ASHRAE Transactions: Symposia 491 ABSTRACT Harris and McQuiston (1988) developed conduction transfer function (CTF) coefficients corresponding to 41 repre- sentative wall assemblies and 42 representative roof assem- blies for use with the transfer function method (TFM). They also developed a grouping procedure that allows design engi- neers 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 —Funda- mentals (1989, 1993, 1997) and the ASHRAE Cooling and Heating Load Calculation Manual, second edition (McQuis- ton and Spitler 1992). More recently, a new, simplified design cooling load calculation procedure, the radiant time series method (RTSM), has been developed (Spitler et al. 1997). The RTSM uses peri- odic response factors to model transient conductive heat trans- fer. While not a true manual load calculation procedure, it is quite feasible to implement the RTSM in a spreadsheet. To be useful in such an environment, it would be desirable to have a pre-calculated set of periodic response factors. 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 build- ing 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 multi- layer wall or roof with boundary conditions that can be repre- sented 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 infi- nite 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 McQuis- ton 1988) corresponding to 41 representative roof types and 42 representative wall types. In addition, a grouping proce- dure 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 mate- rial (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 repre- sentative wall or roof type has been identified, the CTF coef- ficients may be “unnormalized” so that the U-factor of the 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 Jeffrey D. Spitler is associate professor at Oklahoma State University, Stillwater. Daniel E. Fisher is senior research engineer at the University of Illinois at Urbana-Champaign. SE-99-1-1
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ABSTRACT
Harris and McQuiston (1988) developed conductiontransfer function (CTF) coefficients corresponding to 41 repre-sentative wall assemblies and 42 representative roof assem-blies for use with the transfer function method (TFM). Theyalso developed a grouping procedure that allows design engi-neers to determine the correct representative wall or roofassembly that most closely matches a specific wall or roofassembly. The CTF coefficients and the grouping procedurehave been summarized in the ASHRAE Handbook —Funmentals (1989, 1993, 1997) and the ASHRAE Cooling aHeating Load Calculation Manual, second edition (McQuiton and Spitler 1992).
More recently, a new, simplified design cooling loacalculation procedure, the radiant time series method (RTShas been developed (Spitler et al. 1997). The RTSM uses odic response factors to model transient conductive heat trafer. While not a true manual load calculation procedure, it quite feasible to implement the RTSM in a spreadsheet. Tuseful in such an environment, it would be desirable to havpre-calculated set of periodic response factors. Accordinglyset of periodic response factors has been calculated anpresented in this paper.
INTRODUCTION
The transfer function method (TFM) for design coolingload calculations has been in use for a number of years. Thismethod uses conduction transfer functions (CTFs) to calculatethe transient, one-dimensional conduction through the build-ing wall and roof elements. Conduction transfer functions area closed form representation of a conduction response factorseries. Conduction response factors, as derived by Mitalas and
ASHRAE Transactions: Symposia
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s-
dM),peri-ns-iso bee a, a
d is
Stephenson (1967) and Hittle (1981), are an exact solution tothe transient conductive heat transfer problem for a multi-layer wall or roof with boundary conditions that can be repre-sented by a piecewise linear profile. The response factor seriesis infinite, so in practice it must be truncated, resulting in someminor, but controllable, loss of accuracy. Stephenson andMitalas (1967) compared a response factor method to ananalog computer simulation and showed that for the one-hourtime steps commonly used in building energy and thermal loadcalculations, the expected error due to truncation of the infi-nite response factor series is small. Procedures for developingconduction transfer functions from response factors aredescribed by Peavy (1978) and Hittle (1981). The methods arenecessarily inexact but have been compared to both analyticaland numerical solutions with excellent results. Maloney(1985) showed that for a one-dimensional, transient slab withlinearly varying surface temperatures, the differences betweena CTF solution and an analytical solution based on Duhammethod are negligible.
ASHRAE research project RP-472 provided a set conduction transfer function coefficients (Harris and McQuiton 1988) corresponding to 41 representative roof types 42 representative wall types. In addition, a grouping procdure was developed so that, theoretically, any wall or rocould be mapped into one of the representative wall types. walls are mapped using four parameters: primary wall marial (the most thermally massive element), secondary wmaterial, R-value, and the thermal mass location (in, out,integral). Roofs are also mapped using four parameteprimary roof material, thermal mass location, R-value, apresence or absence of a suspended ceiling. Once the resentative wall or roof type has been identified, the CTF coficients may be “unnormalized” so that the U-factor of th
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
Jeffrey D. Spitler is associate professor at Oklahoma State University, Stillwater. Daniel E. Fisher is senior research engineer at the Universityof Illinois at Urbana-Champaign.
actual wall or roof is preserved in the CTF coefficients. Oncethe coefficients have been determined, they are applied usingthe conduction transfer function:
(1)
Since originally appearing in the 1989 ASHRAE Hand-book—Fundamentals, the tabulated CTF coefficients haveproved to be useful to designers. The radiant time seriesmethod (RTSM) uses periodic response factors to model tran-sient conductive heat transfer:
(2)
Since the RTSM is amenable to spreadsheet-level imple-mentation, a tabulated set of periodic response factors is auseful addition to the literature. The objective of this paper isto present a set of periodic response factors that correspondexactly to the CTFs previously published. If desired, they canbe used with exactly the same grouping procedure and aslightly different “unnormalization” procedure.
METHODOLOGY
Periodic response factors can be derived directly fromset of CTF coefficients. The procedure is discussed in gredetail in another paper (Spitler and Fisher 1999), butpresented here to document the method to generate the odic response factors directly from the tabulated CTF coecients. The periodic response factors developed with tprocedure yield exactly the same results as the conductransfer functions, if the boundary conditions (sol-air tempatures) are steady periodic. Table 1 defines the terms usethe discussion.
Noting that a fundamental property of conduction tranfer functions is that Σbn = Σcn, the general CTF equation (1can be written out for each hour to form a set of 24 houequations, as follows:
(3a)
(3b)
(3c)
The 24 hourly equations can be rearranged and writtea matrix form, as shown in Equation 4.
qθ″
bnTe θ nδ–, dnqθ nδ–″
Trc–n 1=
6
∑ cn
n 0=
6
∑–n 0=
6
∑=
qθ″
YPjTe θ jδ–, Trc– YPj
j 0=
23
∑j 0=
23
∑=
q1″
bnTe 1 nδ–, dnq1 nδ–″
Trc–n 1=
6
∑ bn
n 0=
6
∑–n 0=
6
∑=
q2″
bnTe 2 nδ–, dnq2 nδ–″
Trc–n 1=
6
∑ bnn 0=
6
∑–n 0=
6
∑=
q24″
bnTe 24 nδ–, dnq24 nδ–″
Trc–n 1=
6
∑ bn
n 0=
6
∑–n 0=
6
∑=
492
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s-)rly
n in
(4)
Equation 4 can be rearranged and represented msimply as:
q´´ = the column vector containing the conductive heat fluxes,
B = the right-hand side coefficient matrix,
Te = the column vector containing the sol-air temperatures,
Trc = a column vector containing Trc in every row.
Similarly, the response factor equations can be writtena matrix formulation, as in Equation 6.
(6)
Equation 6 can be represented as:
q´´ = Y Te − Y Trc = Y(Te − Trc) (7)
where
q´´ = the column vector containing the conductive heat fluxes,
Y = the periodic response factor matrix,
Te = the column vector containing the sol-air temperatures,
Trc = a column vector containing Trc in every row.
Together, Equations 5 and 7 yield the following relatioship between the conduction transfer functions and the podic response factors:
Y = D-1 B. (8)
Consequently, the periodic response factors can be demined from the matrix, D-1 B. In fact, the first column of D-1
B is a column vector containing YP0,YP1,YP2, …, YP23.
q1″
q2″
q3″
…
q24″
YP0 YP23 YP22 …
YP2 YP1
YP1 YP0 YP23 YP22 … YP2
YP2 YP1 YP0 YP23 YP22 …
…
YP2 YP1 YP0
=
•
Te 1,
Te 2,
Te 3,
…
Te 24,
TrcΣYPn
…
TrcΣYPn
–
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as
n-eri-
ter-
In order to generate the periodic response factors, following algorithm was implemented in a FORTRANprogram: • Read wall and roof CTF coefficients from the ASHRA
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.
RESULTS AND DISCUSSION
Periodic response factors were calculated for the representative wall types summarized in Table 2 and therepresentative roof types summarized in Table 3. The layare described using the code numbers detailed in Tablechapter 28, 1997 ASHRAE Handbook—Fundamentals.
The periodic response factors are given in Tables 4through 9. They are given to accuracy of six decimal places,which should be more than sufficient for any practical appli-cation. SI versions of the periodic response factors are tabu-lated in the appendix in Tables A-1 through A-6.
APPLICATION
The representative wall types and roof types may beselected using exactly the same procedure as originally givenby Harris and McQuiston (1988) and later described inASHRAE Fundamentals (ASHRAE 1989, 1993, 1997) andCooling and Heating Load Calculation Manual, secondedition (McQuiston and Spitler 1992). However, if this proce-dure is used, a slightly different “unnormalization” proceduis necessary. (The “unnormalization” procedure modifies tCTF coefficients so that they reflect the U-factor of the actuwall rather than the U-factor of the typical wall.) The CTcoefficients are “unnormalized” by multiplying the b coeffi-cients by the ratio of the actual U-factor to the typical wall roof’s U-factor. To “unnormalize” the periodic responsfactor, each Y coefficient is multiplied by the ratio of the actuaU-factor to the typical wall or roof’s U-factor.
Finally, the error associated with using the tabulated peodic response factors is the same as that associated with uthe tabulated CTF coefficients. The grouping procedudeveloped by Harris and McQuiston (1988) was designed wtwo criteria:
1. The typical wall or roof has a peak heat gain within ± ohour of the actual wall or roof.
2. The typical wall or roof has a peak heat gain at least as has, but no more than 20% higher than, the actual walroof.
If users of the RTSM desire a higher degree of accuraperiodic response factors may be calculated for the actual or roof type using software developed as part of ASHRA875-RP. (Pedersen et al. 1998).
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494 ASHRAE Transactions: Symposia
TABLE 2 Wall Types
Layers (Inside to Outside) Description U (Btu/h⋅ft2 °F) U (W/m2 K)
1 E0 A3 B1 B13 A3 A0 Steel siding with 4 in. (100 mm) insulation 0.066 0.375
2 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 E0 E1 C8 B6 A1 A0 8 in. (200 mm) h.w. concrete block with 2 in. (50 mm) insulation 0.109 0.619
The periodic response factor tables presented in this papersupport wall and roof conductive heat gain calculations by theperiodic response factor method. Heat gains calculated usingthe tables are equivalent to heat gains calculated using CTFsfor the same wall type. The Harris and McQuiston groupingprocedure is carried through to the periodic response factortables. As a result an “unnormalization” procedure similarthe CTF procedure is required to obtain response factors reflect correct wall and roof U-factors.
Although the tables were generated specifically support the RTSM, the periodic response factor methodgenerally applicable to any conductive heat gain calculatwith steady periodic inputs. As such, it is useful for all pedesign day cooling load calculations that typically assume tprevious days were identical to the design day.
REFERENCES
ASHRAE. 1989. 1989 ASHRAE Handbook—Fundamentals. Atlanta: American Society of Heating, Refrigerat-ing and Air-Conditioning Engineers, Inc.
ASHRAE. 1993. 1993 ASHRAE Handbook—Fundamentals. Atlanta: American Society of Heating, Refrigerat-ing and Air-Conditioning Engineers, Inc.
ASHRAE. 1997. 1997 ASHRAE Handbook—Fundamentals. Atlanta: American Society of Heating, Refrigerat-ing and Air-Conditioning Engineers, Inc.
Falconer, D.R., E.F. Sowell, J.D. Spitler, and B. Todovorich.1993. Electronic tables for the ASHRAE Load Calcula-tion Manual. ASHRAE Transactions 99 (1): 193-200.
Harris, S.M., and F.C. McQuiston. 1988. A study to catego-rize walls and roofs on the basis of thermal response.ASHRAE Transactions 94 (2): 688-714.
Hittle, D.C. 1981. Calculating building heating and coolingloads using the frequency response of multilayeredslabs. Ph.D. thesis, University of Illinois at Urbana-Champaign.
Maloney, D. 1985. A verification of the use of heat conduc-tion 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 heat-ing load calculation manual, 2d ed. Atlanta: AmericanSociety of Heating, Refrigerating and Air-ConditioningEngineers, Inc.
502
tothat
to isionakhat
-
-
-
Mitalas, G.P., and D.G. Stephenson. 1967. Cooling load cal-culations by thermal response factor method. ASHRAETransactions 73, pp. III 2.1-2.10.
Peavy, B.A. 1978. A note on response factors and conduc-tion 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, Refrigeratingand Air-Conditioning Engineers, Inc.
Spitler, J.D., and D.E. Fisher. 1999. On the relationshipbetween the radiant time series and transfer functionmethods for design cooling load calculations. Interna-tional Journal of HVAC&R Research, 5 (2): 125-138.
Spitler, J.D., D.E. Fisher, and C.O. Pedersen. 1997. The radi-ant time series cooling load calculation procedure.ASHRAE Transactions 103 (2): 503-515.
Stephenson, D.G., and G.P. Mitalas. 1967. Room thermalresponse factors. ASHRAE Transactions 73, pp. III 1.1-1.7.
DISCUSSION
Byron Jones, Associate Dean of Research, KansasState University, Manhattan, Kansas: The calculationmethods used in this paper make the implicit assumption thatthe cooling system is air-based. That is, they evaluate theenergy gain of the air. Can these calculations be applied to loadcalculations when part or all of the cooling is provided by radi-ant cooling systems? How will the loads differ for a radiantcooling 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 oughtbe performed using a load calculation procedure which expitly accounts for surface-to-surface radiant interchangTherefore, we recommend using a heat balance procedrather than the radiant time series procedure for which periodic response factors in this paper were developed. Immentation of a radiant heating and/or cooling model in a hbalance procedure has been described by Strand and Ped(1997). (Reference: Strand, R.K., and C.O. Pedersen. Immentation of a radiant heating and cooling model into an ingrated building energy analysis program. ASHRAETransactions, 103 (1): 949-958.)
After the review process and very near to press time, itwas determined that the conduction transfer function coeffi-cients originally published by Harris and McQuiston (1988)were inaccurate for a few of the very high mass walls androofs. Furthermore, these CTF coefficients have beenpublished in the 1989, 1993, and 1997 editions of theASHRAE Handbook—Fundamentals. Presumably, this is dueto precision problems in the CTF calculation program,although this has not been investigated in detail by theauthors. The errors may be quantified by checking whether ornot the CTF coefficients satisfy a fundamental relationshipbetween the U-factor and the CTF coefficients:
(B-1)
The surfaces for which the discrepancy between theactual U-factor and the U-factor determined from the CTF
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 CTFcoefficients is lower than the actual U-factor. Hence, theerrors in the CTF coefficients will result in under-predictingthe cooling loads.
Therefore, new CTF coefficients were determined usingSeem’s method (Seem et al. 1989). For each of the massivesurfaces, either 12 or 13 temperature history terms (b coeffi-cients) and 12 or 13 flux history terms (d coefficients)resulted. Periodic response factors were determined fromthese CTF coefficients using the procedure described in thepaper. Surfaces for which new CTF coefficients were used todetermine the periodic response factors are marked with anasterisk in Tables 6, 9, A-3, and A-6.
REFERENCES
Seem, J.E., S.A. Klein, W.A. Beckman, J.W. Mitchell. 1989.Transfer functions for efficient calculation of multidi-mensional transient heat transfer. Journal of Heat Trans-fer, 11: 5-12, February.