Transactions on Engineering Sciences vol 5, 1994 WIT Press, ISSN

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1 Generalize heat conuction transfer functions in two imensional linear systems of assigne geometry F. Marcotullio & A. Ponticiello Dipartimento i Energetica- Universit i L 'Aquila Monteluco i Roio L'Aquila, Italy ABSTRACT The present paper eals with the search for transfer functions vali for the whole class of object of an assigne two imensional geometry. The basic criterion was to turn the governing conuction equation an the transient bounary conitions as well into imensionless variables. The conclusion was that the same the TFs hol for all systems provie the Nusselt number (Nu) an the Fourier moulus (Fo) are equal. An appropriate TF's coefficients table has been set-up for the central an the corner points of a homogeneous soli square profile. The imensionless time interval AFo was selecte accoring to the sampling theorem, i.e. to the cut-off frequency of the system which epens on Nu. In aition, the Nu incremental steps were set in orer to provie "equal" percent eviation (10%) an thus avoi the necessity of temperature interpolations. The generalize TF's coefficients enables a wier use of the TF approach which is very useful for thermal transient analysis. INTRODUCTION The Z-transfer function approach represents a very powerful tool for preicting temperature histories in multiimensional thermal fiels, e.g. Mascheroni [1]. It is particularly useful, for instance, in simulating the ynamic behaviour of linear systems uner ifferent transient bounary conitions, e.g. Stephenson an Mitalas [2]. The search for the transfer functions (TF) in multiimensional heat conuction by a finite-element technique was envisage by the authors [3]. The basic criterion was that of rearranging an conensing the matrix finite-element equation to reuce the egrees of freeom to only one. The conensing proceure will yiel, in general, aitional terms (higher orer time-enva-

2 36 Heat Transfer tives) in the resulting "lumpe" matrix equation. At this stage, the numerical solution in the time-omain iscretization was obtaine explicitly using an appropriate recurrence formula. The coefficients of the Z-transfer functions were thereby compute from the cite recurrence scheme by matching the system free response in orer to set the correct recurrence parameters. In this stuy, the matrix conuction equation is turne into imensionless variables reaching the conclusion that the same TFs hol for the whole class of objects having the same geometric shape (plates, ros, bars, cyliners...) provie the Nusselt number (Nu) an the Fourier moulus (Fo) are equal. This expecte but relevant result enables a wier use of the TF approach, otherwise relegate to rare applications since the TF's ientification presupposes a certain amount of skilfulness which is infrequent in common practice. So far, it will suffice to set-up, at preselecte points an for iffering geometric shapes of practical interest, appropriate tables listing the series of the TF's coefficients for pairs of the inepenent imensionless variables Nu an Fo BASIC CONSIDERATIONS The governing conuction equation in a uniform thermal conuctivity soli ii (x,y,z) free of sources is given by: x* Let our task be the search for the general temperature history within the soli unergoing the following initial an transient bounary conitions: To turn Equation (1) into imensionless variables let: ' - - where L<. is a characteristic length an Fo is the Fourier moulus. By substituting in Equation (1) we obtain:

3 Heat Transfer 37 with the bounary conition expresse as: T (A\ ul ~. XT,, /nr T \ = Q (5) vv < where Nu=hLJk is the Nusselt number. It follows that the analytical solution T will be a function of the imensionless variables %,T,^, Fo, Nu an will not epen explicitly upon the soli thermophysical properties which, instea, are inclue in Fo (iffusivity a) an Nu (conuctivity *). The same conclusions can be rawn by performing a numerical analysis base, for example, on the finite-element metho. In this case, the iscretization of the whole omain Q (%,T,Q in a certain number M of finite elements leas to the well-known matrix equation Ca+Ka-f = 0 (6) here a is the vector of the unknown noal temperature of the system an a the first time-erivative with respect to the Fourier moulus Fo. All the matrices are assemble from the element submatnces in the stanar manner with submatnces K, CT an f given by: K = " j\$ $ T) ri t c^q ; (7a) (7b) (7c) where a is the ambient flui temperature varying with time (forcing function) an Ni, N^ the interpolation an weighting functions. Now, since the

4 38 Heat Transfer transfer functions (TF) at any selecte noe (,T,Q of the fiel are obtaine from the matrix Equation (6) by means of the proceure escribe in [3], it follows that exactly the same TF's hol for any assigne geometry provie the interpolation functions N(,r,Q, the Nusselt number Nu an the imensionless time interval AFo=Aia/L/ are equal. The transfer functions are given as sets of coefficients each relate to its own forcing function represente in general by m inepenent surrouning temperatures. The time-series of the preselecte noal temperature # +, are linke to each timeseries of the input # +, by means of a convolution equation such as: m ^ Nu AND AFo SELECTION To set-up a table listing the series of TF's coefficients for pairs of the inepenent imensionless variables Nu an AFo to be use without interpolation, some appropriate criteria shoul be followe. Firstly, the Fourier moulus iscretization AFo is selecte at least equal to X/15 accoring to the sampling theorem, X being the ominant eigenvalue of the matrix K C obtaine from Equation (6). The above conition implies that the system frequency response to the unitary step-function of the ambient flui temperature is ampe out at approximately 2% of its maximum value. In fact, the temperature history at any arbitrary point is given by: which, as time progresses, can be aequately represente by: T o) The Laplace transfer function is: an by setting the Laplace parameter s=27wi the moulus IW(s)l takes the form:

5 Heat Transfer 39 Table 1 - TF's coefficients for a soli homogeneous square profile 2^ i Nu AFo xlo* L;= t L ^~ I 4-/Vi-y+ tf-/v,-/ = <> 2=0 /= * b* b~ b> b^ b^ b- n n n n n n

6 40 Heat Transfer 1 (12) If a cut-off frequency v, is chosen so as to obtain for Iwl a 2% attenuation, then v,«50/27txi an, corresponingly, AFo=l/2v«.=A,i/15. As it clearly appears from the set of Equations (7), since A,, epens upon Nu then the above time interval selection criterion allows us to set-up the TF's coefficients table with only one entry (Nu). Another problem that arose was the search for the appropriate Nusselt incremental steps ensuring "equal" percent temperature variations. For general engineering purposes, a 10% sprea will suffice to hol the transient response within 5% maximum eviation an thus avoi teious temperature interpolations. As an application example, the TF's coefficients for the central an the corner points of an infinite homogenous plate 0.6m by 0.6m of a soli square profile subject to transient convective flow are reporte in Table 1. Assuming the characteristic length L<=L=0.6m, the unit surface conuctance h=5.8wm^k"'an the thermal conuctivity k=0.65wm"*k% then Nu=hLJ( ^=5.35 against 4.8, which is the nearest Nu value in Table 1. Temperature histories for both noes 1 an 2 were plotte by using the series of coefficients corresponing to Nu=4.8 an AFo=6.510^ an assuming (Figure 1) an ambient flui linear rise temperature (3 C/h) an (Figure 2) a triangular pulse 10 C amplitue an 2Ai uration (AT=AFo-L//a=1.42h). The plots were checke in terms of actual time intervals against the results eriving from the use of the exact sets of TF's coefficients (Table 2). It is confirme that all eviations are within the chosen 5% maximum error range. Table 2 - Exact FT's Coefficients ( Ai=1.42h) Nu n+1 n n n-2 n-3 n b* CONCLUSIONS The following comments can be emphasize from the above stuy: the availability of the transfer function coefficients for the whole class of objects of assigne geometry in terms of imensionless variables such as the Nusselt number Nu an the time interval AFo, enables a wier use of the

7 Heat Transfer =0.65 Wm'iK'1 = 1700kgm" =0.60m = " =5.8 Wm'^K Noe! - Nu=4.8 Noel-Nu=5.35 Noe2 - Nu=4.8 Noe2-Nu= ,81 15,66 18,5 21,36 5,69 ' 8,54 11,39 14,23 17,08 19,93 Time (h) Figure 1 - Temperature histories at noes 1 an 2 with an ambient flui linear rise temperature (3 C/h) '"t" k=0.65 Wnr'K = 1700kgm'* c=836jkg-'k-' L =0.60 m a = ,85 5,69 8,54 11,39 Time (h) 14,23 19,93 21,35 Figure 2 - Temperature histories at noes 1 an 2 with an ambientfluitriangular pulse 2Ar=2.85 h uration an 10 C amplitue

8 42 Heat Transfer transfer function approach. appropriate tables listing the TF's coefficients have been set-up by choosing for the Nusselt number incremental steps of "equal" percent eviation (10%). This criterion permits the irect use of the sets of coefficients by referring to the closest Nu value, without the necessity of teious temperature interpolations. Moreover, by choosing the time interval AFo=l/2Vc=X/15, which epens on Nu, the TF's coefficients table turns into a more suitable one-entry table. REFERENCES 1. Mascheroni, R.H., Sanz, P.D. an Dominguez, M. 'A New Way to Preict Thermal Histories in Multiimensional Heat Conuction: the Z- Transfer Function Metho' Int. Comm. Heat & Mass Transfer, Vol.14, pp , Stephenson, D.G. an Mitalas, G.P. 'Calculation of Heat Conuction Transfer Function for Multilayer Slabs' ASHRAE Transactions, Marcotullio, F an Ponticiello, A. "Determination of Transfer Functions in Multiimensional Heat Conuction by Means of a Finite Element Technique' (Eite by R.W. Lewis, University of Swansea U.K.), Vol.VIII, part 1, sect. 2, pp , Proceeings of the Eighth International Conference on Numerical Methos in Thermal Problems, Swansea, Wales, Salvaori, U.O., Reynoso, R.O. an Mascheroni, R.H. 'The Use of Z- Transfer Functions to Evaluate Temperature an Quality Changes in Refrigerate Storage' Proceeings of meetings of Comm. B2, C2, Dl, D2/3, Int. Institute of Refrigeration, Paris, pp , ASHRAE Hanbook of Founamentals, 1977, Chap Zienkiewicz, O C The Finite Element Metho McGraw-Hill, Lonon, Huebner, K.H. The Finite Element Metho for Engineers J Wiley&Sons, New York, Rao, S.S. The Finite Element Metho in Engineering Pergamon Press, Oxfor, Schneier, P.J. Conuction Heat Transfer Aison-Wesley Publishing Company, Inc, 1974