OPTIMIZATION OF CONTROLLING PARAMETERS OF SMALL SIZED CARBON FOAM HEAT STORAGE

ARPN Journal o Engineering and Applied Sciences OPTIMIZATION OF CONTROLLING PARAMETERS OF SMALL SIZED CARBON FOAM HEAT STORAGE Amal El Berry, M. A. Ziada, M. El-Bayoumi and A. M. Abouel-Fotouh Department o Mechanical Engineering, National Research Centre, Egypt E-Mail: mahmoud.a.el.bayoumi@gmail.com ABSTRACT Heat storages are essential elements o renewable solar energy systems as well as conventional industry applications. Their perormance and cost are detrimental to the concerned systems. Carbon oam material excels as heat storage medium due to its high thermal conductivity coupled with its high heat capacity, and light weight, allowing design o smaller, more eicient, cheaper, and lighter heat storages. To optimize the design o small sized carbon oam heat storage, ANOVA analysis was employed to determine signiicances o heat transer controlling actors such as pore size, porosity, length and diameter o the bed, and mass low rate o charging hot air. Also, Surace Response Method was employed to construct a statistical model that describes the energy/volume as a unction o the above parameters. The model employs one-dimensional continuity, momentum and energy equations to simulate the heat transer process within a volume o carbon oam material with pore diameter o 0.003 m, and porosity (ε) ranging rom 0.08 to 0.385. The investigation shows that coeicient o determination (R 2 )=0.84 o the statistical model. With ambient and charging temperatures o 25ºC and 85ºC, bed length o 0.4 m, bed diameter o 0.1 m, mass low rate per area o 0.25 kg/sm 2, and ε o 0.19, the maximum heat energy/volume (82000kj/m 3 ) is achieved at about 1.3 hours o charging time. At mass low rate per area o 0.125 kg/sm 2 and ε o 0.175 the maximum energy stored is achieved at about 1.7 hours o charging time. Keywords: luid low, heat transer, numerical analysis, spongy-porous media, thermal perormance, regression analysis. INTRODUCTION In many applications o renewable solar energy systems as well as conventional industry processes, there is a luid that needs to be heated or cooled and an exhaust luid having thermal energy that needs to be reclaimed. Thermal energy storage systems (heating and cooling) plays a very important role in reclaiming thermal energy rom one luid (usually heat rom exhaust gas) and transerring it to another luid, allowing systems to cut its energy consumption and cost and to reduce its environmental impact. In heat storages, higher thermal conductivity o storage medium leads to shorter charging and discharging times, which guarantees better storage perormance. A research by Yanping and Yulong, [1] ound that embedding metal oams in a phase change material enhances the charging and discharging rate by up to 8 times. While Zhu et al, [2] investigated the cost and eect o copper oam pore size and illing height ratio on the phase change heat storage eectiveness. Wu et al, [3] investigated copper oam eect on SAT heat storage and ound that copper oam increased the storage s thermal conductivity about 11 times and increased its energy density. Graphite oam has attracted research eorts because o its high thermal conductivity. Ladi et al. [4] investigated thermal properties o graphite oams impregnated with phase change materials or energy storage applications. Also, Py et al [5] investigated experimentally with a composite o Compressed Expanded Natural Graphite Matrix and parain as heat storage medium with up to 95% parain by weight. They concluded that thermal conductivity o the composite was ound to be equal to that o the graphite matrix. Mills et al, [6] investigated enhancement o phase change materials thermal conductivity using porous graphite matrix. Zhou and Zhao, [7] compared graphite oam matrix with metal oam matrix and ound that metal oam delivered higher perormance. Research about thermal and hydraulic properties o porous ceramics and metallic oams has attracted a lot o recent eorts, as reviewed by Zhao [8]. Barari et al, [9], investigated experimentally with Aluminum metal oam as heat storage medium and investigated the eect o pore size on its perormance. Carbon oam was numerically studied or heat transer, luid low and dynamics o thermal energy storing to evaluate its thermal perormance in packed bed systems, [10]. Also, the eect o dierent variables and parameters, such as porosity, air mass velocity, and porosity, on the amount o heat stored/released in/rom selected porous medium was discussed, [10]. Carbon oam material is currently considered as one o the most promising heat storage materials. It can enhance heat transer and energy storage processes through increasing heat transer surace area without compromising structural strength. It also exhibits many multiunctional capabilities such as better tortuosity, high area to volume ratio, and excellent heat transer properties. Accordingly, carbon oam is considered promising medium that would replace conventional thermal energy storage materials. The aim o the current study is to provide optimization means or design o small sized packed bed heat storages illed with carbon oam. The objectives are as ollowing: Employ ANOVA analysis to determine signiicances o heat transer controlling parameters namely pore diameter, charging time, porosity, length and diameter 3037

ARPN Journal o Engineering and Applied Sciences o the packed bed and mass low rate o charging hot air. Surace Response Method is used to construct a statistical model that describes the energy/volume as a unction o above mentioned parameters. Finally, the maximum energy stored per unit volume, which is inluenced by above mentioned parameters, is determined. THE CURRENT STUDY Computational luid dynamics simulations were carried out or luid low as well as heat transer to obtain the maximum quantity o thermal energy stored inside the carbon oam heat storage spongy porous media or the working luid and the medium. Simulations were perormed or oams with pore diameter o 0.003 m, and porosity ranging rom 0.08 to 0.385. The regression analysis results are veriied using theoretical results obtained using solid, mathematical and metal oam eective conductivity models. Physical model The physical model o this system is described as a cylindrical enclosure illed with carbon oam. The model employs air as the working luid. The model acts as heat exchanger in which hot air heats carbon oam during charging and cold air cools it during discharge. The air lows in the axial direction rom top downward under a steady state, one-dimensional, condition. Figure-1 shows the present physical model. The set o equations o the model evaluates the dynamics o temperature distribution o the air and the carbon oam. It also estimates the amount o thermal energy stored within the system at any instant o time. Working luid in Z = 0 Control volume o spongyporous medium i-1 L ΔZ i Thermal insulation i+1 Z = L r i r o Working luid out Figure-1. Physical model o the present system. Solid model The microstructure o carbon oam is quite complex due to the nature o its manuacturing process. Thereore, study o the relationship between its microstructure and bulk properties is quite diicult. In the current study, a solid model is employed. It aims to simulate the carbon oam by a volumetric segment (C) obtained by subtracting a bed o spheres (B) rom a solid volume (A), Figure-2 where C = A - B. The proposed solid model allows evaluation o the geometrical and thermalphysical characteristic parameters, including equivalent pore diameter, hydraulic diameter o the oam, pore window diameter, ligament diameter, internal surace area to volume ratio, permeability, thermal conductivity, and Forchheimer coeicient. Mathematical model The current model is based on the ollowing assumptions: a) Carbon oam is homogeneous and isotropic. b) Inside the carbon oam, natural convection eects are negligible. c) Thermo physical properties change due to temperature is ignored. d) Radiation heat transer is neglected as maximum operation temperature is 100 o C (relatively low). e) Steady state o luid low and heat transer inside the heat storage. Given the current assumptions, the volumeaveraging technique is applied to establish the governing equations o both velocity and temperature inside the heat storage. 3038

ARPN Journal o Engineering and Applied Sciences A B C Solid cube Bed o spheres Volumetric segment o carbon oam Figure-2. Construction o solid model, [10]. The governing equations: Continuity equation. t v 0. 0 Momentum equation 1 C C u k h at T T t T z z (2) Working luid phase energy equation: e T z s ~ s (1) At t=0 T s = T o or all Z The carbon oam s temperature boundary conditions (upstream and downstream) satisy the energy equations or both upstream and downstream. Accordingly:- At Z=0 or all t T s(0, t) T s(0, t) 1 k h ~ scs se sat s(0, t) T (0, t) t z z At Z=L or all t 2 T s(l, t) T s(l, t) 1 scs kse hs T (L, t) T s(l, t) (6) 2 t z (5) T T T C C u h ~ sa(ts T ) (3) t z z k z Energy equation or the solid phase Ts Ts 1 ) C (k ) h ~ s s se sa(ts T ) (4) t z z ( The physical model equations or both luid and solid regions are solved as one-dimensional steady state case. Initial and boundary conditions or the luid: The luid s initial and boundary conditions descries the situations in which the luid temperature T is known (T o and T h ). Where, T o is the ambient temperature and T h is the hot charging air temperature at the inlet o the storage tank. They are given as: - At Z=0 T = T h or all t At t=0 T = T o or Z 0 The initial condition o the carbon oam temperature is given as: Metal oam eective conductivity modeling Calmidi and Mahajan, [11] investigated the permeability (K) and inertial variable (F I ) o the momentum equation (2). They gave speciic ormulations or K and inertial variable F I o metal oam. Their ormulation is base on experimental study results. In the current work the ormulation proposed by Calmidi, [12] is employed, accordingly:- K / d p 2 = 0.00073 ( ε ) -0.224 (d /d p ) -1.11 F I =0.00212 (1-ε) -0.132 (d /d p ) -1.63 / (K) 0.5 Boomsma and Poulikakos, [13] have analytically investigated eective conductivities k se, k e and dispersion conductivity k d o open-celled metal oams. The study is based on three-dimensional cellular morphology. In the current work the ormulation o k se, k e and k d proposed by Boomsma and Poulikakos, [13] is employed, accordingly:- k e = 2/ ( 2 ( R A + R B + R C + R D )) R A = 4 / (2 e 2 + (1 - e))k s + (4-2 e 2 - (1 - e)) k R B = (e - 2 ) 2 / ((e - 2 ) e 2 k s + 3039

ARPN Journal o Engineering and Applied Sciences (2 e 4 - (e - 2 ) e 2 ) k ) R C = ( 2-2e) 2 / (2 2 (1-2e 2)k s + 2( 2-2e - 2 (1 2e 2 2))k ) R D = 2e / (e 2 k s + (4 - e 2 )k ) Where:- = ( 2(2 0.625 e 3 2 2ε) / ((3-4e 2-e))) 0.5 The evaluated k se and k e using the above conductivity model are developed or solid struts metal oams (e.g., ERG oams). On the other hand, the current carbon oam has hollow struts cell ligaments, similar to Porvair oams. Accordingly, the porosity and eective solid conductivity have to be treated to relect those o Porvair oams. I Porvair oam has the same relative density o ERG, then its porosity could be evaluated as ollowing:- ε porvair = (ε r 2 ) / (1 r 2 ) Where: r is the ratio between inner and outer diameters o the hollow struts. Similarly, eective solid conductivity could be evaluated as ollowing:- k se, porvair = k se (1 r 2 ) For the evaluation o dispersion conductivity, k d, the ollowing, widely employed, expression is adopted:- k d = C D ( Re k Pr e )u k e / u m Where:- u m is the average low velocity at the entrance CD is the thermal dispersion coeicient ( 0.1) Where the Reynolds number is evaluated as ollowing: - Re k = u m K / V The Prandtl number based on eective conductivity is evaluated as ollowing: - Pr e = μ C / k e The interacial surace between solid and luid is normally considered as the area o three parallel arrays o cylinders intersecting perpendicularly. It is given as: - ã = 3d / d p 2 When the 3D microstructure o metal oam s open cells (dodecahedra pores shape and noncircular cross section o ibers) is taken into account, the pore diameter d p and solid strut diameter d, are multiplied by actors o 0.59 and (1 e -((1-ε) / 0.04) ), respectively, [12]. The interstitial heat transer coeicient (h s ) in the current study is evaluated using Zukauskas s correlation o staggered cylinders in cross low, [14]. Thus, evaluated as ollowing: - h s =0.52Re 0.5 Pr 0.37 k / d Where:- Re = u d / (εν) To solve partial dierential equations (3) and (4), governing thermal behavior o the oam thermal storage, they are initially reduced to their inite dierence versions along with their initial and boundary conditions. Finally, a specially designed computer program is employed to solve them numerically, using a numerical scheme. Regression modeling The Response surace technique (RSM) by Abouel-otouh et al, [15] has been adopted to construct a multivariate statistical model that describes the energy stored per volume as a unction o controlling actors. These actors are the bed length, the bed diameter, storage time, porosity o the bed, pore diameter and the hot air mass low rate. The data employed to generate such a statistical model is the output results o energy stored per volume o the physical model which is described by continuity, momentum and energy equations 1-6 with speciic boundary conditions as unction o the controlling actors. The values o these actors have been coded according to the ollowing equation: x i Xi 0.5 Xih X 0.5 X X ih il il Where:- x i Controlling actor X i Natural value o the controlling actor X il Lowest values o a given actor Highest values o a given actor X ih The regression equation is: Y b n n n 0 bixi bijxix j i1 j1 i1 Where Y is the energy/volume and x i is controlling actor, i and j are the parameters index. The e is the error between the model predicted values and physical model predictions. The analysis o errors using ANOVA technique tests the adequacy o the statistical model and the signiicance o each term o the statistical model through the t-test and the -test. I the terms o the statistical signiicance, (p-values), are greater than 5% they are dropped rom the model terms due to its nonsigniicance. The term representing the linear eects is given as: e 3040

ARPN Journal o Engineering and Applied Sciences n i1 b i x i The term representing the interaction between the controlling actors and the quadratic eects on the predictions o the model is given as: b n n ij j1 i1 x x i j The model coeicients b 0, b i, and b ij are determined using multivariate linear regression technique. The physical model data output, (energy / volume) were transormed using the Box-Cox transormation that transorms a set o non-normally distributed data to a set o data that approximately has a normal distribution. The Box-Cox transormation is described by the ollowing equation:- data(l)=(data lb -1)/l b. B, [16] Where l is the Box-Cox parameter. RESULTS AND DISCUSSIONS The multivariate linear regression results analysis o the model, which describes the energy/volume data as unction o the prescribed controlling parameters, Table (1), is as ollowing:- Y b0 b2*x2 b3*x3 b4*x4 b6*x6 b21*x2 * x1* b33*x3 *x3 b34*x3 *x4 b44*x4 *x4 Table-1. The parameters and their symbols in the regression equation. Parameters BL BD t ε Equation symbol Pore Diameter G x 1 x 2 x 3 x 4 x 5 x 6 Table-2 shows the values o the model coeicients and their statistical signiicance (p-values). The p-values greater than 5%, have no tangible eect on the model output. The linear trend o normal probability o the error between the model predictions and physical model predictions indicates that the model is not biased. Table-3 illustrates the statistical characteristics o the model which conirm the adequacy o the model. Figure-3 shows that the errors are approximately normally distributed since the errors ollow a trend o a straight line. Figure-4 shows the regression predictions plotted against physical model results. It shows that most o the points are close to the original physical model results and prediction is in good agreement. Table-2. The values o the model constants and their p-values. Model coeicients values p-value b 0 140045 7.45 e -60 b 2-25016 0.0005 b 3-20274 1.01 e -6 b 4 38322 1.8 e -9 b 6-33188 7.4 e -18 b 21-33188 7.4 e -18 b 33-17392 0.05 b 34-12295 0.05 b 44-60018 3.9 e -16 Table-3. The main statistical characteristics o the model. R 2 Adjust R 2 F-test p-value l b 84% 82% 62 3.3e-42 1.07 A multiple variant linear regression model with two independent variables, carbon oam porosity and charging time, is employed to predict the stored thermal energy per unit volume, Figures (5-6a-6b). The bed length, bed diameter, and pore diameter (0.4m, 0.1m and 0.003m respectively) are the controlling actors under which the results are presented throughout this investigation. This regression model describes a surace in the threedimensional space o thermal energy per unit volume, porosity and charging time. Figure-5 shows that, the statistical model results are in good agreement with the physical model results. Figures 6(a) and 6(b) show the corresponding contour plot or the multiple regression model represented by the three dimensional plot shown in Figure-5. These contour plots are or dierent mass low rates per unit area, namely 0.125 and 0.25 kg/m 2 s. These plots indicate that the expected change in the stored energy per volume is a unction o both charging time and porosity. 3041

ARPN Journal o Engineering and Applied Sciences Figure-3. Normal probability plot. Figure-4. Model predictions against physical model results. Figure-5. Three-dimensional plot o the regression model. Figures 6(a) and 6(b) show that at mass low rate per unit area (G) equal to 0.125 kg/m 2 s, the maximum energy stored (82000kj/m 3 ) is achieved at ε = 0.175 in 1.7 hr charging time. For mass low rate per unit area (G) equal to 0.25 kg/m 2 s, the maximum energy stored is achieved at 0.19 and 1.3 hr charging time. This demonstrate the time required to reach the maximum energy stored is minimized at certain porosity at a given mass low rate. They also show that, the minimum charging time at which the maximum energy is stored is inversely proportional to the mass low rate. They also show that, dierent levels o energy stored is reached aster at dierent porosities at the same storage design, charging temperature and mass low rate. This allows employing porosity to optimize charging time according to average amount o energy to be stored. CONCLUSIONS The above analysis shows that the results obtained rom both regression model and mathematical model are in agreement. As the true unctional relationship between energy stored per unit volume, porosity and charging time is unknown, the model can be used as empirical model to predict uture observations o the rate o stored energy per volume. Model results show that, or ambient and charging temperatures o 25ºC and 85ºC and G o 0.25 kg/m 2 s, ε o 0.19 is required to achieve maximum energy stored by volume (82000 kj/m 3 ) in shortest time o 1.3 hr. At G o 0.125 kg/sm 2, ε o 0.175 is required to reach the maximum energy stored in shortest time o 1.7 hr. The study demonstrated the model capacity to simulate and optimize the energy storage process in carbon oam. 3042

ARPN Journal o Engineering and Applied Sciences Z Axial direction Geek symbols Viscosity o luid e Eective viscosity ã Wetted area per volume ε Porosity o the porous medium, ε Eective thermal conductivity b Box-Cox lambda α e Eective thermal diusivity Kinematic viscosity o luid, (m 2 /sec) ρ Density o luid, (kg/m 3 ) Δ Increment o variable Subscripts d Dispersion Fluid in Inlet h Hot s solid phase REFERENCES Figure-6. The variation o the stored energy (kj/m 3 ) (a) at G = 0.125 kg/m 2 s, (b) at G = 0.25 kg/m 2 s Nomenclature a Interacial surace area per unit volume. Bd Bed diameter, (m) BL Bed length, (m) C Heat capacity o luid, (W/kg K) d Solid strut diameter, (m) d p Pore diameter, (m) F I Inertial variable with unit, (m -1 ) G Mass low rate per unit cross section, (kg/m 2 s) h External heat transer coeicient, (W/m 2 K) H Height o the channel, (m) k Eective thermal conductivity o luid, (kg/mk) k d Thermal dispersion conductivity k s Eective thermal conductivity o solid, (kg/mk) K Permeability o porous medium, (m 2 ) L Length o the channel, (m) Pe Peclet number Pr Prandtl number Pr e Eective Prandtl number Q Energy stored, (J) Re Reynolds number T h Hot charging air temperature, ( o C) T o Ambient temperature, ( o C) T s Solid phase temperature, ( o C) t Heating or cooling times, (h) u Average velocity, (m/sec) u m Average low velocity, (m/sec) V Velocity vector X Horizontal direction [1] Du Y., Ding Y. 2016. Towards improving charge/discharge rate o latent heat thermal energy storage (LHTES) by embedding metal oams in phase change materials (PCMs). Chemical Engineering and Processing. 108, pp. 181-188. [2] Zhu Z. Q., Huang Y. K., Yi Zeng N. H., and Fan L. W. 2018. Transient perormance o a PCM-based heat sink with partially illed metal oam: Eects o the illing height ratio. Applied Thermal Engineering. 128, pp. 966-972. [3] Wu D.L., Li T.X., He F., and Wang R.Z. 2017. Experimental investigation on copper oam/hydrated salt composite phase change material or thermal energy storage. International Journal o Heat and Mass Transer. 115, pp. 148-157. [4] Ladi K., Mesalhy O. and Elgay A. 2008. Graphite oams iniltrated with phase change materials as alternative materials or space and terrestrial thermal energy storage applications. Carbon. 46, pp. 15-168. [5] Py X., Olives R. and Mauran S. 2001. Parain/porous-graphite-matrix composite as a high and constant power thermal storage material. International Journal o Heat and Mass Transer. 44, pp. 2727-2737. [6] Mills, A. Farid, M. Selman J. R. and Al-Hallaj S. 2006. Thermal conductivity enhancement o phase change materials using a graphite matrix. Applied Thermal Engineering. 26, pp. 1652-1661. 3043

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