Graphite Composite

R. Pokhrel Department of Mechanical Engineering, Santa Clara University, Santa Clara, CA 95053

J. E. González1 Department of Mechanical Engineering, City College of New York, New York, NY, 10031 e-mail: [email protected]

T. Hight Department of Mechanical Engineering, Santa Clara University, Santa Clara, CA 95053

T. Adalsteinsson Department of Chemistry, Santa Clara University, Santa Clara, CA 95053

1

Analysis and Design of a Paraffin/Graphite Composite PCM Integrated in a Thermal Storage Unit The addition of latent heat storage systems in solar thermal applications has several benefits including volume reduction in storage tanks and maintaining the temperature range of the thermal storage. A phase change material (PCM) provides high energy storage density at a constant temperature corresponding to its phase transition temperature. In this paper, a high temperature PCM (melting temperature of 80° C) made of a composite of paraffin and graphite was tested to determine its thermal properties. Tests were conducted with a differential scanning calorimeter and allowed the determination of the melting and solidification characteristics, latent heat, specific heat at melting and solidification, and thermal conductivity of the composite. The results of the study showed an increase in thermal conductivity by a factor of 4 when the mass fraction of the graphite in the composite was increased to 16.5%. The specific heat of the composite PCM (CPCM) decreased as the thermal conductivity increased, while the latent heat remained the same as the PCM component. In addition, the phase transition temperature was not influenced by the addition of expanded graphite. To explore the feasibility of the CPCM for practical applications, a numerical solution of the phase change transition of a small cylinder was derived. Finally, a numerical simulation and the experimental results for a known volume of CPCM indicated a reduction in solidification time by a factor of 6. The numerical analysis was further explored to indicate the optimum operating Biot number for maximum efficiency of the composite PCM thermal energy storage. 关DOI: 10.1115/1.4001473兴

Introduction

Phase change materials have been used in thermal energy storage systems for temperature control in various applications since the last decade. Applications of PCM can be divided into different groups such as thermal storage for heating/cooling 共of buildings, electronic goods, automobile engines, and space craft兲, applications in the food industries, in medical applications, waste heat recovery, heat pump systems, coolsuits, and cold storage. All storage systems should have a high thermal diffusivity to ensure rapid redistribution of the stored energy. Slow diffusion 共discharging time兲 and slow charging rate often limit the application of PCM. PCM have been applied in several modalities that all have shown considerable improvement in the heat transfer rates including 关1兴 the use of finned tubes, PCM dispersed in conductive particles, use of high conductive carbon brushes, PCM infused in an aluminum metal matrix, and expanded graphite. The lower the bulk density, the higher the pore volume and thus the higher the impregnation, with improved thermal properties, for PCM infused in an expanded graphite matrix 关2兴. The configuration of the storage container also plays an important role in heat transfer; it was shown that a spherical container transfers heat faster than the cylindrical and flat plate containers 关3兴. It has also been shown that the thermal stability and the thermal cycling of the paraffin are not affected by the contact metal 关4兴. This paper explores the development of a composite made of a high temperature PCM and expanded graphite. This paper reports the fabrication of the composite, thermophysical properties, and 1 Corresponding author. Contributed by the Solar Energy Division of ASME for publication in the JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received November 18, 2009; final manuscript received February 2, 2010; published online September 3, 2010. Assoc. Editor: Gilles Flamant.

Journal of Solar Energy Engineering

the design of a thermal storage unit for practical applications. The end application is solar thermal systems requiring constant temperatures above 70° C such as space cooling or heating 关5兴.

2

PCM/Composite Preparation

For an application of heat storage at high temperature, RT-80 paraffin 共from Rubitherm technologies GmbH, Berlin兲, melting temperature of 80° C with high latent and specific heat, was chosen as the PCM. Expanded graphite powder with controllable thermal conductivity depending on the bulk density of the matrix was provided by SGL Technic Inc., Valencia, CA, in the bulk density of 0.15 gm/ cm3. Graphite matrices of bulk densities 200 kg/ m3, 300 kg/ m3, 400 kg/ m3, and 500 kg/ m3 were prepared by pressing the graphite powder in a known volume of die and controlling the thickness of the matrix depending on the density required. Each prepared matrix was placed in an aluminum box and an excess mass of PCM of known weight was placed on top of the matrix. Both were heated in an oven at 250° F for 3 h and 30 min. The mass and density were measured both before and after the composite preparation. Two different CPCMs, 210 kg/ m3 共CPCM 1兲 and 417 kg/ m3 共CPCM 2兲, were chosen for differential scanning calorimeter 共DSC兲 tests because they represented desired impregnations of PCM of 83.5% and 65%.

3 Measurement Methods in DSC for Thermal Properties The relevant thermal properties to be evaluated for the latent heat storage system, as seen from Eq. 共1兲, consist of the phase transition temperature, the specific enthalpy, and specific heat at different phases. The thermal conductivity and the density were

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Table 1 Measured properties of PCM and CPCM Material

PCM

CPCM 1

CPCM2

Mass of sample 共mg兲 Solid density 共kg/ m3兲 Bulk density of graphite matrix 共kg/ m3兲 Mass fraction of PCM in composite 共%兲 Latent heat 共J/g兲 Specific heat capacity 共J / g ° C兲 Solid 共35° C兲/liquid 共90° C兲 Rule of mixture for latent heat 共%兲

4.5 920 100 187.9

24.9 788 210 83.5 144.9

32.9 733 417 65 87.7

2.25/2.3 100

1.35/1.4 77

1.35/1.4 44

measured as well. The density was determined by direct mass and volume measurements whereas all other properties were evaluated using a METTLER TOLEDO DSC823e. 3.1 Measurement of Latent Heat and Specific Heat. The total heat flow in a sample in a DSC cell is generally written as the sum of time rate of change of sensible and latent heat as shown in Eq. 共1兲,

⳵T ⳵L ⳵Q = mc p + m ⳵t ⳵t ⳵t

3.2 Measurement of Thermal Conductivity. Fourier’ law for heat diffusion equation 共2兲 was used to estimate the 1D heat transfer in a small DSC sample, and from there the thermal conductivity was determined. dQ = − ␭A dt

冕

dT dx

The direction of heat flow is normal to an isothermal surface, and given the fact that the height of the sample is small, a linear temperature gradient results in the sample 关7兴. With the above conditions, Eq. 共2兲 is reduced to 关8,9兴

⌬x → 0

共2兲

⌬Q ⌬x 1 ⌬q ⌬x = ⌬t A ⌬T ⌬T A

共3兲

⌬Q ⌬x 1 ⌬q ⌬x = 共CF兲 ⌬t A ⌬T ⌬T A

共4兲

␭=

共1兲

where the sensible heat, Q, is proportional to the heating rate 共°C / min兲 whereas the latent heat, L, is independent of the heating rate because heat is absorbed or released when there is physical change of the state. Measurements of the specific heat through a wider temperature range is possible if one uses a periodic succession of short, linear heating and cooling rates, which is often known as temperature modulated differential scanning calorimeter 共TMDSC兲. We use the TOPEM method, which uses the TMDSC technology 关6兴 for experimentation/evaluation. This method measures reversible and nonreversible heats 共specific heat and latent heat兲, as well as quasistatic specific heat, at different temperatures. The temperature program used in the experiment involved a dynamic segment of heating from 25° C to 95° C, with a heating rate of 2 deg/min followed by an isothermal segment for 1 min and then another dynamic segment of cooling to 25° C with a rate of ⫺2 deg/min. The temperature perturbation, i.e., the frequency of linear short heating and cooling phases was selected as ⫾0.5 deg. The minimum and maximum pulses of the perturbation remained at the standard values of 15 s and 30 s, respectively. The TOPEM evaluation of latent heat 共shown in Fig. 2兲 and specific heat 共shown in Fig. 3兲 were compared with PCM and CPCM. Properties of PCM and CPCM are condensed in Table 1.

冕

Fig. 1 Sample setup in DSC for axial and radial thermal conductivities depending on direction of heat flow with respect to fiber direction. Fiber is oriented normal to the direction of applied force while preparing the graphite matrix.

␭=

where ⌬q / ⌬T is the slope of the curve 共Fig. 4兲 of the linear temperature gradient, Q is the heat flow, T is the sample temperature, ⌬x is the height of the sample, ␭ is the thermal conductivity of the sample, and CF in these experiments is a correction factor applied to composite solids with pores that is evaluated as q共reference兲 / q共measured兲. The values tabulated in Table 2 for CF clearly indicate the difference of heat flow in radial and axial directions depending on fiber orientation in the sample. A sample of constant melting temperature gallium 共30° C兲 of known mass was placed in an aluminum crucible of 60° C such that the bottom of the testing pan is fully covered by gallium. To avoid a chemical reaction between aluminum and gallium, the inner surface of the testing pan was coated with a solution of nitrocellulose liqueur and thinner 关8兴. A typical DSC curve for gallium is shown in Fig. 4. Thermal paste was applied between the sample and crucible interface to ensure good contact. The direction of heat flow with respect to fiber direction 共Fig. 1兲 is dependent on the direction of the force applied during the graphite matrix preparation. It can be used to measure the directional thermal conductivity of the CPCM 关10,11兴. If the heat flow and applied force 共while preparing the graphite matrix兲 are in the same direction, then measurement is done for axial thermal conductivity, and if the heat flow and applied force are normal to each other, then the measurement done is referred to as radial thermal conductivity. Small samples were prepared from two different composites with fiber orientation in the radial and axial directions. Five different experiments were performed on three different materials. The temperature program used in the DSC consisted of heating the sample from 25° C to 45° C at a constant heating rate

Table 2 Summary of the thermal conductivity in both directions for PCM and CPCM

Direction of heat flow in DSC

Material PCM

Perpendicular to fiber direction ␭a Parallel to fiber direction ␭r Perpendicular to fiber direction ␭a Parallel to fiber direction ␭r

041006-2 / Vol. 132, NOVEMBER 2010

CPCM 1 CPCM 2

⌬x / A 共m−1兲

Slope 共mW/K兲

CF

␭ 共Eq. 共3兲兲共mW/ m K兲

␭ 共Eq. 共4兲兲共mW/ m K兲

69.862 72.690 80.050 68.260 85.733

2.8 9.159 9.2629 12.6365 13.8859

1.0632 1.109 0.8385 0.9085

195.614 665.768 741.495 862.567 1190.477

707.844 822.318 720.123 1081.548

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Fig. 4 DSC curve for gallium showing the melting temperature of 30° C Fig. 2 DSC curve with heat flow versus sample temperature for „a… PCM, „b… CPCM 1, and „c… CPCM 2, respectively

of 1 deg/min. After each test, the gallium was cooled to 0 ° C. Results for thermal conductivity are condensed in Table 2 and shown in Figs. 5–7.

4

DSC Results

Figure 2 shows the DSC measurements of heat flow versus sample temperature for PCM and the two different composites. The two peaks in the curves show endothermic 共heating兲 and exothermic 共cooling兲 processes. The first change in slope during melting occurs at about 55° C where the material goes from solid to a two phase region. The temperature tends to remain constant regardless of the heat flow when the transition is complete, which is seen at 85° C, and the system is at liquid phase. Another change in slope is seen during the cooling cycle, where the sample goes from a liquid phase to a two phase region at liquidus 共85° C兲 when crystallization finally initiates, thus releasing the latent heat ending in a solidus state 共55° C兲. It is seen that the solidus temperature is lower than the liquidus because of the poor equilibrium within the frozen crystals 关12兴. These observations are indicative that the solidification and the melting occur over a wide temperature range. Both solidus and liquidus, and the melting temperature, are observed to be the same in all three materials. Latent heat is measured as the integration of the area under the DSC solidus and liquidus curves. Latent heat of fusion varies extensively with the mass fraction of the PCM in the composite as clearly seen in the DSC curves. Applying the rule of mixture to CPCM 1 and CPCM 2 共for the DSC results of the latent heat兲 results in mass fractions of 77% and 44%, respectively. For CPCM 1, with a graphite matrix density of 217 kg/ m3, the value

is close to the value measured 共by weight/volume measurements兲; however, for CPCM 2, the calculated value is much lower than the measured value 共Table 1兲. Figure 3 shows the TOPEM evaluation of heat capacity normalized to the sample size. This method of experiment with temperature perturbation allows the measurement of specific heat at different temperatures, as seen in Fig. 3; RT-80 shows that specific heat of liquid at 90° C 共2.5 J / g ° C兲 is greater than the specific heat of solid at 35° C 共2.33 J / g ° C兲. The exponential increase in specific heat with temperatures is marked clearly at the start of the melting and solidification due to the effect of the change of phase. The increase continues until the transition is completed, marked by a single phase as seen in Fig. 3. The specific heats of PCM, CPCM1, and CPCM 2 are summarized in Table 1. The decrease in specific heat for CPCM 1 and CPCM 2 indicates the presence of the graphite matrix in the composite; as the weight fraction of the graphite matrix increases, the specific heat decreases indicating lower sensible heat. Thus less energy is required to raise/lower the temperature of the CPCM in comparison to PCM. Figures 5共a兲–5共c兲 show the thermal conductivity for the materials investigated. The primary y-axis indicates heat flow and the secondary y-axis is the slope of the curves presented in Eqs. 共3兲 and 共4兲. The curve in Figs. 5共a兲–5共c兲 represents the heat flow 共in mW兲 with sample temperature 共°C兲 whereas, the horizontal line represents the constant slope 共mW/°C兲. Figures 5共b兲 and 5共c兲 show two different plots. The A curve shows the conductivity in the axial direction, and B curve the conductivity in the radial direction. From Fig. 4 it is clearly seen that the gallium starts melting at 30° C. Addition of heat at this point increases the temperature linearly, until all the mass of the gallium is melted. All plots in Fig. 5 indicate different phase transition temperatures of the gallium indicating a temperature gradient in the samples. This gradient is assumed to be linear. Due to pressure gradient in the axial direction, this would eventually change the porosity along that direction 关10,11兴 in the composite. Due to these changes, 共in axial direction兲 anisotropy in thermal conductivity is seen 共Table 2兲.

␪=

Fig. 3 Specific heat with temperature for both solids and liquids for „a… RT-80, „b… CPCM1, and „c… CPCM2, respectively


⌸ ␭a − tan−1 2 ␭r

共5兲

When ␪ is 90 deg, all the fibers are orientated in the direction perpendicular to the applied force. The values for CPCM1 and CPCM2 are 49 deg and 56 deg, respectively. There is greater anisotropy in the value of thermal conductivity as ␪ increases from 45 deg 关10兴, which can be seen clearly in the mass fraction calculated and measured 共Table 1兲 for CPCM1 and CPCM2. The anisotropic behavior of the composite can be explained by CF as well. For CPCM 2, the heat required to melt the gallium is greater than the reference. This is because of the heat loss in the fiber NOVEMBER 2010, Vol. 132 / 041006-3


Fig. 6 PCM in a copper tube inside a storage tank

Fig. 7 Subdivision of r-t domain with constant ⌬r

direction, which never reached the gallium. For CPCM1, conductivities in the axial and radial directions are almost the same 共isotropic兲.

5 Numerical Formulation and Experimental Procedure

Fig. 5 „a… Thermal conductivity evaluation for PCM showing DSC curve for RT-80. The blue line indicates heat flow and the red line indicates constant slope at the phase transition of gallium. „b… DSC curve for thermal conductivity evaluation for CPCM 1. The axial thermal conductivity is represented by A and the radial thermal conductivity by B, respectively. „c… DSC curve for thermal conductivity evaluation for CPCM 2. The axial and radial thermal conductivities are represented by A and B, respectively. The different phase transition temperatures of the gallium on top of the sample depend on the height of the sample, and specific heat and thermal conductivity of the sample, neglecting the losses at the interface. A small amount of gallium „0.015 g… on top of the PCM required a longer time to melt completely, whereas 0.3 g of gallium melts completely at 5.3 min, 5.0 min, 4.5 min, and 4.5 min for CPCM 1 and CPCM 2, respectively, i.e., total melting time decreases with an increase in thermal conductivity. The values of thermal conductivity are tabulated in Table 2.

041006-4 / Vol. 132, NOVEMBER 2010

To investigate the possible practical use of the CPCM for solar thermal applications, the design of a cylindrical unit containing the CPCM embedded in the storage tank containing water, as shown in Fig. 6, was explored using numerical analysis. The storage tank was assumed to be well mixed with very low laminar flow. The variables of interest are size of the unit, outlet fluid temperature, the heat transfer coefficient, and total solidification or melting time. The one dimensional phase change problem with a convective boundary condition 共BC兲 was considered in the numerical formulation following the numerical analysis strategies outlined in Ref. 关13兴. The PCM was assumed to be initially in liquid phase, and it is suddenly subjected to a convective BC 共T⬁兲 lower than the melting temperature 共Tm兲 of the PCM. The governing equations for the solid phase and the interface are as follows. For solid phase,

冉冊

1 ⳵ Ts 1 ⳵ ⳵ Ts , r = r ⳵r ⳵r ␣s ⳵ t − ␭s

⳵ Ts = hw共Tw − Ts兲, ⳵r

s共t兲 ⬍ r ⬍ R,

r = R,

t⬎0

t ⬎ 0共BC兲

共6兲共7兲

For interface energy balance, ␭s

⳵ Ts ␳L ⳵ s共t兲 = ⳵r N ⳵t

Tl = Ts = Tm

at

r = s共t兲,

共8兲 t⬎0

共9兲

where N is the number of segments i.e., N = R / ⌬r. Furthermore, let ␭s = ␭l and ␳s = ␳l. The region R ⬍ s共t兲 ⬍ 0 is subdivided into n equal steps and it is assumed that the interface moves exactly a Transactions of the ASME


position ⌬r at a time step ⌬t 共Fig. 7兲 such that the interface moves inward with time toward the center of the cylinder from r = R to r = 0. To solve the above problem numerically, the modified variable time step 共MVTS兲 method 关14兴 was used. An implicit finite difference scheme was used with a forward in time and central in space discretization for the diffusion equation, with boundary and interface conditions using forward discretization. The notation for space and time are adopted in the form T共r,tn兲 = T共i⌬r,tn兲 For boundary condition, n+1 共 p兲关Tn+1 = − H⌬rTw 1 − 共1 + H⌬r兲T0 兴

where hw / ␭ = H. Equation 共6兲 reduces to

冋冉冊

n+1 Tin = − Ti−1 rn −

冉冊册

rn rn n+1 + 共1 + 2rn兲Tin+1 − rn + T 2i 2i i+1

共10兲

共 p兲

共11兲 p refers to the pth iteration and rn is defined as rn =

␣⌬tn 共⌬r兲2

and

␣= At the interface,

n+1 Tn+1 = Tm

␭ ␳c p

is the temperature,

冋

␳共1 − ␧兲L 共⌬r兲2 关⌬tn兴 p+1 = n+1 ␭N 共Tn+1 − Tn+1 n 兲

冋

␳共1 − ␧兲L 共⌬r兲2 n+1 ␭N 共Tn+1 − Tn+1 n 兲

册

p

共12兲

册

共13兲

The rest of the time steps are calculated by setting n = 1 , 2 , 3 , . . . , and solving Eqs. 共10兲–共12兲 simultaneously. In this method, an accuracy test is done at the interface with the iteration continuing until a convergence criterion is met 关13兴. After the interface has reached r = R from r = 0, it was assumed that the temperature decay would converge to a one dimensional transient conduction in cylindrical coordinates without phase change effects. The formulation was considered as an infinite length PCM rod of radius R having the uniform initial temperature To suddenly plunged into a quiescent air at temperature T⬁. The formulation now becomes 关15兴

冉冊

⳵␪ ␣ ⳵ ⳵␪ r = ⳵t r ⳵r ⳵r

␪ = T i − T ⬁,

␪共r,0兲 = ␪0,

to

test

numerical

melted PCM and CPCM 1 were allowed to solidify at the quiescent temperature of 25° C. Thermocouples were used to monitor the temperature of the PCM in the container at the center and the surface of the container, as shown in Figs. 9共a兲 and 9共b兲. The void fractions of the container containing PCM and CPCM were calculated as 23% and 12.5% respectively. An average Nusselt number and heat transfer coefficient was calculated using Cebecci correlation as described in Refs. 关16兴 since 共L / D兲 ⫻ 共1 / Gr1/4L兲 = 0.06⬎ 0.025 did not fall within the limit to use the flat plate correction. 共16兲

This resulted in a heat transfer coefficient of 6.23 W / m2 K. Physical properties used for the numerical analysis are listed in Table 3. Figure 10 shows the experimental results for both melting and solidification for the PCM and CPCM. As can be seen from the experimental data, the surface temperature for both materials followed the water bath temperature. The phase transition range for both materials was seen to lie between 70° C and 82° C. Both the PCM and CPCM were maintained at 90° C to ensure that all the mass of the PCM was melted completely. The materials were then allowed to cool down to ambient air. The solidification curve is marked clearly by the improvement in the temperature gradient in the composite PCM. This is mainly due to improvement in the thermal conductivity of the material.

i = 0,1,2,3 共15兲

⳵ ␪共R,t兲 = h␪共R,t兲 ⳵r

Fig. 9 „a… Melting PCM in a constant water bath of 90° C. „b… Solidifying PCM in an quiescent air of 25° C.

For simplicity, we used four nodes, as shown in Fig. 8.

6

case

共14兲

⳵ ␪共R,t兲 =0 ⳵r −␭

semi-infinite

共Nuave兲cylinder = 1.13 共Nuave兲flatplate

The starting time step can be determined explicitly by setting n = 0 and p = 0 in Eq. 共12兲, which reduces to ⌬t0 =

Fig. 8 Idealized formulation

Table 3 Properties used for numerical analysis

Material

L 共kJ/g兲

␳ 共kg/ m3兲

cp 共kJ/ g ° C兲

␭ 共W / m K兲

␧ void fraction

PCM CPCM 1

187.9 164

770 605

2.5 2.18

0.195 0.8223

0.23 0.12

Experimental Procedure/Results

The experimental procedure involved containment of 550 g of PCM and CPCM in a 0.0508 m diameter copper tube of 0.4572 m length. Both the PCM and CPCM 1 containers were melted at the constant water bath temperature of 90° C one at a time. The Journal of Solar Energy Engineering

NOVEMBER 2010, Vol. 132 / 041006-5


Fig. 12 Experimental results versus numerical analysis for PCM with the phase change, assuming h = 6 W / m2 K for the numerical analysis phase change Fig. 10 Experiment data for heating and solidification of PCM

The nondimensional temperatures as function time for centerline and surface locations of the PCM and the composite are plotted in Fig. 11 for the case of solidification in quiescent air. The curves also show an improvement in the solidification rate of 40%. The experimental result shows that the decay in temperature proceeds faster as the interface reaches the centerline from the surface. After the interface has reached the centerline, it was assumed that the PCM would behave as a hot cylindrical rod with no latent heat. Using this assumption and utilizing the experimental results and numerical formulation, results are shown in Figs. 12 and 13 for PCM, and in Figs. 14 and 15 for composite PCM. To avoid the uncertainty associated with the prediction of the heat transfer coefficient from the heat transfer correlation, we explored the values of 6 W / m2 K and 8 W / m2 K in the numerical formulation for the same experimental data. From Figs. 12–15, it is seen that with an increase in heat transfer coefficient by 2 W / m2 K the change in the solidification rate was seen to vary by 18–20% for PCM and CPCM in the numerical analysis. The deviation of the temperature in numerical results at the phase transition temperature 关共Ts − T⬁兲 / 共Tm − T⬁兲 = 1兴 from that of experimental result is due to the phase change occurring at a certain temperature range instead of a constant value as it is assumed for the numerical analysis. It is very interesting to note that after the interface has reached r = 0 共i.e., when the solidification is complete兲, the whole PCM behaves as a solid cylinder of constant specific heat as indicated by the decay in temperature in all curves. Using the same numerical formulation, it was explored the total solidification rate with heat transfer coefficients of 6 W / m2 K, 8 W / m2 K, 15 W / m2 K, 25 W / m2 K, 35 W / m2 K, and

45 W / m2 K having the corresponding Biot numbers of 0.18, 0.25, 0.46, 0.77, 1.08, and 1.39 for CPCM 1. Figure 16 shows the plot of the solidification time of the composite PCM with Biot number at different radial locations. From the figure it is seen that there is an optimum range of a heat transfer coefficient beyond which there is very insignificant changes in the solidification rate. With the same Biot number the solidification rate is seen to be reduced by a factor of 5 for the composite PCM. This would provide an advantage of having lower pumping power to extract the same amount of heat as compared with PCM. Thus, while designing the heat exchanger and defining the pumping power needed for a phase change storage the optimum value of the solidification rate should be explored.

Fig. 13 Experimental results versus numerical analysis for PCM with the phase change, assuming h = 8 W / m2 K for the numerical analysis phase change

Fig. 11 Nondimensional temperature of the PCM and CPCM during solidification in air

041006-6 / Vol. 132, NOVEMBER 2010

Fig. 14 Experimental results versus numerical analysis for CPCM 1 with the phase change, assuming h = 6 W / m2K for the numerical analysis



perature were calculated as ⫾8.4% and ⫾22.6% for axial and radial thermal conductivities, respectively.

8

Fig. 15 Experiment results versus numerical analysis for CPCM 1 with the phase change, assuming h = 8 W / m2K for the numerical analysis

7

Measurement Uncertainty

Most of the measurements carried out in DSC 823e involved direct measurement from the instrument. The uncertainty associated with measurement of latent heat and specific heat depends on the heating rate 共of instrument兲 and a single observation variable 共mass兲. Considering minimum uncertainty related to direct measurement from well calibrated instrument and dealing with measurement carried by observation, the uncertainty is mainly due to mass measurement, height, and diameter measurement. Thus, with a random variable for specific heat and latent heat measurement, we neglect the error associated with it. The percentage uncertainty associated with measuring the axial thermal conductivity with two random variables and one direct measurement variable from instrument can be computed as by the quadratic sum 关17兴 to give ␭ ␭best

=

冑冉冊冉冊冉冊 ⳵m m

2

+

⳵x x

2

+

⳵r r

2

␭ = 冑共共1兲2 + 共3.35兲2 + 共6.7兲2兲 = 7.6% ␭best

共17兲共18兲

where

⳵m ⳵x ⳵r + + m x r is the fractional uncertainty in the measurement of mass, height and radius of the sample of which the thermal conductivity is to be determined. Similarly, the uncertainty associated with the measurement of radial thermal conductivity is calculated as 22.4%, as a result of significant discrepancy in the measurement of the average value. This is mainly due to uneven sample. The value is still higher for radial thermal conductivity at higher graphite bulk density because the method used in sample preparation might have broken the adhesion bond between the molecules. The total propagation uncertainties on measuring the nondimensional tem-

Fig. 16 Numerical results for solidification rate for CPCM 1 with the phase change at different Biot numbers


Conclusions

The DSC results of the tested composite consisting of 83.5% mass fraction of RT-80 paraffin 共melting point equal to 80° C兲 and 16.5% mass fraction of expanded graphite powder showed an increment in effective thermal conductivity by a factor of 4. The experimental results showed a phase transition range of 72– 81° C, both in molecular and bulk levels for both PCM and composite PCM, thus providing a prospect for solar thermal space cooling and heating applications that require the same temperature range. A numerical model of the phase change process of a small cylinder made of the CPCM embedded in the storage tank allowed exploring different design parameters. The numerical results provided the measurement of the solidification rate within the approximately predicted uncertainty of 20. The numerical simulation and the experimental results for a known volume of CPCM indicated a reduction in solidification time by a factor of 6. The numerical analysis was further explored to indicate the optimum operating Biot number for maximum efficiency of the composite PCM thermal energy storage. These numerical results provided optimum operating points for a PCM or composite PCM; beyond this point the change in solidification or melting rate is very insignificant. Optimum operating points would facilitate design of heat exchangers to extract the heat from PCM storage tanks at significant reduced pumping powers.

Acknowledgment The authors would like to thank the Department of Mechanical Engineering of Santa Clara University for providing the financial assistanship to a MS student and also providing all the support to move this research project forward. The research project was carried according to the task specified in EISG Project 05-18 and the authors would also acknowledge the financial support from the California Energy Commission. The authors also recognize the Departments of Chemistry and of Civil Engineering, Santa Clara University, for allowing the use of their laboratory facilities for the research.

Nomenclature Q Cp m T L t x A R D N h r s q Gr Nu Fo Bi N

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

Greek Symbols ⌬ ⫽ ␭ ⫽ ␳ ⫽ ␪ ⫽

heat flux 共W兲 specific Heat 共kJ/ g ° C兲 mass 共kg兲 temperature 共K兲 latent heat 共kJ/kg兲 time 共s兲 height 共m兲 area 共m2兲 radius 共m兲 diameter 共m兲 number of time steps. heat transfer coefficient 共W / m2 K兲 radial direction 共m兲 interface position total heat 共J兲 Grashof number= g␤共Ts − T⬁兲x3 / ␯2 Nusselt number= hR / k Fourier number= ␣⌬t / ⌬r2 Biot number= h⌬r / k number of nodes difference or change thermal conductivity 共W / m K兲 density 共kg/ m3兲 fiber direction 共degrees兲 NOVEMBER 2010, Vol. 132 / 041006-7


␧ ⫽ void fraction= ␳container / ␳PCM Superscripts s l n w a r

⫽ ⫽ ⫽ ⫽ ⫽ ⫽

solid liquid time step water axial direction radial direction

References 关1兴 Sharma, S. D., 2005, “Latent Heat Storage Materials and Systems: A Review,” Int. J. of Green Energy, 2, pp. 1–56. 关2兴 Py, X., Olives, R., and Mauran, S., 2001, “Paraffin/Porous-Graphite-Matrix Composite as a High and Constant Power Thermal Storage Material,” Int. J. Heat Mass Transfer, 44, pp. 2727–2737. 关3兴 Barba, A., and Spiga, V., 2003, “Discharge Mode for Encapsulated PCMs in Storage Tanks,” Sol. Energy, 74, pp. 141–148. 关4兴 Zalba, B., Marin, J. M., Cabeza, L. F., and Mehling, H., 2003, “Review on Thermal Energy Storage With Phase Change: Material, Heat Transfer Analysis and Applications,” Appl. Therm. Eng., 23, pp. 251–283. 关5兴 Alva, L. H., González, J. E., and Dukham, N., 2006, “Solar Air Conditioning Systems With PCM Solar Collectors,” ASME J. Sol. Energy Eng., 128, pp. 173–177.

041006-8 / Vol. 132, NOVEMBER 2010

关6兴 Schawe, J. E., Hutter, T., Heitz, C., Alig, I., and Lellinger, D., 2006, “Stochastic Temperature Modulation: A New Technique in Temperature-Modulated DSC,” Thermochim. Acta, 446, pp. 147–155. 关7兴 Incopera, P. F., and DeWitt, D. P., 1996, Fundamentals of Heat and Mass Transfer, 4th ed., Wiley, New York. 关8兴 http://us.mt.com/mt_ext_files/Editorial/Generic/3/TA_UserCom22_EditorialGeneric_1166701704723_files/tausc22e.pdf 关9兴 Hakvoort, G., van Reijen, L. L., and Aartsen, A. J., 1985, “Measurement of the Thermal Conductivity of Solid Substances by DSC,” Thermochim. Acta, 93, pp. 317–320. 关10兴 Bonnissel, M., Luo, L., and Tondeur, D., 2001, “Compacted Exfoliated Natural Graphite as Heat Conduction Medium,” Carbon, 39, pp. 2151–2161. 关11兴 Han, J. H., Cho, K. W., Lee, K. H., and Kim, H., 1998, “Porous Graphite Matrix for Chemical Heat Pump,” Carbon, 36, pp. 1801–1810. 关12兴 Lane, G. A., 1983, Solar Heat Storage: Latent Heat Material, Vol. 1: Background and Scientific Principles, CRC Press, Boca Raton, FL. 关13兴 Özisik, M. N., 1993, Heat Conduction, 2nd ed., Wiley, New York. 关14兴 Gupta, R. S., and Kumar, D., 1981, “Variable Time Step Methods for OneDimensional Stefan Problem With Mixed Boundary Condition,” Int. J. Heat Mass Transfer, 24, pp. 251–259. 关15兴 Arpaci V. S., 1966, Conduction Heat Transfer, Addison-Wesley, New York. 关16兴 Rolle, K. C., 2000, Heat and Mass Transfer, Prentice-Hall, Englewood Cliffs, NJ. 关17兴 John, T. R., 1996, An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements, University Science Book, 2nd ed., Maple-Vail Book Manufacturing Group.



Graphite Composite

Graphite Composite

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