Study of pressure drop-flow rate and flow resistance characteristics of heated porous materials under local thermal non-equilibrium conditions

International Journal of Heat and Mass Transfer 102 (2016) 528–543 Contents lists available at ScienceDirect International Journal of Heat and Mass ...

Download PDF

6MB Sizes 247 Downloads 225 Views

Report

Full Text

International Journal of Heat and Mass Transfer 102 (2016) 528–543

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Study of pressure drop-flow rate and flow resistance characteristics of heated porous materials under local thermal non-equilibrium conditions Yuxuan Liao a, Xin Li a,⇑, Wei Zhong b, Guoliang Tao a a b

The State Key Lab of Fluid Power and Mechatronic Systems, Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang 310027, PR China The School of Mechanical Engineering, Jiangsu University of Science and Technology, 2 Mengxi Road, Zhenjiang, Jiangsu 212003, PR China

a r t i c l e

i n f o

Article history: Received 18 November 2015 Received in revised form 11 May 2016 Accepted 24 May 2016

Keywords: Porous material Pressure drop Flow resistance Temperature-based pumping power Temperature distribution

a b s t r a c t Porous materials are used as pneumatic components in a wide range of industrial applications. Such porous materials contain thousands of interconnected irregular micro-pores that produce a large pressure drop (DP) between the upstream and downstream sides of the porous material when a fluid flows through it. The relationship between the pressure drop and flow rate (i.e., the DP G characteristic) and the relationship between flow resistance (c) and flow rate (the c G characteristic) are two very important basic characteristics. One factor affecting them is temperature, whose variation changes the viscosity and density of the fluid. In this study, we experimentally and theoretically analyzed the effect of temperature on DP G and flow resistance characteristics of porous materials by heating them under constant electric heating power. The resulting experimental DP G and flow resistance curves shift upward relative to their counterparts at room temperature owing to the increase in fluid temperature, but remain within the adiabatic and room temperature curves. The temperature-effect ratio g at constant heating power increases from 1.3 to 1.7 as the flow rate decreases from 21.53 105 kg/s to 5.80 105 kg/s, indicating that DP G and flow resistance characteristics and pumping power change significantly when a porous material is heated. Furthermore, temperature distributions were obtained numerically to gain deeper understanding of the temperature effect. The effects of heating power values, characteristic porous material coefficients, and average fluid density on DP G and flow resistance characteristics were also investigated. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction A porous material contains thousands of interconnected irregular micro-pores that produce a large pressure difference between the upstream and downstream sides of the material when a fluid flows through it. Such porous materials are used in a wide range of industrial applications. For example, on a large LCD glass production line, porous material is placed in the air flotation rail system, allowing compressed air to flow out of the porous material uniformly; this significantly reduces stress in the thin glass substrate caused by the air flow and enhances suspension stability [1–3]. Another typical application of porous material is in air bearings. Porous materials, fabricated from sintered metal or ceramic powders, replace conventional orifices in applications where high load capacity, stiffness, and stability are critical [4,5]. To study the use of porous materials as pneumatic components, we focus on the relationship between the pressure difference (DP) and flow ⇑ Corresponding author. E-mail address: [email protected] (X. Li). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.05.101 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

rate (G), i.e., the DP G characteristics, of porous materials [6–10]. Many studies on the DP G characteristics of porous materials have been conducted. Lage et al. investigated two types of nonlinear pressure-drop-versus-flow-rate relations [11]. Meanwhile, Mancin and Simone et al. focused on the experimental and theoretical analysis of pressure losses during air flow in aluminum foams with different numbers of pores per inch and different porosity values [12]. Wei et al. proposed a new charge method to determine permeability and inertia coefficients from DP G characteristics [13]. Calamas studied flow behavior and pressure drops in porous disks with bifurcating flow passages. The effects of the bifurcation angle, porosity, and pore size on the pressure drop across a porous disk were examined computationally [14]. Naaktgeboren investigated inlet and outlet pressure drop effects on permeability and form coefficient of a porous medium [15]. Jin studied porous medium flow of highly compressible gas near a wellbore; in the study, a model with an acceleration effect was used to predict DP G characteristics [16]. Dukhan and Nihad et al. presented a new set of experimental data for water flow in metal foam to establish various flow regimes (from pre-Darcy to turbulent), and to assess the

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

529

Nomenclature Afs a/2 b Cpf Da dp G H hfs K Keff kf,eff ks,eff ks kf m P P1 P2 DP DPT

DP T Pr qw R R1 R2 Rep Rep,in ReT

specific surface area, m1 viscosity flow resistance, Pa/(kg s1) inertia flow resistance, Pa/(kg s1) specific heat capacity, J K1 kg1 Darcy number diameter of particles, m mass flow rate, kg/s length of the material, m fluid-to-solid heat transfer coefficient, W m2 K1 permeability coefficient, m2 effective permeability coefficient, m2 effective conductivity of the solid, W m1 K1 effective conductivity of the fluid, W m1 K1 conductivity of the solid phase, W m1 K1 conductivity of the fluid phase, W m1 K1 mass of the material, kg pressure of air, Pa inlet pressure, Pa outlet pressure, Pa pressure drop, Pa pressure drop at constant heating power at temperature T, Pa pressure drop at isothermal conditions at temperature T, Pa Prandtl number heating power, W gas constant, J kg1 K1 inner radius of the material, m outer radius of the material, m particle Reynolds number inlet particle Reynolds number inlet Reynolds number at a given temperature T

behavior of pressure drops in each regime [17]. However, these studies were conducted under room temperature conditions. As the temperature of a fluid changes, the viscosity and density of the fluid change accordingly. Many researchers have studied temperature variations and heat transport phenomena in porous materials. Odabaee et al. experimentally examined the heat transfer enhancement from a thin metal foam layer sandwiched between two bipolar plates of a cell. Effects of the key parameters including the free-stream velocity and metal foam characteristics such as porosity, permeability, and form drag coefficient on temperature distribution, heat and fluid flows were investigated [18]. Dukhan and Nihad Direct et al. measured actual air temperatures inside a commercial aluminum foam cylinder whose wall was heated by a constant heat flux and cooled by forced air flow using a specially designed technique. The volume-averaged analytical method was used to analyze temperatures of the solid and fluid phases inside the foam. Comparison showed good agreement between experimental and analytical air temperatures [19]. Zhang discussed compressed air energy storage and described a program in which it is applied to a wind turbine system for leveling power supplied to the electrical grid. Pressure drop, heat transfer, and temperature distribution characteristics were presented [20]. Aboelsoud et al. numerically studied hydraulic and thermal characteristics of V-shape corrugated carbon foams in air flows. The pressure drop, overall heat transfer coefficient, and temperature contours across the foam wall for four geometries were calculated [21]. Carpenter et al. experimentally investigated aluminum foam test sections with discrete pore-size gradients. The pressure drop, thermal performance, pumping power, etc. were discussed [22]. However, while they dis-

r T Tf Ts Tf,in Tf,out Ts,in Ts,out T0 T Ta V_ v W WT

radius, m temperature of air, K temperature of the fluid phase, K temperature of the solid phase, K temperature of inlet air flow, K temperature of outlet air flow, K temperature of inlet surface of porous material, K temperature of outlet surface of porous material, K room temperature, K average temperature of air, K adiabatic temperature of air, K volumetric flow rate, m3/s radial mean flow velocity, m/s pumping power, W pumping power at temperature T, W

Greek symbols b inertia coefficient c flow resistance, Pa/(kg s1) cT flow resistance at constant heating power at temperature T, Pa/(kg s1) cT flow resistance at isothermal conditions at temperature T, Pa/(kg s1) g temperature-effect ratio of DP, c, and W gT temperature-effect ratio of DP, c, and W at temperature T l air viscosity at a given temperature T, Pas l0 viscosity at room temperature, Pas q air density, kg/m3 q0 air density at room temperature, kg/m3 qs density of steel alloy, kg/m3 u porosity

cussed pressure drop-flow rate characteristics, they aimed to analyze heat transfer between porous materials and the fluid rather than the effect of temperature variation on pressure drop-flow rate characteristics. In practical applications, the effects of temperature variation on pressure drop-flow rate characteristics is of great importance, for example, in the pumping power of metal foam heat exchangers designed for fuel cells or other cooling systems [18,23,24]. The pumping power is determined by temperature-dependent pressure drop-flow rate characteristics, indicating that pumping power is another parameter affected by temperature. Thus, research into temperature effects on pressure drop-flow rate characteristics is very important to cooling systems. On the other hand, temperature effects on pressure drop-flow rate characteristics are also farreaching. For instance, Yeo studied the effect of gas temperature on the flow rate characteristics of an averaging pitot-tube-type flow meter [25,26]. Based on the principle of using two hot wires, Lange investigated the characteristics of a micro-mechanical thermal flow sensor operating at constant-temperature in a small channel and considered temperature-based DP G characteristics [27]. However, few studies have dealt with the effects of temperature variation on pressure drop-flow rate characteristics of porous materials. Narasimhan performed a theoretical analysis to predict the effects of a fluid with temperature-dependent viscosity flowing through an isoflux-bounded porous medium channel. The pressure drop, heat transfer, and temperature profiles of this system were obtained by solving numerically the differential balance equations [28,29]. Vanderlaan studied heat-flow-induced pressure drops using superfluid helium (He II) contained in porous media. In his

530

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

experiment, heat was applied to one side of a He II column containing a random pack of uniform-size polyethylene spheres. Measured results included steady-state pressure drops across the random packs of spheres for different heat inputs [30,31]. However, the study analyzed mainly the effect of the temperature-based liquid viscosity on pressure drops. As for gases, temperature also changes their viscosity and density, which has not been addressed. On the other hand, temperature changes not only pressure dropflow rate characteristics but also flow resistance of porous materials. Temperature effects on flow resistance are also very important. For example, flow resistance is one of the most important parameters determining sound absorption properties. Many studies have investigated flow resistance of porous materials [32], but these studies were also conducted at room temperature. Wu and Jiu Hui et al. presented a quantitative theoretical model to investigate the sound absorbing properties of porous metal materials at high temperature and high sound pressure based on Kolmogorov turbulence theory [32], but they focused on temperature effects on sound absorption properties and not flow resistance. Another motivation for the present study is to propose a new design for a variable flow resistance (VFR) device. The new VFR device is made from a tight, porous material. The porous material is first heated, and heat is then transferred from the porous material to the fluid in order to raise the fluid’s temperature. The flow resistance changes because of changes in viscosity and density caused by temperature variation. Compared with conventional VFR devices (poppet-type, spool-type, etc.) that adjust the flow rate or pressure by changing the flow area using complicated mechanical structures and moving parts, the new VFR device has a very simple structure without any moving parts; thus, its fabrication is easier and it can suppress noise and operate in high-vibration environments. To realize the new design, it is essential to study the effects of temperature on hydraulic and flow resistance characteristics. Therefore, in this study we discuss the effects of temperature on hydraulic and flow resistance characteristics of porous materials, as this has scarcely been addressed in previous studies. 2. Experimental setup and methods

2.1. Test porous materials The dimensions, weights, mean pore size, and porosity of the test materials are listed in Table 1. Three porous materials with the same dimensions were selected and designated as #1, #2, and #3, as shown in Fig. 1. Porosity is defined as the fraction of the total volume occupied by void space and is calculated by Eq. (1):

m

pqs H R22 R21

2.2. Packing shell, heating, and temperature measurement system for porous materials In order to connect the porous material to the pneumatic circuit, we designed a packing shell for the test material (as shown in Fig. 2) consisting mainly of three polytetrafluoroethylene (PTFE) shells. The porous material was placed between two shells and sealed with sealing glue. The sealing glue can withstand temperatures up to 600 K. The two PTFE shells were then tightened using joint bolts. Tests were performed to confirm that no leakage occured even at relatively high pressures (e.g., 600 kPa). Two air pipes connected the packing shell to the experimental pneumatic circuit. An electric heating system was used to heat the porous material (see Fig. 2). The system was composed of a DC source (ARRAY 3663A, Range: 0–80 V, 0–6.5 A) and a heating coil with an electrically insulated surface. The heating coil was coiled tightly around the outer surface of the porous material; thus, the porous material could be heated rapidly. The working voltage and heating power ranges of the heating coil were 0–36 V and 0–40 W, respectively. By adjusting the input voltage, the heating power could be regulated. Fourteen thermocouples (OMEGA, range: 223–673 K, resolution: 0.1 K) were fixed onto the PTFE shells by glue to measure the inlet and outlet temperatures of both the fluid and the solid. Those thermocouples in contact with the porous material were glued to the solid by a silver colloid to reduce contact thermal resistance and to prevent fluid flow from affecting temperature measurements. 2.3. Pneumatic circuit

In this section, we present details of the experimental setup and methods, including the test porous materials, packaging method, heating method, and pneumatic circuit.

u¼1

Fig. 1. Test porous materials.

;

ð1Þ

where qs = 8.03 103 kg/m3 is the density of the steel alloy, m is the mass of the porous material, and H is the length of the test material. R1 and R2 are the inner and outer radius, respectively.

We used air as the test fluid in these experiments. The pneumatic circuit is shown in Fig. 3. Compressed air was supplied and regulated to obtain a selected pressure. A high-precision pressure gauge (YOKOGAWA, CA700) was used to measure the upstream and downstream pressures. The regulator set the pressure to a selected value, and a ball valve was used to change the upstream pressure. A flow rate sensor (KEYENCE, FD-A10) was placed upstream of the porous material. The downstream pressure could be switched to positive, atmospheric, or negative pressures by controlling switch valves. Fourteen thermocouples were connected to a temperature sensor recorder to display the measured temperature values. 3. Theoretical analysis 3.1. Theoretical model The fluid flow through porous media is schematically shown in Fig. 4. The black dots represent solid grains and the void space rep-

Table 1 Parameters of test porous materials. No.

Pore size [lm]

Inner radius [mm]

Outer radius [mm]

Length [mm]

Weight [g]

Porosity [–]

#1 #2 #3

0.5 5 30

9.5 9.5 9.5

12.7 12.7 12.7

23.5 23.5 23.5

7.57 6.85 6.35

0.281 0.349 0.397

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

(a) Schematic

531

(b) Photograph

Fig. 2. Packing shell, heating, and temperature measurement systems for porous materia.

Fig. 3. Pneumatic circuit.

3.2. Governing equation for temperature distribution At steady-state conditions and neglecting the work that is done by the pressure difference, the temperature distribution of the fluid and solid can be calculated by the equations of conservation of mass and energy. Because of the complexity of flow through the porous material (Fig. 1), the following assumption is made to simplify the theoretical derivation: (1) The inner geometry of the porous material is isotropic and homogeneous; (2) Temperature distribution only exists along the radial direction.

Fig. 4. Schematic of fluid flow through porous material.

resents micro-pores through which the fluid flows. The outer surface of the porous material was heated by a constant heat flux qw . An air flow with a mass flow rate of G was driven by the inlet pressure (denoted by P1 ), and flowed from the outer surface to the inner surface of the porous material. The outlet pressure is denoted by P 2 . Meanwhile, a heat transfer process took place between the fluid and solid phases when air flowed through the porous material. The temperatures of the inlet and outlet air flows were T f ;in and T f ;out , respectively, and the temperatures of the inner and outer surfaces were T s;in and T s;out , respectively. Next, based on this model, we establish and solve the governing equations.

Based on the above assumptions, the equation of conservation of mass for steady fluid flow takes the form:

1 dðr qv Þ ¼ 0: r dr

ð2Þ

Eq. (2) can be integrated directly to give the radial mean flow velocity:

v¼

G ; 2prHuq

ð3Þ

where G is the mass flow rate, r is the radius, u is the porosity, H. the length of the porous material, and q is the density of air. Assuming that air is a perfect gas, density q is expressed as

q¼

P ; RT

ð4Þ

532

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

Fig. 5. Calculated finite-difference mesh for fluid flow through the porous material.

where P is the pressure, R is the gas constant, and T is the temperature of air. As for the energy equations, there are two primary models: local thermal equilibrium (LTE) and local thermal nonequilibrium (LTNE) [33–40]. The LTE model based on one energy equation assumes that heat exchange between the solid and fluid phases is fast enough so that the local temperature difference between the two phases is negligible [41–45]. However, in applications where a substantial temperature difference exists between the two phases, the LTE model does not hold. In these situations, the two-equation LTNE model must be utilized; it can be expressed as follows [33]. The energy conservation equation of the fluid phase is

@ Tf C pf 1 @T f hfs Afs G þ 1 ðT s T f Þ ¼ 0; þ 2pH kf ;eff r @r @r 2 kf ;eff

ks;eff ¼ ð1 uÞks ;

where ks and kf are the conductivities of the solid and fluid phases, respectively. The specific surface area appearing in the energy equations is expressed as [33]:

Afs ¼

6ð1 uÞ ; dp

ð9Þ

where dp is the diameter of the particles. The following correlation is used for the fluid-to-solid heat transfer coefficient [33]:

hfs ¼

2

i 1 kf h 2 þ 1:1Pr3 Re0:6 ; p dp

ð10Þ

ð5Þ

where Pr is the Prandtl number. As for gases, Pr is a constant that is independent of temperature; its value is about 0.71. Rep is the particle Reynolds number, which is defined as [46]:

ð6Þ

Rep ¼

and the energy conservation equation of the solid phase is

@ 2 T s 1 @T s hfs Afs þ ðT f T s Þ ¼ 0; þ r @r @r 2 ks;eff

ð8Þ

qf v dp ; l

ð11Þ

where T f and T s are the fluid and solid temperatures, respectively. C pf is the specific heat capacity of the fluid. hfs and Afs are the fluid-to-solid heat transfer coefficient and specific surface area, respectively. The effective conductivities of the solid and fluid phases are ks;eff and ks;eff , and are expressed as follows [33]:

where l is the air viscosity. The value of air viscosity l at a given temperature T is governed by Sutherland’s formula as follows [32]:

kf ;eff ¼ ukf ;

where l0 is the viscosity at room temperature T 0 (298 K). Conductivity of the fluid phase is also a temperature-dependent parameter, and can be expressed as:

ð7Þ

l ¼ l0

kf ¼

T T0

lC pf Pr

3=2

¼

T 0 þ 110 ; T þ 110

l0 C pf Pr

T T0

3=2

ð12Þ

3=2 T 0 þ 110 T T 0 þ 110 ¼ kf 0 ; T þ 110 T0 T þ 110

ð13Þ

where kf 0 is the thermal conductivity of air at room temperature T 0 . Boundary conditions of the governing equations were determined by experiments: five thermocouples (5 mm apart) were used to measure the temperature of the outer surface of the porous material, and the average value of the five thermocouple measurements was taken as the inlet temperature of the solid. Similarly, we obtained the outlet temperature of the solid and the fluid. The inlet

Table 2 K and b for test porous materials.

Fig. 6. Determination of Darcy and Forchheimer regimes using test material #1 as an example.

No.

K [1012 m2]

b [–]

#1 #2 #3

0.718 3.08 14.40

0.90 0.75 1.23

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

dP lv qv 2 ¼ þ b 1=2 ; dr K K

533

ð14Þ

where K is the permeability coefficient, b is a dimensionless coefficient called the inertia coefficient, and l is the air viscosity. v2 In Eq. (14), lKv is the viscosity term and b Kq1=2 is the inertia term.

When the flow velocity is sufficiently low, viscous effects are dominant and inertia term can be neglected. This flow pattern is said to be in the Darcy regime. As the flow velocity increases, inertial effects become more significant and finally become dominant as they prevail over viscous effects; this is called the Forchheimer regime, in which an inertia term is added to the Darcy equation to describe the flow pattern. The permeability and inertia coefficient can be calculated by experimental data for the Darcy and Forchheimer regimes, respectively. The pressure drop DP is defined as

DP ¼ P 1 P 2 ;

ð15Þ

where P1 is the inlet pressure and P 2 is the outlet pressure. By substituting Eqs. (3) and (4) into Eqs. (14) and (15) and integrating dP/dr with respect to r with the boundary conditions (P = P1 where r = R2, P = P2 where r = R1), the temperature-based pressure drop at constant heating power DP T is obtained:

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u Z R2 u GR Z R2 lT G2 bR T dr þ DP T ¼ t dr þ P22 P 2 : 1 2 pK uH R1 r 2p 2 K 2 H 2 u 2 R1 r

ð16Þ

Fig. 7. Pressure drop-flow rate and flow resistance characteristics of porous materials at room temperature.

temperature of the fluid was room temperature (298 K). Then, based on the experimental boundary conditions, the governing equations were solved using the finite-difference method [47]. The calculated mesh is shown in Fig. 5.

Based on the calculated temperature distribution in Section 3.2, we obtained the theoretical pressure drop at constant heating power. On the other hand, in the isothermal condition, temperature T and viscosity l are parameters that are independent of radius r; therefore, the pressure drop in the isothermal condition can be expressed as

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u u GRlT R2 G2 bRT 1 1 þ þ P22 P2 : DP T ¼ t ln 1 pK uH R1 2p2 K 2 H2 u2 R1 R2

ð17Þ

Analogously to Ohm’s Law, we define flow resistance as the pressure drop in units of mass flow rate:

DP : G

3.3. Pressure drop and flow resistance

c¼

The flow pattern in porous material is characterized by the Darcy–Forchheimer equation, i.e., the equation of conservation of momentum [48]:

Applying Eqs. (16) and (17) to Eq. (18), we obtained the flow resistance both at constant heating power and in the isothermal condition, which are expressed respectively as follows:

(a) P-G curves of porous material #1 at given temperatures

ð18Þ

(b) Relation between K and temperature

Fig. 8. Temperature effects on porous material permeability coefficients.

534

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

Flow resistance in constant heating power condition:

ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Z R2 Z R2 R lT GbR T 2 dr þ cT ¼ dr þ P2 P2 1 2 pK uH R1 r 2p2 K 2 H2 u2 R1 r

ð19Þ

and

cT ¼

ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ RlT R2 GbRT 1 1 2 þ þ P ln 1 2 P2 : pK uH R1 2p 2 K 2 H 2 u 2 R1 R2

ð20Þ

Eqs. (16) and (17) determine the relationship between pressure drop DP and mass flow rate G at constant heating power and in the isothermal condition, respectively. Thus, the two equations represent the hydraulic or DP G characteristic of the heated porous material. Similarly, Eqs. (19) and (20) describe the flow resistance or c G characteristic of the heated porous material. Rearranging Eq. (20), we have:

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !2 !ﬃ u 1u a 2 2 t þ ; cT ¼ þ 4b 1 2 2 þ DP2P 2 þ DP2P 2 2 þ DP a

ð21Þ

P2

where a ¼ pKluTRHP2 ln

R2 R1

;

b¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ bRT 1 R12 . 2p2 K 1=2 u2 H2 R1

From Eq. (21), we make the following conclusions. In the case of a fixed P 2 , when the mass flow rate G is very small, i.e., DP is very small, the term DP2P can be neglected compared to 2; 2 i.e., 2 þ DP2P ! 2. Thus, the inertia term 4b 1 2þ2DP ! 0; in this P2

case, we have cT ! a=2. The parameter a=2 is independent of the inertia term; therefore it is defined as the viscosity flow resistance. Whereas the mass flow rate is sufficiently large, the term 2 þ DP2P ! 1, i.e., the viscosity term 2þaDP ! 0, and we obtain P2

cT ! b. The parameter b is determined only by the inertia term and is defined accordingly as inertia flow resistance. In summary, we obtain cT ! a=2 and cT ! b. G!0

G!1

4. Results and discussion 4.1. Determination of K and b To obtain the theoretical equation for the hydraulic and flow resistance characteristics, i.e., Eqs. (16) and (17) and Eqs. (19) and (20), respectively, we first determine the following coefficients for the test porous materials at room temperature (298 K): porosity u, permeability K, and inertia coefficient b. u is calculated by Eq. (1), and its values are listed in Table 1. K and b can be calculated according to the experimental data for the upstream and downstream pressures and the corresponding flow rates. Rearranging Eq. (17), we obtain the effective permeability coefficient K eff , which can be expressed by the following equation:

0 1 R2 1 1 1 P21 P22 1@ R ¼ 1 þ bD2a 1 Rep;in A ¼ K eff lRTG ln R2 K ln R2 puH

R1

R1

1 1 þ f ðDa ; b; R1 =R2 Þ Rep;in ; ¼ K

ð22Þ Gd

where Da ¼ dK2 . the Darcy number, Rep;in ¼ 2plR2pHu, is the inlet particle p 1

R2 1 R1

Reynolds number, and f ðDa ; b; R1 =R2 Þ ¼ bD2a , which is a paramln

Fig. 9. Relations between pressure difference, flow resistance, temperature effect ratio, average temperature, boundary conditions, and flow rate at 10.8 W of heating power for porous material #1.

R2 R1

eter determined by the characteristics of the porous material. Taking 1=K eff as the vertical axis and Rep;in as the horizontal axis (in logarithmic coordinates), the experimental data can be plotted as shown in Fig. 6. When the flow rate is very small, i.e., Rep;in is

very small, the term f ðDa ; b; R1 =R2 ÞRep;in . Eq. (22) can be neglected; thus, K eff ! K, which is represented by the horizontal broken blue line in Fig. 6. As the flow rate increases, the experimental data deviates from the line of 1=K because of the effect of inertia.

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

Accordingly, we divide the data into two regimes: Darcy and Forchheimer regimes. Therefore, we can determine K using the data from the Darcy regime and b using the data from the Forchheimer regime. By this method, the coefficients for the three test materials were determined (Table 2). 4.2. Pressure drop-flow rate and flow resistance characteristics at room temperature The pressure drop-flow rate and flow resistance characteristics of porous material at room temperature are shown in Fig. 7. From Fig. 7, it is clear that DP increases with flow rate and that the smaller the K and the larger the b, the larger the value of DP. The flow resistance for the three porous materials begins at the value of viscous flow resistance a/2 and converges to inertia flow resistance b as the mass flow rate increases despite noticeable differences in average pore diameter. This experimental tendency is consistent with the theoretical result expressed in Eq. (21). As for porous material #1, K is very small; thus, a/2 > b, i.e., viscous flow resistance is larger than inertia flow resistance. Consequently, flow resistance increases with increasing flow rate. As for porous material #3, K is relatively large; thus, a/2 < b. As a result, the flow resistance decreases with increasing flow rate. As for porous material #2, a=2 b; therefore, flow resistance remains almost constant.

535

The relation of a/2 and b also determines the shape of the DP G curves: for porous materials with a character of a/2 > b, the DP G curves are convex curves; for those where a/2 < b, the DP G curves are concave; and for those where a=2 b, the DP G curves are straight lines (see Fig. 7). The values of K and b in Table 2 are used to draw the theoretical curves for the DP G and flow resistance characteristics. Fig. 7 shows that the theoretical curves agree very well with the plotted experimental values. The error is less than 2% in both the smallflow-rate and large-flow-rate ranges and less than 6% in the transitional region. 4.3. Temperature effects on porous material properties K and b are determined by the properties of the porous material, i.e., porosity, pore size, and shape. Therefore, porous material properties can be revealed by K or b. In this section, we use the discussion of temperature effects on K as an example to study the properties of heated porous materials. By experiments, we obtained the DP G curves of porous materials #1, #2, and #3 at four different isothermal temperatures. As an example, we plotted the DP G curves of porous material #1 in Fig. 8(a). From the figure, it is clear that the fitting curves for these experimental data at different temperatures (298 K,

Fig. 10. Temperature distribution at 10.8 W of heating power for porous material #1.

536

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

339.3 K, 383.5 K, and 434 K) are straight lines that pass through the origin, indicating that the flow pattern is in the Darcy regime. Thus, we calculated the value of K in four different isothermal temperatures using Eq. (17). Based on this method, we obtained the values of K for materials #1, #2, and #3 at four different isothermal temperatures, and plotted them in Fig. 8(b). It is clear that K values remain constant over the range 298–450 K. Therefore, the effect of thermal expansion on porous material properties can be neglected in this temperature range. Hence, temperature does not affect K and b in this temperature regime. Thus, in the process of theoretical analysis, we take K and b as temperatureindependent constants. 4.4. Hydraulic and flow resistance characteristics of heated porous materials We used the electric heating system shown in Fig. 2 to heat porous material #1 and to obtain the experimental data for DP G and flow resistance characteristics under the condition of constant heating power (10.8 W). The downstream side of the porous material was opened to the atmosphere; i.e., P2 was atmospheric pressure. The temperature boundary conditions were measured by thermocouples. First, based on Eq. (17), we draw a group of DP T G isothermal curves for different T values. T was varied from 298 to 498 K with temperature intervals of 20 K between two adjacent curves, as the black dotted curves shown in Fig. 9. From the DP T G isothermal curves we observe that, at the same flow rate, the higher the temperature T, the larger the pressure drop DPT . This is because the increase in temperature T increases the two temperature terms lT and T. Eq. (17). From a physical perspective, the parameter l. 3=2 T 0 þ110 ) in the term lT is the temperature-based vis(i.e., l0 TT0 Tþ110 cosity while T in the above two terms is the temperature term in P density (i.e., q ¼ RT ). The increase in air temperature leads to an increase in viscosity and a decrease in density; both of these changes result in increased DP T . The unfilled blue dots in Fig. 9 represent experimental values measured at room temperature (298 K). The filled blue dots are measured values under the condition of 10.8 W of heating power. Fig. 9 shows that the experimental data at room temperature coincide perfectly with the isothermal curve for 298 K while the experimental data for 10.8 W of heating power intersect with the isothermal curves. As the flow rate increases, the intersection points move from high-temperature isothermal curves to lowtemperature ones. This is because the heating power was kept constant in the experiment, so an increase in flow rate results in a decrease in the temperature of the flowing air. In order to determine the temperatures at the intersection points, we let DPT ¼ DPT :

DPT ¼ DP T vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u u GRlT R2 G2 bRT 1 1 þ þ P 22 P2 ; ¼t ln 1 pK uH R1 2p2 K 2 H2 u2 R1 R2

ð23Þ

where DP T is the pressure drop at constant heating power. Here, DP T can be determined from the measured data. DPT is the pressure drop at isothermal conditions. T is the average temperature. Using Eq. (23), we transform the DP G curves at constant heating power into those at isothermal conditions. Thus, the intersection point temperature T is defined as the average temperature of air. We show the calculated average temperature data in Fig. 9. The figure shows that the average temperature of the flowing air

Fig. 11. Relations between pressure difference, flow resistance, temperature effect ratio, average temperature, and flow rate at different heating powers for porous material #1.

tends to approach room temperature gradually as the flow rate becomes sufficiently large. On the other hand, we also measured temperature boundary conditions. Based on the boundary conditions, we obtained a theoretical curve at 10.8 W of heating power shown as the blue solid

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

curve in Fig. 9. It is clear that the theoretical curve agrees very well with the plotted experimental values. The error is within 6%. The blue dotted line in Fig. 9 represents the adiabatic curve obtained with the assumption that the wall is adiabatic and that the fluid absorbs all heating power. Firstly, we noticed that the inlet temperature of the porous material was slightly higher than the outlet temperature while the temperature of the outlet air was equal to that of the solid. On the other hand, average temperature was approximately equal to solid temperatures. The temperature difference between them was within 7 K (<2%). Thus, to obtain the adiabatic curve, we further assumed that the solid and fluid are isothermal and that the adiabatic temperature of the air was equal to the outlet air temperature, which can be expressed as

Ta ¼

qw þ T f ;in : C pf G

ð24Þ

Applying Eq. (24) to Eq. (17), we have the adiabatic DP G curve at 10.8 W of heating power. We can see that the adiabatic curve also intersects with isothermal curves. The difference lies in that, in the small-flow-rate regime, the intersection point temperature of the adiabatic curve is higher than that of the experimental curve while, in the large-flow-rate regime, they are equal to each other. The reason for this is that, in the small-flow-rate regime, the temperatures of both the solid and the air are higher, which leads to greater heat loss while, in the large-flow-rate regime, the temperature is relatively lower, which results in lower heat loss. We also noticed that the experimental DP G curves at constant heating power remain within the adiabatic and room temperature curves. We deduce that experimental DP G curves at constant heating power with lower heat loss shift closer to the

537

adiabatic curve while those with greater heat loss shift closer to the curve at room temperature. Next, we discuss temperature effects on the flow resistance of porous material. Based on DP G experimental and theoretical data, we obtained isothermal c G curves, experimental c G curves both at room temperature and constant heating power, and adiabatic c G curves using Eq. (18) (shown in Fig. 9). From the isothermal c G curves, it is clear that the higher temperature, the larger the flow resistance. Moreover, the experimental data at constant heating power also intersect with the theoretical isothermal curves. As the flow rate increases, the intersection points move from high-temperature isothermal curves to low-temperature ones. Similarly, the experimental c G curves at constant heating power remain within the adiabatic and room temperature curves. The hydraulic characteristics and flow resistance are relevant to pumping power, which is defined as [22]

G W T ¼ DPT V_ ¼ DPT ;

q0

ð25Þ

where W T is the pumping power at a given temperature T, V_ is the volumetric flow rate, and q0 is the density of air in room temperature. The relationship between pressure drop, flow resistance, and pumping power are shown in the following equation:

gT ¼

cT DPT DPT V_ WT ¼ ¼ ¼ ; cT 0 DPT 0 DP T 0 V_ W T 0

ð26Þ

where cT 0 ; DPT 0 , and W T 0 are flow resistance, pressure drop, and pumping power at room temperature T 0 . gT is the temperature-

Fig. 12. Temperature distribution for different heating powers in porous material #1.

538

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

effect ratio for the pressure drop, flow resistance, and pumping power at a given temperature T. The relation between gT and G is shown in Fig. 9, in which it is clear that the isothermal gT G curve shifts upward as temperature increases, indicating that a higher temperature results in a larger. The tendency of the gT G curves at constant heating power is similar to that of c G curves. The values of gT (the temperatureeffect ratio for DP, c, and W) are approximately 1.3–1.7 and can reach as much as 2.2 at the adiabatic condition, indicating that temperature variation produces considerable changes in pressure drop, flow resistance, and pumping power. Furthermore, based on the measured temperature boundary conditions, we obtained the temperature distribution along the radial direction for both the fluid and solid at different mass flow rates at 10.8 W of heating power, as shown in Fig. 10(a) and (b). From Fig. 10(a), it is evident that, at a given mass flow rate, the fluid temperature increases rapidly from room temperature (298 K) to its maximum when the fluid flows into the solid at the inlet, and then it falls slightly until the fluid flows out of the porous material. As the mass flow rate decreases, the temperature of the fluid increases accordingly. In Fig. 10(b), the temperature of the solid also rises with decreasing mass flow rate but the temperature difference is very small in the radial direction. To glean a deeper understanding of the temperature distribution, we set the mass flow rate at intervals of about 5 105 kg/s to 21.53, 16.60, 11.64, and 6.57 105 kg/s and obtained corresponding temperature distribution curves for both the fluid and solid at these given flow rates. The resulting curves are plotted in Fig. 10(c), from which it is clear that the inlet temperature of the solid is slightly higher than the outlet temperature, and the temperature difference is within 7 K. The reason for this is that the heating coil is rolled around the outer surface (inlet) of the porous material and heat flux flows inward; therefore, the inlet temperature is higher than the outlet temperature. However, the conductivity of the metal porous material is very large; thus, the temperature difference is small. The temperature distribution of the fluid can be divided into two regimes: the thermal non-equilibrium regime, in which fluid temperature rises rapidly until it reaches the temperature of the solid; and the thermal equilibrium regime, where the temperature of the fluid is equal to that of the solid. The thermal non-equilibrium regime is rather small. This is because the average pore diameter of the porous material is very small; therefore, both the specific surface area and the fluid-to-solid heat transfer coefficient (see Eq. (9)– (11)) are sufficiently large so that heat transfer between the solid and fluid is extremely rapid. This also explains why the average temperature of the air is approximately equal to the temperature of the solid (see Fig. 9). On the other hand, at a given radius, for every 5 105 kg/s decrease in mass flow rate, the resulting change in temperature is more obvious. This is the reason why DP, c, and g change more rapidly in the small-flow-rate regime. Furthermore, from the perspective of pressure drop-flow rate characteristics, the temperature effect on the thermal conductivity of air can be ignored. The reason for this is as follows. First, in the regime of local thermal equilibrium, we have: T s ¼ T f . Based on Eqs. (5), (6), and (12), a governing equation for the thermal equilibrium regime is obtained:

2 kf ;eff @ Tf kf ;eff GC pf 1 @T f þ1 þ þ 1 ¼ 0: ks;eff @r2 ks;eff 2pH r @r Since

kf ;eff ks;eff

uk

¼ ð1ufÞks

0:02410:281 16ð10:281Þ

ð27Þ

¼ 0:00059 1, the effect of tem-

perature on the thermal conductivity of air will not influence the temperature distribution in the thermal equilibrium regime. Then, in a local thermal non-equilibrium regime, there exists the term

hfs Afs kf ;eff

¼

hfs Afs ukf

in Eqs. (5) and (6); therefore, the effect of tem-

Fig. 13. Relations between pressure difference, flow resistance, temperature effect ratio, average temperature, and flow rate at different outlet pressures and 10.8 W of heating power in porous material #1.

perature on the thermal conductivity of air influences the temperature distribution in the thermal non-equilibrium regime. However, the value of hfs Afs is very large, so the regime of thermal

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

non-equilibrium is negligibly small compared to the thermal equilibrium regime despite the fact that the term

hfs Afs kf ;eff

changes with the

variation in the temperature-dependent thermal conductivity of air, kf . According to Eq. (16), we know that the effect of temperature distribution on pressure drop-flow rate characteristics is determined by two temperature-dependent integral terms: R R2 lT R R2 T dr and dr. The negligibly small thermal nonR1 r R1 r 2 equilibrium regime contributes a negligible value in the two inteRR RR gral terms: R12 lrT dr and R12 rT2 dr. Thus, the variation in the thermal non-equilibrium regime has very little influence on the theoretical pressure drop-flow rate characteristic curves. 4.5. Comparison at different heating powers We performed a test comparing three different heating powers (5.3, 10.8, and 19 W). The experimental results are shown in Fig. 11. We observe that the higher the heating power, the greater the heat generated in the porous material and, consequently, the higher the temperature (at the same flow rate); the higher temperature results in a larger pressure drop DP, flow resistance c, and temperature-effect ratio g. At the same time, we also note that the experimental curves are within the corresponding adiabatic and room temperature curves and that all curves eventually converge to the curve for room temperature; this tendency does not change with changes in heating power. We obtained average temperature data and plot them in Fig. 11. The figure shows that the higher the heating power, the higher the average temperature of the air; the average temperature of the air converges to room temperature with increasing flow rate. Based on the measured temper-

539

ature boundary conditions we calculated theoretical curves at different heating powers as shown in Fig. 11. It is evident that the theoretical curves agree very well with the experimental data. Similarly, we calculated the temperature distributions of the solid and fluid at three different heating powers. For simplification, we omit the temperature of the solid and plot only the fluid temperature distribution in a two-dimensional form as shown in Fig. 12(a). Additionally, we set the mass flow rate to 21.53, 16.60, 11.64, and 6.57 105 kg/s (the interval is about 5 105 kg/s) and obtained corresponding temperature distribution curves for the fluid and solid as shown in Fig. 12(b). From the fluid temperature color map at different heating powers in Fig. 12(a), it is clear that the temperature of the fluid increases with increasing heating power. Meanwhile, from Fig. 12(b) we observe that the inlet temperature of the solid is higher than the outlet temperature. The temperature difference between them increases with increasing heating power, indicating that the heat transfer intensity also increases with heating power. Meanwhile, at a given radius, for every 5 105 kg/s decrease in mass flow rate, the temperature differences between two adjacent curves increase as the heating power increases. The results indicate that the higher the heating power, the more rapidly the temperature changes. This is illustrated in Fig. 11 by the relation between the average temperature and flow rate. From Fig. 11 it is clear that, for the same flow rate, a larger heating power results in a greater slope of the temperatureflow rate curve; i.e., temperature rises more rapidly. In the thermal equilibrium regime, the fluid temperature is equal to the solid temperature. However, in the thermal non-equilibrium regime we noticed that, although fluid temperature increases extremely rapidly, the temperature of the solid remains almost constant.

Fig. 14. Temperature distribution for different outlet pressures in porous material #1.

540

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

The reason for this is that the effective conductivity of the solid is almost 1560 times larger than that of the fluid phase. Thus, the dramatic variation in fluid temperature has a negligible effect on the temperature of the solid. 4.6. Effect of average density In Eq. (19), the outlet pressure P 2 is one of the parameters that affects the pressure drop and flow resistance. The peculiarity of the outlet pressure P2 lies in that it can be controlled by a pneumatic circuit (see Fig. 3). This quality may have important applications in practical use. To illustrate the physical meaning of P2 , we rearrange Eq. (23) to the form:

DPT ¼ DP T " # 1 P1 P2 GR R2 G2 b 1 1 þ þ ¼ ln : 1 2 RT RT 2pK uH R1 4p2 K 2 H2 u2 R1 R2 ð28Þ From Eq. (28), it is evident that the outlet pressure P 2 only exists 1 P1 þ RPT2 , which represents the average fluid density 2 RT

in the term

in the porous material. Thus, in fact, P 2 affects the pressure drop and flow resistance by influencing the average density of the fluid. Therefore, in this section, the effect of average density is discussed. To study the effect of average density, we used porous material #1 as the test sample and set P2 successively to positive (75 kPa (g)), atmospheric (0 kPa (g)), and negative pressures (45 kPa (g)). At heating powers of 0 and 10.8 W, we measured DP and G, calculated flow resistance c, and plotted the data in Fig. 13. At 298 K, the change in P 2 changes the DP G and c G curves considerably, as is evident from Eqs. (28) and (18). The higher the value of P 2 , the lower the value of DP needed for the same G, indicating that a higher outlet pressure results in a lower flow resistance. This is because, at the same flow rate, the decrease in the average density leads to an increase in the air velocity, leading in turn to an increase in the flow resistance caused by viscosity and inertia effects. When a constant power is used for heating, the DP G and c G curves deviate from the curve at 298 K. We observe that the increments in the pressure drop and flow resistance, which can be expressed by DPT DPT 0 and DcT DcT 0 , respectively, increase as the average density (i.e., the lower outlet pressure) declines. The average temperature curves at different outlet pressures are very close because average temperature is determined by heating power and flow rate, as stated in the energy equation. As for the temperature-effect ratio, the values for 0 kPa (g) and 45 kPa (g) nearly coincide while the value for 75 kPa (g) is the smallest. To explain this, based on Eqs. (23) and (26), we write the temperature effect ratio as:

gT ¼ 1

5=2 DPT DPT 0 T T 0 þ 110 1 þ ReT ; T0 T þ 110 1 þ ReT 0 ðDPT DP T 0 Þ þ DPT 0 þ 2P2 ð29Þ

where, ReT ¼

pﬃﬃﬃ Gb K 2plHuR2

ðR2 =R1 1Þ lnðR2 =R1 Þ

is the inlet Reynolds number at a given

temperature T. In Eq. (29), we deduce that the temperature-effect ratio g is determined by the pressure environment, temperature, and inlet Reynolds number. For the same porous material at a given flow 5=2 T 0 þ110 1þReT rate, TT0 is determined only by temperature. In genTþ110 1þReT

Fig. 15. Relationship between pressure difference, flow resistance, temperature effect ratio, average temperature, and flow rate at different outlet pressures and 10.8 W of heating power in different porous materials.

0

eral, the term DP T DP T 0 . n be neglected compared to DPT 0 þ 2P2 . DP DP T 0 ÞþDP T þ2P 2

Thus, we have 1 ðDP DPTT T

0

0

DP DP T 0 ; þ2P2

1 DPTT

therefore, g

0

decreases with decreasing P2 if average temperature is kept at

the same value, just as the adiabatic and experimental g G curves show in Fig. 13. We plot the two-dimensional fluid temperature distribution at different outlet pressures and the solid and fluid temperature dis-

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

tributions at given flow rates in Fig. 14(a) and (b), respectively. From Fig. 14, it is evident that the temperature distributions of both the fluid and solid are not affected by the outlet pressure. The reason for this is that temperature distribution is determined by heating power and flow rate as well as the external environment; the outlet pressure does not change any of these, and thus it does not affect the temperature distribution. 4.7. Effect of porous material characteristic parameters In this section, we discuss effect of porous material characteristic parameters on hydraulic behavior and flow resistance of heated porous materials. For heating powers of 0 and 10.8 W, we measured DP and G of porous materials #1, #2, and #3, calculated the corresponding flow resistance c, and plotted the data in Fig. 15. At 298 K, the smaller the average pore diameter of the porous material, i.e., the smaller the K and b. Values, the higher the pressure drop and flow resistance. When a heating power of 10.8 W is used, the DP G and c G curves shift upward from that at 298 K. The increment of the pressure drop DPT DPT 0 and that of flow resistance DcT DcT 0 increase with decreasing K and b. The average temperatures of the three porous materials overlapped for the same flow rate, resulting in an approximately equal temperature-effect ratio. The temperature-effect ratio differences of porous materials #1, #2, and #3 originate from pressure environment and inlet Reynolds number, as shown in Eq. (29). It must be mentioned that, although the values of DP T DPT 0 and DcT DcT 0 for material #3 are the smallest, the temperatureeffect ratio for material #3 is the largest.

541

The temperature distributions of porous materials #1, #2, and #3 at 10.8 W of heating power are shown in Fig. 16. It is clear that temperature distributions of different porous materials in the thermal equilibrium regime are very close to one another. However, the thermal non-equilibrium regime grows wider as the average pore size increases. This is because the larger the pore size, the smaller the specific surface area and the fluid-to-solid heat transfer coefficient; thus, the temperature difference narrows per unit distance; in other words, more distance is needed in the radial direction for the same temperature rise. 5. Conclusion In this study, we theoretically and experimentally analyzed the hydraulic characteristics, i.e., DP G characteristics, and flow resistance of heated porous materials under local thermal non-equilibrium conditions using air as the test fluid. The temperature-effect ratio g was defined to evaluate the extent of the effect of heating on hydraulic characteristics, flow resistance, and pumping power. Also, based on the experimental temperature boundary conditions, temperature distributions were obtained and discussed using numerical methods. Firstly, at room temperature, we rearranged the isothermal Forchheimer equation in Eq. (17) to obtain Eq. (22). From this we concluded that, in the Darcy regime, the term 1=K eff is a constant (i.e., 1/K) whereas, in the Forchheimer regime, it is approximately proportional toRep;in . Based on this characteristic, we determined the boundary of the Darcy and Forchheimer regimes and then calculated K using the data for the Darcy regime and b using the data

Fig. 16. Temperature distributions for different porous materials.

542

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543

Table 3 Uncertainty in the pressure drop measurement. Source of uncertainty

Value

Sensitivity coefficient

Converted value

Probability distribution

Divisor

Standard uncertainty

Calibration uncertainty of pressure sensor Resolution of pressure sensor

0–0.04 kPa 0.01 kPa

1 1

0–0.04 kPa 0.01 kPa

Normal Rectangular

0–0.02 kPa 006 kPa

Noise of pressure sensor

0.04 kPa

1

0.04 kPa

Rectangular

Calibration uncertainty of flow rate sensor Resolution of flow rate sensor

0.05 L/min 0.01 L/min

0–5.74 kPa/(L/min) 0–5.74 kPa/(L/min)

0–0.287 kPa 0–0.057 kPa

Normal Rectangular

Calibration uncertainty of temperature sensor Resolution of temperature sensor

1K 0.1 K

0–0.2 kPa/K 0–0.2 kPa/K

0–0.2 kPa 0–0.02 kPa

Normal Rectangular

Combined standard uncertainty Expanded uncertainty

– –

– –

– –

Assumed normal Assumed normal

2 pﬃﬃﬃ 3 pﬃﬃﬃ 3 2 pﬃﬃﬃ 3 2 pﬃﬃﬃ 3 – 2

for the Forchheimer regime. Based on the resulting K and b, flow resistance was studied experimentally and theoretically. It was seen that, in the case of a fixed P2 , cT ! a=2 (viscous flow resisG!0

tance) and cT ! b (inertial flow resistance). Secondly, we took the discussion of the effect of temperature on K as an example in studying the properties of heated porous materials. The results show that the effect of thermal expansion on porous material properties can be neglected in the range 298–450 K; therefore, we took K and b as temperature-independent constants. We analyzed theoretically the effects of temperature on the pressure drop and flow resistance at given constant temperatures and found that, for the same flow rate, DP, c, and g increase with the temperature of the flowing air. We then performed an experiment in which we kept the input heating power at 10.8 W to study the effect of heating on DP G characteristics and flow resistance. It was seen that the experimental DP G and c G curves at constant heating power shifted upward from room temperature curves owing to the increase in fluid temperature but remain within the adiabatic and room temperature curves. The temperature-effect ratio g. 10.8 W of heating power increased from 1.3 to 1.7 as the mass flow rate decreased from 21.53 105 kg/s to 5.80 105 kg/s, indicating that the hydraulic characteristics, flow resistance, and pumping power change significantly when a porous material is heated. Based on the temperature distributions, theoretical curves were obtained. The results show that the theoretical curves agree very well with the experimental data: error was within 6%. Furthermore, we studied the effect of the heating power, the average density of air in the porous material, and characteristic porous material parameters on DP G characteristics. It was found that (a) the greater the heating power, the larger the values of DP, c, and g; (b) the smaller the average density, the larger the effect of temperature on DP and c; and (c) the smaller the values of K and b, the larger the effect of temperature on DP and c. However, g remained approximately the same despite changes in K, b, and average density. G!1

Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 51375441) and the Science Fund for Creative Research Groups of National Natural Science Foundation of China (Grant No. 51221004).

0231 kPa 0–0.144 kPa 0.033 kPa 0–0.1 kPa 0.012 kPa 024–0.181 kPa 0.048–0.362 kPa

mated to be 0.01% of the reading value, which was between 0 kPa and 400 kPa in our experiment. Thus, the calibration uncertainty of the pressure sensor was 0–0.04 kPa. Given that the pressure sensor was digital, the resolution was 0.01 kPa. Moreover, the analog noise signal was experimentally confirmed to be around 0.04 kPa. (2) The supply flow rate was set as the experimental condition and was measured by the thermal flow rate sensor (KEYENCE, FD-A10). The flow rate error due to the calibration uncertainty and resolution affected the pressure measurement. The flow rate sensor was calibrated, and the calibration uncertainty was estimated to be 0.5% of the maximum reading (10 L/min or 21.53 105 kg/s). Thus, the calibration uncertainty was 0.05 L/min. Given that the flow rate sensor was digital, the resolution was 0.01 L/min. The sensitivity coefficient of the flow rate sensor can be calculated by differentiating the pressure drop versus flow rate from the experimental pressure result of Figs. 7–9, 11, 13 and 15. The value of the sensitivity coefficient was between 0–5.74 kPa/(L/min). (3) Temperature sensors (OMEGA, range: 223–673 K; resolution: 0.1 K) were used to confirm the stable thermal condition of the system. Thus, the calibration uncertainty and resolution of the temperature sensor affected the pressure measurement. The calibration uncertainty was estimated to be 0.5% of the full scale. According to the experimental results shown in Figs. 7–9, 11, 13 and 15, we know that the maximum sensitivity coefficient was 0.2 kPa/K, at which point increasing the temperature from 298 K to 338 K increased the pressure drop by 8 kPa (see Fig. 13). Meanwhile, Fig. 15 shows that the pressure drop of porous material #3 almost did not change in spite of the flow rate change. Therefore, the sensitivity coefficient is variable over a range of 0–0.2 kPa/K. Further detailed information on the above possible error sources, e.g., specific values, probability distribution, and divisor, are listed in Table 3. Next, we convert these individual errors into standard uncertainty and calculate the combined standard uncertainty by squaring the individual uncertainties, adding them, and then taking the square root of the total. Finally, we multiplied the result by a coverage factor (equal to 2) to obtain the expanded uncertainty [49]. The uncertainty of the pressure measurement was 0.048–0.362 kPa. This is based on the standard uncertainty multiplied by the coverage factor of 2, providing a level of confidence of approximately 95%.

Appendix A In order to calculate the uncertainty in the pressure drop measurement experiment (see Section 4), we first list the following factors that may cause measurement errors: (1) The high-precision pressure sensor (YOKOGAWA, CA700) was calibrated, and the calibration uncertainty was esti-

References [1] T. Hosokawa, K. Somaya, M. Miyatake, S. Yoshimoto, Static characteristics of aerostatic thrust bearings with multiple porous inlet ports, J. Tribol.-Trans. ASME 137 (2) (2015). [2] T.-Y. Huang, S.-C. Shen, S.-C. Lin, S.-Y. Hsu, Pressure distribution in the air film and the porous conveyor air bearing, Appl. Sci. Precis. Eng. Innovation Pts 1 and 2, 479–480 (2014) 380–384.

Y. Liao et al. / International Journal of Heat and Mass Transfer 102 (2016) 528–543 [3] N. Oiwa, M. Masuda, T. Hirayama, T. Matsuoka, H. Yabe, Deformation and flying height orbit of glass sheets on aerostatic porous bearing guides, Tribol. Int. 48 (2012) 2–7. [4] Y.B.P. Kwan, J. Corbett, Porous aerostatic bearings – An updated review, Wear 222 (2) (1998) 69–73. [5] R. Nicoletti, B.d.M. Purquerio, Z.d.C. Silveira, The effect of permeability distribution on the numerical analysis of aerostatic ceramic porous bearings, Lubr. Sci. 25 (2) (2013) 185–194. [6] K. Kuroshita, Y. Sekiguchi, K. Oshiki, N. Oneyama, Development of new test method for flow-rate characteristics of pneumatic components, Power Transm. Motion Control (2004) 243–256. [7] J. Ma, C. Juan, Z. Ke, S. Mitsuru, IEEE, Flow-rate characteristics parameters of pneumatic component, in: 2008 IEEE International Conference on Automation and Logistics, vols. 1–6, 2008, pp. 2946–2949. [8] J. Ma, B. Wu, J. Chen, Z. Tang, N. Oneyama, Error comparison between ISO/WD 6358-2 and ISO 6358:1989, in: Proc. 7th International Symposium on Instrumentation and Control Technology, 2008. [9] T. Wang, G. Peng, T. Kagawa, Determination of flow-rate characteristics for pneumatic components using a quasi-isothermal tank with temperature compensation, Meas. Sci. Technol. 21 (6) (2010). [10] Z.-P. Xu, X.-Y. Wang, Pneumatic resistance network analysis and dimension optimization of high pressure electronic pneumatic pressure reducing valve, J. Central South Univ. Technol. 18 (3) (2011) 666–671. [11] J.L. Lage, B.V. Antohe, D.A. Nield, Two types of nonlinear pressure-drop versus flow-rate relation observed for saturated porous media, J. Fluids Eng.-Trans. ASME 119 (3) (1997) 700–706. [12] S. Mancin, C. Zilio, A. Cavallini, L. Rossetto, Pressure drop during air flow in aluminum foams, Int. J. Heat Mass Transfer 53 (15–16) (2010) 3121–3130. [13] W. Zhong, G. Tao, X. Li, K. Kawashima, T. Kagawa, Determination of flow rate characteristics of porous media using charge method, Flow Meas. Instrum. (2011) 201–207. [14] D. Calamas, J. Baker, M. Sharif, Flow behavior and pressure drop in porous disks with bifurcating flow passages, J. Fluids Eng.-Trans. ASME 135 (10) (2013). [15] C. Naaktgeboren, P.S. Krueger, J.L. Lage, Inlet and outlet pressure-drop effects on the determination of permeability and form coefficient of a porous medium, J. Fluids Eng.-Trans. ASME 134 (5) (2012). [16] Y. Jin, K.P. Chen, M. Chen, Highly compressible porous media flow near a wellbore: effect of gas acceleration, J. Fluids Eng.-Trans. ASME 134 (1) (2012). [17] N. Dukhan, O. Bagci, M. Ozdemir, Metal foam hydrodynamics: flow regimes from pre-Darcy to turbulent, Int. J. Heat Mass Transfer 77 (2014) 114–123. [18] M. Odabaee, S. Mancin, K. Hooman, Metal foam heat exchangers for thermal management of fuel cell systems – An experimental study, Exp. Thermal Fluid Sci. 51 (2013) 214–219. [19] N. Dukhan, M.A. Al-Rammahi, A.S. Suleiman, Fluid temperature measurements inside metal foam and comparison to Brinkman–Darcy flow convection analysis, Int. J. Heat Mass Transfer 67 (2013) 877–884. [20] C. Zhang, B. Yan, J. Wieberdink, P.Y. Li, J.D. Van de Ven, E. Loth, T.W. Simon, Thermal analysis of a compressor for application to compressed air energy storage, Appl. Therm. Eng. 73 (2) (2014) 1402–1411. [21] W. Aboelsoud, W. Wu, L.C. Chow, B.A. Saarloos, D.P. Rini, Analysis of thermal and hydraulic performance of V-shape corrugated carbon foam, Int. J. Heat Mass Transfer 78 (2014) 1114–1125. [22] K.P. Carpenter, A.K. da Silva, A combined hydro-thermal characterization of high-porosity metal foam test sections with discrete pore-size gradients, Int. J. Heat Mass Transfer 77 (2014) 770–776. [23] K. Boomsma, D. Poulikakos, F. Zwick, Metal foams as compact high performance heat exchangers, Mech. Mater. 35 (12) (2003) 1161–1176. [24] L. Chen, H. Feng, Z. Xie, F. Sun, ‘‘Volume-point” mass transfer constructal optimization based on flow resistance minimization with cylindrical element, Int. J. Heat Mass Transfer 89 (2015) 1135–1140. [25] S.H. Yeo, S.R. Lee, C.H. Lee, Effect of gas temperature on flow rate characteristics of an averaging pitot tube type flow meter, J. Mech. Sci. Technol. 29 (1) (2015) 241–249. [26] D.-S. Oh, C.-H. Lee, A comparative study of flow rate characteristics of an averaging Pitot tube type flow meter according to H parameters based on two kinds of differential pressure measured at the flow meter with varying air temperature, J. Mech. Sci. Technol. 25 (8) (2011) 1961–1967. [27] P. Lange, M. Weiss, S. Warnat, Characteristics of a micro-mechanical thermal flow sensor based on a two hot wires principle with constant

[28]

[29]

[30] [31] [32]

[33]

[34]

[35]

[36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

[49]

543

temperature operation in a small channel, J. Micromech. Microeng. 24 (12) (2014) 12. A. Narasimhan, J.L. Lage, Forced convection of a fluid with temperaturedependent viscosity flowing through a porous medium channel, Numer. Heat Transfer Part A-Appl. 40 (8) (2001) 801–820. A. Narasimhan, J.L. Lage, D.A. Nield, New theory for forced convection through porous media by fluids with temperature-dependent viscosity, J. Heat Transfer-Trans. ASME 123 (6) (2001) 1045–1051. M.H. Vanderlaan, S.W. Van Sciver, He II heat transfer through random packed spheres: pressure drop, Cryogenics 63 (2014) 37–42. M.H. Vanderlaan, S.W. Van Sciver, Steady state He II heat transfer through random packed spheres, Cryogenics 57 (2013) 166–172. J.H. Wu, Z.P. Hu, H. Zhou, Sound absorbing property of porous metal materials with high temperature and high sound pressure by turbulence analogy, J. Appl. Phys. 113 (19) (2013). Y. Mahmoudi, Effect of thermal radiation on temperature differential in a porous medium under local thermal non-equilibrium condition, Int. J. Heat Mass Transfer 76 (2014) 105–121. P. Wang, K. Vafai, D.Y. Liu, C. Xu, Analysis of collimated irradiation under local thermal non-equilibrium condition in a packed bed, Int. J. Heat Mass Transfer 80 (2015) 789–801. Y. Mahmoudi, Constant wall heat flux boundary condition in micro-channels filled with a porous medium with internal heat generation under local thermal non-equilibrium condition, Int. J. Heat Mass Transfer 85 (2015) 524–542. Y. Mahmoudi, M. Maerefat, Analytical investigation of heat transfer enhancement in a channel partially filled with a porous material under local thermal non-equilibrium condition, Int. J. Therm. Sci. 50 (12) (2011) 2386– 2401. Y. Mahmoudi, N. Karimi, K. Mazaheri, Analytical investigation of heat transfer enhancement in a channel partially filled with a porous material under local thermal non-equilibrium condition: effects of different thermal boundary conditions at the porous-fluid interface, Int. J. Heat Mass Transfer 70 (2014) 875–891. Y. Mahmoudi, N. Karimi, Numerical investigation of heat transfer enhancement in a pipe partially filled with a porous material under local thermal non-equilibrium condition, Int. J. Heat Mass Transfer 68 (2014) 161– 173. M. Dehghan, M.S. Valipour, A. Keshmiri, S. Saedodin, N. Shokri, On the thermally developing forced convection through a porous material under the local thermal non-equilibrium condition: an analytical study, Int. J. Heat Mass Transfer 92 (2016) 815–823. F. Wu, W. Zhou, X. Ma, Natural convection in a porous rectangular enclosure with sinusoidal temperature distributions on both side walls using a thermal non-equilibrium model, Int. J. Heat Mass Transfer 85 (2015) 756–771. M.K. Alkam, M.A. Al-Nimr, M.O. Hamdan, Enhancing heat transfer in parallelplate channels by using porous inserts, Int. J. Heat Mass Transfer 44 (5) (2001) 931–938. B.I. Pavel, A.A. Mohamad, An experimental and numerical study on heat transfer enhancement for gas heat exchangers fitted with porous media, Int. J. Heat Mass Transfer 47 (23) (2004) 4939–4952. K. Hooman, A.A. Merrikh, Analytical solution of forced convection in a duct of rectangular cross section saturated by a porous medium, J. Heat TransferTrans. ASME 128 (6) (2006) 596–600. K. Hooman, H. Gurgenci, A.A. Merrikh, Heat transfer and entropy generation optimization of forced convection in porous-saturated ducts of rectangular cross-section, Int. J. Heat Mass Transfer 50 (11–12) (2007) 2051–2059. M.E. Nimvari, M. Maerefat, M.K. El-Hossaini, Numerical simulation of turbulent flow and heat transfer in a channel partially filled with a porous media, Int. J. Therm. Sci. 60 (2012) 131–141. B. Alazmi, K. Vafai, Constant wall heat flux boundary conditions in porous media under local thermal non-equilibrium conditions, Int. J. Heat Mass Transfer 45 (15) (2002) 3071–3087. D.-P. Tan, L.-B. Zhang, A WP-based nonlinear vibration sensing method for invisible liquid steel slag detection, Sens. Actuators B-Chemical 202 (2014) 1257–1269. W. Zhong, X. Li, F. Liu, G. Tao, B. Lu, T. Kagawa, Measurement and correlation of pressure drop characteristics for air flow through sintered metal porous media, Transp. Porous Media 101 (1) (2014) 53–67. X. Li, M. Horie, T. Kagawa, Study on the basic characteristics of a vortex bearing element, Int. J. Adv. Manuf. Technol. 64 (1–4) (2013) 1–12.

Study of pressure drop-flow rate and flow resistance characteristics of heated porous materials under local thermal non-equilibrium conditions

Study of pressure drop-flow rate and flow resistance characteristics of heated porous materials under local thermal non-equilibrium conditions

Recommend Documents