Effectiveness NTU performance of finned PCM storage unit K.A.R. Ismail, M.M. Gongalves

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1 Effectiveness NTU performance of finned PCM storage unit K.A.R. Ismail, M.M. Gongalves Campinas, SP, Brazil ABSTRACT This paper presents a mathematical model based upon two dimensional formulation of the phase change heat transfer problem around externally finned tube immersed in the PCM while the working fluid flows through it. The energy equation is written in its enthalpic form and the heat and flow processes are coupled by an energy balance on the fluid element flowing inside the tube. The numerical solution is based upon the control volume technique and ADI finite difference representation. The results obtained show the effect of the geometrical parameters such as number of fins, fin length, compactness ratio and the fin thickness on the solidified mass, NTU and effectiveness. INTRODUCTION Analytical solutions to the heat transfer problem with phase change although very important are limited to very special cases of little practical interest. Many semi-analytical methods were proposed including the integral method, moving heat source method and perturbation methods. Various numerical methods were proposed for the solution of heat transfer problem with phase change. These methods can be conveniently divided into strong numerical solutions where finite difference or finite element techniques are used in the strong formulation of the process, localizing the interface and the temperature distribution in each step or using special transforms to immobilize the interface. The other group called weak numerical methods in which the explicit attention in following the interface is avoided. The main advantage of the enthalpic formulation is that one

2 280 Free and Moving Boundary Problems basic equation is used to represent the whole region solid and liquid and no modification in the numerical scheme is necessary to handle the interface region. Again the formulation is equaly applicable for the case of PCM fixed and unique fusion temperature as well as for the case of fusion over a temperature range. Because of these advantages these methods are widely. Of particular interest to the present study is the work due to Sparrow etal [2] on a tube with four fins in which they observed the presence of natural convection in the liquid phase which can lead to interrupting or delaying the solidification process. Bathelt and Viskanta [3], studied the case of storage in PCM using a tube with three fins and studied the influence of the geometrical arrangement of the fins on the fusion front and the presence of natural convection. Shamsundar and Srimiver_ san [4], presented charts of NTU and e for a storage unit of the shell-tube type. Ismail and Alves [5] and [6] presented various studies on phase change problems with finned tubes including NTU and e charts for engineering calculations. Ismail [7] presented numerical and experimental results for various finned tube configurations suitable for PCM storage. Padmanabhan and Krishma [8] studied the heat transfer process around a cylinder with axial fins. In the present work the mathematical model is based upon two dimensional formulation of the phase change problem around externally finned cylinder immersed in the PCM as in Fig. 1. The heat conduction equation is written in its enthalpic form and the heat and flow problems were coupled by an energy balance on the fluid element flowing inside the tube. The numerical solution of the energy equation was based upon the control volume technique and the ADI scheme. FORMULATION OF THE PROBLEM In this formulation it is assumed that the heat transfer process is dominated by conduction, the sensible heat of the PCM is negligible compared to its latent heat, the fluid temperature at inlet is constant, constant heat transfer coefficient between the fluid and the tube wall, the phase change occurs over a range of temperature. Because of the symmetry of the problem the representative domain can be considered as in Fig. 2. Considering the above simplification and following Bonacina* s method [1], it is possible to write the heat conduction equation. - 3T 1 9 _ 3T 1 9 k(t) 3T C(T) = - (rk(t) ) + - (-^ ) (1) 3t r 3r 3r r 86 r 36 applicable for the solid, liquid and fin region by substituting the appropriate values for C\T) and Tc(T) as in Bonacina [1].

3 Free and Moving Boundary Problems 281 TUBE TUBE FINNED INSULATION f <Wy vvx\v Figure 1. PCM storage unit. Figure 2. Symmetry region. The boundary, initial and final conditions associated with the above equations are: 3r r = 9T _ r = r " ^ 90" (2) T(r,6,0) = T+ j T(r,6,tf) = where ij - T^ = 2AT, T^ is fluid bulk temperature, T^ is the phase change temperature, T^ is the phase change temperature for the liquid phase, T^ is the phase change temperature for the solid, 2AT is the phase change range and h is convective heat transfer coefficient. The system of equations (1) and (2) represent the conduction dominated phase change problem. In the case of a working fluid flowing in the finned tube, the phase change problem must be coupled to the flow problem by considering

4 282 Free and Moving Boundary Problems energy balance between the fluid and the phase change problem. Figure 3. Energy balance on a fluid element. Performing an energy balance between the fluid and the PCM one can obtain q mb dx where mb = Pb^^i^ ^ and dx (3) Hence equation (3) can be written as = -2 B-f Cef F d% (4) where Cgf = Cg/(pb Cb) = ratio of specific heats, Cb = the fluid specific heat and F = q/%iax ~ the dimensionless heat flux. Note that q = 2-rr r^ kg 3T/8r = heat flux and qmax ~ 2ir r± h (Tm - TQ) = maximum heat flux. Integrating equation (4) one can obtain the local bulk temperature of the Tb(T) = T + - exp [-2BjCsf / T di -f n(lj - 1^(0)9] (5) where the dimensionless time T is determined from T = y* x*/(re Pr p*) (6) where Pr = Cg y^/kg is the Prandtl number for the solid phase; Re = pb u r-^/yb is the Reynold number;

5 Free and Moving Boundary Problems 283 u* = y^/yb is the relative viscosity; P* = Pg/Pb is the relative density; x* = x/r-l is the dimensionless axial position along the tube measured from the entry position of the tube. Define the effectiveness as the ratio of real heat flux to the maximum heat flux one can write c ~ q/4max ~ r (7) where mh Cb '- ' ' -Tb(0)] [Tm " Tb(0)] with x as the position of the fluid element along the axis of the tube. Hence Tm - Tb(0) and the (NTU) number of thermal units as = the effectiveness (8) NTU = 2Trrixh/(mb Cb) (9) NUMERICAL TREATMENT The system of equations and the associated boundary conditions were solved numerically by using the control volume finite difference method due to Patankar [18] and the ADI numerical scheme. Various numerical tests were realized to establish the optimum grid in both the radial and circumferential directions. It was found that 30 radial points and 30 circumferential ones gave best results. DISCUSSIONS Figure 4 shows the variation of solidified mass for the time increment T = 0,1 as function of the number of fins. As can be seen the increases in the number of fins leads to increase in the solidified mass. The effects on the effectiveness and NTU of the storage unit are shown in Fig.5. % ^r. The compactness ratio CR defined as has an adverse effect on the solidified mass as can be verified from Fig. 6.

6 284 Free and Moving Boundary Problems SOLIDIFICATION TIME NUMBER OF FINS Figure 4. Effect of number of fins. Figure 5. Effect of number of fins on the effectiveness and NTU. The increase of CR above about 5 does not seem to effect the solidification any further. Figure 7 shows the effect of CR on the effectiveness and NTU of the storage unit as compared to the finless one. Variation of the solidified mass with the variation of the fin length is shown in Fig.8. The figure indicates that for values of dimensionless fin length above 10, the solidified mass seems to be affected very little. The thickness of the fin seems to affect very little the solidified mass, e and NTU of the storage unit and the curves are omitted for brevity.

7 Free and Moving Boundary Problems COMPACTNESS RATIO Figure 6. Effect of compactness ratio. i rrirnttmi MTim rtiirn^ttrprtttttttrrrmrnn I x I I o.f o«o* o u» ujjjuj.lujjlu.iif i. o DlMf NtlONLff *f T(MC f I» I Figure 7. Effect on the compactiness ratio on e and NTU. CONCLUSIONS This study indicates that the proposed model and the numerical approach are adequate to simulate the effects of the geometrical parameters on the performance of a latent heat storage unit with finned tubes. The curves of NTU and effectiveness are helpful for design purposes. REFERENCES 1. Bonacina et alii. 'Numerical Solution of Phase Change Problems'. International Journal of Heat and Mass Transfer, 16 ( ), 1973.

8 286 Free and Moving Boundary Problems f = FIN LENGTH Figure 8. Effect of fin length. Sparrow, E.M. et alii. 'Freezing on a Finned Tube for Either Conduction-Controlled on Natural Convection Controlled Heat Transfer, Int. J. Heat Mass Transfer, 24, ( ), Bathelt, A.G. and Viskanta, R. 'Heat Transfer and Interface Motion During Melting and Solidification Around a Finned Heat Source/Sink., Shamsundar, N. and Srimivesan, R. 'Effectiveness NTU Charts for Heat Recovery from Latent Heat Storage Units', J. Solar Engineering, 19^( ), Ismail, K.A.R. and Alves, C.L. 'Analysis of Shell Tube PCM Storage System', 8^ Int. Heat Transfer Conf., S. Francisco, California, USA, 1986, paper n FM-10, pp Ismail, K.A.R. and Alves, C.L. 'Numerical Solutions of Finned Geometries Immersed in Phase Change Material', ASME 26^h Nat. Heat Transfer Conf., Philadelphia, USA, HTD, 109(31-36), Ismail, K.A.R. 'Possible Finned Geometries for PCM Thermal Storage Units', Heat Transfer 90, July, Southampton, UK, pp , Padmanabhan, P.V. and Krishma, M.V. 'Outward Phase Change in a Cylindrical Annulus with Axial Fins on the Inner Tub.', Int. J. Heat Mass Transfer, 29(12), pp , Patankar, S.V. 'Numerical Heat Transfer and Fluid Flow', Hemisphere Publishing Corporation, 1980.