1、NA-04-9-1 Application of Conduction Transfer Functions and Periodic Response Factors in Cooling Load Calculation Procedures Ipseng lu Student Member ASHRAE ABSTRACT This paper presents an overview of the conduction trans- fer function (CTF) and periodic response factor (PRF) meth- ods of calculating
2、 conductive heat transfer. Diferent forms of the equations used in cooling load calculations are compared and contrasted. Particular attention is given to the methods included in the ASHRAE Loads Toolkit. The toolkit contains the source code for ASHMES new load calculation methods, the heat balance
3、method (HBM) and the radiant time series method (RTSM). Each method uses a similal; but diferent, conduction calculation technique. The HBMuses CTFs and the RTSMuses PRFs. Since there are limited numbers of CTFs and PRFs in the literature, the toolkit algorithms provide a means of calculating CTFs a
4、nd PRFs for stand-alone computer programs orforgenerating CTFandPRFlibraries. This paper describes the CTF and PRF algorithms in the toolkit and demonstrates implementation of the toolkit modules in a program that calculates CTFs and PRFs. INTRODUCTION In order to effectively use the ASHRAE cooling
5、load procedures, it is necessary to understand and correctly apply conduction transfer functions (CTFs) or periodic response factors (PRFs) to the conductive heat transfer calculation. Although response factor and transfer function methods are well established in the literature (Stephenson and Mital
6、as 1971; Hittle 1979; Ceylan and Myers 1980; Seem 1987; Ouyang and Haghighat 199 i), misconceptions persist concerning their application to cooling load procedures. Several forms of the equations relating to different boundary conditions are shown in the literature. Methods of calculating the coeffi
7、cients differ, and their accuracy is not easily checked. D.E. Fisher, Ph.D., P.E. Member ASHRAE The objective of this paper is to reconcile the various forms of the transfer function equations, discuss implicit assumptions associated with each form, and illustrate by way of an example calculation th
8、e use of the various methods. Particular attention is given to the conduction transfer function methods presented in the ASHRAE Loads Toolkit. An algorithm that uses the toolkit CTF module is presented along with a simple program to generate CTFs and PRFs for use in cooling load procedures. The heat
9、 balance method (HBM) is the standard ASHRAE load calculation method as described in the ASHRAE Handbook-Fundamentals (2001). This method is based on simultaneously satisfying a system of equations that includes a zone air heat balance and a set of outside and inside heat balances at each surfacelai
10、r interface. The system of equations may be solved in a computer program using succes- sive substitution, Newton techniques, or (with linearized radi- ation) matrix methods. The radiant time series method (Spitler et al. 1997) is a simplified method that does not solve the heat balance equa- tions.
11、The method is “heat-balance based” to the extent that the storage and release of energy in the zone is approximated by a predetermined zone response, called “radiant time factors” (RTFs). By incorporating these simplifications, the RTSM calculation procedure becomes explicit, avoiding the require- m
12、ent to solve the simultaneous system of heat balance equa- tions. The method is useful not only for peak load calculations but also for estimating component contributions to the hourly cooling loads. If the radiant time factors and the periodic response factors for a particular zone configuration ar
13、e known, the RTSM may be implemented in a spreadsheet. In both the HBM and the RTSM, two simplifiing assump- tions are made in solving the wall heat conduction problem. Ipseng Iu is a graduate student and D.E. Fisher is an assistant professor in the Department of Mechanical Engineering, Oklahoma Sta
14、te Univer- sity, Stillwater, Okla. 02004 ASHRAE. 829 First, heat conduction is assumed to be one-dimensional. Two- dimensional effects due to corners and nonuniform boundary conditions are neglected. Second, materials are assumed to be homogeneous and have constant thermal properties. As a result, t
15、he difision equation of conductive heat transfer prob- lem is simplified as shown in Equation 1. With the use of Fouriers law (Equation 2) for calculating conductive heat flux, Equations 1 and 2 are the governing equations of conduc- tive heat transfer problems in cooling load calculation. 2T(x,z) -
16、 1 T(x,T) dx2 a t Although the one-dimensional, transient conduction problem can be solved analytically, the analytical solution is immediately complicated when the analysis is extended to multi-layered constructions. Analytical solutions for multi- layered slabs require special mathematic functions
17、 and complex algebra. Ultimately, numerical methods must be employed at some level to solve the problem. Solution tech- niques include lumped parameter methods, frequency response methods, finite difference or finite element methods, and Z-transform methods (McQuiston et al. 2000). The toolkit imple
18、ments Laplace and state-space methods for calculating conduction transfer functions (CTFs) and provides an algo- rithm to derive periodic response factors (PRFs) from a set of conduction transfer functions. CTFs and PRFs are dependent only on material properties and reflect the transient response of
19、 a given construction for any set of environmental boundary conditions. Since material properties are typically assumed to be constant in HVAC ther- mal load calculations, it is possible to pre-calculate these coef- ficients. Although CTF and PRF coefficients for typical constructions are available
20、in the ASHRAE Handbook- Fundamentals (2001) and Spitler and Fisher (1999b), the ASHRAE Loads Toolkit (Pedersen 2001), makes it possible to quickly and accurately construct a stand-alone computer program that will calculate CTFs and PRFs for any arbitrary wall configuration. This paper presents an al
21、gorithm for pre- calculating these coefficients using the toolkit modules. FORMULATIONS OF TRANSFER FUNCTION EQUATIONS The transfer function equations for conduction calcula- tion are formulated differently in load calculation methods. The HBM uses conduction transfer functions (CTFs), while the RTS
22、M uses periodic response factors (PRFs). In the HBM, the instantaneous conduction flux is represented by a simple linear equation that relates the current rate of conductive heat transfer to temperature and flux histories, while in RTSM, the conduction flux is a linear function of temperatures only.
23、 Conduction Transfer Function (CTF) Formulations The CTF formulation of the surface heat fluxes involves four sets of coefficients. Following Spitlers nomenclature (McQuiston et al. 2000) X, Z, and Y are used to represent the exterior, interior, and cross terms, respectively. Equation 3a shows the z
24、eroth outside and cross terms operating on the current hours surface temperatures. Ho, is the flux history term as shown in Equation 3b. Together the current hours surface temperatures and the history term yield the total flux at the outside surface. where NY Nx N+ , Hou, = - C YnTis,o-nF+ C XnTos,-
25、n+ C +nqko,-n n=l n= 1 n= 1 (3b) Likewise, Equations 4a and 4b show the flux at the inside surface. where Nz NY N+ I, H. in = - nTis,0-n+ YnTos,0-n6+ $nqki,O-nF n= 1 n=l n= I (4b) As indicated in Equations 3 and 4, the current heat fluxes are closely related to the flux histories. The flux histories
26、, shown as constant terms in Equations 3 and 4, are not only related to previous surface temperatures but also related to previous heat fluxes. Equations 3a and 3b or Equations 4a and 4b are usually solved iteratively with an assumption that all previous heat fluxes are equal at the beginning of the
27、 iteration. The converged solution produces flux history terms (Hout and Ifin) that correctly account for the thermal capacitance of a given construction. The temperatures operated on by the conduction transfer functions may be either surface or air temperatures. “Surface- to-surface” CTFs, which op
28、erate on surface temperatures and are required by the heat balance method, have the advantage of allowing for variable convective heat transfer coefficients. “Air-to-air” CTFs operate between either the sol-air tempera- ture or the air temperature on the outside and the air setpoint temperature on t
29、he inside. Air-to-air CTFs include the appro- priate film coefficients as resistive layers in the wall assembly. As shown in Figure 1, surface-to-surface CTFs are represented by the thermal circuit between To, and Ti, while air-to-air CTFs are represented by the thermal circuit between To and 7; For
30、 constructions with the same material layer arrangement and properties, the surface-to-surface CTFs are always the same, while air-to-air CTFs differ depending on the selected values of the film coefficients. 830 ASHRAE Transactions: Symposia Figure I CTF schematic diagram. The 1997 ASHRAE Handbook-
31、Fundamentals presents an “air-to-air” conduction equation that includes additional simplifications. The b and c terms shown in Equation 5 operate on the sol-air temperature and the constant room air tempera- ture, respectively. 6 6 6 qe, = bn*e,-n- C nqre,-n6-*rc C cn (5) n=O n= 1 n=O It should be n
32、oted that Equation 5 is suitable only for load calculations. Historically, it was used in the Transfer Function Method (TFM) (McQuiston and Spitler 1992) and can be used without loss of generality in the Radiant Time Series Method (RTSM). Although Equations 3 through 5 are solutions to the tran- sie
33、nt, one-dimensional conduction problem, it is useful to consider the steady-state limit of these equations. Under steady-state conditions, the exterior and interior heat fluxes are equal and the following identities are readily apparent (Equation 6): NI NY Nz 6 6 c Xn = Y, = 1 Zn or 1 bn = c, (6) In
34、 combination with the standard formulation for steady- state heat transfer through a wall (9 = UAT), an expression for U, the overall heat transfer coefficient, in terms of conduc- tion transfer functions can be derived as shown in Equation 7. n=O n=O n=O n=O n=O NY 6 c y, C bn u= n=O or Uf = n=O 1-
35、 C +n c dn (7) Nb 6 n=l n= 1 Periodic Response Factor (PRF) Formulations As formulated in the ASHRAE Loads Toolkit, the Radi- ant Time Series Method for design load calculations uses peri- odic response factors (PRFs) (also called “conduction time factors”) rather than CTFs to calculate conductive h
36、eat trans- fer through walls and roofs. PRFs operate only on tempera- tures; the current surface heat flux is a function only of temperatures and does not rely on previous heat fluxes, as shown in Equation 8. This formulation is premised on the steady, periodic nature of the sol-air temperature over
37、 a 24-hour period (Spitler et al. 1997). Although the number of PRFs may vary, the 24 PRFs shown in Equation 8 correspond to 24 hourly changes in the sol-air temperature for a single diurnal cycle. It is clear from Equation 8 that the overall heat transfer coefficient, U, is represented by the sum o
38、f the periodic response factors as shown in Equation 9. 23 u= CPj j=O (9) The periodic response factor directly scales the contribu- tion of previous fluxes (in the form of temperature gradients) to the current conductive heat flux. As a result, the periodic response factor series provides a visual
39、representation of the thermal response ofthe wall. As shown in Figure 2, wall 17 has a slower thermal response then roof 10 because it is a more thermally massive construction. PRFs are directly related to CTFs as shown in Equation 10 (Spitler and Fisher 1999a) and may be derived directly from CTFs.
40、 The toolkit uses this method to calculate periodic response factors. where ASHRAE Transactions: Symposia 831 0.18 , 0.16 . - 0.14 : E 0.1 - 2 0.08 - Y -7 $ 0.12 - Po . P, P4 P, P, P, P, Po . P, P, P, P, P, P, Po . P, P, P, P, P, P, Po . P, P4 P, P, P, P, P, . P, P, . P, P, P, P, P, p = . . . . . .
41、. . . . . . . . . . . . . . . i - +ROOF 10 -Ei- WALL1 7 (1 1) 0.06 0.04 0.02 O k O 5 10 15 20 25 Hour Figure 2 Periodic response factors for Roof 1 O and Wall 17. d, d, I o o . d= (12) I . O O . b, b, bj b, bo o o . b, As shown in Equation 1 O, the PRFs are related to the cross and flux CTF terms. T
42、he first column of the P matrix is the resulting PRFs, Po, Pl, P, , PZ3. Since the sol-air temper- ature is used in RTSM conduction calculations, the b and d matrices must be filled with air-to-air CTFs. This eliminates the surface heat balance calculations in HBM. However, if conductive heat transf
43、er is an isolated concern, the PRFs can be calculated from surface-to-surface CTFs. This reflects the actual conduction response of a construction without consid- ering the outside and inside film coefficients. IMPLICIT ASSUMPTIONS OF TRANSFER FUNCTION EQUATIONS The assumptions behind the transfer f
44、unction equations come from the CTF calculation methods. Two widely used CTF calculation methods are the Laplace method (Stephenson and Mitalas 1971; Hittle 1979) and the state-space method (Ceylan and Myers 1980; Seem 1987; Ouyang and Haghighat 1991). A brief overview of these two methods is includ
45、ed in the following sections. Laplace Transform Method Hittle (1 979) introduced a procedure to solve the conduc- tive heat transfer governing Equations 1 and 2 by using the Laplace transform method. The system in the Laplace domain is shown in Equation 14. Response factors are generated by applying
46、 a unit trian- gular temperature pulse to the inside and outside surface of the multi-layered slab. The response factors are defined as an infi- nite series of discretized heat fluxes on each surface due to both an outside and inside temperature pulse. Hittle also described an algebraic operation to
47、 group response factors into CTFs, and to truncate the infinite series of response factors by the introduction of flux history coefficients. A convergence criterion shown in Equation 15 is used in the Laplace method to determine whether the numbers of CTFs and flux history coefficients are sufficien
48、t such that the resulting CTFs accu- rately represent the response factors. Nx NY Nz N+ c x, = y, = zn = u n (1 -4J (15) n=O n=O n=O n= 1 where N, = Ny = N, The X, Y,l, and Zn are exterior, cross, and interior CTFs, respectively. They are equivalent to CTFs shown in Equations 3 and 4. The number of
49、CTF terms will increase to satisfi the criteria shown in Equation 15. Heavyweight (long thermal response time) constructions require more CTFs than light- weight constructions. The number of CTF terms can be determined in different ways. Mitalas (1978) suggests that the number of CTF terms should be N XY =N =N z = 1+N, (16) 832 ASHRAE Transactions: Symposia and there is no limited number of CTF terms in this approach. Peavy (1978) suggests that the number of flux CTF terms should always be less than or equal to 5, even for thermally massive wails. State-Space
