L. Malinowski, S. Bielski. To cite this version: HAL Id: hal

Transcription

1 Transient temperature field in a parallel-flo three-fluid heat echanger th the thermal capacitance of the alls and the longitudinal alls conduction L. Malinoski, S. Bielski To cite this version: L. Malinoski, S. Bielski. Transient temperature field in a parallel-flo three-fluid heat echanger th the thermal capacitance of the alls and the longitudinal alls conduction. Applied Thermal Engineering, Elsevier, 2009, 29 (5-6), pp.877. <0.06/j.applthermaleng >. <hal > HAL Id: hal Submitted on 27 Feb 20 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, hether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Accepted Manuscript Transient temperature field in a parallel-flo three-fluid heat echanger th the thermal capacitance of the alls and the longitudinal alls conduction L. Malinoski, S. Bielski PII: S359-43(08) DOI: 0.06/j.applthermaleng Reference: ATE 2494 To appear in: Applied Thermal Engineering Received Date: 0 November 2006 Revised Date: 4 April 2008 Accepted Date: 20 April 2008 Please cite this article as: L. Malinoski, S. Bielski, Transient temperature field in a parallel-flo three-fluid heat echanger th the thermal capacitance of the alls and the longitudinal alls conduction, Applied Thermal Engineering (2008), doi: 0.06/j.applthermaleng This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers e are providing this early version of the manuscript. The manuscript ll undergo copyediting, typesetting, and revie of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered hich could affect the content, and all legal disclaimers that apply to the journal pertain.

3 Transient temperature field in a parallel-flo three-fluid heat echanger th the thermal capacitance of the alls and the longitudinal alls conduction L. Malinoski* and S. Bielski Faculty of Maritime Technology Szczecin University of Technology Al. Piastó Szczecin, Poland Abstract Transient temperature response of a parallel-flo three-fluid heat echanger th the thermal capacitance of the alls and the longitudinal heat conduction through the alls is investigated numerically, by the implicit MacCormack method, for a step change in flo rate of one fluid. The impact of thermal properties of the alls on temperature field is eamined. The results of calculations sho that as the thermal diffusivity of the alls decreases, the effect of the alls increases. Keyords: Three fluid heat echanger; Transient heat transfer; Implicit MacCormack method; Transient temperature field *Corresponding author: Tel ; fa address: lmal@ps.pl (L. Malinoski)

4 Nomenclature A - diagonal matri of thermal diffusivity, 2 m s A i - cross-sectional area of channel no. i, m 2 a - thermal diffusivity, 2 m s b - interconnection vector, / s c - specific heat at constant pressure, J ( kgk ) l L q T T t V - characteristic dimension dependent on the shape of the all, m - length of channels, m - heat flu vector, 2 W m - temperature vector, o C - temperature, o C - time, s - diagonal matri of fluid velocity, m s v i - fluid velocity in channel no. i, m s u - interconnection vector, / s - coefficient resulting from replacing the heat conducted in the y direction by an equivalent heat source, / s X y - spatial co-ordinate, m - dimensionless spatial co-ordinate - spatial co-ordinate, m α - heat transfer coefficient, W ( m 2 K ) δ - implicit temporal difference operator δ - all thickness, m 2

5 Θ - eplicit temporal difference operator - dimensionless temperature λ - thermal conductivity, W ( mk ) ρ - density, 3 kg m φ - stability parameter, m / s τ Ω - dimensionless time - heat transfer perimeter, m subscripts - all superscripts a, b - labels of the nodes situated on the opposite surfaces of the all o y - initial steady state distribution of temperature - in the direction - in the y direction. Introduction In the general design process of multi-fluid heat echangers, the stationary operating conditions are of decisive importance, hoever, in reality heat echangers frequently undergo transients resulting from eternal load variations and regulations. Knoledge of dynamic behaviour of heat echangers is necessary for designing control and regulation systems of different industrial processes and operations, such as in nuclear reactors, cryogenic and petrochemical process plants and HVAC systems. A lot of attention has been given to mathematical modelling of three- and multifluid heat echangers in the literature. Hoever, in most cases, the solutions presented 3

6 refer to the determination of steady-state temperature fields [ - 7]. According to the best knoledge of the authors, the available literature gives little information regarding mathematical modelling of transient behaviour of such echangers [8-2]. Sekulic and Herman [8] solved numerically the set of partial differential equations describing temperature field in a counter-flo three-fluid heat echanger. They used the Wendroff implicit finite difference approimation and steady state initial conditions. In Reference [9] Sekulic et al. studied eperimentally and numerically transient temperature fields in a three-fluid heat echanger th to thermal connections. The eperiments confirmed the accuracy of the method used in Reference [8]. A semi-analytical solution for a three-fluid parallel-flo heat echanger th to thermal communications and the steady state initial conditions as formulated by Bielski and Malinoski [0]. The authors applied the Laplace transform technique th numerical inversion. The same authors [] derived fully analytical epressions for temperatures in a parallel-flo three-fluid heat echanger th to heat connections beteen the fluids, constant temperature in one channel, the uniform temperature initial conditions, and a step increase in the inlet temperature of one fluid. Luo et al. [2] investigated semianalytically and numerically dynamic responses to temperature and flo transients in multi-stream parallel- and counter-flo heat echangers taking into account the heat capacities of alls. They used the uniform temperature initial conditions or steady-state conditions. For linear and linearized cases, the governing equations ere solved by the semi-analytical Laplace transform method th numerical inversion. Non-linear cases ere solved numerically. Malinoski [3] formulated equations for transient behaviour of multi-fluid heat echangers and solved numerically, by the MacCormack eplicit predictor-corrector method, a four-fluid case. He analysed the response of the echanger 4

7 to a step change in temperature or flo rate of one fluid using the uniform temperature initial condition or steady state initial condition. It is orth mentioning that an n-fluid heat echanger characterizes itself by n separate channels in hich can flo up to n different fluids. A to-fluid n-channel heat echanger, e.g. multipass heat echanger, can be considered mathematically as an n- fluid heat echanger, i.e. the temperature field is described by n differential equations. Dynamic behaviour of n-channel to-fluid heat echangers is studied in References [4-2]. Among References [-2], only the authors of Reference [9] have accounted for the thermal capacity of the alls and the conduction of heat along the alls in their mathematical model of transient temperature field in a plate-type heat echanger (they have, hoever, omitted the impact of heat resistance on the heat transfer across the alls). In the present paper e solve numerically, by the implicit MacCormack predictor-corrector method, the system of partial differential equations describing the transient temperature field in a parallel-flo three-fluid heat echanger th three heat connections beteen the fluids. The model accounts for the thermal capacitance of the alls, the longitudinal heat conduction through the alls, and the conductive resistance across the alls. Eemplary calculations are carried out to determine the unsteady response of the heat echanger, initially at steady-state, to a step change in flo rate of one fluid. 2. Mathematical model A parallel-flo three-fluid heat echanger th three heat connections beteen the fluids is illustrated schematically in Fig.. For deriving the governing partial 5

8 differential equations describing the transient temperature field in such an echanger, the follong assumptions are made: physical properties of the fluids and alls materials are constant, the heat transfer coefficients are independent on temperature, but they depend on the flo rate, mass flo and fluid temperature in each channel are taken to be uniform over the cross-section perpendicular to the flo direction, no heat is lost to the ambient. With these assumptions, the model equations for heat transfer can be stated as T T + V + b = 0 () t 2 y T T T + A + A + u = 0 (2) t y here T (, (, (, = T2 T (3) T 3 T (, y, (, y, (, y, = T2 T (4) T 3 v 0 0 V = 0 v2 0 (5) 0 0 v3 a 0 0 A = 0 a2 0 (6) 0 0 a 3 6

9 y a 0 0 y y A = 0 a2 0 (7) y 0 0 a 3 a i λ = (8) ρ c a y i y λ = (9) ρ c (, ) (,, ) (, ) (,, ) (, ) (,, ) (, ) (,, ) (, ) (,, ) (, ) (,, ) b T t T y t + b 3 T t T3 y t b = b2 T2 t T y t + b22 T2 t T2 y t (0) b3 2 T3 t T2 y t + b33 T3 t T3 y t (,, ) (, ) (,, ) (, ) (,, ) (, ) (,, ) (, ) (,, ) (, ) (,, ) (, ) u T y t T t + u2 T y t T2 t u = u22 T 2 y t T2 t + u32 T2 y t T3 t () u 3 T3 y t T t + u33 T3 y t T3 t b u ij ij αiωij = (2) ρ c A i j i i αiωij = (3) ρ c A j j Anisotropic alls are assumed for calculations in order to allo for the impact of heat conduction across or/and along the alls being accounted for or not. In this study a transient response of the heat echanger is determined for a step change in flo rate of fluid no. For the case under consideration, the initial conditions are given by (,0) = ( ) T ϕ (4) (, y,0 ) = (, y) T ϕ (5) here 7

10 ( ) ϕ ( ) ( ) ( ) = ϕ2 ϕ (6) ϕ 3 ( y) ϕ (, y) ( y) (, y) = ϕ2 ϕ,, (7) ϕ 3 The ϕ ( ) and ( y) i ϕ are functions of spatial co-ordinates outlining the steady-state, temperature field in the analysed heat echanger at a start time t = 0 s. The boundary conditions to be applied to set () (2) are for fluids in T in T ( 0, =T2 (8) in T 3 here the for alls in T i, for i =, 2, 3, are the constant inlet temperatures of the fluids ( 0, y, ( 0, y, λ T q = = 0 (9) ( L, y, ( L, y, λ T q = = 0 (20) Conditions (9) and (20) mean that the edges of the alls are thermally insulated. 3. Numerical solution The set of partial differential equations given by Egs. () (3) is solved numerically by means of the MacCormack implicit predictor-corrector method [22-24]. This to-step finite difference method of second order accuracy is very effective 8

11 for solving classical transient heat transfer problems in heat echangers, both for a step change in flo rate and for a step change in inlet temperature [0,, 3]. While meshing the alls e used only to node points in the y direction, situated on the surfaces of the alls, because δ << L. These nodes are designated as a and b (see Fig. 2). To simplify the solution of the problem given by Eqs. () and (2), e replaced the to-dimensional equation of heat conduction (2) th a onedimensional equation in hich the heat conducted in the y direction is considered as the heat generated by an equivalent heat source. T T + V + b * = 0 t (2) 2 T T * + A + u = 0 2 t (22) Vector T and matri A have no the follong form, respectively a T b T a T 2 T = b (23) T 2 a T 3 b T3 a a a A = (24) a a a3 The source term, u *, in Eq. (22) takes into account the heat echanged in the y direction both by conduction and by convection. A numerical scheme for solving Eqs. (2) and 9

12 (22) is developed on the basis of epressions formulated by MacCormack [22]. The finite-difference representations of Eqs. (2) and (22) are predictor ( ) n t n n * n Tj = V Tj Tj + tb j n t n t n δ Tj = I + Tj + δ Tj n+ n n T j = Tj + δ Tj n t n n n * n T = A 2 ( T 2 j T j j j ) t + + T + u j n t n t n δ T = + + δ j I T j T j n+ n n T = T + δ j T j j corrector (final value) ( + ) n+ t n+ n+ n+ * T j = V T j T j + tb j n+ t n+ t n+ δ T j = I + T j + δ T j+ n+ n n+ n+ Tj = 0.5( Tj + T j + δ T j ) t T = A 2 ( T 2T + T ) + u + n+ t n+ t n+ δ T = + + δ j I T j T j+ n+ n n+ n+ T = 0.5 T + T + δ T n+ n+ n+ n+ n+ * t j j j j j ( ) j j j j (25) (26) In equations (25) and (26), n is a time step, j is a grid point, δ and are implicit and eplicit temporal difference operators, respectively. The entries of matries and are determined from the conditions for the method s stability formulated by MacCormack [22]. 0

13 φ 0 0 = 0 φ2 0 (27) 0 0 φ3 φ φ φ = (28) φ φ φ3 φ i ma0.5 v i, 0 i =, 2, 3 (29) t 2a φ ma0.5 i, 0 i =, 2, 3 (30) t Interconnections vectors * b and * u have the form a (, ) (, ) + (, ) (, ) a b T t T t b 3 T3 t T t * b a b = b 2 T (, T2 (, b 22 T2 (, T2 (, + (3) b b b 32 T2 (, T3 (, + b 33 T3 (, T3 (, u * a b a (, ) (, ) + (, ) (, ) u T t T t T t T t b a b u 2 T2 (, T (, T (, t ) T (, + a b a u 22 T2 (, T2 (, + 2 T2 (, t ) T2 (, = b a b u 32 T3 (, T 2 (, t ) + 2 T2 (, T2 (, a u 3 T (, T3 (, + b a 3 T 3 (, T3 (, b a b u 33 T3 (, T3 (, + 3 T3 (, T3 (, (32) The value of coefficient i results from the equality of heat conducted to a cell and the heat generated in the cell by an equivalent volumetric heat source th capacity q vi. qvi ρ c = i b a ( T T ) (33)

14 b T and for a T are the temperatures of nodes a and b, respectively. The general epression i is as follos y λ = (34) ρ c δ l here l is a characteristic dimension dependent on the shape of the all. For a flat all l = δ / 2. The n+ n+ j, j T T are predicted, and the n j + T, n+ j T are final values of temperatures at time step n +. In the predictor equation e use a backard difference for, hile in the corrector equation a forard difference is used. A second derivative, 2 2, is approimated th a central difference both in the case of predictor and corrector equations. In the implicit MacCormack method the calculation for each time step is divided into to stages. In the first stage, the eplicit MacCormack method is used to calculate the changes in temperatures. In the second stage these changes are used in the implicit difference formula to determine the final temperatures for a given time step. Such to stages take place both at the predictor and corrector steps. During all calculations performed in this paper, e use the spatial step equal to = 0.00 m to achieve good accuracy of the results. 4. Validation of the model and method The correctness of the mathematical model adopted and the method of solution used is verified by comparing the results obtained from the numerical method th those from a semi-analytical solution for the case of a co-current three-fluid heat echanger th to thermal couplings [0]. The present model th three thermal 2

15 couplings, accounting for the thermal capacity of the alls and heat conduction along the alls, is reduced to the model analysed in the ork [0] by the adoption of the follong data for calculations: c, λ 0, for i =, 2, 3 (heat is neither transferred y along the alls nor accumulated in them), λ i, for i =, 2 (heat resistance of alls no. and no. 2 in the direction perpendicular to the ais of the heat echanger equals y zero) and λ 3 0 (all no. 3 also does not conduct heat in the y direction). Verifying calculations are carried out assuming that the system, initially at steady-state, undergoes an eponential rise of the inlet temperature of the fluid flong through channel no.. The temperature profiles calculated using the semi-analytical method and the implicit MacCormack method are compared pictorially in Fig. 3. Very good consistency of the results is observed for the to methods applied. 5. Sample calculations Sample calculations for the heat echanger presented schematically in Fig. are carried out using the implicit MacCormack method. The calculations account for the thermal capacity of the alls, longitudinal heat conduction through the alls, as ell as the alls thermal resistance in the direction perpendicular to the heat echanger ais. We consider a case here heat is transferred from a hot fluid to to cold fluids. Such a case is encountered in various technological processes here one heating agent is used to arm to different fluids (or the same fluids to different temperatures). Our calculations do not concern one specific practical process. The date as selected to be representative and to enable us to best illustrate the influence of alls on temperatures of fluids. 3

16 o The steady-state distributions of fluids temperatures, ϕ ( ) = temperatures, ( ) o T i T i, and alls ϕ =, for i =, 2, 3, are assumed to be knon. The follong values of fluids temperatures at the echanger inlet are chosen: in o T = 60 C, in o T2 = 20 C, in o T3 = 0 C. The remaining data taken for the calculations are: D = 0.05 m, D2 = 0.09 m, L = 0.50 m, δ = 0.00 m, ν = 0.20 m s, ν 2 = 0.0 m s, ν 3 = 0.5 m s, 2 K 2 2 K 2 3 K α = 3692W /( m ), α = 227W /( m ), α = 350W /( m ). The thermophysical parameters of the fluids correspond th the values for ater. The thermophysical parameters of the alls have values corresponding to typical materials used for construction of heat echangers: copper alloy C24000, stainless steel AISI 40 and titanium alloy Ti6Al4V. The thermal properties of these constructional materials are presented in Table. In Figures 4-7 the transient response of the sample heat echanger is presented for the case of a step-like tofold increase of fluid inlet velocity in channel no.. In our calculations e took into account the increase of heat transfer coefficient in channel no. resulting from the increase in the fluid velocity. Figure 4 shos the dimensionless eit temperatures of fluids, = [ ( L, t ) T ] ( T T ) Θ, for i =, 2, 3, versus i T i min / ma min dimensionless time τ = t / t. T min and T ma are the minimum and maimum ma temperatures in the echanger, respectively, tma = 0 s is the observation time. Time transients of the dimensionless average alls temperatures at the echanger outlet, Θ = [ ( L, t ) T ] ( T T ) T min / ma a b temperature is calculated as T =. 5( T + T ) min, are demonstrated in Fig. 5. The average all 0. It is seen in Figs. 4 and 5 that as the time elapses, the fluids temperatures at the outlet and the alls temperatures at the outlet cross-section approach the steady-state values. It appears from Figs. 4 and 5 that 4

17 the durations of transients depend on the all material. The echanger reaches the steady state in about 8-0 s. Figure 6 presents unsteady fluids temperature profiles, ( T i,, for i =, 2, 3, for time t = s. Figure 7 presents unsteady average alls temperature profiles, T (,, for i =, 2, 3, for the same time t = s. The results of research into the impact of alls on the steady-state temperature field (unsteady temperature field for observation time t ) in the analysed heat echanger are shon in Fig. 8. The calculated temperature profiles are compared th those determined thout taking into account the heat capacity of alls and heat conduction along and across the alls (solid lines). It is found that for the values of thermal diffusivity, a λ ( ρ c ) =, characteristic for good heat conductors, such as copper alloy C24000 and stainless steel AISI 40, the impact of alls on the temperature field in the heat echanger is negligibly small. The loer the value of thermal diffusivity (e.g. for titanium alloy Ti6Al4V), the higher the impact of alls on the temperature field in the heat echanger. 6. Conclusions Transient and steady state temperature fields have been determined, by the MacCormack implicit method, in a parallel-flo three-fluid heat echanger th regard to the transverse resistance and thermal capacitance of the alls, as ell as the longitudinal alls conduction. Impact of the alls material on the transient temperature profiles has been eamined. This impact is not very large. The alls influence the fluids temperatures in three ays: () they resist heat conduction in the transverse direction, (2) they accumulate heat during transient process, (3) they conduct heat longitudinally. Good thermal conductivity of the alls increases the influence of longitudinal heat 5

18 conduction and at the same time it decreases the effect of resistance in transverse direction. An increase in specific heat alays enlarges the effect of the alls on the transient temperature field in the echanger. Lo thermal conductivity and high specific heat result in lo thermal diffusivity. It has been observed that the impact of alls on the temperature of fluids flong in the channels of the echanger is the higher, the loer is the value of thermal diffusivity of alls. Obviously, this impact increases th the increase of the all thickness. 6

19 References [] D.D. Aulds, R.F. Barron, Three-fluid heat echanger effectiveness, International Journal of Heat and Mass Transfer 0 (967) [2] J.C. Chato, R.J. Laverman, J.M. Shah, Analyses of parallel flo, multi-stream heat echangers, International Journal of Heat and Mass Transfer 4 (97) [3] D.P. Sekulic, R.K. Shah, Thermal design theory of three-fluid heat echangers, in: J.P. Harnett, T.F. Irvine (eds.), Advances in Heat Transfer, Vol. 26, Academic Press, Ne York, 995. [4] D. Shrivastava, T.A. Ameel, Three-fluid heat echangers th three thermal communications. Part A: general mathematical model, International Journal of Heat and Mass Transfer 47 (2004) [5] D. Shrivastava, T.A. Ameel, Three-fluid heat echangers th three thermal communications. Part B: effectiveness evaluation, International Journal of Heat and Mass Transfer 47 (2004) [6] X. Luo, M. Li, W. Roetzel, A general solution for one-dimensional multistream heat echangers and their netorks, International Journal of Heat and Mass Transfer 45 (2002) [7] L. Malinoski, Application of nonlinear programming methods for calculation of temperature profiles in parallel-flo multi-channel heat echangers, International Communications in Heat and Mass Transfer 28 (200) [8] D.P. Sekulic, C.V. Herman, Transient temperature fields in a three fluid heat echanger, Proceedings of the XVIIth International Congress of Refrigeration B, IIF, Vienna, Austria 987,

20 [9] D.P. Sekulic, M. Dzolev, I. Kmecko, Dynamic behaviour of a three fluid heat echanger: The eperimental study, in: J.F. Keffer, R.K. Shah, E.N. Ganic (eds.), Eperimental Heat Transfer, Fluid Mechanics, and Thermodynamics, Elsevier, Ne York, 99, [0] S. Bielski, L. Malinoski, A semi-analytical method for determining unsteady temperature field in a parallel-flo three-fluid heat echanger, International Communications in Heat and Mass Transfer 30 (2003) [] S. Bielski, L. Malinoski, An analytical method for determining transient temperature field in a parallel-flo three-fluid heat echanger, International Communications in Heat and Mass Transfer 32 (2005) [2] X. Luo, X. Guan, M. Li, W. Roetzel, Dynamic behaviour of one-dimensional flo multistream heat echangers and their netorks, International Journal of Heat and Mass Transfer 46 (2003) [3] L. Malinoski, Equations for transient behaviour of parallel-flo multichannel heat echangers, Heat and Mass Transfer 39 (2003) [4] M.N. Roppo, E.N. Ganic, Time-dependent heat echanger modelling, Heat Transfer Engineering 4 (983) [5] D.J. Correa, J.L. Marchetti, Dynamic simulation of shell-and-tube heat echangers, Heat Transfer Engineering 8 (987) [6] C.C. Lakshmanan, O.E. Potter, Dynamic simulation of plate heat echangers, International Journal of Heat and Mass Transfer 33 (990) [7] W. Roetzel, Y. Xuan, Transient behaviour of multipass shell-and-tube heat echangers, International Journal of Heat and Mass Transfer 35 (992)

21 [8] W. Roetzel, Y. Xuan, Analysis of transient behaviour of multipass shell and tube heat echangers th the dispersion model, International Journal of Heat and Mass Transfer 35 (992) [9] S.K. Das, W. Roetzel, Dynamic analysis of plate heat echangers th dispersion in both fluids, International Journal of Heat and Mass Transfer 38 (995) [20] S.K. Das, B. Spang, W. Roetzel, Dynamic behaviour of plate heat echangers eperiments and modeling, Transactions of the ASME. Journal of Heat Transfer 7 (995) [2] W. Roetzel, S.K. Das, Hyperbolic aial dispersion model: concept and its application to a plate heat echanger, International Journal of Heat and Mass Transfer 38 (995) [22] R.W. MacCormack, A numerical method for solving the equations of compressible viscous flo, AIAA Journal 20 (982) [23] D.A. Anderson, J.C. Tannehill, R.H. Pletcher, Computational fluid mechanics and heat transfer, Hemisphere Publishing Corporation, Washington, 984. [24] C.A.J. Fletcher, Computational techniques for fluid dynamics, Springer Verlag, Ne York,

22 Figure captions Fig. Schematic representation of the three-fluid heat echanger under consideration. Fig. 2 Cross section of the three-fluid heat echanger. Fig. 3 Comparison of transient temperature profiles calculated by the numerical and semianalytical methods for t =.25 s. Fig. 4 Dimensionless outlet temperatures of fluids versus time in the sample three-fluid heat echanger. = [ ( L, t ) T ] ( T T ) Θ, τ = t / tma, X = / L. i T i min / ma min Fig. 5 Dimensionless average temperatures of alls at the echanger outlet versus time. [ ( L, t ) T ] ( T T ) Θ =, τ = t / tma, X = / L. T min / ma min Fig. 6 Transient temperature profiles of fluids for the sample three-fluid heat echanger for t = 0.75 s. Fig. 7 a b Transient average temperature of alls, T =. 5( T + T ) 0, versus co-ordinate along the echanger for the sample three-fluid heat echanger for t = 0.75 s. Fig. 8 Steady-state temperature profiles of fluids for the sample three-fluid heat echanger (transient model for t ). The solid lines represent the case here the heat capacity of the alls and heat conduction along and across the alls are not taken into account. 20

23 Table Selected thermophysical properties of typical materials used for construction of heat echangers Material ρ 3 kg m J c ( kgk ) W λ ( mk ) 6 a 0 Copper alloy C Stainless steel AISI Titanium alloy Ti6Al4V m 2 s 2

24 FIG. FIG. 2 22

25 90 Numerical solution MacCormack's implicit method Semi-analytical solution Tzou's method T (,t =.25 s) T 2 (, ( o C) T (, ( o C) 75 T 2 (,t =.25 s) T 3 (,t =.25 s) T 3 (, ( o C) (m) FIG. 3 23

26 Θ (X=,τ) Θ 2 (X=,τ) Θ 2 (X,τ) Θ (X,τ) Copper alloy C24000 Stainless steel AISI 40 Titanium 6Al4V Θ 3 (X=,τ) Θ 3 (X,τ) τ FIG

27 Θ (X,τ) Θ 3 (X=,τ) Θ (X=,τ) Θ 2 (X=,τ) Copper alloy C24000 Stainless steel AISI 40 Titanium 6Al4V Θ 3 (X,τ) Θ 2 (X,τ) τ FIG

28 T (,t=0.75 s) T 2 (, ( o C) T (, ( o C) T 2 (,t=0.75 s) Copper alloy C24000 Stainless steel AISI 40 Titanium 6Al4V T 3 (, ( o C) T 3 (,t=0.75 s) (m) 0 FIG. 6 26

29 46.7 Copper alloy C24000 Stainless steel AISI 40 Titanium 6Al4V T 3 (, ( o C) T 3 (,t=0.75 s) T (, ( o C) 46.3 T (,t=0.75 s) T 2 (,t=0.75 s) T 2 (, ( o C) (m) 4.2 FIG. 7 27

30 60 T () Copper alloy C24000 Stainless steel AISI 40 Titanium 6Al4V T 2 () ( o C) T () ( o C) 59.6 T 2 () T 3 () ( o C) T 3 () (m) FIG. 8 28