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Thermal response in thermal energy storage material around heat transfer tubes: effect of additives on heat transfer rates
Solar Energy 75 (2003) 317–328 www.elsevier.com/locate/solener
Thermal response in thermal energy storage material around heat transfer tubes: effect of additives on heat transfer rates Yuichi Hamada, Wataru Ohtsu, Jun Fukai
*
Department of Chemical Engineering, Kyushu University 6-10-1, Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan Received 24 February 2003; accepted 24 July 2003
Abstract The effects of carbon-fiber chips and carbon brushes as additives on the thermal conductivity enhancement of phase change materials (PCMs) using in latent heat thermal energy storage are investigated experimentally and numerically by considering the wall effect of the additives. The carbon-fiber chips are effective for improving the heat transfer rate in PCMs. However, the thermal resistance near the heat transfer surface is higher than that for the carbon brushes. As a result, the overall heat transfer rate for the fiber chips is lower than that for the carbon brushes. Consequently, the carbon brushes are superior to the fiber chips for the thermal conductivity enhancement under the present experimental conditions. The carbon brushes are moreover applied to the packed beds of particles to overcome their low thermal conductivity in chemical heat pump/storage. The carbon brushes essentially improve the heat transfer characteristics in the packed beds, though the thermal resistance is observed because the particles obstruct contact between the fibers and the heat transfer surfaces. 2003 Elsevier Ltd. All rights reserved.
1. Introduction Thermal energy storage is one of the key technologies for the efficient use of thermal energy. That is, the seasonally as well as diurnally fluctuating thermal energy such as the solar energy and the wasted heat in industry is temporarily stored, and then rapidly is supplied when it is demanded. Especially, latent heat thermal energy storage using phase change materials (PCMs) has attracted a great deal of attention due to their high energy density. However, the heat transfer rate between the PCMs and heat transfer fluid is generally unacceptably low due to the low thermal conductivities of the PCMs (Hasnain, 1998). To overcome the low thermal conductivities of the PCMs, thin aluminum strips (Hoogendoorn and Bart,
*
Corresponding author. Tel.: +81-92-642-3515; fax: +81-92642-3519. E-mail address: [email protected] (J. Fukai).
1992), thin wall rings made of steel (Velraj et al., 1999), porous aluminum (Weaver and Viskanta, 1986), porous graphite matrices (Py et al., 2001), copper chips (Tayeb, 1996) and carbon fibers (Fukai et al., 1997, 1999) have been proposed as additives to enhance the heat transfer rate in PCMs. For instance, Hoogendoorn and Bart (1992) reported that the effective thermal conductivity of a paraffin/1 vol.% thin-aluminum-strip composite is seven times as high as the thermal conductivity of the paraffin. However, the effective thermal conductivities of the composites near the heat transfer surface are definitively lower than those of the bulk due to the wall effect or contact thermal resistance, thus reducing the overall heat transfer rate. This fact was demonstrated by Fukai et al. (2002), who quantitatively discussed the effective thermal conductivity of paraffin/carbon-brush composites by considering the wall effect. However, there are few studies that discuss the heat transfer enhancement of additives by considering the wall effect or contact thermal resistance. Chemical heat pump/storage systems are also notable thermal energy storage techniques because reaction heat
0038-092X/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2003.07.028
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Y. Hamada et al. / Solar Energy 75 (2003) 317–328
Nomenclature Bi cp cp d d f h k k k k ltp Nu Pr r Re t T u x, y X, Y Xf Xfa Xtot
Biot number, hltp =km specific heat (J/kg K) dimensionless specific heat, cp =cp;m diameter (m) dimensionless diameter, d=ltp friction factor heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) thermal conductivity tensor (W/m K) dimensionless thermal conductivity, k=km dimensionless thermal conductivity tensor, k=km tube pitch (m) Nusselt number, hh di =kh Prandtl number, ðcp lÞh =kh radial distance (m) Reynolds number, qh di uh =lh time (s) temperature (K) mean fluid velocity (m/s) x- and y-coordinates (m) dimensionless X - and Y -coordinates, x=ltp , y=ltp local volume fraction of fibers volume ratio of fibers to the composite volume ratio of fibers to the shell side of heat exchanger
Greek symbols d thickness of a control volume (m) D dimensionless thickness of a control volume, d=ltp e void fraction / parameter representing contact heat transfer rate between particles l viscosity (Pa s) h angle (rad)
is much higher than latent heat. However, the low effective thermal conductivities of the packed beds are an impediment to practical use (Wongsuwan et al., 2001). Groll (1993) reviewed the effective thermal conductivities of several additives/metals and the additives/inorganic-materials composites as well as the heat transfer coefficients between the heat transfer surface and the composites. According to his report, the effective thermal conductivities of the composites are 0.3–0.7 W/m K (25 wt.%) for graphite binder and 2–9 W/m K for consolidated activated carbon and zeolite beds including highly porous metallic foams (Ni, Cu), while the heat transfer coefficients for these additives are 35–180 W/ m2 K. The effective thermal conductivities are 5–30 W/
H q q s n n f f
dimensionless temperature, ðT T0 Þ= ðTh;in T0 Þ density (kg/m3 ) dimensionless density, q=qm dimensionless time, km t=fðcp qÞm l2tp g coordinate normal to the boundary surface (m) dimensionless coordinate normal to the boundary surface, n=ltp coordinate tangential to the boundary surface (m) dimensionless coordinate tangential to the boundary surface, f=ltp
Subscripts 0 initial c composite ct composite-tube wall interface cal calculation eff effective exp experiment f fiber g gas h heat transfer fluid i inner in inlet m phase change material or packed bed o outer p parallel model par particle s series model t tube v container w wall Superscript 0 including wall effect
m K (>10 wt.%) for porous graphite matrices, while the heat transfer coefficients are 400–1000 W/m2 K. In this case, however, the porous graphite matrices are compressed in the reactor container, which results in obstructing the uniform gas flows in the packed beds due to low gas permeability (Han et al., 1998). Dellero et al. (1999a,b) enhanced the effective thermal conductivity using 30-mm long carbon fibers. In this case, the heat transfer coefficient between the carbon fibers and heat transfer surface is 137 W/m2 K. The authors have proposed the brushes made of carbon fibers as additives (Fukai et al., 2000) and investigated their effect on the heat transfer rate of the brush/PCM composite in shell-and-tube heat exchangers
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
(Fukai et al., 2003). Similar to other additives, the wall effect was realized in this system. However, it has not been concluded if the brush as additives is inferior to other carbon fiber configurations. The authors focus on the heat transfer in the shelland-tube type, while most of the previous studies used cylindrical capsule/reactor, because the apparent energy density stored in the shell-and-tube type is higher. The present paper is composed of two parts. The purpose of the first part is to compare the effect of carbon-fiber chips and the carbon brushes, as additives, on the heat transfer rate in the PCM. That of the second part is to examine the thermal effect of the carbon brushes on the packed bed of the glass particles as alternative reactants in the shell side. In both parts, the wall effect between the composite and the heat transfer surface is investigated.
2. Thermal conductivity enhancement in PCMs 2.1. Experimental procedure Fig. 1 shows the experimental apparatus. Four steel tubes are vertically arranged in a cylindrical container
319
made of acrylic resin. Carbon-fiber chips (fiber diameter ¼ 10 lm, fiber length ¼ 5 mm) are packed in the container. The container is filled with n-octadecane (km ¼ 0:34 W/m K) as a PCM up to a height of about 50 mm. The container is insulated using glass wool. Two methods are used to place the fiber chips/PCM composite in the container. In method I, the container is filled with the melted PCM after the fiber chips are packed into the container. In method II, the composites are similarly made using a cylindrical vessel whose inner diameter is less than that of the container. The composite is placed in the center of the container in Fig. 1, and then the melted PCM fills the space around the composite. When the diameter of the composite is larger than that of the tube pitch (ltp ¼ 108 mm), four holes are provided in the composite to adjust them in the tubes. Three kinds of fiber chips are used. They are referred to as CFL (carbon fiber with low thermal conductivity, kf ¼ 5 W/m K), CFM (carbon fiber with middle thermal conductivity, 190 W/m K) and CFH (carbon fiber with high thermal conductivity, 500 W/m K). The fiber volume fractions to the composite Xfa in the experiments are shown in Table 1. M0:8 and M1:2 are provided to investigate the effect of CFM on the thermal
164 mm
80.8 mm
Tube (Steel)
Container
23.6 mm
Thermocouple 1.8 mm
24
60 mm
E D
A
m
B
m
(a)
C
.6 24
Water
m
Water
m
35 mm
Insulator
.6
Chip/n-Octadecane Composite
l1 l1 l1 l1
108 mm
108 mm
l2
l2
(l1 = 10.1 mm, l2 = 19.1 mm) (b)
Fig. 1. Experimental apparatus.
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Table 1 Experimental conditions Volume fraction Xfa N M0:8 M1:2 L1:4 L1:4 H0:3 L1:4 H0:5
CFL
CFM
CFH
kc;eff
0 0 0 0.014 0.014 0.014
0 0.008 0.012 0 0 0
0 0 0 0 0.003 0.005
0.34 0.80 1.03 0.36 0.82 1.12
conductivity enhancement. L1:4 H0:3 and L1:4 H0:5 are provided to examine whether the same effective thermal conductivities as M0:8 and M1:2 are obtained with a smaller volume of higher conductive fibers. In this case, CFH is mixed with CFL because the volume fraction of CFH is too low to be solely packed into the container up to the height of n-octadecane. L1:4 is the base case to investigate the effect of CFH on the thermal conductivity enhancement. Three experimental apparatuses are provided for each composite to confirm the experimental reproducibility. Fig. 2 shows the photograph of the fiber chips packed into the container before the PCM is poured into the container. It also shows the carbon brush settled in the container, which is compared to the fiber chips in Section 2.3. The effective thermal conductivities of bulk in the fiber chips/PCM composite in Table 1 are estimated using the following semi-empirical equation (Fukai et al., 1999, 2000). kc;eff =km ¼ 0:456f1 ð1 Xf Þ2=3 gðkf =km Þ þ ð1 Xf Þ1=3 ð1Þ For L1:4 H0:3 and L1:4 H0:5 , kc;eff of the CFL/PCM composite is estimated, and then kc;eff of the CHL/CFL/PCM composite is again estimated using Eq. (1) based on the
assumption that the CFL/PCM composite is thermally homogeneous. The estimated effective thermal conductivities of L1:4 H0:3 and L1:4 H0:5 are almost the same as those of M0:8 and M1:2 , respectively. In the experiment, the apparatus was placed in a room at 18 C overnight to maintain the container at a uniform temperature. Next, water at 23 C was provided from a thermostatic bath to the tubes. The mean flow velocity of water in the tubes is uh ¼ 0:35 m/s. The transient thermal responses were measured at points A– E as shown in Fig. 1(b). A digital recorder recorded the temperatures at intervals of 30 s. It should be noted that the experimental temperature range was below the melting point of n-octadecane (28 C) in order to investigate the conductive heat transfer excluding latent heat. The experiment was finished when the temperature fluctuations were within ±0.1 C in 300 s. 2.2. Numerical procedure Fig. 3 shows the computational domain, which consists of the fiber-chips/PCM composite and the tube wall. At t < 0, the composite, including the PCM at the periphery, and the tube wall are initially kept at temperature T0 At t ¼ 0, the heat transfer fluid at temperature Th;in flows in the tubes. Choosing T0 and Th;in as the characteristic temperatures and the tube pitch ltp as the characteristic length leads to the dimensionless expressions for the two-dimensional heat conduction equations in the composite (c) and the tube wall (t), boundary conditions and initial conditions as shown in Table 2(a). The definitions of the dimensionless are shown in nomenclature. kc in the heat conduction equation is equivalent to kc;eff =km in Eq. (1). The heat capacity is given by ðcp qÞc ¼ Xf ðcp qÞf þ ð1 Xf Þðcp qÞm
ð2Þ
At the interfaces between the tube wall and the composite, the wall effects are considered using the heat
Fig. 2. The photographs of carbon-fiber chips and carbon brushes in the containers.
Y. Hamada et al. / Solar Energy 75 (2003) 317–328 y
ζ
ζ
d
ltp/2
ξ
dv/2
ξ
ltp/2
dv/2
y
321
y'
x'
θ
2 c/
x
x
ltp/2
ltp/2
dv/2
dv/2
(b) Carbon-fiber brushes
(a) Carbon-fiber chips
Fig. 3. Computational domains corresponding to the experimental apparatuses.
Table 2 Mathematical description of the composite (a) Carbon-fiber chips
(b) Carbon-fiber brushes
c ¼ r ðkc r Hc Þ ðcp qÞc oH os oHt ðcp qÞt os ¼ r ðkt r Ht Þ
c ðcp qÞc oH ¼ r ðkc r Hc Þ os
t kt oH on
Hc ¼ Ht ¼ 0 t k oH ¼ Bih ð1 Ht Þ on c t ¼ Bict ðHt Hc Þ ¼ kc oH kt oH ¼ Bict ðHt Hc Þ on on oHc oHc ¼ knn;c on þ knf;c oH of ¼ 0 on
transfer coefficient hct . hct ¼ 1 (Bict ¼ 1) means no wall effect. The mathematical model is numerically solved using the control volume method (Patankar, 1980). To consider the boundary conditions at the composite-tube wall interface, the dimensionless thermal conductivity of the control volume by the interface is given by 1 1 1 1 k ¼ þ þ ð3Þ 2kt BiD 2kc where D is the dimensionless thickness of the control volume. The curvature surfaces are approximated using square grids. The 50 · 50 uniform meshes and Ds ¼ 0:0002 are needed to have numerical results independent of the mesh size and the time step. The mesh size and the time step independences of the numerical results are discussed in Appendix A. The effective thermal conductivity of the composite 0 keff including the wall effect is estimated when the composite is packed into a shell-and-tube heat exchanger as shown in Fig. 4. Carbon brushes are also drawn in the
ltp
I.C. B.C.
In composite In tube wall In the whole region At inner surface of tube At tube wall––composite interface At exposed surface and iteration boundary
y
o
x
ltp
Fig. 4. Computational domain corresponding to a heat exchanger.
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
figure. The shadowed area in the figure is chosen as the computational domain considering the periodicity of the tube arrangement. The mathematical model in Table 0 2(a) is also applied to this system. keff is estimated using a previous reported technique (Fukai et al., 2002). That is, the time variation in the heat transfer rate on the inner surface of the tubes is evaluated from the calculated temperature field. Separately, the equation where 0 kc is substituted by keff is numerically solved assuming Bict ¼ 1 to evaluate the time variation in the heat 0 transfer rate on the inner surface. The value of keff is determined by fitting the latter heat transfer rate with the former.
Point B
1
Θ
322
0.5
N M0.8 M1.2 L1.4 L1.4H0.3 L1.4H0.5
0 Point D
1
The properties of the materials used in the experiments are km ¼ 0:34 W/m K, kt ¼ 80 W/m K, cp;m qm ¼ 1:53 106 J/m3 K, cp;f qf ¼ 2:12 106 J/m3 K and cp;t qt ¼ 3:48 106 J/m3 K. The heat transfer coefficient hh in the tubes is calculated with the Petukhov– Gnielinski equation (Gnielinski, 1976; Petukhov, 1970): Nu ¼
Θ
2.3. Results
0.5
ðf =2ÞðRe 1000ÞPr pffiffiffiffiffiffiffiffi 1 þ 12:7 f =2ðPr2=3 1Þ 6
ð3000 < Re < 10 ; 0:5 < Pr < 2000Þ
0
0
0.05
where f is the friction factor, Re the Reynolds number and Pr the Prandtl number. According to Eq. (4), hh is 1.7 · 103 W/m2 K. It is numerically confirmed that this value is sufficiently high for the estimation error of hh not to affect the calculated temperature distribution in the composite. Fig. 5 shows the experimental results for the nondimensional time variations in the non-dimensional temperatures. The experimental data are plotted every fifteen recorded data. The reproducibility of the experimental results is shown in Appendix B. The time required for H to reach a value is calculated from the experiments. The time factor is defined as the ratio of the times with no fibers to with the fibers, and shown in Table 3. A comparison among the results for N, M0:8 and M1:2 shows that the CFM chips essentially improve the thermal responses as the volume fraction of the fibers increases. It should be noted that H for M1:2 reaches 0.5 in nearly half the time of N. From the results for N and L1:4 , CFL is not effective for heat transfer enhancement, as expected from the effective thermal conductivities in Table 1. Moreover, the response for L1:4 H0:3 is not as sensitive as that for M0:8 though the estimated effective thermal conductivity of L1:4 H0:3 is the same as that of M0:8 as shown in Table 1. The CFH chips have to be placed in contact with each other to improve the effective thermal conductivity. In this case, the CFH chips may have few chances to meet other CFH chips because of a low density of CFH, though
0.1
τ
ð4Þ
Fig. 5. The non-dimensional time variations in non-dimensional temperatures in the fiber chips/PCM composite using method I (experimental results).
Table 3 The average time factor for three runs H ¼ 0:5 M0:8 M1:2 L1:4 L1:4 H0:3 L1:4 H0:5
H ¼ 0:7
Point B
Point D
Point B
Point D
1.63 1.86 1.08 1.44 1.77
1.65 1.95 1.08 1.34 1.95
1.56 1.81 1.09 1.34 1.63
1.60 1.80 1.07 1.31 1.80
they have many chances to meet the CFL chips. This is a reason for the low effective thermal conductivity of L1:4 H0:3 . However, the thermal response for L1:4 H0:5 is close to that for M1:2 . Accordingly, there is a critical value above which the CFH is effective for heat transfer enhancement. Fig. 6 shows a comparison between the experimental and calculated transient thermal responses using the composite dc ¼ 0:65. The calculated temperatures agree well with the experimental. This fact shows the accuracy of Eq. (1). Fig. 7 shows a comparison between the calculated transient thermal responses for hct ¼ 1 (Bict ¼ 1) and
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
1
1
dc* = 1.52
0.5
Θ
Θ
dc* = 0.65
Point Exp. A B C D E
0
0
0.02
τ
0.04
1
Θ
dc* = 1.52
0.5 Point Exp. A B C D E
0
0.02
τ
0.5 Point Exp. A B C D E
Cal.
Fig. 6. Comparison between the experimental and calculated transient thermal responses (M0:8 ). The experimental apparatus is constructed using method II.
0
323
Cal.
0.04
Fig. 7. Comparison between the experimental and calculated transient thermal responses (M0:8 ). The experimental apparatus is constructed using method I.
the experimental thermal responses. The composite was made using method I as mentioned above. The experimental temperatures are obviously lower than the calculated ones. In this case, the fiber chips near the tubes were visually parallel to the tubes though those of the bulk composite were random. This wall effect results in a lower overall heat transfer rate in the experiment than in the calculation. To confirm this issue, the experimental results from the apparatus made using method II are compared with the calculated results in Fig. 8. In this case, the directions of the carbon fibers must be random
0
0
0.02
τ
Cal.
0.04
Fig. 8. Comparison between the experimental and calculated transient thermal responses (M0:8 ). The experimental apparatus is constructed using method II.
near the tube wall as well as in the bulk. The experiments are certainly closer to the calculated results than those in Fig. 7. The value of hct is evaluated by the non-linear leastsquare method. As a result, hct ¼ 150 W/m2 K (Bict ¼ 47:6) is determined. In Fig. 9, the calculated results for Bict ¼ 47:6 are compared with the experimental results. The calculated results agree with the experimental results irrespective of the volume fraction of the fibers. The prediction error is within ±10% in the whole region of H. Considering the facts that the effective thermal conductivity and the transient thermal responses for L1:4 H0:5 are identical to those for M1:2 (Table 1 and Fig. 5), the value of hct for CFH is expected to be nearly 150 W/m2 K. Fig. 10 shows the estimated effective thermal con0 ductivity keff in the fiber chips/PCM composite using method I. Xtot is defined as the volume fraction of the fibers added to the heat exchangers. The effective thermal conductivity of the bulk in the composite keff is also plotted. The results for the carbon brushes (brush diameter ¼ 110 mm, axial length ¼ 60 mm) are referred to from the previous paper (Fukai et al., 2002). hct for the carbon brush/PCM composite is 340 W/m2 K (Bict ¼ 108). The reason for the higher value of hct in the carbon brush system is that the fibers are pressed on the tube wall due to their elastic forces. As a re0 sult, keff is higher for the carbon brush than for the fiber chips though keff is lower for the former than for the latter. The heat transfer rate in the brush/PCM composite is heterogeneous, while that for the fiber-chip/PCM is macroscopically homogeneous. Although the former is inferior to the latter in terms of the heat transfer rate
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Y. Hamada et al. / Solar Energy 75 (2003) 317–328
1 Chip Brush
M0.8
5
keff / km
0% +1 %
0.5
10
Θ exp
keff
Point A B C D E
3
hct = 150 W/m2- K
0 1
1
0% +1 %
0.5
0
0.005
0.01 Xtot
0.015
0.02
Fig. 10. The variation in the predicted effective thermal conductivities with the volume fraction of carbon fibers (kf =km ¼ 560). A comparison between fiber chips and carbon brushes.
10
Θ exp
M1.2
0
hct = 340 W/m2- K
k' eff
Point A B C D E
particles (dpar ¼ 0:1 mm, kpar ¼ 0:8 W/m K, qpar ¼ 2410 kg/m3 ) are packed in the container. The experimental procedure is the same as described above. 3.2. Mathematical model
0
0.5
1
Θ cal Fig. 9. Comparison between the calculated results for Bict ¼ 47:6 and experimental transient thermal responses.
of bulk, this figure shows that the former is superior in terms of the overall heat transfer rate.
3. Thermal conductivity enhancement in packed beds 3.1. Experimental setup As described above, carbon brushes are superior to carbon-fiber chips as additives. Moreover, it is difficult to uniformly pack the fiber chips in a packed bed due to the difference between the densities of carbon fibers and particles. Accordingly, the carbon brushes are used for thermal conductivity enhancement in packed beds of particles. A carbon brush made of CFM is inserted in the center of the container as shown in Fig. 1. The diameter of the brush and the volume fraction of the fibers are 110 mm and 0.008, respectively. Instead of the PCM, glass
The mathematical model is briefly described because the model for the brush/PCM composite has been reported (Fukai et al., 2002). The mathematical model shown in Table 2(b) is used into the computational domain in Fig. 3(b). kc in the heat conduction equation is the dimensionless thermal conductivity tensor. The local x0 –y 0 -coordinate system, where the x0 -axis is along the fiber orientation, is introduced (Fig. 3(b)). The thermal conductivity tensor kc is given by 1 cos h sin h kp 0 cos h sin h kc ¼ ð5Þ sin h cos h sin h cos h 0 ks where kp and ks are the thermal conductivity in the x0 and y 0 -directions, respectively, and h the angle between the x0 -axis and y 0 -axis. If the matrix is a continuous body such as paraffin, kp and ks are given by the well-know parallel and series models. However, the parallel model is not applied to the fibers/solid particles composites because the fibers theoretically undergo point contact with the particles. Therefore, the effective thermal conductivity of such composites in the fiber orientation is lower than that calculated using the parallel model (Fukai et al., 2000). In the present study, a semiempirical regression is derived based on the experimental results (Fukai et al., 2000):
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
kp ¼ 0:66kf Xf þ ð1 0:66Xf Þkm
ð6Þ
325
1
ks is given by the series model: ks ¼ fXf =kf þ ð1 Xf Þ=km g1
ð7Þ
km ¼
Θ
The thermal conductivity model for the packed bed km is given by the model of Kunii and Smith (1960): e 1 e=ð1 Xf Þ kg kg þ 1 Xf / þ ð2kg Þ=ð3kpar Þ
0.5
ð8Þ Point Exp. A B C D E
where e is the void fraction of particles, the parameter representing the heat transfer rate between particles, and kg the thermal conductivity of gas, or air in the present study. The heat capacity is given by e ðcp qÞc ¼ Xf ðcp qÞf þ ð1 Xf Þðcp qÞpar 1 ð9Þ 1 Xf The heat capacity of the glass particles ðcp qÞpar is 1.81 · 106 J/m3 K. 3.3. Results First of all, the carbon brushes essentially improve the experimental transient thermal responses as shown in Fig. 11. For the carbon brush/solid particle composite, the wall effect representing by hct in the present model is contributed by that due to a low density of the particles near the tube wall and that due to the contact conditions between the fibers and the tube wall. The heat transfer coefficient for the former is modeled by Yagi and Kunii (1961): hw dpar 1 ¼ 0:5f1=ðkw =kg Þ 1=ðkm =kg Þg kg
ð10Þ
0
0
Cal.
0.05
0.1
τ
Fig. 12. Comparison between the calculated results for Bict ¼ 443 and the experimental transient thermal responses with no carbon brushes (e ¼ 0:36).
the wall. As a result, the value of hw ¼ 780 W/m2 K is obtained. Fig. 12 compares the calculated thermal responses to the experimental ones in the composite with no carbon brushes. The calculated results for hct ð¼ hw Þ ¼ 780 W/ m2 K (Bict ¼ 443) agree well with the experimental results. In addition, the calculated results for hct ¼ 1 (Bict ¼ 1) are identical to those for 780 W/m2 K (Bict ¼ 443), though they are not plotted in the figure because they are apparently overlapped with those for hct ¼ 780 W/m2 K. This fact shows that the wall effect does not dominate the heat transfer rate in the packed beds. Fig. 13 shows the experimental thermal responses in the composite with carbon brushes. The value of hct was
kw is estimated by substituting Xf ¼ 0 and e ¼ 0:7 into Eq. (10). e ¼ 0:7 is a value using the sphere particles near
1
1 Point B
Θ
Θ
dc* = 1.0
0.5
0.5
Point D
0
0
0.05
τ
Point Exp. A B C D E
Xfa 0 0.008
0.1
Fig. 11. The effect of the carbon brush on the experimental transient thermal responses in the packed bed of the glass particles.
0
0
0.05
τ
Cal.
0.1
Fig. 13. Comparison between the calculated results for Bict ¼ 74:3 and the experimental transient thermal responses (CFM, Xfa ¼ 0:008, e ¼ 0:39).
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Y. Hamada et al. / Solar Energy 75 (2003) 317–328
3
k'eff / km
Brush/PCM
2
Brush/glass particle
1
0
0.005
0.01
0.015
0.02
The carbon-fiber chips are effective for improving the heat transfer rate in PCMs. However, the thermal resistance near the heat transfer surface is higher than that for the carbon brushes. As a result, the overall heat transfer rate for the fiber chips is lower than that for the carbon brushes even though the effective thermal conductivity of the bulk of the former is higher. Consequently, the carbon brushes are superior to the carbon-fiber chips under the present experimental conditions. The effect of the carbon brushes is discussed when they are applied to the packed beds of particles in the shell side of heat exchangers. A higher thermal resistance is observed probably because the particles obstruct contact between the fibers and the heat transfer surfaces. However, the estimated heat transfer coefficients are the same as those for the other additives.
Xtot
4. Conclusions This study experimentally and numerically compares the effects of carbon-fiber chips and carbon brushes as additives on the overall heat transfer rates in the composites.
The first important issue on the discretization of the computation domain, shown in Fig. 3, is that the area of each region must be same as that of exact one. At this point, the differences between them are within less than 1%. The second important issue is that the length of the numerical boundary surface is longer than the exact one because the curvature surface is approximated using zigzag lines as shown in Fig. 15. This fact results in the overestimate of the heat flow rate between the heat transfer fluid and the inner tube wall. To solve this problem, the heat transfer coefficient hh used in the calculation is corrected by multiplying the heat transfer
y
r
ltp
identified using the non-linear least-square method (Fukai et al., 1985). As a result, the value of hct ¼ 110 W/m2 K (Bict ¼ 74:3) is identified. Considering that hw is high enough not to affect the thermal responses, the low value of hct is due to the wall effect of the fibers. The hct value is much lower than the carbon-brush/PCM composite shown in Fig. 10. This is probably because the particles obstruct contact between the fibers and the tube walls. The estimated hct in the present system is on the same order as that for graphite binder and porous metallic foams (Groll, 1993). The effective thermal conductivity of the carbon brush/glass particles composite is estimated using the numerical technique described above. The estimated effective thermal conductivity including the wall effect of fibers is compared with that of the brush/PCM composite in Fig. 14. The effective thermal conductivity in the brush/glass particles composite is lower than that in the brush/PCM composite because of the lower hct value. However, the effective thermal conductivity at Xtot ¼ 0:015 is twice as high as that with no additives. This fact shows that the reaction time for the charge and discharge process with the brush can be half that with no additives even if the particles are reactive.
Appendix A
do/2
Fig. 14. The comparison between the predicted effective thermal conductivities of the brush/glass-particle composite and that of the brush/PCM (kf ¼ 190 W/m K).
x
do/2 ltp Fig. 15. The discretization of a computational domain.
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
Θ
1
For the computation domain in Fig. 3(a), the effects of the mesh size and time step on the calculated result are shown in Fig. 17. Obviously, 50 · 50 mesh and Ds ¼ 0:0002 lead to the results being independent of them.
r/ltp = 0.334
0.630
0.5
Appendix B
0.926
Fig. 18 compares the results measured using three different apparatuses for typical composites. This figure shows good reproducibility in the present experiments.
Numerical Analytical 0
327
0
0.5
1
τ
1
Fig. 16. The comparison between the numerical solution and the analytical one (Bih ¼ 50).
Θ
coefficient using Eq. (3) by the ratio of the exact surface area to the numerical one. The numerical and analytical solutions are compared to examine the accuracy of the present numerical technique. The problem considered is a homogeneous hollow cylinder whose inner surface is uniformly heated by fluid and outer surface is adiabatic (Fig. 15). The analytical solution in the radial direction is shown by Carslaw and Jaeger (1986). 50 · 50 mesh is used in the calculation. Fig. 16 demonstrates that the numerical results agree well with the analytical ones. The numerical error for Bih ¼ 1000 is less than that for Bih ¼ 50 though the result is not shown here.
M0.8
0.5
Point C D 0
0
Exp.
0.05
0.1
τ
(a) Carbon fiber chips/PCM composite 1
1
Xfa = 0.008
Θ
Θ
Point C
0.5
Point A
Point C D
mesh ∆τ 3 30 × 30 2×10 4 2×10 50 × 50 5 100 × 100 2×10
0
0.5
0
0.02
τ
0 0.04
Fig. 17. The effect of the mesh size and time step on the calculated result. (M0:8 , dc ¼ 1:52).
0
Exp.
0.05
0.1
τ
(b) Carbon brush/glass particles composite Fig. 18. Reproducibility of the experimental results.
328
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References Carslaw, H.S., Jaeger, J.C., 1986. Conduction of Heat in Solids. Clarendon press, Oxford. Dellero, T., Sarmeo, D., Touzain, P., 1999a. A chemical heat pump using carbon fibers as additive. Part I: Enhancement of thermal conductivity. Applied Thermal Engineering 19, 991–1000. Dellero, T., Sarmeo, D., Touzain, P., 1999b. A Chemical heat pump using carbon fibers as additive. Part II: Study of constraint parameters. Applied Thermal Engineering 19, 1001–1011. Fukai, J., Watanabe, M., Miura, T., Ohtani, S., 1985. Simultaneous estimation of thermophysical properties by nonlinear least squares. Kagaku Kogaku Ronbunshu 11, 84–92, translated into International Chemical Engineering 27, 1987, 455–465. Fukai, J., Omori, H., Oishi, A., Miyatake, O., 1997. Effect of carbon fibers on thermal response within heat storage material. Kagaku Kogaku Ronbunshu 23, 82–87. Fukai, J., Oishi, A., Kodama, Y., Kanou, M., Miyatake, O., 1999. Improvement of discharge characteristics of latent heat thermal energy storage unit by using carbon fibers. In: Proc. 5th ASME/JSME Joint Thermal Engineering Conference (AJTE99-6338), San Diego. Fukai, J., Kanou, M., Kodama, Y., Miyatake, O., 2000. Thermal conductivity enhancement of energy storage media using carbon fibers. Energy Conversion and Management 41, 1543–1556. Fukai, J., Hamada, Y., Morozumi, Y., Miyatake, O., 2002. Effect of carbon-fiber brushes on conductive heat transfer in phase change materials. International Journal of Heat and Mass Transfer 45, 4781–4792. Fukai, J., Hamada, Y., Morozumi, Y., Miyatake, O., 2003. Improvement of thermal characteristics of latent heat thermal energy storage units using carbon-fiber brushes: experiments and modeling. International Journal of Heat and Mass Transfer 46, 4513–4525.
Gnielinski, V., 1976. New equations for heat and mass transfer in turbulent pipe and channels flow. International Chemical Engineering 16, 359–368. Groll, M., 1993. Reaction beds for dry sorption machines. Heat Recovery Systems & CHP 13, 341–346. Han, J.H., Cho, K.W., Lee, K.-H., Kim, H., 1998. Porous graphite matrix for chemical heat pump. Carbon 36, 1801–1810. Hasnain, S.M., 1998. Review on sustainable thermal energy storage technologies. Part I: Heat storage materials and techniques. Energy Conversion and Management 39, 1127–1138. Hoogendoorn, C.J., Bart, G.C.J., 1992. Performance and modeling of latent heat stores. Energy 48, 53–58. Kunii, D., Smith, J.M., 1960. Heat transfer characteristics of porous rocks. AIChE Journal 6, 71–78. Patankar, S.V., 1980. Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York. Petukhov, B.S., 1970. Heat transfer and friction in turbulent pipe flow with variable physical properties. Advances in Heat Transfer 6, 503–564. Py, X., Olives, R., Mauran, S., 2001. Paraffin/porous-graphitematrix composite as a high and constant power thermal storage material. International Journal of Heat and Mass Transfer 44, 2727–2737. Tayeb, A.M., 1996. Use of some industrial wastes as energy storage media. Energy Conversion and Management 37, 127–133. Velraj, R., Seeniraj, R.V., Hafner, B., Faber, C., Schwarzer, K., 1999. Heat transfer enhancement in a latent heat storage system. Solar Energy 65, 171–180. Weaver, J.A., Viskanta, R., 1986. Melting of frozen, porous media contained in a horizontal or a vertical, cylindrical capsule. International Journal of Heat and Mass Transfer 29, 1943–1951. Wongsuwan, W., Kumar, S., Neveu, P., Meunier, F., 2001. A review of chemical heat pump technology and applications. Applied Thermal Engineering 21, 1489–1519. Yagi, S., Kunii, D., 1961. Studies on heat transfer in packed beds. International Development of Heat Transfer, Part III, ASME, 750.
Thermal response in thermal energy storage material around heat transfer tubes: effect of additives on heat transfer rates Yuichi Hamada, Wataru Ohtsu, Jun Fukai
*
Department of Chemical Engineering, Kyushu University 6-10-1, Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan Received 24 February 2003; accepted 24 July 2003
Abstract The effects of carbon-fiber chips and carbon brushes as additives on the thermal conductivity enhancement of phase change materials (PCMs) using in latent heat thermal energy storage are investigated experimentally and numerically by considering the wall effect of the additives. The carbon-fiber chips are effective for improving the heat transfer rate in PCMs. However, the thermal resistance near the heat transfer surface is higher than that for the carbon brushes. As a result, the overall heat transfer rate for the fiber chips is lower than that for the carbon brushes. Consequently, the carbon brushes are superior to the fiber chips for the thermal conductivity enhancement under the present experimental conditions. The carbon brushes are moreover applied to the packed beds of particles to overcome their low thermal conductivity in chemical heat pump/storage. The carbon brushes essentially improve the heat transfer characteristics in the packed beds, though the thermal resistance is observed because the particles obstruct contact between the fibers and the heat transfer surfaces. 2003 Elsevier Ltd. All rights reserved.
1. Introduction Thermal energy storage is one of the key technologies for the efficient use of thermal energy. That is, the seasonally as well as diurnally fluctuating thermal energy such as the solar energy and the wasted heat in industry is temporarily stored, and then rapidly is supplied when it is demanded. Especially, latent heat thermal energy storage using phase change materials (PCMs) has attracted a great deal of attention due to their high energy density. However, the heat transfer rate between the PCMs and heat transfer fluid is generally unacceptably low due to the low thermal conductivities of the PCMs (Hasnain, 1998). To overcome the low thermal conductivities of the PCMs, thin aluminum strips (Hoogendoorn and Bart,
*
Corresponding author. Tel.: +81-92-642-3515; fax: +81-92642-3519. E-mail address: [email protected] (J. Fukai).
1992), thin wall rings made of steel (Velraj et al., 1999), porous aluminum (Weaver and Viskanta, 1986), porous graphite matrices (Py et al., 2001), copper chips (Tayeb, 1996) and carbon fibers (Fukai et al., 1997, 1999) have been proposed as additives to enhance the heat transfer rate in PCMs. For instance, Hoogendoorn and Bart (1992) reported that the effective thermal conductivity of a paraffin/1 vol.% thin-aluminum-strip composite is seven times as high as the thermal conductivity of the paraffin. However, the effective thermal conductivities of the composites near the heat transfer surface are definitively lower than those of the bulk due to the wall effect or contact thermal resistance, thus reducing the overall heat transfer rate. This fact was demonstrated by Fukai et al. (2002), who quantitatively discussed the effective thermal conductivity of paraffin/carbon-brush composites by considering the wall effect. However, there are few studies that discuss the heat transfer enhancement of additives by considering the wall effect or contact thermal resistance. Chemical heat pump/storage systems are also notable thermal energy storage techniques because reaction heat
0038-092X/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.solener.2003.07.028
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Y. Hamada et al. / Solar Energy 75 (2003) 317–328
Nomenclature Bi cp cp d d f h k k k k ltp Nu Pr r Re t T u x, y X, Y Xf Xfa Xtot
Biot number, hltp =km specific heat (J/kg K) dimensionless specific heat, cp =cp;m diameter (m) dimensionless diameter, d=ltp friction factor heat transfer coefficient (W/m2 K) thermal conductivity (W/m K) thermal conductivity tensor (W/m K) dimensionless thermal conductivity, k=km dimensionless thermal conductivity tensor, k=km tube pitch (m) Nusselt number, hh di =kh Prandtl number, ðcp lÞh =kh radial distance (m) Reynolds number, qh di uh =lh time (s) temperature (K) mean fluid velocity (m/s) x- and y-coordinates (m) dimensionless X - and Y -coordinates, x=ltp , y=ltp local volume fraction of fibers volume ratio of fibers to the composite volume ratio of fibers to the shell side of heat exchanger
Greek symbols d thickness of a control volume (m) D dimensionless thickness of a control volume, d=ltp e void fraction / parameter representing contact heat transfer rate between particles l viscosity (Pa s) h angle (rad)
is much higher than latent heat. However, the low effective thermal conductivities of the packed beds are an impediment to practical use (Wongsuwan et al., 2001). Groll (1993) reviewed the effective thermal conductivities of several additives/metals and the additives/inorganic-materials composites as well as the heat transfer coefficients between the heat transfer surface and the composites. According to his report, the effective thermal conductivities of the composites are 0.3–0.7 W/m K (25 wt.%) for graphite binder and 2–9 W/m K for consolidated activated carbon and zeolite beds including highly porous metallic foams (Ni, Cu), while the heat transfer coefficients for these additives are 35–180 W/ m2 K. The effective thermal conductivities are 5–30 W/
H q q s n n f f
dimensionless temperature, ðT T0 Þ= ðTh;in T0 Þ density (kg/m3 ) dimensionless density, q=qm dimensionless time, km t=fðcp qÞm l2tp g coordinate normal to the boundary surface (m) dimensionless coordinate normal to the boundary surface, n=ltp coordinate tangential to the boundary surface (m) dimensionless coordinate tangential to the boundary surface, f=ltp
Subscripts 0 initial c composite ct composite-tube wall interface cal calculation eff effective exp experiment f fiber g gas h heat transfer fluid i inner in inlet m phase change material or packed bed o outer p parallel model par particle s series model t tube v container w wall Superscript 0 including wall effect
m K (>10 wt.%) for porous graphite matrices, while the heat transfer coefficients are 400–1000 W/m2 K. In this case, however, the porous graphite matrices are compressed in the reactor container, which results in obstructing the uniform gas flows in the packed beds due to low gas permeability (Han et al., 1998). Dellero et al. (1999a,b) enhanced the effective thermal conductivity using 30-mm long carbon fibers. In this case, the heat transfer coefficient between the carbon fibers and heat transfer surface is 137 W/m2 K. The authors have proposed the brushes made of carbon fibers as additives (Fukai et al., 2000) and investigated their effect on the heat transfer rate of the brush/PCM composite in shell-and-tube heat exchangers
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
(Fukai et al., 2003). Similar to other additives, the wall effect was realized in this system. However, it has not been concluded if the brush as additives is inferior to other carbon fiber configurations. The authors focus on the heat transfer in the shelland-tube type, while most of the previous studies used cylindrical capsule/reactor, because the apparent energy density stored in the shell-and-tube type is higher. The present paper is composed of two parts. The purpose of the first part is to compare the effect of carbon-fiber chips and the carbon brushes, as additives, on the heat transfer rate in the PCM. That of the second part is to examine the thermal effect of the carbon brushes on the packed bed of the glass particles as alternative reactants in the shell side. In both parts, the wall effect between the composite and the heat transfer surface is investigated.
2. Thermal conductivity enhancement in PCMs 2.1. Experimental procedure Fig. 1 shows the experimental apparatus. Four steel tubes are vertically arranged in a cylindrical container
319
made of acrylic resin. Carbon-fiber chips (fiber diameter ¼ 10 lm, fiber length ¼ 5 mm) are packed in the container. The container is filled with n-octadecane (km ¼ 0:34 W/m K) as a PCM up to a height of about 50 mm. The container is insulated using glass wool. Two methods are used to place the fiber chips/PCM composite in the container. In method I, the container is filled with the melted PCM after the fiber chips are packed into the container. In method II, the composites are similarly made using a cylindrical vessel whose inner diameter is less than that of the container. The composite is placed in the center of the container in Fig. 1, and then the melted PCM fills the space around the composite. When the diameter of the composite is larger than that of the tube pitch (ltp ¼ 108 mm), four holes are provided in the composite to adjust them in the tubes. Three kinds of fiber chips are used. They are referred to as CFL (carbon fiber with low thermal conductivity, kf ¼ 5 W/m K), CFM (carbon fiber with middle thermal conductivity, 190 W/m K) and CFH (carbon fiber with high thermal conductivity, 500 W/m K). The fiber volume fractions to the composite Xfa in the experiments are shown in Table 1. M0:8 and M1:2 are provided to investigate the effect of CFM on the thermal
164 mm
80.8 mm
Tube (Steel)
Container
23.6 mm
Thermocouple 1.8 mm
24
60 mm
E D
A
m
B
m
(a)
C
.6 24
Water
m
Water
m
35 mm
Insulator
.6
Chip/n-Octadecane Composite
l1 l1 l1 l1
108 mm
108 mm
l2
l2
(l1 = 10.1 mm, l2 = 19.1 mm) (b)
Fig. 1. Experimental apparatus.
320
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
Table 1 Experimental conditions Volume fraction Xfa N M0:8 M1:2 L1:4 L1:4 H0:3 L1:4 H0:5
CFL
CFM
CFH
kc;eff
0 0 0 0.014 0.014 0.014
0 0.008 0.012 0 0 0
0 0 0 0 0.003 0.005
0.34 0.80 1.03 0.36 0.82 1.12
conductivity enhancement. L1:4 H0:3 and L1:4 H0:5 are provided to examine whether the same effective thermal conductivities as M0:8 and M1:2 are obtained with a smaller volume of higher conductive fibers. In this case, CFH is mixed with CFL because the volume fraction of CFH is too low to be solely packed into the container up to the height of n-octadecane. L1:4 is the base case to investigate the effect of CFH on the thermal conductivity enhancement. Three experimental apparatuses are provided for each composite to confirm the experimental reproducibility. Fig. 2 shows the photograph of the fiber chips packed into the container before the PCM is poured into the container. It also shows the carbon brush settled in the container, which is compared to the fiber chips in Section 2.3. The effective thermal conductivities of bulk in the fiber chips/PCM composite in Table 1 are estimated using the following semi-empirical equation (Fukai et al., 1999, 2000). kc;eff =km ¼ 0:456f1 ð1 Xf Þ2=3 gðkf =km Þ þ ð1 Xf Þ1=3 ð1Þ For L1:4 H0:3 and L1:4 H0:5 , kc;eff of the CFL/PCM composite is estimated, and then kc;eff of the CHL/CFL/PCM composite is again estimated using Eq. (1) based on the
assumption that the CFL/PCM composite is thermally homogeneous. The estimated effective thermal conductivities of L1:4 H0:3 and L1:4 H0:5 are almost the same as those of M0:8 and M1:2 , respectively. In the experiment, the apparatus was placed in a room at 18 C overnight to maintain the container at a uniform temperature. Next, water at 23 C was provided from a thermostatic bath to the tubes. The mean flow velocity of water in the tubes is uh ¼ 0:35 m/s. The transient thermal responses were measured at points A– E as shown in Fig. 1(b). A digital recorder recorded the temperatures at intervals of 30 s. It should be noted that the experimental temperature range was below the melting point of n-octadecane (28 C) in order to investigate the conductive heat transfer excluding latent heat. The experiment was finished when the temperature fluctuations were within ±0.1 C in 300 s. 2.2. Numerical procedure Fig. 3 shows the computational domain, which consists of the fiber-chips/PCM composite and the tube wall. At t < 0, the composite, including the PCM at the periphery, and the tube wall are initially kept at temperature T0 At t ¼ 0, the heat transfer fluid at temperature Th;in flows in the tubes. Choosing T0 and Th;in as the characteristic temperatures and the tube pitch ltp as the characteristic length leads to the dimensionless expressions for the two-dimensional heat conduction equations in the composite (c) and the tube wall (t), boundary conditions and initial conditions as shown in Table 2(a). The definitions of the dimensionless are shown in nomenclature. kc in the heat conduction equation is equivalent to kc;eff =km in Eq. (1). The heat capacity is given by ðcp qÞc ¼ Xf ðcp qÞf þ ð1 Xf Þðcp qÞm
ð2Þ
At the interfaces between the tube wall and the composite, the wall effects are considered using the heat
Fig. 2. The photographs of carbon-fiber chips and carbon brushes in the containers.
Y. Hamada et al. / Solar Energy 75 (2003) 317–328 y
ζ
ζ
d
ltp/2
ξ
dv/2
ξ
ltp/2
dv/2
y
321
y'
x'
θ
2 c/
x
x
ltp/2
ltp/2
dv/2
dv/2
(b) Carbon-fiber brushes
(a) Carbon-fiber chips
Fig. 3. Computational domains corresponding to the experimental apparatuses.
Table 2 Mathematical description of the composite (a) Carbon-fiber chips
(b) Carbon-fiber brushes
c ¼ r ðkc r Hc Þ ðcp qÞc oH os oHt ðcp qÞt os ¼ r ðkt r Ht Þ
c ðcp qÞc oH ¼ r ðkc r Hc Þ os
t kt oH on
Hc ¼ Ht ¼ 0 t k oH ¼ Bih ð1 Ht Þ on c t ¼ Bict ðHt Hc Þ ¼ kc oH kt oH ¼ Bict ðHt Hc Þ on on oHc oHc ¼ knn;c on þ knf;c oH of ¼ 0 on
transfer coefficient hct . hct ¼ 1 (Bict ¼ 1) means no wall effect. The mathematical model is numerically solved using the control volume method (Patankar, 1980). To consider the boundary conditions at the composite-tube wall interface, the dimensionless thermal conductivity of the control volume by the interface is given by 1 1 1 1 k ¼ þ þ ð3Þ 2kt BiD 2kc where D is the dimensionless thickness of the control volume. The curvature surfaces are approximated using square grids. The 50 · 50 uniform meshes and Ds ¼ 0:0002 are needed to have numerical results independent of the mesh size and the time step. The mesh size and the time step independences of the numerical results are discussed in Appendix A. The effective thermal conductivity of the composite 0 keff including the wall effect is estimated when the composite is packed into a shell-and-tube heat exchanger as shown in Fig. 4. Carbon brushes are also drawn in the
ltp
I.C. B.C.
In composite In tube wall In the whole region At inner surface of tube At tube wall––composite interface At exposed surface and iteration boundary
y
o
x
ltp
Fig. 4. Computational domain corresponding to a heat exchanger.
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
figure. The shadowed area in the figure is chosen as the computational domain considering the periodicity of the tube arrangement. The mathematical model in Table 0 2(a) is also applied to this system. keff is estimated using a previous reported technique (Fukai et al., 2002). That is, the time variation in the heat transfer rate on the inner surface of the tubes is evaluated from the calculated temperature field. Separately, the equation where 0 kc is substituted by keff is numerically solved assuming Bict ¼ 1 to evaluate the time variation in the heat 0 transfer rate on the inner surface. The value of keff is determined by fitting the latter heat transfer rate with the former.
Point B
1
Θ
322
0.5
N M0.8 M1.2 L1.4 L1.4H0.3 L1.4H0.5
0 Point D
1
The properties of the materials used in the experiments are km ¼ 0:34 W/m K, kt ¼ 80 W/m K, cp;m qm ¼ 1:53 106 J/m3 K, cp;f qf ¼ 2:12 106 J/m3 K and cp;t qt ¼ 3:48 106 J/m3 K. The heat transfer coefficient hh in the tubes is calculated with the Petukhov– Gnielinski equation (Gnielinski, 1976; Petukhov, 1970): Nu ¼
Θ
2.3. Results
0.5
ðf =2ÞðRe 1000ÞPr pffiffiffiffiffiffiffiffi 1 þ 12:7 f =2ðPr2=3 1Þ 6
ð3000 < Re < 10 ; 0:5 < Pr < 2000Þ
0
0
0.05
where f is the friction factor, Re the Reynolds number and Pr the Prandtl number. According to Eq. (4), hh is 1.7 · 103 W/m2 K. It is numerically confirmed that this value is sufficiently high for the estimation error of hh not to affect the calculated temperature distribution in the composite. Fig. 5 shows the experimental results for the nondimensional time variations in the non-dimensional temperatures. The experimental data are plotted every fifteen recorded data. The reproducibility of the experimental results is shown in Appendix B. The time required for H to reach a value is calculated from the experiments. The time factor is defined as the ratio of the times with no fibers to with the fibers, and shown in Table 3. A comparison among the results for N, M0:8 and M1:2 shows that the CFM chips essentially improve the thermal responses as the volume fraction of the fibers increases. It should be noted that H for M1:2 reaches 0.5 in nearly half the time of N. From the results for N and L1:4 , CFL is not effective for heat transfer enhancement, as expected from the effective thermal conductivities in Table 1. Moreover, the response for L1:4 H0:3 is not as sensitive as that for M0:8 though the estimated effective thermal conductivity of L1:4 H0:3 is the same as that of M0:8 as shown in Table 1. The CFH chips have to be placed in contact with each other to improve the effective thermal conductivity. In this case, the CFH chips may have few chances to meet other CFH chips because of a low density of CFH, though
0.1
τ
ð4Þ
Fig. 5. The non-dimensional time variations in non-dimensional temperatures in the fiber chips/PCM composite using method I (experimental results).
Table 3 The average time factor for three runs H ¼ 0:5 M0:8 M1:2 L1:4 L1:4 H0:3 L1:4 H0:5
H ¼ 0:7
Point B
Point D
Point B
Point D
1.63 1.86 1.08 1.44 1.77
1.65 1.95 1.08 1.34 1.95
1.56 1.81 1.09 1.34 1.63
1.60 1.80 1.07 1.31 1.80
they have many chances to meet the CFL chips. This is a reason for the low effective thermal conductivity of L1:4 H0:3 . However, the thermal response for L1:4 H0:5 is close to that for M1:2 . Accordingly, there is a critical value above which the CFH is effective for heat transfer enhancement. Fig. 6 shows a comparison between the experimental and calculated transient thermal responses using the composite dc ¼ 0:65. The calculated temperatures agree well with the experimental. This fact shows the accuracy of Eq. (1). Fig. 7 shows a comparison between the calculated transient thermal responses for hct ¼ 1 (Bict ¼ 1) and
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
1
1
dc* = 1.52
0.5
Θ
Θ
dc* = 0.65
Point Exp. A B C D E
0
0
0.02
τ
0.04
1
Θ
dc* = 1.52
0.5 Point Exp. A B C D E
0
0.02
τ
0.5 Point Exp. A B C D E
Cal.
Fig. 6. Comparison between the experimental and calculated transient thermal responses (M0:8 ). The experimental apparatus is constructed using method II.
0
323
Cal.
0.04
Fig. 7. Comparison between the experimental and calculated transient thermal responses (M0:8 ). The experimental apparatus is constructed using method I.
the experimental thermal responses. The composite was made using method I as mentioned above. The experimental temperatures are obviously lower than the calculated ones. In this case, the fiber chips near the tubes were visually parallel to the tubes though those of the bulk composite were random. This wall effect results in a lower overall heat transfer rate in the experiment than in the calculation. To confirm this issue, the experimental results from the apparatus made using method II are compared with the calculated results in Fig. 8. In this case, the directions of the carbon fibers must be random
0
0
0.02
τ
Cal.
0.04
Fig. 8. Comparison between the experimental and calculated transient thermal responses (M0:8 ). The experimental apparatus is constructed using method II.
near the tube wall as well as in the bulk. The experiments are certainly closer to the calculated results than those in Fig. 7. The value of hct is evaluated by the non-linear leastsquare method. As a result, hct ¼ 150 W/m2 K (Bict ¼ 47:6) is determined. In Fig. 9, the calculated results for Bict ¼ 47:6 are compared with the experimental results. The calculated results agree with the experimental results irrespective of the volume fraction of the fibers. The prediction error is within ±10% in the whole region of H. Considering the facts that the effective thermal conductivity and the transient thermal responses for L1:4 H0:5 are identical to those for M1:2 (Table 1 and Fig. 5), the value of hct for CFH is expected to be nearly 150 W/m2 K. Fig. 10 shows the estimated effective thermal con0 ductivity keff in the fiber chips/PCM composite using method I. Xtot is defined as the volume fraction of the fibers added to the heat exchangers. The effective thermal conductivity of the bulk in the composite keff is also plotted. The results for the carbon brushes (brush diameter ¼ 110 mm, axial length ¼ 60 mm) are referred to from the previous paper (Fukai et al., 2002). hct for the carbon brush/PCM composite is 340 W/m2 K (Bict ¼ 108). The reason for the higher value of hct in the carbon brush system is that the fibers are pressed on the tube wall due to their elastic forces. As a re0 sult, keff is higher for the carbon brush than for the fiber chips though keff is lower for the former than for the latter. The heat transfer rate in the brush/PCM composite is heterogeneous, while that for the fiber-chip/PCM is macroscopically homogeneous. Although the former is inferior to the latter in terms of the heat transfer rate
324
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
1 Chip Brush
M0.8
5
keff / km
0% +1 %
0.5
10
Θ exp
keff
Point A B C D E
3
hct = 150 W/m2- K
0 1
1
0% +1 %
0.5
0
0.005
0.01 Xtot
0.015
0.02
Fig. 10. The variation in the predicted effective thermal conductivities with the volume fraction of carbon fibers (kf =km ¼ 560). A comparison between fiber chips and carbon brushes.
10
Θ exp
M1.2
0
hct = 340 W/m2- K
k' eff
Point A B C D E
particles (dpar ¼ 0:1 mm, kpar ¼ 0:8 W/m K, qpar ¼ 2410 kg/m3 ) are packed in the container. The experimental procedure is the same as described above. 3.2. Mathematical model
0
0.5
1
Θ cal Fig. 9. Comparison between the calculated results for Bict ¼ 47:6 and experimental transient thermal responses.
of bulk, this figure shows that the former is superior in terms of the overall heat transfer rate.
3. Thermal conductivity enhancement in packed beds 3.1. Experimental setup As described above, carbon brushes are superior to carbon-fiber chips as additives. Moreover, it is difficult to uniformly pack the fiber chips in a packed bed due to the difference between the densities of carbon fibers and particles. Accordingly, the carbon brushes are used for thermal conductivity enhancement in packed beds of particles. A carbon brush made of CFM is inserted in the center of the container as shown in Fig. 1. The diameter of the brush and the volume fraction of the fibers are 110 mm and 0.008, respectively. Instead of the PCM, glass
The mathematical model is briefly described because the model for the brush/PCM composite has been reported (Fukai et al., 2002). The mathematical model shown in Table 2(b) is used into the computational domain in Fig. 3(b). kc in the heat conduction equation is the dimensionless thermal conductivity tensor. The local x0 –y 0 -coordinate system, where the x0 -axis is along the fiber orientation, is introduced (Fig. 3(b)). The thermal conductivity tensor kc is given by 1 cos h sin h kp 0 cos h sin h kc ¼ ð5Þ sin h cos h sin h cos h 0 ks where kp and ks are the thermal conductivity in the x0 and y 0 -directions, respectively, and h the angle between the x0 -axis and y 0 -axis. If the matrix is a continuous body such as paraffin, kp and ks are given by the well-know parallel and series models. However, the parallel model is not applied to the fibers/solid particles composites because the fibers theoretically undergo point contact with the particles. Therefore, the effective thermal conductivity of such composites in the fiber orientation is lower than that calculated using the parallel model (Fukai et al., 2000). In the present study, a semiempirical regression is derived based on the experimental results (Fukai et al., 2000):
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
kp ¼ 0:66kf Xf þ ð1 0:66Xf Þkm
ð6Þ
325
1
ks is given by the series model: ks ¼ fXf =kf þ ð1 Xf Þ=km g1
ð7Þ
km ¼
Θ
The thermal conductivity model for the packed bed km is given by the model of Kunii and Smith (1960): e 1 e=ð1 Xf Þ kg kg þ 1 Xf / þ ð2kg Þ=ð3kpar Þ
0.5
ð8Þ Point Exp. A B C D E
where e is the void fraction of particles, the parameter representing the heat transfer rate between particles, and kg the thermal conductivity of gas, or air in the present study. The heat capacity is given by e ðcp qÞc ¼ Xf ðcp qÞf þ ð1 Xf Þðcp qÞpar 1 ð9Þ 1 Xf The heat capacity of the glass particles ðcp qÞpar is 1.81 · 106 J/m3 K. 3.3. Results First of all, the carbon brushes essentially improve the experimental transient thermal responses as shown in Fig. 11. For the carbon brush/solid particle composite, the wall effect representing by hct in the present model is contributed by that due to a low density of the particles near the tube wall and that due to the contact conditions between the fibers and the tube wall. The heat transfer coefficient for the former is modeled by Yagi and Kunii (1961): hw dpar 1 ¼ 0:5f1=ðkw =kg Þ 1=ðkm =kg Þg kg
ð10Þ
0
0
Cal.
0.05
0.1
τ
Fig. 12. Comparison between the calculated results for Bict ¼ 443 and the experimental transient thermal responses with no carbon brushes (e ¼ 0:36).
the wall. As a result, the value of hw ¼ 780 W/m2 K is obtained. Fig. 12 compares the calculated thermal responses to the experimental ones in the composite with no carbon brushes. The calculated results for hct ð¼ hw Þ ¼ 780 W/ m2 K (Bict ¼ 443) agree well with the experimental results. In addition, the calculated results for hct ¼ 1 (Bict ¼ 1) are identical to those for 780 W/m2 K (Bict ¼ 443), though they are not plotted in the figure because they are apparently overlapped with those for hct ¼ 780 W/m2 K. This fact shows that the wall effect does not dominate the heat transfer rate in the packed beds. Fig. 13 shows the experimental thermal responses in the composite with carbon brushes. The value of hct was
kw is estimated by substituting Xf ¼ 0 and e ¼ 0:7 into Eq. (10). e ¼ 0:7 is a value using the sphere particles near
1
1 Point B
Θ
Θ
dc* = 1.0
0.5
0.5
Point D
0
0
0.05
τ
Point Exp. A B C D E
Xfa 0 0.008
0.1
Fig. 11. The effect of the carbon brush on the experimental transient thermal responses in the packed bed of the glass particles.
0
0
0.05
τ
Cal.
0.1
Fig. 13. Comparison between the calculated results for Bict ¼ 74:3 and the experimental transient thermal responses (CFM, Xfa ¼ 0:008, e ¼ 0:39).
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Y. Hamada et al. / Solar Energy 75 (2003) 317–328
3
k'eff / km
Brush/PCM
2
Brush/glass particle
1
0
0.005
0.01
0.015
0.02
The carbon-fiber chips are effective for improving the heat transfer rate in PCMs. However, the thermal resistance near the heat transfer surface is higher than that for the carbon brushes. As a result, the overall heat transfer rate for the fiber chips is lower than that for the carbon brushes even though the effective thermal conductivity of the bulk of the former is higher. Consequently, the carbon brushes are superior to the carbon-fiber chips under the present experimental conditions. The effect of the carbon brushes is discussed when they are applied to the packed beds of particles in the shell side of heat exchangers. A higher thermal resistance is observed probably because the particles obstruct contact between the fibers and the heat transfer surfaces. However, the estimated heat transfer coefficients are the same as those for the other additives.
Xtot
4. Conclusions This study experimentally and numerically compares the effects of carbon-fiber chips and carbon brushes as additives on the overall heat transfer rates in the composites.
The first important issue on the discretization of the computation domain, shown in Fig. 3, is that the area of each region must be same as that of exact one. At this point, the differences between them are within less than 1%. The second important issue is that the length of the numerical boundary surface is longer than the exact one because the curvature surface is approximated using zigzag lines as shown in Fig. 15. This fact results in the overestimate of the heat flow rate between the heat transfer fluid and the inner tube wall. To solve this problem, the heat transfer coefficient hh used in the calculation is corrected by multiplying the heat transfer
y
r
ltp
identified using the non-linear least-square method (Fukai et al., 1985). As a result, the value of hct ¼ 110 W/m2 K (Bict ¼ 74:3) is identified. Considering that hw is high enough not to affect the thermal responses, the low value of hct is due to the wall effect of the fibers. The hct value is much lower than the carbon-brush/PCM composite shown in Fig. 10. This is probably because the particles obstruct contact between the fibers and the tube walls. The estimated hct in the present system is on the same order as that for graphite binder and porous metallic foams (Groll, 1993). The effective thermal conductivity of the carbon brush/glass particles composite is estimated using the numerical technique described above. The estimated effective thermal conductivity including the wall effect of fibers is compared with that of the brush/PCM composite in Fig. 14. The effective thermal conductivity in the brush/glass particles composite is lower than that in the brush/PCM composite because of the lower hct value. However, the effective thermal conductivity at Xtot ¼ 0:015 is twice as high as that with no additives. This fact shows that the reaction time for the charge and discharge process with the brush can be half that with no additives even if the particles are reactive.
Appendix A
do/2
Fig. 14. The comparison between the predicted effective thermal conductivities of the brush/glass-particle composite and that of the brush/PCM (kf ¼ 190 W/m K).
x
do/2 ltp Fig. 15. The discretization of a computational domain.
Y. Hamada et al. / Solar Energy 75 (2003) 317–328
Θ
1
For the computation domain in Fig. 3(a), the effects of the mesh size and time step on the calculated result are shown in Fig. 17. Obviously, 50 · 50 mesh and Ds ¼ 0:0002 lead to the results being independent of them.
r/ltp = 0.334
0.630
0.5
Appendix B
0.926
Fig. 18 compares the results measured using three different apparatuses for typical composites. This figure shows good reproducibility in the present experiments.
Numerical Analytical 0
327
0
0.5
1
τ
1
Fig. 16. The comparison between the numerical solution and the analytical one (Bih ¼ 50).
Θ
coefficient using Eq. (3) by the ratio of the exact surface area to the numerical one. The numerical and analytical solutions are compared to examine the accuracy of the present numerical technique. The problem considered is a homogeneous hollow cylinder whose inner surface is uniformly heated by fluid and outer surface is adiabatic (Fig. 15). The analytical solution in the radial direction is shown by Carslaw and Jaeger (1986). 50 · 50 mesh is used in the calculation. Fig. 16 demonstrates that the numerical results agree well with the analytical ones. The numerical error for Bih ¼ 1000 is less than that for Bih ¼ 50 though the result is not shown here.
M0.8
0.5
Point C D 0
0
Exp.
0.05
0.1
τ
(a) Carbon fiber chips/PCM composite 1
1
Xfa = 0.008
Θ
Θ
Point C
0.5
Point A
Point C D
mesh ∆τ 3 30 × 30 2×10 4 2×10 50 × 50 5 100 × 100 2×10
0
0.5
0
0.02
τ
0 0.04
Fig. 17. The effect of the mesh size and time step on the calculated result. (M0:8 , dc ¼ 1:52).
0
Exp.
0.05
0.1
τ
(b) Carbon brush/glass particles composite Fig. 18. Reproducibility of the experimental results.
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Y. Hamada et al. / Solar Energy 75 (2003) 317–328
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