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Experimental and numerical investigation of forced convection heat transfer of air in non-sintered porous media Pei-Xue Jiang *, Guang-Shu Si, Meng Li, Ze-Pei Ren Department of Thermal Engineering, Tsinghua University, Beijing 100084, China Received 1 February 2003; accepted 20 July 2003

Abstract Forced convection heat transfer of air in plate channels ﬁlled with glass or non-sintered steel spherical particles was investigated experimentally and numerically. The eﬀects of thermal dispersion, variable properties caused by the pressure variation, particle diameter, particle thermal conductivity and ﬂuid velocity were studied. The experimental results and numerically calculated values for the friction factor in porous media agree well with established formula. The porous media signiﬁcantly increased the pressure drop in the plate channel compared to an empty channel. The non-sintered porous media enhanced the heat transfer by 4–8 times for the conditions studied, which was much less than the enhancement due to the sintered porous media due to the contact thermal resistance in the non-sintered porous media. The heat transfer coeﬃcient decreased with smaller glass particle diameters and increased with the particle thermal conductivity. The inﬂuence of the solid particle thermal conductivity on the convection heat transfer of air in porous media decreased as the solid particle thermal conductivity decreased. Numerical simulation results with the local thermal non-equilibrium model with consideration of thermal dispersion corresponded well to the experimental data for both glass and metallic porous structures. The eﬀects of pressure variation on the convection heat transfer was less important. 2003 Elsevier Inc. All rights reserved. Keywords: Convection heat transfer; Porous media; Thermal non-equilibrium

1. Introduction Forced convection heat transfer in porous media has many important applications in packed bed reactors, catalytic and chemical particle beds, transpiration cooling, solid matrix heat exchangers, packed-bed regenerators, heat transfer enhancement devices, microthrusters, industrial furnaces, combustors, ﬁxed-bed nuclear propulsion systems, and many others. Therefore, ﬂuid ﬂow and heat transfer in porous media has received much attention during the past ﬁve decades. Many researches have shown that porous media are an eﬀective heat transfer augmentation technique because the porous structures intensify mixing of the ﬂuid ﬂow and increase the contact surface area [1–10]. Cooling systems using porous structures have been applied to cool microelectronic chips [1,2] and mirrors in *

Corresponding author. Tel.: +86-10-62772661; fax: +86-1062770209. E-mail address: [email protected] (P.-X. Jiang). 0894-1777/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermﬂusci.2003.07.006

powerful lasers [3,4]. Zeigarnick et al. [3] and Haritonov et al. [4] studied convection heat transfer of water in plate channels ﬁlled with metallic porous materials. Their results showed that the porous media increased the heat transfer coeﬃcient 5–10 times relative to an empty channel. Koichi and Takeshi [5] investigated the eﬀect of porous media on the local heat transfer and ﬂuid ﬂow in forced convection heat transfer. They concluded that an optimum Darcy number exist for a porous media with a high ratio of the thermal conductivity of the solid particle to that of the ﬂuid which can enhance the heat transfer with a smaller increase of the ﬂow resistance. Hwang and Chao [6,7] studied convection heat transfer of air in sintered porous channels experimentally and numerically and showed that the heat transfer increased with the solid particle thermal conductivity. Jiang et al. [8–10] studied convection heat transfer of water in non-sintered porous plate channels experimentally and numerically and showed that the packed beds greatly intensiﬁed the convection heat transfer (up to 10 times). A modiﬁed criterion was

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Nomenclature cp dp De fe G h hx hsf hv L M q Re ReD

ﬂuid speciﬁc heat [J/(kg K)] particle diameter [m] equivalent hydraulic diameter of the porous plate channel [m] friction factor mass ﬂow rate [kg/s] channel height [m] local heat transfer coeﬃcient ð¼ qw =ðTwðxÞ Tf ðxÞÞÞ [W/(m2 K)] heat transfer coeﬃcient between solid particles and the ﬂuid [W/(m2 K)] volumetric heat transfer coeﬃcient between solid particles and the ﬂuid [W/(m3 K)] channel length [m] mass ﬂux (¼ qu) [kg/(m2 s)] heat ﬂux [W/m2 ] Reynolds number (¼ equp dp =l) Reynolds number based on the plate channel hydraulic diameter (qf uf De =lf )

developed to judge the eﬀect of particle diameter on the heat transfer coeﬃcient in [10]. Jiang et al. [11] investigated convection heat transfer of water and air in sintered porous media experimentally and numerically. Their results showed that the heat transfer in sintered porous media was much more intensive than in nonsintered porous media. More and more studies have shown that the local thermal equilibrium assumption between the ﬂuid and the particles is inadequate for a number of problems such as ﬁxed bed nuclear propulsion systems, catalytic reactors, storage of heat energy, convection heat transfer in metallic porous structures and nuclear reactor modeling. In recent years, the local thermal non-equilibrium model has been used more frequently in the energy equation in theoretical and numerical research on convection heat transfer in porous media [9,12–20]. With the thermal non-equilibrium model, the treatment of the boundary conditions for the energy equation signiﬁcantly aﬀects the numerical simulation results. A number of papers have analyzed this problem, e.g. Amiri et al. [15], Jiang et al. [16,19], Quintard [17], Lee and Vafai [18], and Alazmi and Vafai [20]. For the constant heat ﬂux boundary condition there are several diﬀerent methods to treat the boundary conditions of the energy equations. The main diﬀerences among them are whether the ﬂuid temperature is equal to the solid phase temperature on the wall surface and how to calculate the heat ﬂux at the wall. Jiang et al. [19] investigated the various treatments of the constant heat ﬂux boundary conditions by comparing numerical results with experimental data for water in porous media. They

Ree T up vp x; y

equivalent Reynolds number (2Mdp =ð3lf ð1 eÞÞ) temperature [K] pore velocity in the x-direction [m/s] pore velocity in the y-direction [m/s] coordinates in the ﬂow region [m]

Greek symbols k thermal conductivity [W/(m K)] km stagnant eﬀective thermal conductivity of the ﬂuid and porous media [W/(m K)] l ﬂuid absolute viscosity [N s/m2 ] q ﬂuid density [kg/m3 ] e1 porosity [dimensionless] em mean porosity [dimensionless] Subscripts d dispersion f ﬂuid w wall

concluded that the convection heat transfer in non-sintered porous media can be best predicted numerically using the thermal non-equilibrium model with the ideal constant wall heat ﬂux boundary condition that assumes the heat ﬂuxes into the liquid and into the solid phase are the same. Alazmi and Vafai [20] analyzed eight different forms of the constant wall heat ﬂux boundary conditions in the absence of local thermal equilibrium conditions in porous media. They found that the different boundary condition treatment methods may lead to substantially diﬀerent results. At the same time, the authors pointed out that selecting one model over the others is not an easy issue since previous studies validated each of the two primary models. In addition to that, the mechanics of splitting the heat ﬂux between the two phases is not yet resolved. In the present work, forced convection heat transfer of air is studied experimentally and numerically in plate channels ﬁlled with glass and metallic particles having diﬀerent particle diameters (1.43–1.60, 1.60–2.00, 2.00– 2.50 mm). The numerical simulation results of the local thermal equilibrium model and the local thermal nonequilibrium model are compared with the experimental data. The eﬀects of thermal dispersion, variable properties caused by pressure variation, particle diameter, particle thermal conductivity and ﬂuid velocity on convection heat transfer and heat transfer enhancement in porous structures are examined experimentally and numerically. The experimental data for convection heat transfer of air in non-sintered porous media in the present paper will enrich the database of convection heat transfer in porous media.

P.-X. Jiang et al. / Experimental Thermal and Fluid Science 28 (2004) 545–555

547

Fig. 1. Schematic of the physical system.

ﬁne mesh stainless steel screens at the inlet and outlet of the test section. Air was chosen as the working ﬂuid. The upper plate was heated by a 0.2 mm thick plate heater using low-voltage alternating current to simulate a heat sink with constant heat ﬂux. A mica sheet was placed between the heater and the plate channel surface. The small air gap that might exist between the mica sheet and the plate channel surface was ﬁlled with a high thermal conductivity paste to minimize the contact resistance. The heater voltage and current were measured by digital multimeters. The electric power input to the heater was calculated from the measured current and voltage readings. The local temperature of the plate channel was measured with 20 copper–constantan thermocouples. Fig. 3 shows the locations of the 15 thermocouples inserted into the upper plate of the test section (0.8 mm deep) along the centerline and of the ﬁve thermocouples inserted into the upper plate of the test section (0.8 mm deep) along a line 2.5 cm away from the centerline to monitor the temperature variations across the test section. The inlet air temperature was measured by two thermocouples located at the inlet, approximately 6 cm upstream from the heated section. Three thermocouples were located at the plate channel outlet, approximately 6 cm downstream from the heated section to measure the bulk exit temperature. Prior to installation, the thermocouples were calibrated using a constant-temperature oil bath. The overall accuracy was within ±0.2 C. The inlet and outlet pressures were measured using U-type manometers. The mass ﬂow rate was obtained from the volumetric ﬂow meters or by measuring the average velocity at the outlet using a hot-wire anemometer (TSI8385). For each test, the ﬂow rate, input power and inlet ﬂuid temperature were ﬁxed. The temperatures were

Fig. 2. Experimental system: (1) compressor; (2) water ﬁlter; (3) pressure regulator; (4) ﬁlter; (5) by-pass; (6,7) volumetric ﬂow meters; (8) test section; (9) mixer; (10) hot-wire; (11) multiplexer; (12) digital multimeter; (13) computer; (14) voltage stabilizer; (15) voltage regulator; (16) transformer; (17) current meter.

Fig. 3. Distribution of thermal couples on the test section.

2. Experimental apparatus and data reduction The test section geometry is depicted schematically in Fig. 1. The size of the heated test section was 100 mm · 100 mm · 10 mm. The upper plate of the channel received a constant heat ﬂux, qw , while the bottom and the side plates were adiabatic. The air entered the channel with an average velocity, u0 , and constant temperature, T0;f . Adiabatic sections (13 mm long) were placed before and after the heated section. The experimental system, shown schematically in Fig. 2, consisted of a compressor, a test section, two volumetric ﬂow meters, and instrumentation to measure temperatures, pressures, velocities and electrical power inputs. The test section was made from a stainless steel plate by wire machining. The upper wall of the test section was 2.0 mm thick. The plate channel was ﬁlled with packed steel or glass particles. The average bead diameters were 1.20, 1.515, 1.80 and 2.25 mm. The spheres were supported by two perforated plates and

qw x

O

u0

h

L y

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P.-X. Jiang et al. / Experimental Thermal and Fluid Science 28 (2004) 545–555

measured by the 25 thermocouples connected through a multiplexer to a digital multimeter (HP34401A) and a personal computer. The temperatures were monitored and recorded after steady-state conditions were reached. The ﬂow rate, outlet velocity, inlet and outlet ﬂuid bulk temperatures, and electric current and voltage across the heater were also recorded. The local bulk mean ﬂuid temperature at each measuring section was calculated from the inlet temperature, ﬂow rate and power input, or from the inlet and outlet temperatures using linear interpolation. The ﬂuid enthalpy increase was checked against the electric power input. The experimental uncertainty of the heat balance was ±5%. The data reduction method was similar to that used in previous experimental research on forced convection heat transfer of water in porous plate channels described by Jiang et al. [10]. Preliminary tests were performed for data calibration and error estimates. The maximum error was within ±0.2 C for the temperature measurement. The maximum error in the ﬂow rate was less than 1.5%. In the experiments the system required a long time to reach steady state (e.g. 45 min). The temperature and ﬂow rate variations were monitored continuously. The system was determined to be at steady state when the variations of the wall temperatures and the inlet and outlet temperatures were all within ±0.2 C for at least 15 min. The experimental uncertainty of the convection heat transfer coeﬃcients was mainly caused by the experimental errors in the heat balance, the contact of the plate heater to the test section plate surface, axial thermal conduction in the stainless steel plate test section, temperature measurement errors and the calculation of the heat transfer surface temperature. The maximum experimental uncertainty in the convection heat transfer coeﬃcient was estimated to be ±17.1% with a rootmean-square uncertainty of 10.5%. The experimental uncertainties in the pressure drop were estimated to be ±3.0%.

3. Mathematical formulation and numerical method The physical model and the coordinate system are shown in Fig. 1. The model assumes that the plate channel is ﬁlled with homogeneous and isotropic solid particles; the ﬂuid is single-phase and the ﬂow is twodimensional, steady, non-Darcian ﬂow. Because only high Peclet number ﬂow is treated, longitudinal conduction in the ﬂuid and the pressure variation along the y-direction are negligibly small. The assumption of local thermal equilibrium between the ﬂuid and solid phases at any location in the porous media is inadequate for a number of problems such as catalytic reactors, nuclear reactor modeling, ﬁxed bed nuclear propulsion systems, storage of heat energy, and convection heat transfer in metallic porous structures

where the diﬀerence between the thermal conductivities of the solid and ﬂuid phases is very large. In these cases the temperature diﬀerence between the coolant and the solid phase becomes crucial; therefore, the two-phase model with ﬂuid phase and solid phase energy equations (non-thermal equilibrium model) must be used [9]. The steady state, two-dimensional governing equations and boundary conditions for single-phase ﬂuid ﬂow in an isotropic, homogeneous porous medium based on the Brinkman–Darcy–Forchheimer model and the local thermal equilibrium model or local thermal non-equilibrium model with consideration of variable properties, variable porosity and thermal dispersion have been given by Jiang et al. [9,16]. Some important parameters used in the governing equations were obtained from Hsu and Cheng [21], Achenbach [22] and Dixon and Cresswell [23] as e ¼ e1 ð1 þ 1:7e6y=dp Þ e ¼ e1 ð1 þ 1:7e

6ðhyÞ=dp

ð0 6 y 6 h=2Þ; Þ

ðh=2 6 y 6 hÞ

ð1Þ

hv ¼ hsf 6ð1 eÞ=dp 4

1=4

Nu ¼ hsf dp =kf ¼ ½ð1:18Re0:58 Þ þ 10:23Re0:75 h ðAchenbach’s formula ½22 Þ

ð2Þ

1 dp dp 0:255 1=3 2=3 Pr Re ; ¼ þ ; Nusf ¼ hsf Nusf kf bks e b ¼ 10 ðDixon and Cresswell’s formula ½23 Þ

ð3Þ

Re ¼ equp dp =l;

Reh ¼ Re=ð1 eÞ

The mean porosity of the porous structures, em , was used as the porosity in the experimental measurements. The variable porosity model uses e1 , which can be derived according from the variable porosity model: e1 ¼

em 1 þ 0:567dp =hð1 e3h=dp Þ

ð4Þ

The eﬀect of thermal dispersion, resulting from the mixing of local ﬂuid streams and the temperature and velocity ﬂuctuation as the ﬂuid ﬂows through the solid particles, on forced convection heat transfer in packed beds plays an important role on forced convection heat transfer in porous media. Hsu and Cheng [21] presented a model for the additional thermal conductivity resulting from thermal dispersion as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ kd ¼ Cðqcp Þf dp u2p þ v2p ð1 eÞ ð5Þ The constant C was determined by comparing calculated results and experimental data. Hsu and Cheng [21] recommended that the coeﬃcient C was 0.04 if ReD ¼ qudp =l 10. Jiang et al. [8] found that C ¼ 0:025 was needed for the numerical simulation results to match the experimental data. Here we assume that C ¼ 0:025.

P.-X. Jiang et al. / Experimental Thermal and Fluid Science 28 (2004) 545–555

Jiang [24] numerically analyzed the heat transfer boundary characteristics on the surface between a thin plate and a solid particle in contact with each other with convection heat transfer due to water or air and a constant heat ﬂux on the outer plate wall. The analysis studied the inﬂuence of the thermal contact resistance between the plate wall and the solid particle and the wall thickness on the surface thermal characteristics. As shown by Jiang [24], if the thermal contact resistance on the wall surface between the solid particles and the wall is negligible as would occur in sintered porous media and the wall is relatively thick (e.g. 1–2 mm), the heat ﬂuxes through the solid particles and the ﬂuid near the wall are signiﬁcantly diﬀerent, while the temperatures of the solid particles and the ﬂuid near the wall are fairly close. If the thermal contact resistance on the wall surface between the solid particles and the plate wall is considered as would occur in non-sintered porous media, the solid particle temperature on the contact surface and the ﬂuid temperature on the convection heat transfer surface of the thin plate near the particle are signiﬁcantly diﬀerent, while the heat ﬂuxes transferred through the solid particles and the ﬂuid adjacent to the surface are similar. Therefore, the following boundary conditions were used in the analysis for the non-sintered porous media [9,16,19]: x¼0: y¼0:

up ¼ up ðyÞ; vp ¼ vp ðyÞ; Tf ¼ Tf0 up ¼ vp ¼ 0; qw ¼ kf oTf =oy ¼ ks oTs =oy

y¼h:

up ¼ vp ¼ 0;

549

nominal particle sizes (1.515, 1.80 and 2.25 mm) while one size of steel particles was used (1.2 mm). The thermal conductivities of the glass and steel are 0.744 W/ (m K) and 40.06 W/(m K), respectively. The porous plate porosities are listed in Table 1. The experimental setup was evaluated by comparing the results obtained for convection heat transfer in an empty plate channel with established correlations and numerical simulations. The range of mass ﬂow rates for the experiments in the empty plate channel was 0.49– 7.78 g/s, which is equivalent to Reynolds numbers of 499–7710. For laminar ﬂow (ReD < 2300), the convection heat transfer in an empty plate channel was calculated numerically. The local Nusselt numbers for turbulent and transition ﬂow convection heat transfer in an empty plate channel were calculated using the formula proposed by Petukhov et al. [25]: ! 1=4 Nux x 3600 0:4 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1 þ 0:416Pr 1þ De Nu1 ReD x=De x exp 0:17 4000 6 ReD 6 106 ; De x 0:7 6 Pr 6 100; > 0:5 ð7Þ De where Nu1 ¼

ðf=8ÞReD Pr pﬃﬃﬃﬃﬃﬃﬃ 1 þ 9000=ReD þ 12:7 f=8ðPr2=3 1Þ

kf oTf =oy ¼ ks oTs =oy ¼ 0 ð6Þ

A detailed description of the numerical method can be found in Jiang et al. [9]. The diﬀerence between the numerical methods of the present paper and that used by Jiang et al. [9] is that the present analysis also considers the variation of the air properties resulting from variations of the temperature and pressure in the channel. Analysis of the grid independence showed that for an adequate distribution of grid points, the numerical results did not depend on the number of grid points (e.g., 202 · 52, 202 · 102 and 202 · 202). Therefore, a nonuniform grid was employed with 102 grid points normal to the wall and 202 grid points in the axial direction.

4. Experimental results and discussion The experimental and numerical investigations used glass beads and steel particles. The glass beads had three

f ¼ ð1:821 log ReD 1:64Þ

2

Fig. 4 compares the experimental results, the values given by Eqs. (7) and (8) and the numerical results for various Reynolds numbers. The standard deviation between the experimental results and the predictions of Eqs. (7) and (8) was 18.3%. The experimental uncertainties near the inlet and outlet were relatively large due to longitudinal thermal conduction along the test section. In general, the accuracy of the experimental system is acceptable. In the test section the small particles were supported by two perforated plates and ﬁne mesh stainless steel screens at the inlet and outlet, so that the experimentally measured pressure drop between the inlet and outlet included the additional pressure drop resulting from the perforated plates and the screens. This component of the pressure drop was measured experimentally using the empty plate channel with the perforated plates and the screens at the inlet and outlet. Fig. 5 presents the

Table 1 Porosities of the porous plate test section dp [mm] Material Porosity

ð8Þ

1.43–1.6 Glass beads 0.393

1.6–2.0 Glass beads 0.388

2.0–2.5 Glass beads 0.392

1.2 Steel particles 0.382

P.-X. Jiang et al. / Experimental Thermal and Fluid Science 28 (2004) 545–555

60

50

50

40

∆ p (kPa)

hx (W/(m2 °C))

550

40 3 30

30 20

2 10

20

1 0

10

0

1

2

3

4

5

0.000

6

0.002

x/De Fig. 4. Comparison of measurements with predictions for the empty plate channel (, }, ) experimental data; (––) predicted values. ReD : 1 ()––495; 2 (})––2771; 3 ( )––4130.

∆ p (kPa)

1.5

1.0

0.5

0.002

0.004

0.006

0.008

G (kg/s) Fig. 5. Pressure drop with the perforated plates and screens.

ð9Þ

The pressure drop for the porous plate channel was then the measured pressure drop minus the additional pressure loss resulting from the perforated plates and the ﬁne mesh stainless steel screens. Fig. 6 shows the pressure drop in the porous plate channel as a function of mass ﬂow rate. The pressure drop in the porous media greatly increased with increasing ﬂow rate and decreasing particle diameter. The porous media signiﬁcantly increased the pressure drop in the plate channel compared with the empty channel (several thousand fold). The pressure drop can be decreased by increasing the porosity.

160 Re=699.8 Re=986.4 Re=1297.3 Re=1718.3 Re=2411.3

140 120

hx (W/(m2 °C))

½kPa

0.008

Fig. 6. Pressure drop along the porous plate channel: () dp ¼ 2:25 mm, (j) dp ¼ 1:80 mm, ( ) dp ¼ 1:515 mm, () dp ¼ 1:20 mm, (– – –) in the empty channel.

pressure drop for air in the empty plate channel with the perforated plates and the screens at the inlet and outlet as a function of mass ﬂow rate. The data was correlated by Dp ¼ 0:1377 19:6222G þ 32765G2

0.006

Figs. 7–10 present the local heat transfer coeﬃcients for convection heat transfer of air in the glass and steel packed beds. The local heat transfer coeﬃcients increased as the Reynolds number increased and decreased along the axial direction for most of the channel length except near the outlet section where the heat transfer coeﬃcient increased due to the edge eﬀect. Fig. 11 compares the experimental local heat transfer coeﬃcients in the various plate channels ﬁlled with glass beads or steel particles. The higher particle thermal conductivity caused higher heat transfer coeﬃcients. For a porous plate channel ﬁlled with glass particles, the heat transfer coeﬃcients increased as the particle diameter decreased, which diﬀers from the heat transfer of water in glass packed beds where the heat transfer coeﬃcients increased with increasing particle diameter [10]. The reasons for this phenomenon were analyzed by

2.0

0.0 0.000

0.004

G (kg/s)

100 80 60 40 20 0

0

1

2

3

4

5

x/De Fig. 7. Local heat transfer coeﬃcient for air in glass beads packed bed (dp ¼ 1:515 mm).

P.-X. Jiang et al. / Experimental Thermal and Fluid Science 28 (2004) 545–555

160

150 Re=389.5 Re=854.5 Re=1407.4 Re=1957.4 Re=2723.1

hx (W/(m2 °C))

120

3

hx (W/(m2 °C))

140

100 80 60

100 1 2

50 4

40 20

0 0

0 0

1

2

3

Fig. 8. Local heat transfer coeﬃcient for air in glass beads packed bed (dp ¼ 1:80 mm).

h x (W/(m2 °C))

Re=715.4 Re=2879.2 Re=5637.1 Re=7661.7

200

100

0

1

2

3

4

5

x/De Fig. 9. Local heat transfer coeﬃcient for air in glass beads packed bed (dp ¼ 2:25 mm).

3

4

5

6

x/De Fig. 11. Local heat transfer coeﬃcients for diﬀerent packed beds (ReD 1800). (1–3) Experimental data for porous media; (4) predicted values for empty plate channel: 1 ()––glass, dp ¼ 1:515 mm; 2 (})–– glass, dp ¼ 1:80 mm; 3 ( )––steel, dp ¼ 1:20 mm.

Fig. 12 presents the criterion with the additional experimental data. According to this criterion, the convection heat transfer coeﬃcient of air in the glass packed bed should increase with decreasing particle diameter

Re=550.4 Re=930.3 Re=1462.7 Re=1629.7 Re=2155.7

150

hx (W/(m2 ˚C))

2

Jiang et al. [10] who found that for convection heat transfer in non-sintered porous media, if q0 l0 dp > 0:065k4:82 k5:82 f m =cpf ðem =ð1 em ÞÞ the convection heat transfer coeﬃcients increased with increasing particle diameter. Here, km is the stagnant eﬀective thermal conductivity of ﬂuid and porous media, which can be calculated from [21]: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ " pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 e ð1 rÞB km 1 ¼ ½1 1 e þ ln 1 rB ð1 rBÞ2 rB kf # Bþ1 B1 ð10Þ 2 1 rB

400

300

1

5

4

x/De

0

551

100

50

0 0

1

2

3

4

5

6

x/De Fig. 10. Local heat transfer coeﬃcient for air in steel particles packed bed (dp ¼ 1:2 mm).

-4.82

0.065λ f

5.82

λm

c pf (ε m (1 – ε m ))

Fig. 12. Experimental data for judging the eﬀect of particle diameter: (I) water–bronze; () water–stainless steel; ( ) water glass; (}) water–glass; (+) water–bearing steel; () air–glass; (D) air–bearing steel.

552

P.-X. Jiang et al. / Experimental Thermal and Fluid Science 28 (2004) 545–555 sintered: ReD: non-sintered: ReD:

433; 382;

916; 930;

1637; 1629;

4

5985 5782

1000

fe

Nux

3

100

2

2

1 10

0

1

2

3

4

5

6

1

x/De

0

Fig. 13. Comparison of the convection heat transfer in non-sintered porous media and sintered porous media.

0

200

400

Fig. 14. Friction factor in porous media: ( (11); (2) numerical.

which corresponds well with the experimental data shown in Fig. 11. The porous media increased the heat transfer coeﬃcients 4–8 times for the conditions studied here. It should be pointed out that the heat transfer enhancement in non-sintered porous media is much less than in sintered porous media due to the larger contact thermal resistance and the larger porosity at the wall in the nonsintered porous media, especially for low thermal conductivity ﬂuids such as air [11]. Fig. 13 compares the convection heat transfer in non-sintered porous media and sintered porous media [11] to show that the heat transfer coeﬃcients in sintered porous media are much higher than those in non-sintered porous media. The diﬀerence between the heat transfer in sintered and nonsintered porous media was analyzed by Jiang et al. [11].

) experimental; (1) Eq.

merically calculated values of fe agree well with the experimental data and Eq. (11). Figs. 15 and 16 compare the numerical results for convection heat transfer of air in plate channels ﬁlled with glass or steel particles calculated using the local thermal equilibrium model and a local thermal nonequilibrium model with the experimental data. The local heat transfer coeﬃcient decreased along the axial direction and increased as the mass ﬂow rate increased. The numerical results using the local thermal nonequilibrium model with consideration of the thermal dispersion eﬀects with C ¼ 0:025 correspond well to most of the experimental results in the present research. The numerical simulations showed that if the thermal dispersion eﬀect is not considered, the predicted heat

200

5. Numerical results and discussion

150

h x (W/(m2 °C))

The numerical results were checked in numerous ways to verify the reliability of the physical–mathematical model, the solution procedures and the numerical simulation program [9]. Previous work [9,10] showed that the friction factor, fe , calculated from the experimental data for water ﬂow in non-sintered porous media can be well predicted by numerical simulation or by the formula given by Aerov and Tojec [26]: e3m qf dp Dp 36:4 ¼ fe ¼ þ 0:45 Ree 1 em 3M 2 L

600

Ree

4

3

100 2

50

1

4'

2'

ðfor Ree < 2000Þ

0

0

1

2

ð11Þ Fig. 14 compares the friction factors, fe , predicted by the numerical simulation and calculated from the experimental data for air in the non-sintered packed beds with the friction factor predicted by Eq. (11). The nu-

3

4

5

6

x/De Fig. 15. Local heat transfer coeﬃcients for glass packed beds (dp ¼ 1:515 mm, em ¼ 0:392). (, }, D, ) experimental data, (– – –) predicted with thermal equilibrium, (––) predicted with thermal non-equilibrium; (1–4) kd 6¼ 0, (20 ,40 ) kd ¼ 0. ReD : 1 ()––438; 2,20 (})––993; 3 (D)––1311; 4,40 ( )––2512.

P.-X. Jiang et al. / Experimental Thermal and Fluid Science 28 (2004) 545–555

200 4 3

hx (W/(m2 °C))

150

2

100

1

50 4'

2'

0

0

1

2

3

4

5

6

x/De Fig. 16. Local heat transfer coeﬃcients for steel packed beds (dp ¼ 1:20 mm, em ¼ 0:382). (; }; D; ) experimental data, (– – –) predicted with thermal equilibrium, (––) predicted with thermal non-equilibrium; (1– 4) kd 6¼ 0, (20 ,40 ) kd ¼ 0. ReD : 1 ()––562; 2,20 (})––949; 3 (D)––1667; 4,40 ( )––2200.

transfer coeﬃcients are much lower than the experimental data. The predicted heat transfer coeﬃcients using the local thermal equilibrium model with consideration of the thermal dispersion eﬀects with C ¼ 0:025 are much higher than the experimental results. Therefore, for convection heat transfer of air in glass or non-sintered metallic packed beds, the local thermal non-equilibrium and thermal dispersion eﬀects must be simultaneously taken into consideration in numerical simulations. Fig. 17 shows the distribution of the local heat transfer coeﬃcient with and without consideration of the variable thermophysical properties of air caused by the variable pressure along the channel. For small ﬂow rates, the eﬀect of the variable thermophysical properties of air caused by the pressure variation along the channel

553

on the convection heat transfer in porous media was negligibly small. However, for large ﬂow rates, the eﬀect of the pressure variation along the channel on the convection heat transfer in porous media became more evident. For example, as shown in Fig. 17, for ReD ¼ 5778, the diﬀerence between the heat transfer coeﬃcients with and without consideration of pressure variation was about 1.2%. In the present paper, the numerical simulations of forced convection heat transfer of air in the porous media included the eﬀect of the pressure variation along the channel. Fig. 18 shows the numerical calculation results for convection heat transfer of air in non-sintered porous media with diﬀerent solid materials. The inﬂuence of solid particle thermal conductivity on the convection heat transfer of air in the non-sintered porous media decreased with the thermal conductivity of the solid particles. For example, in a non-sintered porous media with solid particle thermal conductivities of 0.0259 and 0.744 W/(m K), the convection heat transfer coeﬃcient of air on the heat transfer surface in the porous plate channel increased from 74 to 185 W/(m2 K) at x=De ¼ 5:57. When the solid particle thermal conductivity increased more from 75.35 to 398 W/(m K), the convection heat transfer coeﬃcient of air on the heat transfer surface in the porous plate channel only increased from 275 to 276 W/(m2 K) at x=De ¼ 5:57. Jiang et al. [10] showed that convection heat transfer in porous plate channels is controlled by two factors, the convection heat transfer between the ﬂuid and the channel surface, and the ‘‘ﬁn eﬀect’’ of the particles which intensiﬁes as the particle thermal conductivity and Reynolds number increase. For convection heat transfer of air in porous media ﬁlled with solid particles having large thermal conductivities, the solid phase transfers

400

250 300

hx (W/(m2 ˚C))

hx (W/(m2 °C))

200 3

150

100 2 50

2 100 1

1 0

0

3,4,5

200

0

1

2

3

4

5

6

x/De Fig. 17. Local heat transfer coeﬃcients for glass packed beds (dp ¼ 1:515 mm, em ¼ 0:392): (– – –) predicted with constant pressure; (––) predicted with variable pressure. ReD : 1––1311; 2––2512; 3––5778.

0

1

2

3

4

5

6

x/De Fig. 18. Inﬂuence of solid particle thermal conductivity on convection heat transfer G ¼ 0:00244 kg/s, ReD ¼ 2455, em ¼ 0:36. Material and k (W/(m K)): (1) assumed material, 0.0259; (2) glass, 0.744; (3) bearing steel, 40.064; (4) bronze, (75.35); (5) pure copper, 398.

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much more heat from the upper surface to the volume than is transferred by the air in the porous media. Therefore, the overall heat transfer intensity is mainly limited by the convection heat transfer capability of the air. The inﬂuence of the solid particle thermal conductivity on the convection heat transfer of air in porous media decreases with increasing solid particle thermal conductivity. The numerically predicted convection heat transfer for air in porous media with and without consideration of viscous dissipation showed that for the conditions in the present paper, the viscous dissipation had very little inﬂuence on the convection heat transfer.

6. Practical signiﬁcance Porous structures greatly enhance convection heat transfer. However, the porous media also signiﬁcantly increases the pressure drop in the plate channel compared with an empty channel. Therefore, porous structures shall be used only in some special cases to enhance the heat transfer when high heat transfer rates are of critical importance. Convection heat transfer in sintered porous media is much more intensive than that in non-sintered porous media due to the reduced thermal contact resistance and porosity at the wall in sintered porous media. Therefore, for enhanced heat transfer, sintered porous structures should be used. The present research analyzes the diﬀerence between the transport phenomena in sintered and non-sintered porous media. The experimental data in the present paper can also be used to check physical–mathematical models and numerical results for convection heat transfer in nonsintered porous media.

7. Conclusions (1) The experimental results and numerically calculated values of the friction factor in porous media agreed well with established formula. The porous media signiﬁcantly increased the pressure drop in the plate channel compared with an empty channel (several thousand fold). (2) For a porous plate channel ﬁlled with glass particles, the heat transfer coeﬃcients increased with decreasing particle diameter. Increasing the particle thermal conductivity caused the heat transfer coeﬃcients to increase. (3) The non-sintered porous media increased the heat transfer coeﬃcients 4–8 times for the conditions studied here, which is much less than the increase in sintered porous media. (4) The numerical results for non-sintered metallic or glass porous structures using the local thermal

non-equilibrium model with consideration of thermal dispersion agree well with the experimental results. (5) For large ﬂow rates, the pressure variation along the channel aﬀects the convection heat transfer in porous media. (6) The inﬂuence of solid particle thermal conductivity on the convection heat transfer of air in non-sintered porous media decreases with increasing solid particle thermal conductivity.

Acknowledgements The project was supported by the National Outstanding Youth Fund from the National Natural Science Foundation of China (no. 50025617) and the Major State Basic Research Development Program (no. G1999033106).

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