# ANALYTICAL APPROACH TO TRANSIENT HEAT CONDUCTION IN ...

ANALYTICAL APPROACH TO TRANSIENT HEAT CONDUCTION IN ...

ANALYTICAL APPROACH TO TRANSIENT HEAT CONDUCTION IN ...

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<strong>ANALYTICAL</strong> <strong>APPROACH</strong> <strong>TO</strong> <strong>TRANSIENT</strong> <strong>HEAT</strong> <strong>CONDUCTION</strong><strong>IN</strong> COOL<strong>IN</strong>G LOAD CALCULATIONSMichal Duška 1 , Martin Barták 1 , František Drkal 1 and Jan Hensen 21 Department of Environmental Engineering, CTU in Prague166 07 Prague 6, Czech Republic2 Center for Building & Systems TNO - TU/e, TU Eindhoven5600 MB Eindhoven, Netherlandse-mail: michal.duska@fs.cvut.czABSTRACTThe paper briefly reviews four existing analyticalmethods for the solution of the Fourier’s equation incooling load calculations. The performance of ninedifferent procedures (the four methods and theirmodifications) is presented on an example of heattransmission through a heavy wall under realisticexternal conditions. For the current case it wasproved that the admittance method gives resultssimilar to those obtained by the periodic responsefactors (PRF) approach, which is normallyconsidered to be more advanced. Possible errorswere found in the previously published coefficientsfor one of the PRF modifications.<strong>IN</strong>TRODUCTIONCurrent pressure to lower the consumption ofprimary energy sources has recently emphasized theissue of cooling load calculation accuracy in theCzech Republic and other European countries. Thisleads to revisions of existing methods and efforts todefine standards for required accuracy in coolingload calculations (e.g. prEN 15255).Cooling load calculations account for a number ofinterrelated processes that are difficult to describeprecisely. One of the most complicated componentsis the thermal storage effect which is important whenconverting instantaneous heat gains for a given roominto its cooling load. The thermal storage capacity ofconstructions that enclose a room affects (a) the heatconduction of radiant energy absorbed on internalsurfaces and (b) the heat conduction due to externaltemperature and radiant excitation. The latter is asubject of the present study.The Fourier's equation is linear and time invariantwhich enables to find its analytical solutions even forcomplicated boundary conditions.Four analytical methods for the solution of theFourier’s equation are reviewed and comparedregarding their accuracy and complexity in thecalculation of heat transmission through an externalwall:• thermal response factors (TRF),• conduction transfer functions (CTF),• periodic response factors (PRF),• admittance method (AM).The first two methods can be found in computeraddedenergy performance analyses while the lattertwo are usually applied in manual cooling loadcalculations.METHODOLOGYThermal Response FactorsThe TRF concept is a simplification of wallboundary conditions (i.e. temperature variations) tothe series of linear functions, for which the Fourier'sequation can be solved analytically. The mostly usedfunctions in the form of triangular pulses (Mitalasand Stephenson, 1967) can be combined into a piecewiselinear approximation of the requiredtemperature profile as presented in Fig. 1.The spatial and temporal distribution of temperaturet ( x,τ ) in a homogeneous wall subject to onedimensional heat flow is given by the Fourier'sequation:2∂tλ ⎛ ∂ t ⎞=⎜⎟2∂τρ ⋅ c ⎝ ∂x⎠(1)Fig. 1: Linear approximation of input by triangularpulses

The linearity of the equation (1) allows superposingreactions to individual temperature pulses imposedon a wall surface. This means that instead of solvingthe equation (1) directly for a complicated input wecan superpose the solutions for a number of verysimple inputs (e.g. triangular pulses) to evaluate thewall response to continuously changing boundaryconditions. An example of the heat flux responseinduced by a unit triangular temperature step isshown in Fig. 2.where a j and b j are the coefficients of the z-transferfunction K(z) (Stephenson and Mitalas, 1971)K−1−2− Ja0 + a z + a z + ... + a z(6)b + b z + b z + ... + b z1 2J( z) =−1−2−P012where O n-j is the output signal (heat flux) and I n-j isthe input signal (temperature) at time step n–j. Theinput signal is a series of discrete pulses replacing thecontinuous temperature excitation (see Fig. 3).PFig. 2: Response to a unit triangular pulseThe heat flux response q j at time step j to the initialtemperature pulse of an arbitrary magnitude t isqj= r ⋅t(2)jwhere r j are the response factors obtained from theresponse function illustrated in Fig. 2.The heat flux at the surface A and time step n,derived as a response to the series of triangulartemperature pulses acting on both surfaces of thewall, readsqAn=∞∞A−A A∑rj⋅tn−j− ∑j=0 j=0rB−Aj⋅tBn−j(3)where A, B denote the surfaces, n and j are the timeindex variables, t n–j is the surface temperature at timestep j ahead of n, r A-A and r B-A are the response factorsat the surface A corresponding to the unittemperature pulses acting on the surface A and B,respectively.Similarly, the heat flux at the surface B isqBn=∞∞A−BA∑rj⋅ tn−j− ∑j=0 j=0rB−Bj⋅ tBn−j(4)where j determines the number of temperatures andresponse factors which must be considered prior tothe step n. The maximum of j depends on the type ofa wall and required accuracy (ideally j would beinfinite). The identity r A-B = r B-A applies for any wall.Conduction Transfer FunctionsThis method assumes that the relation between inputand output signals for a multi-layer wall can bedescribed asO ⋅b+ O ⋅ b + O ⋅b+ ... + O ⋅bn 0 n−11 n−22n−P= In⋅ a0+ In−1⋅ a1+ In−2⋅ a2+ ... + In−JP⋅ aJ(5)Fig. 3: Approximation of input by discrete pulsesRewriting the equation (5) for a wall with surfaces Aand B yields:qqAnBn==J∑tAn−jj=0J∑j=0tBn−j⋅ a⋅ aAjBj−−J∑j=0J∑j=0tBn−jtAn−j⋅ a⋅ aBjAj−−P∑j=1P∑j=1qqAn−jBn−j⋅ b⋅ bAjBj(7)(8)Finding the z-transfer function coefficients is moredifficult task than solving the response factors. Thez-transfer approach in heat conduction analysis wasprobably introduced by Stephenson and Mitalas(1971). Their direct root finding procedure was laterimproved by Hittle and Bishop (1983) and used forthe calculation of z-transfer function coefficients inHarris and McQuiston (1988) and ASHRAEHandbook of Fundamentals (1989, 1993, 1997).Spitler and Fisher (1999) pointed out that all thepreviously published coefficients for heavy walls hadbeen erroneous. Other possibilities to determine thecoefficients are the time-domain method (Davies,1996), state-space method (Jiang, 1982) andfrequency-domain regression method (Chen andWang, 2001).Periodic Response FactorsThe periodic response factors were defined for use inthe Radiant Time Series method developed by Spitleret al. (1997). This method assumes periodicvariations of external sol-air temperature andconstant internal air temperature for the computationof the heat flux through a wall at time step n:23B A−BA Bq = r ⋅ ( t − t )(9)n∑Pjj=0n−j

where A and B are the external and internal surfaces,A−BArPjis the periodic response factor, tn− j is the sol-airtemperature j hours ago and, tB is the constant roomair temperature. The PRF values can be obtainedusing TRF (Spitler, 1997) or CTF (Spitler and Fisher,1999).Admittance MethodThis approach is based on the assumption that timevariations of temperature or heat flow at a wallsurface are sinusoidal. The response to sinusoidalexcitation on one surface is again sinusoidal withinthe wall and on its other surface. The excitation andresponse signals differ in amplitude and phase.The relations between the variables on surfaces Aand B are usually presented in matrix form:⎡tˆ⎢⎣qˆBB⎤ ⎡a⎥ = ⎢⎦ ⎣a1121aa1222⎤ ⎡tˆ⎥ ⋅ ⎢⎦ ⎣qˆAA⎤⎥⎦(10)where tˆ and qˆ denote the cyclic (sinusoidal)temperatures and heat fluxes. The complex elementsof the transmission matrix are given as:a21a11= a22= cosh( p + ip)a12L ⋅sinh(p + ip)=λ ⋅ ( p + ip)( p + ip) ⋅ sinh( p + ip)(11)(12)⋅= λ (13)Lwhere L is the wall thickness and the parameter p fora 24-hour cycle (24×3600 = 86400 seconds) is:2π ⋅ L ⋅ ρ ⋅ cp =(14)86400 ⋅ λThe wall response to periodic excitations can bedescribed by three factors implemented in AM-basedcooling load calculations: admittance, decrementfactor and surface factor (Milbank and Lynn, 1974).The admittance is the magnitude of the ratio of thecyclic heat flux to the cyclic temperature at the samesurface. The surface factor is the ratio of the cyclicheat flow readmitted to the space from the surface tothe cyclic heat flow absorbed by the same surface.The decrement factor f is the magnitude of the ratioA Bof the cyclic transmittance Uˆ− to the steady-stateU value:fA−BA−BA−BU= f ˆ ˆ=U(15)The cyclic transmittance is defined using theequations (10) and (12) as:Bˆ A−Bqˆ1U = =(16)tˆAa12A BThe time lag ω in hours between tˆ and qˆ is:⎡ ˆ A−B12 Im( f ) ⎤ω = ⋅ arctan⎢⎥ (17)−⎢ˆ A Bπ ⎣Re(f ) ⎥⎦The mean temperatures over 24-hour cycle at thesurfaces A and B can be obtained fromt2323AB∑tn∑tnA n= 0 B n=0= =24andt24(18a,b)The swing in temperature at the surface A and timestep n is~ A A At = t − t(19)nnand the heat flux on the opposite surface B at thesame time step n readsq~ AA⋅ t ω + U ( t − t ) (20)B ~ B BA−BBn= qn+ q = U ⋅ fn−Awhere~ t denotes the swing in temperature at then−ωsurface A and time step ω ahead of n (assuming onehoursteps).EXAMPLE AND ANALYSISThe above methods were applied in the calculationsof transient heat conduction through a heavy externalwall defined as the ASHRAE Wall Group 37(ASHRAE, 1997). The WG37 is 505 mm thick andconsists of four layers (from outside to inside): facebrick, lightweight concrete, insulation and plaster.The thermal resistance at the external and internalsurfaces is 0.06 m 2 ⋅K⋅W -1 and 0.12 m 2 ⋅K⋅W -1 ,respectively.The external boundary conditions were defined bysol-air temperature varying in time. Realistic externalconditions (air temperature, solar and longwaveradiation flux) were taken from the BESTESTdatabase (Judkoff, 1995). The indoor air temperaturewas set constant at 20 °C. The present analysis waselaborated for a single summer day.The thermal response factors were adopted from thestudy by Chen et al. (2006) which provides not onlythe TRF values but also their verification. In thecurrent case of WG37, 144 TRF values are necessaryfor an acceptable accuracy (Chen et al., 2006). Thismeans that input values of the sol-air temperaturemust be provided for a period starting six days priorto the design day. The input sol-air data are presentedin Fig. 4.

Temperature [°C]4035302520151050-144 -120 -96 -72 -48 -24 0 24Time [h]Fig. 4: Sol-air temperature (design day: 0 – 23 h)Two sets of the coefficients for the CTF method wereapplied in this study. Those adopted from Harris andMcQuiston (1988) are indicated as CTF/ASHRAE.The set developed by Chen et al. (2006) is denoted asCTF/FDR. The CTF values for WG37 cover thecurrent time step plus six preceding hours. However,the heat flux at the beginning of the above period iscrucial and not known a priori. Therefore the presentCTF calculations included a start-up period of sixdays to obtain an accurate value of the heat flux sixhours before the design day begins.The version of PRF values published by Spitler andFisher (1999) is indicated as PRF/ASHRAE. Wecalculated another set of periodic response factorsdenoted as PRF/FDR using TRF and the procedureintroduced by Spitler et al. (1997).The PRF- and AM-based manual cooling loadcalculations assume that the design day has beenpreceded by an infinite number of identical days.Therefore the temperatures just from the design dayare needed as input. The results of this standardprocedure are noted as "cyclic" in the current study.In order to use boundary conditions that are realisticand comparable with the TRF and CTF methods,another set of PRF and AM calculations wasproduced using the real sol-air temperatures from the24-hour interval ahead of the design day.RESULTSThe comparison of the wall response characteristicscalculated by different methods is presented as timevariation plots of the heat flux at the internal surfaceand the external sol-air temperature for the designday in Fig. 5. Note that the response curves takeaccount of the preceding period not shown in thegraph, except those denoted with the suffix "cyclic".4035sol-air temperature0.50.25300Temperature [°C]25201510-0.25-0.5-0.75-1Heat flux [W/m 2 ]5-1.2500 6 12 18Time [h]-1.5TRF CTF/FDR CTF/ASHRAEPRF/ASHRAE - cyclic PRF/ASHRAE PRF/FDR - cyclicPRF/FDR AM - cyclic AMFig. 5: Time-variation of the heat flux based on different calculation methods

The results obtained by the TRF technique areconsidered as reference because fully documentedand verified factors for this method were availablefrom Chen et al. (2006). The results from othermethods are compared with the reference in Fig. 6using the standard deviation calculated over thedesign day. The standard deviation for each methodwas defined asmethodSTDmethodq j=23j=0method( q jq )∑ − TRFjTRFq j242(21)where and denote the heat fluxescalculated at the same time step j according to theassessed method and TRF, respectively.D [W/ m 2 ]ST0.70.60.50.40.30.20.10CTF/FDRCTF/ASHRAEPRF/ASHRAE - cyclicFig. 6: Standard deviation from TRF resultsDISCUSSIONComparison of the results in Fig. 6 shows that eventhough the CTF/FDR method requires only 6preceding time steps for the calculation, it performscomparably to the TRF technique which needs inputdata for 143 preceding time steps. The results fromthe CTF/ASHRAE procedure confirm the incorrectcoefficients for WG37 as pointed out by Spitler andFisher (1999).The PRF/ASHRAE procedure produced results withsurprisingly high STD comparing to the PRF/FDRmethod. Further analysis revealed that thePRF/ASHRAE coefficients published by Spitler aFisher (1999, Table A-3) are not correct because theydo not follow the summation rule (Chen et al., 2006):23A−B∑ rPj = Uj=0PRF/ASHRAEwhere U = 0.226 W·m –2·K –1 for WG37.(22)The control sums for the methods where thesummation rule is applicable are listed in Table 1.Another important outcome of the current analysis isthat both PRF and AM techniques produce verysimilar results when assuming identical cycles of thePRF/FDR - cyclicPRF/FDRAM - cyclicAMdesign day, although the AM is based on a rathercrude simplification of boundary conditions.It is obvious that the use of a cyclic design daysubstantially deteriorates the potential of thePRF/FDR and AM methods.Table 1: Control sums of coefficientsMethodTRFCTF/FDRCTF/ASHRAEPRF/ASHRAEPRF/FDRSummation rule143∑j=0rA−Bj=U6 6A∑aj ∑ =j= 0 j=0Controlsum0.226B/ bjU 0.2256 6A∑aj ∑ =j= 0 j=023B/ bjU 0.158A−B∑ rPj = Uj=023A−B∑ rPj = Uj=00.1590.226CONCLUSIONSCoefficients for the TRF, CTF and PRF techniquesshould be carefully revisited and those alreadypublished should be used with caution.Manual cooling load calculations based on simplifiedboundary conditions may produce results comparableto results obtained by more complicated methods.Current attempts to define standards for the coolingload calculation accuracy (e.g. prEN 15255) shouldbe aware of achievable accuracy for individualcomponents participating in cooling load. Thepresent study performed only for a single type ofwall indicates that this could be a difficult issue.Future work in this area is needed to analyseproperly different modes of heat gain in buildings.ACKNOWLEDGEMENTSThis research was supported by the Czech Ministryof Education under the Research Plan MSM6840770011.REFERENCESCIBSE Guide A: Environmental design, CharteredInstitution of Building Services Engineers, London,1999.Davies M. G., “A time-domain estimation of wallconduction transfer function coefficients”, ASHRAETransactions, v. 102, no. l, pp. 328-343, 1996.

ASHRAE Handbook of Fundamentals, AmericanSociety of Heating, Refrigerating and AirconditioningEngineers, Atlanta (GA), 1989.ASHRAE Handbook of Fundamentals, AmericanSociety of Heating, Refrigerating and AirconditioningEngineers, Atlanta (GA), 1993.ASHRAE Handbook of Fundamentals, AmericanSociety of Heating, Refrigerating and AirconditioningEngineers, Atlanta (GA), 1997.Harris, S. M., F. C. McQuiston, “A study tocategorize walls and roofs on the basis of thermalresponse”, ASHRAE Transactions, v. 94, no. 2, pp.688-715, 1988.Hittle D. C., R. Bishop, “An improved root-findingprocedure for use in calculating transient heat flowthrough multilayered slabs”, International Journal ofHeat and Mass Transfer, v. 26, no.1, pp. 1685–1694,1983.Chen Y. M., J. Zhou, J. D. Spitler, “Verification fortransient heat conduction calculation of multilayerbuilding constructions”, Energy and Buildings, v. 38,no. 4, pp. 340-348, 2006Chen Y. M., S. W. Wang, “Frequency domainregression method for estimating CTF model ofbuilding multi-layer walls”, Applied MathematicalModeling, v. 25, no. 7, pp 579–592, 2001.Spitler J. D., D. E. Fisher, “Development of periodicresponse factors for use with the radiant time seriesmethod”, ASHRAE Transactions, v 105, no 2, pp.491-502, 1999.Spitler J. D., D. E. Fisher, C. O. Pedersen, “Theradiant time series cooling load calculationprocedure”, ASHRAE Transactions, v. 103, pp. 503-515, 1997.Stephenson, D. G., G. P. Mitalas, “Calculation ofheat conduction transfer functions for multi-layerslabs”, ASHRAE Transactions, v.77, no.2, pp.117-126, 1971.NOMENCLATUREc specific heat capacity [J·kg –1·K –1 ]L wall thickness [m]q heat flux [W·m –2 ]t temperature [°C]U steady-state heat transmittance [W·m –2·K –1 ]x spatial coordinate [m]λ thermal conductivity [W·m –1·K –1 ]ρ density [kg·m –3 ]τ time [s]Jiang Y., “State-space method for the calculation ofair-conditioning loads and the simulation of thermalbehavior of the room”, ASHRAE Transactions v. 88,no. 2, pp. 122–138, 1982.Judkoff R., J. Neymark. “International EnergyAgency building energy simulation test (BESTEST)and diagnostic method”, National Renewable EnergyLaboratory, Golden (CO), 1995.Milbank, N. O., J. H. Lynn, “Thermal response andthe admittance procedure”, Building ServicesEngineer, pp. 38-50, 1974.Mitalas, G. P., D. G. Stephenson, “Room thermalresponse factors”, ASHRAE Transactions, v.73,no.1, pp. 1-10, 1967.prEN 15255 “Thermal performance of buildings –sensible room cooling load calculation - Generalcriteria and validation procedures”, EuropeanCommittee for Standardization (under approval).Seem J. E., S. A. Klein, W. A. Beckman, J. W.Mitchell, “Transfer functions for efficient calculationof multidimensional transient heat transfer”, Journalof Heat Transfer, v. 11, pp. 5–12, 1989.