1、2010 ASHRAE 585ABSTRACT This paper presents a new mathematical approach applied to the Conduction Transfer Functions (CTFs) of a multilayered wall to predict the reliability of building simula-tions based upon them. Such a procedure can be used to develop a decision support system that identifies th
2、e best condition to calculate the best CTFs set. This is a critical point at the core of ASHRAE calculation methodology founded on the Transfer Function Method (TFM). To evaluate the perfor-mance of different CTFs sets, the authors built a large amount of data, subsequently employed to train a Neura
3、l Network Classifier (NNC) able to predict the reliability of a simulation without performing it. For this purpose all the multilayered walls included in the HVAC ASHRAE Handbook were used, and moreover many other walls typical of Mediterranean building heritage were added. The results show that the
4、 proposed method to optimize CTFs based on NNC is highly accurate, fast and easy to integrate in other buildings simula-tion tools. INTRODUCTIONSimulation and analysis of the thermal fluxes in a building help the developer to choose the best materials for the local climatic characteristics and conse
5、quently to improve the enve-lope thermal performance and the inner thermal conditions. Many thermal processes are relevant in the assessment of building thermal behavior. These include the following: heat conduction through exterior walls, roofs, ceilings, floors and interior partitions;solar radiat
6、ion through transparent surfaces; latent or sensible heat generated in the space by occu-pants, lights, and appliances; andheat transfer through ventilation and infiltration of out-door air and other miscellaneous heat gains. (ASHRAE Handbook 2005)One of the most important items in the above process
7、 is the thermal conduction through a multilayered wall which can be calculated in several ways, including:Numerical finite differences;Numerical finite elements;Transform methods; andTime series methodsMany software packages for the thermal dynamic simu-lation of buildings employ the Transfer Functi
8、on Method (TFM) or Conduction Transfer Functions (CTFs) to provide a set of coefficients to relate the conductive heat fluxes to the current and past surface temperatures and past heat fluxes. TFM has been selected for the procedure recommended by ASHRAE, and called the Heat Balance Method (HB) (ASH
9、RAE Handbook 2005), mainly because of the following:computational time advantage; andinputs or outputs data are discrete in the time domain such as climatic dataAccurate simulations of thermal systems in the built envi-ronment can be performed using complicated modeling tech-niques and are available
10、 in many software packages. Some of the most used software, as DOE-2, TRNSYS, and ENERGY PLUS, which are employed to perform design cooling load To Assess the Validity of the Transfer Function Method: A Neural Model for the Optimal Choice of Conduction Transfer FunctionsMaurizio Cellura, PhD Valerio
11、 Lo Brano, PhDMarina Mistretta, PhD Aldo OrioliMaurizio Cellura is an associate professor, Valerio Lo Brano is an assistant professor, and Aldo Orioli is a full professor in the Dipartimento di Ricerche Energetiche e Ambientali, Universit degli Studi di Palermo, Viale delle Scienze, Palermo, Italy.
12、Marina Mistretta is an assistant professor in the Dipartimento di Arte, Scienza e Tecnica del Costruire, Universit degli Studi Mediterranea di Reggio Calabria, Salita Melissari, Reggio Calabria, Italy. AB-10-0222010, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (ww
13、w.ashrae.org). Published in ASHRAE Transactions (2010, Vol. 116, Part 2). For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.586 ASHRAE Transactionscalculations, use mathematical mode
14、ls based on transform methods, such as:the response factor method; andTFM or z-transfer function methodThe TFM developed by Stephenson and Mitalas (Stephenson and Mitalas 1971) uses CTFs to calculate the transient one-dimensional heat conduction through the build-ing wall and roof elements.As presen
15、ted in the following section some researchers have shown that the earlier methods can lead to unreliable evaluations especially when the walls have high thermal iner-tia. CTFs represent an efficient method to compute surface heat fluxes because they do not require the knowledge of temperatures and f
16、luxes within the boundary surfaces of the thermal elements. Unfortunately, conduction transfer function series become progressively more unstable as the time step decreases, and eventually this instability can lead the entire simulation to diverge.In this paper, the authors focus on these issues and
17、 also assess the numerical stability and sensitivity analyses of the time step and of the thermal inertia on the prediction of thermal performance of walls.CONDUCTION TRANSFER FUNCTIONThe evaluation of CTFs can be performed with an approx-imate mathematical approach because the exact solution would
18、require an infinite number of calculations. The CTFs method identifies the relationship between the signal that is applied to the system (a multilayered wall), called “input” and the response of the system (the temperature of the wall surfaces or a thermal flux through the wall), called “output”. Th
19、is relation is called the Transfer Function of the system. Following the well-known approach developed by Stephenson and Mitalas (Stephenson and Mitalas 1971), based on the use of the Z-transform (ZT) (Jury 1964), let us consider a thermal system, like a wall, in which is the input signal and is the
20、 correlated output signal. Those signals, which vary with the time , can be the temperatures of the fluids adjacent the wall or the heat fluxes through the surfaces.If and are the corre-sponding z-transformed signals, the transfer function of the system can be written in the formwhere and are polyno
21、mial expressions. The roots of the denominator are called poles, (P) and they are mathe-matically infinite. A signal that is time sampled is called a discrete-time signal. The sampling period is related to the hours of data-collection, usually hour. Excluding approximations linked to physical assump
22、-tions, the weakest point is due to the truncation of the infinite coefficients that constitute the Transfer Functions (TFs). The absolute values of the coefficients n and d of the expression:(1)very quickly decreases when the order of the addendum increases and, for this reason, it is possible to t
23、runcate the terms of and . In order to accomplish a correct trun-cation procedure, which is affected by the choice of the selected number of poles, it is necessary to evaluate the effect on the numerical response linked to the insertion or to the elimination of the coefficient of order R + 1 or M +
24、1. Such an evaluation can be consciously performed only if a large number of coefficients is available.The nonsteady state heat transmission through a multi-layered wall can be described with the following equations:(2)in which and are the z-transform of the temperatures and of the inside and outsid
25、e air surrounding the wall, respectively and and are the z-transform of the heat fluxes and in corresponding of the inside and outside surfaces of the wall, respectively. A, B, C and D are the coefficients of the wall conduction matrix, reached through the product of transmission matrices of each la
26、yer forming the wall; and are the inside and outside convection coefficients.One consequence of the previous equations is:(3)and the quantities:(4)are the transfer functions necessary to solve the problem. If each expression is truncated to the first N terms, it is possible to write:(5)(6)(7)u ()y (
27、)Uz() Zu()= Yz() Zy()=Gz()Zy()Zu()-nz()dz()-=nz() dz() 1=nz()dz()-n0n1z1n2z2 nRzR+d0d1z1d2z2 dMzM+-=nz() dz()toqo1 1 hi0 1ABCD1 1 ho0 1tiqi=tit0TiToqiq0QiQohihoqiz()1Bz()-toz()Az()Bz()-tiz()=H1z()1Bz()-num1z()den z()-b0b1z1b2z2+d0d1z1d2z2+-= =H2z()Az()Bz()-num2z()den z()-c0c1z1c2z2+d0d1z1d2z2+-= =qi
28、z() qi 0,qi 1,z1 qiN,zN+=qoz() qo 0,qo 1,z1. + qoN,zN+=tiz() ti 0,ti 1,z1 tiN,zN+=2010, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions (2010, Vol. 116, Part 2). For personal use only. Additional reproduction, distrib
29、ution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.2010 ASHRAE 587(8)(9)(10)where . Equation 3 can be rewritten in the following form:(11)Performing the products and ordering with respect the variable z, the last equation becomes:(12)wher
30、e(13)(14)(15)In conclusion it is possible to state that:(16)Thus at the discrete instant N, the heat gain per unit area through a wall or a roof can be calculated using a simple, recur-sive equation with constant coefficients; the sol-air tempera-ture Tair-solcan be used to represent outdoor conditi
31、ons. Assuming that the internal temperature is constant, the indoor heat flux through a multilayered wall at the current time is calculated by means of the following ASHRAE formula:(17)where= indoor heat flux at the current timeA = indoor surface area of a walln = summation indexNMAX= number of the
32、coefficients used = time stepTsol-air= sol-air temperature representing outdoor conditionsTindoor= constant indoor temperatureThe maximum number of coefficients that can be obtained from the polynomial transfer function depends on the number of poles P.(18)Therefore the CTFs calculation has two degr
33、ees of free-dom. One is due to the choice of the number of poles P, while the other is due to the choice of the number N of coefficients which cannot be greater than P 1.Reliability of Simulations Using CTFsThe reliability of CTFs has been addressed in the litera-ture and is still an interesting res
34、earch field due to the wide dissemination of Buildings Simulation Programs that use this algorithm. Although the methods above are widely used for predicting the hourly energy performance of the building envelope, in many commercial software the reliability of such an approximated mathematical appro
35、ach often cannot be checked. Other researchers aimed to assess the numerical problem of the z-transfer functions related to the influence of time step and thermal inertia. Dos Santos et al. (2004) applied different numerical methods to integrate the governing differ-ential equations in the air domai
36、n, and assessed the results in terms of accuracy and time computing. Moreover they showed the influence of simulation time step on the room air temper-ature and humidity profiles within the building envelope. Chen et al. (2006) used dynamic thermal parameters, including CTFs coefficients, thermal re
37、sponse factors and periodic response factors, in order to calculate the transient heat conduction in building constructions. They also highlighted that computational inaccuracy could occur in calculating CTF coefficients and response factors, and they introduced a method for verifying CTF coefficien
38、ts and response factors over the whole frequency range. To improve the performance of simulations some authors have proposed a multidimensional approach or new proce-dures. Kosny et al. (2002) asserted that most of the whole building thermal modeling software, such as DOE-2, BLAST, and ENERGY PLUS,
39、apply simplified one-dimensional descriptions of the building envelopes. However, they showed that one-dimensional analysis may generate serious errors in building loads estimation, especially for several structural and material configurations of envelope components with high thermal mass. toz() to
40、0,to 1,z1 toN,zN+=1Bz()-b0b1z1 bNzN+d0d1z1 dNzN+-=Az()Bz()-c0c1z1 cNzN+d0d1z1 dNzN+-=d01=qi 0,qi 1,z1 qiN,zN+()d0d1z1 dNzN+() =to 0,to 1,z1 toN,zN+ b0b1z1 bNzN+= +ti 0,ti 1,z1 tiN,zN+ c0c1z1 cNzN+()qi 0,D0+()qiN,DN+()zN+B0 BNzN+ C0 CNzN+=Dndjqin j,j 1=n=Bnbjton j,j 1=n=Cncjtin j,j 1=n=qi0qi1z1 qiNzN
41、+()B0D0C0 B1D1C1()z1+= BNDNCN()zN+TiTindoor=q () = A bnTsol air n()dnq n()A-n 1=NMAXn 0=NMAX Tindoorcnn 1=NMAXQz()=q ()NMAXP 1=2010, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions (2010, Vol. 116, Part 2). For person
42、al use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission.588 ASHRAE TransactionsLiao et al. (2004) highlighted that the most used software for thermal building simulation are often very difficult to
43、setup because too many parameters have to be defined. They presented a simplified physical model for estimating the aver-age air temperature in multi-zone heating systems.Other authors developed a procedure, based on the frequency-domain regression method, in order to calculate directly and accurate
44、ly the outside, across and inside periodic response factors of a multilayer wall or roof, taking into account the related geometric and thermal properties (Chen et al. 2005). They also highlighted that the results related to the periodic response factors, whose CTF coefficients are tabu-lated in the
45、 ASHRAE Handbook, are inaccurate. The accuracy of simulations also depends on the choice of the time step. It is well-known that analytical methods are equivalent to numerical ones only when the time step tends to zero. Thus, the choice of the time step has to optimize computer run time and results
46、accuracy, but generally most of multi-zone building simulations are performed by using a time step as long as 1 hour (dos Santos et al.2004).Furthermore, massive walls have a strongly time-depen-dent behavior and cannot be accurately represented using hourly simulations. The inaccurate computer mode
47、ls will assess incorrect energy requirements and will led to improp-erly sized heating and cooling equipment (Kosny et al. 2002).The authors have focused on some issues concerning the reliability of CTFs in the case of walls characterized by high thermal inertia, typical of European buildings (Beccali et al. 2005; Beccali et al. 2003; Cellura et al. 2003). Quality Assessment of CTFsThe authors carried out a comparison between simulation data obtained from the Fourier periodic steady state algorithm and those obtained from CT
