk, 2 in. (50 mm) ins, 2 in. (50 mm) h.w. RTS
0.082
0.466
25
E0 E5 E4 C13 B13 E3 E2 A0
6 in. (150 mm) h.w. conc. deck with 4 in. (100 mm) ins. and susp. clg.
0.056
0.318
26
E0 E5 E4 B15 C15 E3 E2 A0
6 in. (150 mm) insul. with 6 in. (150 mm) l.w. conc. deck and susp. clg.
0.034
0.193
27
E0 C13 B15 E3 E2 C12 A0
6 in. (150 mm) h.w. deck with 6 in. (150 mm) ins. and 2 in. (50 mm) h.w. RTS
0.045
0.256
28
E0 B9 B14 E3 E2 A0
4 in. (100 mm) wood deck with 5 in. (125 mm) insulation
0.044
0.250
29
E0 E5 E4 C12 B13 E3 E2 C5 A0
Susp. clg, 2 in. (50 mm) h.w. dk, 4 in. (100 mm) ins, 4 in. (100 mm) h.w. RTS
0.056
0.318
30
E0 E5 E4 B9 B6 E3 E2 A0
4 in. (100 mm) wood deck with 2 in. (50 mm) insul. and susp. ceiling
0.064
0.363
31
E0 B27 C13 E3 E2 C13 A0
4.54 in. (115 mm) ins. with 6 in. (150 mm) h.w. deck and 6 in. (150 mm) h.w. RTS
0.057
0.324
32
E0 E5 E4 C5 B20 E3 E2 C13 A0
Susp. clg, 4 in. (100 mm) h.w. dk, 0.76 in. (20 mm) ins, 6 in. (150 mm) h.w. RTS
0.133
0.755
33
E0 E5 E4 C5 B13 E3 E2 C5 A0
Susp. clg, 4 in. (100 mm) h.w. dk, 4 in. (100 mm) ins, 4 in. (100 mm) h.w. RTS
0.055
0.312
34
E0 E5 E4 C13 B23 E3 E2 C5 A0
Susp. clg, 6 in. (150 mm) h.w. dk, 2.42 in. (60 mm) ins, 4 in. (100 mm) h.w. RTS
0.077
0.437
35
E0 C5 B15 E3 E2 C13 A0
4 in. (100 mm) h.w. deck with 6 in. (150 mm) ins and 6 in. (150 mm) h.w. RTS
0.045
0.256
36
E0 C13 B27 E3 E2 CI3 A0
6 in. (150 mm) h.w. deck with 4.54 in. (115 mm) ins. and 6 in. (150 mm) h.w. RTS
0.057
0.324
37
E0 E5 E4 B15 C13 E3 E2 C13 A0 Susp. clg, 6 in. (150 mm) ins, 6 in. (150 mm) h.w. dk, 6 in. (150 mm) h.w. RTS
0.040
0.227
38
E0 E5 E4 B9 B15 E3 E2 A0
0.035
0.199
39
E0 E5 E4 C13 B20 E3 E2 C13 A0 Susp. clg, 6 in. (150 mm) h.w. dk, 0.76 in. (20 mm) ins, 6 in. (150 mm) h.w. RTS
0.131
0.744
40
E0 E5 E4 C5 B26 E3 E2 C13 A0
Susp. clg, 4 in. (100 mm) h.w. dk, 3.64in. (90 mm) ins, 6 in. (150 mm) h.w. RTS
0.059
0.335
41
E0 E5 E4 C13 B6 E3 E2 C13 A0
Susp. clg, 6 in. (150 mm) h.w. deck, 2 in. (50 mm) ins, 6 in. (150 mm) h.w. RTS
0.085
0.483
42
E0 E5 E4 C13 B14 E3 E2 C13 A0 Susp. clg, 6 in. (150 mm) h.w. deck, 5 in. (125 mm) ins, 6 in. (150 mm) h.w. RTS
0.046
0.261
ASHRAE Transactions: Symposia
4 in. (100 mm) wood deck with 6 in. (150 mm) insul. and susp. ceiling
495
CONCLUSIONS The periodic response factor tables presented in this paper support wall and roof conductive heat gain calculations by the periodic response factor method. Heat gains calculated using the tables are equivalent to heat gains calculated using CTFs for the same wall type. The Harris and McQuiston grouping procedure is carried through to the periodic response factor tables. As a result an “unnormalization” procedure similar to the CTF procedure is required to obtain response factors that reflect correct wall and roof U-factors. Although the tables were generated specifically to support the RTSM, the periodic response factor method is generally applicable to any conductive heat gain calculation with steady periodic inputs. As such, it is useful for all peak design day cooling load calculations that typically assume that previous days were identical to the design day.
Mitalas, G.P., and D.G. Stephenson. 1967. Cooling load calculations by thermal response factor method. ASHRAE Transactions 73, pp. III 2.1-2.10. Peavy, B.A. 1978. A note on response factors and conduction transfer functions. ASHRAE Transactions 84 (1): pp. 688-690. Pedersen, C.O., D. Fisher, J.D. Spitler, and R. Liesen. 1998. Cooling and heating load calculation principles . Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. Spitler, J.D., and D.E. Fisher. 1999. On the relationship between the radiant time series and transfer function methods for design cooling load calculations. International Journal of HVAC&R Research , 5 (2): 125-138.
REFERENCES
Spitler, J.D., D.E. Fisher, and C.O. Pedersen. 1997. The radiant time series cooling load calculation procedure. ASHRAE Transactions 103 (2): 503-515.
ASHRAE. 1989. 1989 ASHRAE Handbook—Fundamen-
Stephenson, D.G., and G.P. Mitalas. 1967. Room thermal
CONCLUSIONS The periodic response factor tables presented in this paper support wall and roof conductive heat gain calculations by the periodic response factor method. Heat gains calculated using the tables are equivalent to heat gains calculated using CTFs for the same wall type. The Harris and McQuiston grouping procedure is carried through to the periodic response factor tables. As a result an “unnormalization” procedure similar to the CTF procedure is required to obtain response factors that reflect correct wall and roof U-factors. Although the tables were generated specifically to support the RTSM, the periodic response factor method is generally applicable to any conductive heat gain calculation with steady periodic inputs. As such, it is useful for all peak design day cooling load calculations that typically assume that previous days were identical to the design day.
Mitalas, G.P., and D.G. Stephenson. 1967. Cooling load calculations by thermal response factor method. ASHRAE Transactions 73, pp. III 2.1-2.10. Peavy, B.A. 1978. A note on response factors and conduction transfer functions. ASHRAE Transactions 84 (1): pp. 688-690. Pedersen, C.O., D. Fisher, J.D. Spitler, and R. Liesen. 1998. Cooling and heating load calculation principles . Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. Spitler, J.D., and D.E. Fisher. 1999. On the relationship between the radiant time series and transfer function methods for design cooling load calculations. International Journal of HVAC&R Research , 5 (2): 125-138.
REFERENCES
Spitler, J.D., D.E. Fisher, and C.O. Pedersen. 1997. The radiant time series cooling load calculation procedure. ASHRAE Transactions 103 (2): 503-515.
ASHRAE. 1989. 1989 ASHRAE Handbook—Fundamentals. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.
Stephenson, D.G., and G.P. Mitalas. 1967. Room thermal response factors. ASHRAE Transactions 73, pp. III 1.11.7.
ASHRAE. 1993. 1993 ASHRAE Handbook—Fundamentals. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.
DISCUSSION
ASHRAE. 1997. 1997 ASHRAE Handbook—Fundamentals. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. Falconer, D.R., E.F. Sowell, J.D. S pitler, and B. Todovorich. 1993. Electronic tables for the ASHRAE Load Calculation Manual. ASHRAE Transactions 99 (1): 193-200. Harris, S.M., and F.C. McQuiston. 1988. A study to categorize walls and roofs on the basis of thermal response. ASHRAE Transactions 94 (2): 688-714. Hittle, D.C. 1981. Calculating building heating and cooling loads using the frequency response of multilayered slabs. Ph.D. thesis, University of Illinois at UrbanaChampaign. Maloney, D. 1985. A verification of the use of heat conduction transfer functions as used in the program BLAST. BLAST Support Office Report, University of Illinois. McQuiston, F.C., and J.D. Spitler. 1992. Cooling and heating load calculation manual, 2d ed. Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.
502
Byron Jones, Associate Dean of Research, Kansas State University, Manhattan, Kansas: The calculation methods used in this paper make the implicit assumption that the cooling system is air-based. That is, they evaluate the energy gain of the air. Can these calculations be applied to load calculations when part or all of the cooling is provided by radiant cooling systems? How will the loads differ for a radiant cooling system as compared to an air-based system? Jeffrey D. Spitler: In the author’s opinion, load calculations for systems involving radiant heating or cooling ought to be performed using a load calculation procedure which explicitly accounts for surface-to-surface radiant interchange. Therefore, we recommend using a heat balance procedure rather than the radiant time series procedure for which the periodic response factors in this paper were developed. Implementation of a radiant heating and/or cooling model in a heat balance procedure has been described by Strand and Pedersen (1997). (Reference: Strand, R.K., and C.O. Pedersen. Implementation of a radiant heating and cooling model into an integrated building energy analysis program. A SHRAE Transactions , 103 (1): 949-958.)
ASHRAE Transactions: Symposia
APPENDIX B After the review process and very near to press time, it was determined that the conduction transfer function coefficients originally published by Harris and McQuiston (1988) were inaccurate for a few of the very high mass walls and roofs. Furthermore, these CTF coefficients have been published in the 1989, 1993, and 1997 editions of the ASHRAE Handbook—Fundamentals . Presumably, this is due to precision problems in the CTF calculation program, although this has not been investigated in detail by the authors. The errors may be quantified by checking whether or not the CTF coefficients satisfy a fundamental relationship between the U-factor and the CTF coefficients:
∑b U =
n
n =0
1+
REFERENCES
∑ d
n
n =1
coefficients exceeds 1% are Roof 37 (3%), Roof 38 (8%), Wall 30 (2%), Wall 31 (6%), Wall 35 (16%), Wall 37 (30%), Wall 38 (30%). In every case, the U-factor based on the CTF coefficients is lower than the actual U-factor. Hence, the errors in the CTF coefficients will result in under-predicting the cooling loads. Therefore, new CTF coefficients were determined using Seem’s method (Seem et al. 1989). For each of the massive surfaces, either 12 or 13 temperature history terms (b coefficients) and 12 or 13 flux history terms (d coefficients) resulted. Periodic response factors were determined from these CTF coefficients using the procedure described in the paper. Surfaces for which new CTF coefficients were used to determine the periodic response factors are marked with an asterisk in Tables 6, 9, A-3, and A-6.
(B-1)
The surfaces for which the discrepancy between the
Seem, J.E., S.A. Klein, W.A. Beckman, J.W. Mitchell. 1989. Transfer functions for efficient calculation of multidimensional transient heat transfer. Journal of Heat Trans-
APPENDIX B After the review process and very near to press time, it was determined that the conduction transfer function coefficients originally published by Harris and McQuiston (1988) were inaccurate for a few of the very high mass walls and roofs. Furthermore, these CTF coefficients have been published in the 1989, 1993, and 1997 editions of the ASHRAE Handbook—Fundamentals . Presumably, this is due to precision problems in the CTF calculation program, although this has not been investigated in detail by the authors. The errors may be quantified by checking whether or not the CTF coefficients satisfy a fundamental relationship between the U-factor and the CTF coefficients:
∑b U =
n
n =0
1+
REFERENCES
∑ d
n
n =1
(B-1)
The surfaces for which the discrepancy between the actual U-factor and the U-factor determined from the CTF
ASHRAE Transactions: Symposia
coefficients exceeds 1% are Roof 37 (3%), Roof 38 (8%), Wall 30 (2%), Wall 31 (6%), Wall 35 (16%), Wall 37 (30%), Wall 38 (30%). In every case, the U-factor based on the CTF coefficients is lower than the actual U-factor. Hence, the errors in the CTF coefficients will result in under-predicting the cooling loads. Therefore, new CTF coefficients were determined using Seem’s method (Seem et al. 1989). For each of the massive surfaces, either 12 or 13 temperature history terms (b coefficients) and 12 or 13 flux history terms (d coefficients) resulted. Periodic response factors were determined from these CTF coefficients using the procedure described in the paper. Surfaces for which new CTF coefficients were used to determine the periodic response factors are marked with an asterisk in Tables 6, 9, A-3, and A-6.
Seem, J.E., S.A. Klein, W.A. Beckman, J.W. Mitchell. 1989. Transfer functions for efficient calculation of multidimensional transient heat transfer. Journal of Heat Trans fer, 11: 5-12, February.
509
This paper has been downloaded from the Building and Environmental Thermal Systems Research Group at Oklahoma State University (www.hvac.okstate.edu) The correct citation for the paper is: Spitler, J.D., D.E. Fisher. 1999. Development of Periodic Response Factors for Use with the Radiant Time Series Method. ASHRAE Transactions. 105 (2): 491-509. Reprinted by permission from ASHRAE Transactions (Vol. #105 Part 2, pp. 491-509). © 1999 American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc.
