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Pore-scale study on flow and heat transfer in 3D reconstructed porous media using micro-tomography images
Accepted Manuscript Title: Pore-scale study on flow and heat transfer in 3D reconstructed porous media using micro-tomography images Author: Zhenyu Liu, Huiying Wu PII: DOI: Reference:
S1359-4311(16)30200-9 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.02.057 ATE 7786
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
12-8-2015 20-2-2016
Please cite this article as: Zhenyu Liu, Huiying Wu, Pore-scale study on flow and heat transfer in 3D reconstructed porous media using micro-tomography images, Applied Thermal Engineering (2016), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.02.057. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Pore-scale study on flow and heat transfer in 3D reconstructed porous media using micro-tomography images Zhenyu Liu, Huiying Wu* School of Mechanical Engineering, Shanghai Jiao Tong University, China 200240 * Corresponding author: Tel.: ++86 (021) 34205299; Fax: ++86 (021) 34205299; Email: [email protected] Highlights
The complex porous domain has been reconstructed with the micro CT scan images.
Pore-scale numerical model based on LB method has been established.
The correlations for flow and heat transfer were derived from the predictions.
The numerical approach developed in this work is suitable for complex porous media.
Abstract This paper presents the numerical study on fluid flow and heat transfer in reconstructed porous media at the pore-scale with the double-population thermal lattice Boltzmann (LB) method. The porous geometry was reconstructed using micro-tomography images from micro-CT scanner. The thermal LB model was numerically tested before simulation and a good agreement was achieved by compared with the existing results. The detailed distributions of velocity and temperature in complex pore spaces were obtained from the pore-scale simulation. The correlations for flow and heat transfer in the specific porous media sample were derived based on the numerical results. The numerical method established in this work provides a promising approach to predict pore-scale flow and heat transfer characteristics in reconstructed porous domain with real geometrical effect, which can be extended for the continuum modeling of the transport process in porous media at macro-scale. Keywords: Pore-Scale, Flow, Heat Transfer, Porous Media, Geometrical Reconstruction
1. Introduction The heat storage is one of important technologies for the energy saving, which includes sensible heat storage and latent heat storage[1, 2]. The porous media can be used as heat storage material or to enhance the heat transfer in the heat storage equipment. But there still exists further investigation that should be focused on the detailed characteristics of flow and heat transfer in complex porous media [3, 4]. The transport phenomena in porous media can span several orders in temporal and spatial scales. Modeling transport phenomena in porous media should adopt different assumptions and governing equations for different spatial scales [5, 6]. Normally, the flow and heat transfer of fluid in porous media can be studied at the representative elementary volume (REV) scale, which is much larger than the porescale, but much smaller than the macro-scale [7, 8]. REV scale study is capable of obtaining the macroscopic property values related to macro-scale transport equation, such as porosity, permeability and local heat transfer coefficient, which are not influenced by the REV size any more. So the continuum modeling of the transport process in porous media can be performed based on these REV scale study. For example, the Darcy flow model or its modified ones can describe the relation between pressure drop and velocity as the fluid flows through the porous domain. And the nonequilibrium thermal model can describe the heat transfer between the fluid and the solid structure in porous domain, which is more accurately compared to the equilibrium thermal model [9-11]. As mentioned above, the use of the empirical correlations in the transport equations is necessary in the continuum modeling. These macroscopic parameters and their relation can be determined or derived from experimental and analytical studies. The experimental approach is relatively high cost and time consuming. In fact, the velocity or temperature distribution in the obstructed complex porous geometry is difficult to be obtained accurately in experimental way. The analytical study is suitable for the simple geometry, which is hardly applied to the complex and irregular real structure. So the pore-scale study becomes one innovative and promising approach in investigating the transport phenomena in porous media [12, 13]. The macroscopic parameters can be accurately determined with the detailed velocity and temperature distributions in the pore spaces obtained from pore-scale predictions. The study of transport phenomena in porous media at the pore-scale has attracted more and more attentions in recent years [14-16]. The computational fluid dynamics (CFD) is an effective tool in the fluid flow and heat transfer modeling and simulation [17-19]. The Finite Volume method (FVM) can be employed to solve the Navier-Stokes and energy equations for the fluid flow and heat transfer, which has been proved to be an available numerical method in the cases with idealized geometrical boundaries. At the pore-scale level, Navier-Stokes equations are capable of governing the pore fluid
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motion. Kopanidis et al. [20] presented pore-scale simulation of flow and heat transfer in open cell metal foam. The pore-scale structure was described in a three dimensional simplified geometrical model and it was discretized with tetrahedral volume meshes for the void and solid phases. Dyck and Straatman [21] used FVM based CFD software Ansys CFX to predict the hydraulic and thermal parameters in the digital samples of spherical-void-phase carbon foam and relevant transport properties were obtained from the numerical results. As the porous media structure becomes more complex, the FVM modeling may meet with the meshing or convergence problem. The lattice Boltzmann (LB) modeling has been developed rapidly as a new and promising numerical method over the last decade [22, 23]. Guo et al. [24] developed a thermal lattice BGK (Bhatnagar, Gross and Krook) model, in which the flow and heat transfer process were simulated using two independent lattice BGK equations in a coupled model. In the work of Hosseini et al. [25] , the transport processes in porous media were simulated at the pore-scale using lattice Boltzmann method (LBM), in which the porous structure was simplified as randomly distributed solid bars. Due to the complexity of the pore geometry, most of pore-scale modeling was carried out in the porous domain with geometrical simplifications. The numerical modeling with these idealized porous domains can provide pore-level information on flow and heat transfer processes. But it is difficult to establish simplified porous geometrical model equivalent to the real porous media in both thermal and hydrodynamic performance, so the effect of real porous structure has to be considered in the numerical modeling. Li et al. [26] simulated transport phenomena in a porous wick on pore-scale by LBM, in which the random porous media was numerically reconstructed with the stochastic method. Image-based porescale modeling was proved to be a promising method to predict fluid flow in real porous media [27, 28]. Zhu et al. [29] performed geometrical reconstruction for aluminum foam with MATLAB image processing and CT scanning. Ranut et al. [30] used X-ray computed micro tomography technique to describe the micro-structures of metal foam. In the above work, the velocity and temperature distributions were obtained from FVM based CFD simulation. Very few studies on LB modeling for flow and heat transfer processes with geometrical reconstruction of real porous media can be found in open literature up to now. In this paper, a double-population function thermal LB model was employed in the numerical simulation and geometrical reconstruction of Berea sandstone sample was performed to establish the computational domain. Pore-scale simulation in three-dimensional reconstructed porous media was carried out to study the flow and heat transfer processes considering the effect of real porous structure. The velocity and temperature distributions in pore spaces were obtained from the pore-scale numerical predictions. The correlations for flow and heat transfer in the specific porous media sample were derived based on the numerical results in this work.
2. 3D Double-Population Thermal Lattice Boltzmann Model The double-population function is capable of simulating both flow and heat transfer simultaneously in the LBM modeling [31]. The D3G15 LBM model can be used to describe the fluid flow and the D3Q6 thermal LBM model is available for simulating the temperature variation in the pore spaces. The lattice velocity directions for these two models are shown in Fig. 1. For a particle at position distribution functions
and at time , the probability of this particle with a velocity can be defined as particle , and the Boltzmann equation describes its evolution as [32]: (1)
Ω on the right side of equal sign in Eq. (1) is the collision operator, which describes the particles interaction. The directional particle velocities can be introduced to simplify the Boltzmann equations and the velocity space is discretized with D3Q15 model in the flow simulating: (2) α = 0, …, 14. In D3Q15 model, 14 directional particle velocities point to neighboring nodes and one equals zero. These equations are discretized in both space and time, in which c = Δx/Δt (node spacing Δx, time step Δt), and the lattice Boltzmann equation can be expressed as: (3) is the dimensionless relaxation time for velocity, an
is the equilibrium distribution function described as:
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(4)
In the above equation,
can be defined as: (5)
where = 2/9, = 1/9 for α = 1–6, = 1/71 for α= 7–14, and is the pseudo sound speed ( , and in equilibrium distribution function can be determined by solving Eq. (6)-(8):
). And
(6) (7) (8) It can be obtained that
,
,
.
Macroscopic parameter, such as density ρ, momentum ρ
and pressure
can be obtained from following equations: (9) (10) (11)
And the macroscopic viscosity
can be calculated by: (12)
The thermal LB equation for temperature field is expressed as [24]: (13) where 6), and
is the dimensionless relaxation time for temperature, is the distribution function of temperature ( is the equilibrium distribution function of temperature:
= 1-
(14) The macroscopic temperature can be obtained by the following expression: (15) To validate the present flow model, a comparison of the numerical results using LBM, FVM and analytical method was performed. Poiseuille flow was numerically simulated and the grid independency was tested before simulation. The predicted non-dimensional velocity profiles are shown in Fig. 2(a). The comparison of LBM, FVM and analytical solution Eq. (16) shows a good agreement. (16) To validate the thermal model, the local Nusselt number (Nu) is defined as: (17)
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Fig. 2(b) shows Nusselt number variation along the flow direction (the non-dimensional position is defined as (x/H)/(RePr)). The predicted Nu reaches a constant value of 7.61 as it reaches the fully-developed condition and the analytical value was given as 7.54 in [33], which shows a difference of 0.93%.
3. Porous Media Reconstruction and Boundary Conditions The available and promising geometrical reconstruction approaches is to generate the porous media using the tomographic reconstruction, which is a nondestructive imaging technique. 2D sliced images of the micro CT scanned sample can be obtained and stacked to reconstruct the 3D real geometry. Berea sandstone can be used as a standard porous media material due to its relatively homogenous and well-characterized property, which was adopted in this numerical model to reconstruct the porous computational domain with real porous structure characteristic. A Berea sandstone sample with a diameter of 1 cm was scanned on a XRADIA Micro XCT-200 scanner and the resolution of scanned images is 12.5 μm. Then these images were processed with a Java based open source software ImageJ, which performed noise reduction and thresholding processing. The threshold values were determined for each scan image and the background and foreground were defined as black and white, respectively. Due to the blur area near the edge, the images were trimmed to achieve a more accurately reconstructed geometry. The final binary format images are shown in Fig. 3(a). The 3D geometry model (5mm×5mm×5mm) in Fig. 3(b), which was reconstructed by image processing software Mimics, represents the real porous structure and it was imported into the LBM code with a format of STL. The experimental porosity value of porous sample is 0.61 and the porosity of reconstructed one is 0.58. The difference in porosity is mainly caused by the existing of small particles below the scanner’s resolution and the loss during image processing. The numerical work simulated the water flowing through a cubic domain filled with reconstructed porous media as shown in Fig. 4. The inlet and outlet are set at different pressure values on the left and right side: Pin = 20, 40, 60, 80 Pa and Pout = 0 Pa. The other four walls are set as static and adiabatic wall boundary conditions. The temperature of solid porous structure is set at 320 K, and the temperature of fluid at the inlet is at T = 300 K. The flow is assumed to be three dimensional and laminar. The fluid properties are constant ones and the no-slip boundary condition is applied in flow modeling. The resolution in the LB model should be improved to describe the accurate pore spaces and pore fluid flow characteristics. In LBM modeling, the numerical resolution normally increases by placing more lattice nodes in the computational domain and the independence of resolution should be tested before the simulation. We performed simulations with lattice number of 0.4 × 10 6, 0.8 × 106, 1.6 × 106, 3.2 × 106 and 6.4 × 106, respectively. It is shown that the velocity distribution varies in a very small difference as the lattice number reaches 1.6 × 106. The computational cost is one of important concerning in the numerical modeling. So 1.6 × 10 6 is the number of lattice nodes we adopted in the numerical modeling and simulation in this work.
4. Results and Discussion 4.1 Pore-scale fluid flow Figure 5 shows the predicted averaged velocity (the flow rate divided by the flow area in cross section) along the x axis for different inlet pressure conditions. It shows that an increase in the pressure gradient results in an increase in averaged velocity in the cross section. For a fixed inlet pressure, the averaged velocity varies in a small range along the flow direction, which is due to the difference in the flow area on the different x position. Fig. 6 (a)-(d) show the velocity distribution in the plane of z=2.5 mm for different inlet pressures, it can be found that the velocity distribution varies intensely due to the existence of irregular porous structure in the computational domain. It is obvious that liquid flows through narrow flow area at a high velocity and the velocity of fluid near the static wall is relatively slow due to the static boundary condition. The flow velocity vectors in longitudinal section (z=2.5 mm) and in cross section (x=2.5 mm) are shown in Fig. 7 (a) and (b), respectively. In these two figures, a long vector represents a high velocity and the influence of complex pore structure on the flow characteristics can be well observed. Because the liquid flow should be driven along the direction of the pressure gradient, the water in the pore spaces mainly flows along the x direction due to the pressure drop between inlet and outlet of porous domain. Fig. 7(a) shows the liquid flowing in the confined pore spaces and the flow status is complicated due to the irregular porous structure. In Fig. 7(b), the flow velocity vectors can be clearly observed in the plane of x=2.5 mm, which is not along the main flow direction. The complicated flow field in the real porous domain will have a direct influence on the heat transfer characteristics. Eq. (18) can present the relation between pressure drop and superficial velocity, which is extended from Darcy’s law expression. -dP/dx is the pressure gradient along the x direction in the present model, U is the averaged superficial -4Page 4 of 9
velocity, K is the permeability, and is the inertia coefficient. The first term on the right side of Eq. (18) represents influence of viscous force and the second term represents influence of inertia force. At a low velocity, the viscous force is dominant and pressure drop is mainly caused by the first term, in which the second term can be neglected. However, the inertia force is dominant at a higher velocity and second term can not be neglected any more. So a linear relation between pressure drop and velocity dominates at low velocities, and a nonlinear relation dominates as velocities reach a higher value. It can be obtained from the numerical predictions for the coefficients K and : , , which is applicable for the specific porous media sample adopted in this work. (18)
4.2 Pore-scale fluid heat transfer Temperature distributions in the longitudinal section of z=2.5 mm for different inlet pressures are shown in Fig. 8 (a)(d). It can be observed that increasing the inlet pressure reduces the outlet liquid temperature. In Fig. 8(a), the liquid temperature increases to a high value in a short distance as it enters the porous domain, and it nearly reaches 320 K as it flows out of the domain. For high inlet pressure boundary condition, the liquid temperature keeps at a low value as it flows through the porous domain. In the narrow flow area, the liquid temperature increases quickly due to the small flow rate. For where wide flow area exists, the heat transfer process still undergoes near the outlet of porous domain, as shown in Fig. 8 (d). The heat is transferred mainly by the conduction form as the velocity in the porous domain is very slow. For a higher inlet pressure, the flow velocity in the domain increases and complex flow status occurs due to the irregular porous structure, the heat will be transferred by the convection form under this condition. An increase in inlet pressure leads to a higher convective heat transfer coefficient, but it decreases the amount of heat transferred per unit mass due to the high flow rate. To represent the averaged temperature profile in the porous domain, the non-dimensional temperature in the following form:
＝
is introduced
(19)
where = 320 K is the temperature of the solid porous structure, is the inlet liquid temperature (300 K), and is the local liquid temperature, which is an averaged value over the cross section. The variations of along x direction for different inlet pressures are shown in Fig. 9. The curve with a higher inlet pressure is below the curve with a lower inlet pressure. The liquid needs to flow more distance along the main flow direction to reach thermal equilibrium for a higher inlet pressure, which is mainly because of the low residence time of fluid particles in the pore space. The flow particles with low velocities have a higher residence time and reach the thermal equilibrium in a short distance in the main flow direction. The bulk liquid temperature depends on the location along the main flow direction, which can be calculated by the following expression: (20) The local convective heat transfer coefficient
can be calculated by the following expression: (21)
where A is heat transfer area in cross section and q is the heat flux from solid to liquid. Fig. 10 presents variations along the main flow direction in the porous domain for different inlet pressures. It appears that varies in a limited range at different positions on x-axis. However, its value is mainly determined by the velocity distribution in the pore space. It should be noted that the velocity distribution is mainly influenced by the complex pore structure and the pressure drop across the porous domain. To analytically treat local thermal non-equilibrium phenomenon in porous media, Kuznetsov studied the thermal non-equilibrium effect with the high production of convective heat transfer coefficient and the specific surface area [34]. And several successful applications were performed to analyze the flow and heat transfer in porous media under the local thermal non-equilibrium condition [35-37]. The accurate values of convective heat transfer coefficient and the specific surface area can be obtained from the pore-scale numerical simulation work presented in this paper. So it is available to extend the pore-scale model to analytical model to investigate the transport phenomena in porous media under the local thermal non-equilibrium condition. The three-dimensional temperature profile in the porous media is presented in Fig. 11. It is obvious that the temperature distribution varies intensely in the porous domain. The complex porous structure leads to the flow status varying on the -5Page 5 of 9
different location in the pore space, the heat transfer process will vary accordingly. In order to obtain a better understanding of whole heat transfer characteristics, a relation between Nusselt number and Reynolds number can be derived based on the numerical prediction in this work. The characteristic length used to calculate the Nusselt number (
) and Reynolds number (
) is based on the permeability in this paper. The relation between heat
transfer and flow is presented in Eq. (22), for which . It should be noted that this relation is suitable for the specific porous media adopted in this numerical work. For different porous media, the same procedure can be carried out to obtain the unique correlation between flow and heat transfer, which can be extended for the continuum modeling of the transport process in porous media. (22)
4. Conclusion The flow and heat transfer were successfully predicted in a complex porous medium at the pore-scale by doublepopulation thermal LB method. The porous geometry was reconstructed based on the micro CT scan images of Berea sandstone sample. The thermal LB model was numerically tested before simulation and a good agreement was achieved by compared with the existing CFD and analytical results. The numerical approach established in this work is capable of predicting transport phenomena in porous media at pore-scale by importing 3D reconstructed porous domain into the numerical model. The detailed velocity and temperature distributions in complex pore spaces were obtained from the pore-scale numerical simulation. The correlations for flow and heat transfer in the specific Berea sandstone porous media were derived based on the numerical results and the procedure developed in this paper can be applicable for porous media with different geometrical characteristics.
Nomenclature A c cs Cf f g h H K Nu p Pr q Re t T u U x,y,z x
heat transfer area, m2 velocity, m/s pseudo sound speed, m/s inertia coefficient, particle velocity, m/s particle distribution function of velocity, particle distribution function of temperature, heat transfer coefficient, W/(m2·oC) height, m permeability, md Nusselt number, pressure, N/m2 Prandtl number, heat flow, W Reynolds number, time, s Temperature, K x-velocity, m/s dimensionless velocity, - or superficial velocity, m/s velocity vector, m/s Cartesian coordinates position, (x,y,z)
Greek
Ω
velocity, m/s dimensionless temperature, dynamic viscosity, kg/(m·s) density, kg/m3 dimensionless relaxation time, viscosity, Pa·s collision operator, -
Subscripts b f
boundary fluid
μ ρ
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in lf s T
inlet local fluid solid temperature
Acknowledgements This paper is supported by the National Natural Science Foundation of China through grant no. 51306119 & 51536005, the National Basic Research Program of China (973 Program) through grant no. 2012CB720404 and Key Basic Research Projects of Science and Technology Commission of Shanghai through grant no. 12JC1405100.
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Figure Captions Figure 1 Lattice velocity directions (a) D3G15 (b) D3Q6 Figure 2 Validation for the numerical model (a) Velocity profile (b) Nu variation Figure 3 Reconstruction of porous media (a) 2D scan images (b) Reconstructed porous media (5mm×5mm×5mm) Figure 4 View of geometry model and boundary condition Figure 5 Averaged velocities along the flow direction for different Pin Figure 6 Velocity distribution in the plane of z=2.5 mm (a) Pin=20Pa (b) Pin=40Pa (c) Pin=60Pa (d) Pin=80Pa Figure 7 Velocity vectors in the longitudinal and cross sections (a)z=2.5mm (b)x=2.5mm Figure 8 Temperature distribution at z=2.5 mm (a) Pin=20Pa (b) Pin=40Pa (c) Pin=60Pa (d) Pin=80Pa Figure 9 The dimensionless temperature distribution along the main flow direction for different inlet pressures Figure 10 The local convective heat transfer coefficient along the flow direction for different inlet conditions Figure 11 3D temperature profile in reconstructed porous media (P in=80 Pa)
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S1359-4311(16)30200-9 http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.02.057 ATE 7786
To appear in:
Applied Thermal Engineering
Received date: Accepted date:
12-8-2015 20-2-2016
Please cite this article as: Zhenyu Liu, Huiying Wu, Pore-scale study on flow and heat transfer in 3D reconstructed porous media using micro-tomography images, Applied Thermal Engineering (2016), http://dx.doi.org/doi: 10.1016/j.applthermaleng.2016.02.057. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Pore-scale study on flow and heat transfer in 3D reconstructed porous media using micro-tomography images Zhenyu Liu, Huiying Wu* School of Mechanical Engineering, Shanghai Jiao Tong University, China 200240 * Corresponding author: Tel.: ++86 (021) 34205299; Fax: ++86 (021) 34205299; Email: [email protected] Highlights
The complex porous domain has been reconstructed with the micro CT scan images.
Pore-scale numerical model based on LB method has been established.
The correlations for flow and heat transfer were derived from the predictions.
The numerical approach developed in this work is suitable for complex porous media.
Abstract This paper presents the numerical study on fluid flow and heat transfer in reconstructed porous media at the pore-scale with the double-population thermal lattice Boltzmann (LB) method. The porous geometry was reconstructed using micro-tomography images from micro-CT scanner. The thermal LB model was numerically tested before simulation and a good agreement was achieved by compared with the existing results. The detailed distributions of velocity and temperature in complex pore spaces were obtained from the pore-scale simulation. The correlations for flow and heat transfer in the specific porous media sample were derived based on the numerical results. The numerical method established in this work provides a promising approach to predict pore-scale flow and heat transfer characteristics in reconstructed porous domain with real geometrical effect, which can be extended for the continuum modeling of the transport process in porous media at macro-scale. Keywords: Pore-Scale, Flow, Heat Transfer, Porous Media, Geometrical Reconstruction
1. Introduction The heat storage is one of important technologies for the energy saving, which includes sensible heat storage and latent heat storage[1, 2]. The porous media can be used as heat storage material or to enhance the heat transfer in the heat storage equipment. But there still exists further investigation that should be focused on the detailed characteristics of flow and heat transfer in complex porous media [3, 4]. The transport phenomena in porous media can span several orders in temporal and spatial scales. Modeling transport phenomena in porous media should adopt different assumptions and governing equations for different spatial scales [5, 6]. Normally, the flow and heat transfer of fluid in porous media can be studied at the representative elementary volume (REV) scale, which is much larger than the porescale, but much smaller than the macro-scale [7, 8]. REV scale study is capable of obtaining the macroscopic property values related to macro-scale transport equation, such as porosity, permeability and local heat transfer coefficient, which are not influenced by the REV size any more. So the continuum modeling of the transport process in porous media can be performed based on these REV scale study. For example, the Darcy flow model or its modified ones can describe the relation between pressure drop and velocity as the fluid flows through the porous domain. And the nonequilibrium thermal model can describe the heat transfer between the fluid and the solid structure in porous domain, which is more accurately compared to the equilibrium thermal model [9-11]. As mentioned above, the use of the empirical correlations in the transport equations is necessary in the continuum modeling. These macroscopic parameters and their relation can be determined or derived from experimental and analytical studies. The experimental approach is relatively high cost and time consuming. In fact, the velocity or temperature distribution in the obstructed complex porous geometry is difficult to be obtained accurately in experimental way. The analytical study is suitable for the simple geometry, which is hardly applied to the complex and irregular real structure. So the pore-scale study becomes one innovative and promising approach in investigating the transport phenomena in porous media [12, 13]. The macroscopic parameters can be accurately determined with the detailed velocity and temperature distributions in the pore spaces obtained from pore-scale predictions. The study of transport phenomena in porous media at the pore-scale has attracted more and more attentions in recent years [14-16]. The computational fluid dynamics (CFD) is an effective tool in the fluid flow and heat transfer modeling and simulation [17-19]. The Finite Volume method (FVM) can be employed to solve the Navier-Stokes and energy equations for the fluid flow and heat transfer, which has been proved to be an available numerical method in the cases with idealized geometrical boundaries. At the pore-scale level, Navier-Stokes equations are capable of governing the pore fluid
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motion. Kopanidis et al. [20] presented pore-scale simulation of flow and heat transfer in open cell metal foam. The pore-scale structure was described in a three dimensional simplified geometrical model and it was discretized with tetrahedral volume meshes for the void and solid phases. Dyck and Straatman [21] used FVM based CFD software Ansys CFX to predict the hydraulic and thermal parameters in the digital samples of spherical-void-phase carbon foam and relevant transport properties were obtained from the numerical results. As the porous media structure becomes more complex, the FVM modeling may meet with the meshing or convergence problem. The lattice Boltzmann (LB) modeling has been developed rapidly as a new and promising numerical method over the last decade [22, 23]. Guo et al. [24] developed a thermal lattice BGK (Bhatnagar, Gross and Krook) model, in which the flow and heat transfer process were simulated using two independent lattice BGK equations in a coupled model. In the work of Hosseini et al. [25] , the transport processes in porous media were simulated at the pore-scale using lattice Boltzmann method (LBM), in which the porous structure was simplified as randomly distributed solid bars. Due to the complexity of the pore geometry, most of pore-scale modeling was carried out in the porous domain with geometrical simplifications. The numerical modeling with these idealized porous domains can provide pore-level information on flow and heat transfer processes. But it is difficult to establish simplified porous geometrical model equivalent to the real porous media in both thermal and hydrodynamic performance, so the effect of real porous structure has to be considered in the numerical modeling. Li et al. [26] simulated transport phenomena in a porous wick on pore-scale by LBM, in which the random porous media was numerically reconstructed with the stochastic method. Image-based porescale modeling was proved to be a promising method to predict fluid flow in real porous media [27, 28]. Zhu et al. [29] performed geometrical reconstruction for aluminum foam with MATLAB image processing and CT scanning. Ranut et al. [30] used X-ray computed micro tomography technique to describe the micro-structures of metal foam. In the above work, the velocity and temperature distributions were obtained from FVM based CFD simulation. Very few studies on LB modeling for flow and heat transfer processes with geometrical reconstruction of real porous media can be found in open literature up to now. In this paper, a double-population function thermal LB model was employed in the numerical simulation and geometrical reconstruction of Berea sandstone sample was performed to establish the computational domain. Pore-scale simulation in three-dimensional reconstructed porous media was carried out to study the flow and heat transfer processes considering the effect of real porous structure. The velocity and temperature distributions in pore spaces were obtained from the pore-scale numerical predictions. The correlations for flow and heat transfer in the specific porous media sample were derived based on the numerical results in this work.
2. 3D Double-Population Thermal Lattice Boltzmann Model The double-population function is capable of simulating both flow and heat transfer simultaneously in the LBM modeling [31]. The D3G15 LBM model can be used to describe the fluid flow and the D3Q6 thermal LBM model is available for simulating the temperature variation in the pore spaces. The lattice velocity directions for these two models are shown in Fig. 1. For a particle at position distribution functions
and at time , the probability of this particle with a velocity can be defined as particle , and the Boltzmann equation describes its evolution as [32]: (1)
Ω on the right side of equal sign in Eq. (1) is the collision operator, which describes the particles interaction. The directional particle velocities can be introduced to simplify the Boltzmann equations and the velocity space is discretized with D3Q15 model in the flow simulating: (2) α = 0, …, 14. In D3Q15 model, 14 directional particle velocities point to neighboring nodes and one equals zero. These equations are discretized in both space and time, in which c = Δx/Δt (node spacing Δx, time step Δt), and the lattice Boltzmann equation can be expressed as: (3) is the dimensionless relaxation time for velocity, an
is the equilibrium distribution function described as:
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(4)
In the above equation,
can be defined as: (5)
where = 2/9, = 1/9 for α = 1–6, = 1/71 for α= 7–14, and is the pseudo sound speed ( , and in equilibrium distribution function can be determined by solving Eq. (6)-(8):
). And
(6) (7) (8) It can be obtained that
,
,
.
Macroscopic parameter, such as density ρ, momentum ρ
and pressure
can be obtained from following equations: (9) (10) (11)
And the macroscopic viscosity
can be calculated by: (12)
The thermal LB equation for temperature field is expressed as [24]: (13) where 6), and
is the dimensionless relaxation time for temperature, is the distribution function of temperature ( is the equilibrium distribution function of temperature:
= 1-
(14) The macroscopic temperature can be obtained by the following expression: (15) To validate the present flow model, a comparison of the numerical results using LBM, FVM and analytical method was performed. Poiseuille flow was numerically simulated and the grid independency was tested before simulation. The predicted non-dimensional velocity profiles are shown in Fig. 2(a). The comparison of LBM, FVM and analytical solution Eq. (16) shows a good agreement. (16) To validate the thermal model, the local Nusselt number (Nu) is defined as: (17)
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Fig. 2(b) shows Nusselt number variation along the flow direction (the non-dimensional position is defined as (x/H)/(RePr)). The predicted Nu reaches a constant value of 7.61 as it reaches the fully-developed condition and the analytical value was given as 7.54 in [33], which shows a difference of 0.93%.
3. Porous Media Reconstruction and Boundary Conditions The available and promising geometrical reconstruction approaches is to generate the porous media using the tomographic reconstruction, which is a nondestructive imaging technique. 2D sliced images of the micro CT scanned sample can be obtained and stacked to reconstruct the 3D real geometry. Berea sandstone can be used as a standard porous media material due to its relatively homogenous and well-characterized property, which was adopted in this numerical model to reconstruct the porous computational domain with real porous structure characteristic. A Berea sandstone sample with a diameter of 1 cm was scanned on a XRADIA Micro XCT-200 scanner and the resolution of scanned images is 12.5 μm. Then these images were processed with a Java based open source software ImageJ, which performed noise reduction and thresholding processing. The threshold values were determined for each scan image and the background and foreground were defined as black and white, respectively. Due to the blur area near the edge, the images were trimmed to achieve a more accurately reconstructed geometry. The final binary format images are shown in Fig. 3(a). The 3D geometry model (5mm×5mm×5mm) in Fig. 3(b), which was reconstructed by image processing software Mimics, represents the real porous structure and it was imported into the LBM code with a format of STL. The experimental porosity value of porous sample is 0.61 and the porosity of reconstructed one is 0.58. The difference in porosity is mainly caused by the existing of small particles below the scanner’s resolution and the loss during image processing. The numerical work simulated the water flowing through a cubic domain filled with reconstructed porous media as shown in Fig. 4. The inlet and outlet are set at different pressure values on the left and right side: Pin = 20, 40, 60, 80 Pa and Pout = 0 Pa. The other four walls are set as static and adiabatic wall boundary conditions. The temperature of solid porous structure is set at 320 K, and the temperature of fluid at the inlet is at T = 300 K. The flow is assumed to be three dimensional and laminar. The fluid properties are constant ones and the no-slip boundary condition is applied in flow modeling. The resolution in the LB model should be improved to describe the accurate pore spaces and pore fluid flow characteristics. In LBM modeling, the numerical resolution normally increases by placing more lattice nodes in the computational domain and the independence of resolution should be tested before the simulation. We performed simulations with lattice number of 0.4 × 10 6, 0.8 × 106, 1.6 × 106, 3.2 × 106 and 6.4 × 106, respectively. It is shown that the velocity distribution varies in a very small difference as the lattice number reaches 1.6 × 106. The computational cost is one of important concerning in the numerical modeling. So 1.6 × 10 6 is the number of lattice nodes we adopted in the numerical modeling and simulation in this work.
4. Results and Discussion 4.1 Pore-scale fluid flow Figure 5 shows the predicted averaged velocity (the flow rate divided by the flow area in cross section) along the x axis for different inlet pressure conditions. It shows that an increase in the pressure gradient results in an increase in averaged velocity in the cross section. For a fixed inlet pressure, the averaged velocity varies in a small range along the flow direction, which is due to the difference in the flow area on the different x position. Fig. 6 (a)-(d) show the velocity distribution in the plane of z=2.5 mm for different inlet pressures, it can be found that the velocity distribution varies intensely due to the existence of irregular porous structure in the computational domain. It is obvious that liquid flows through narrow flow area at a high velocity and the velocity of fluid near the static wall is relatively slow due to the static boundary condition. The flow velocity vectors in longitudinal section (z=2.5 mm) and in cross section (x=2.5 mm) are shown in Fig. 7 (a) and (b), respectively. In these two figures, a long vector represents a high velocity and the influence of complex pore structure on the flow characteristics can be well observed. Because the liquid flow should be driven along the direction of the pressure gradient, the water in the pore spaces mainly flows along the x direction due to the pressure drop between inlet and outlet of porous domain. Fig. 7(a) shows the liquid flowing in the confined pore spaces and the flow status is complicated due to the irregular porous structure. In Fig. 7(b), the flow velocity vectors can be clearly observed in the plane of x=2.5 mm, which is not along the main flow direction. The complicated flow field in the real porous domain will have a direct influence on the heat transfer characteristics. Eq. (18) can present the relation between pressure drop and superficial velocity, which is extended from Darcy’s law expression. -dP/dx is the pressure gradient along the x direction in the present model, U is the averaged superficial -4Page 4 of 9
velocity, K is the permeability, and is the inertia coefficient. The first term on the right side of Eq. (18) represents influence of viscous force and the second term represents influence of inertia force. At a low velocity, the viscous force is dominant and pressure drop is mainly caused by the first term, in which the second term can be neglected. However, the inertia force is dominant at a higher velocity and second term can not be neglected any more. So a linear relation between pressure drop and velocity dominates at low velocities, and a nonlinear relation dominates as velocities reach a higher value. It can be obtained from the numerical predictions for the coefficients K and : , , which is applicable for the specific porous media sample adopted in this work. (18)
4.2 Pore-scale fluid heat transfer Temperature distributions in the longitudinal section of z=2.5 mm for different inlet pressures are shown in Fig. 8 (a)(d). It can be observed that increasing the inlet pressure reduces the outlet liquid temperature. In Fig. 8(a), the liquid temperature increases to a high value in a short distance as it enters the porous domain, and it nearly reaches 320 K as it flows out of the domain. For high inlet pressure boundary condition, the liquid temperature keeps at a low value as it flows through the porous domain. In the narrow flow area, the liquid temperature increases quickly due to the small flow rate. For where wide flow area exists, the heat transfer process still undergoes near the outlet of porous domain, as shown in Fig. 8 (d). The heat is transferred mainly by the conduction form as the velocity in the porous domain is very slow. For a higher inlet pressure, the flow velocity in the domain increases and complex flow status occurs due to the irregular porous structure, the heat will be transferred by the convection form under this condition. An increase in inlet pressure leads to a higher convective heat transfer coefficient, but it decreases the amount of heat transferred per unit mass due to the high flow rate. To represent the averaged temperature profile in the porous domain, the non-dimensional temperature in the following form:
＝
is introduced
(19)
where = 320 K is the temperature of the solid porous structure, is the inlet liquid temperature (300 K), and is the local liquid temperature, which is an averaged value over the cross section. The variations of along x direction for different inlet pressures are shown in Fig. 9. The curve with a higher inlet pressure is below the curve with a lower inlet pressure. The liquid needs to flow more distance along the main flow direction to reach thermal equilibrium for a higher inlet pressure, which is mainly because of the low residence time of fluid particles in the pore space. The flow particles with low velocities have a higher residence time and reach the thermal equilibrium in a short distance in the main flow direction. The bulk liquid temperature depends on the location along the main flow direction, which can be calculated by the following expression: (20) The local convective heat transfer coefficient
can be calculated by the following expression: (21)
where A is heat transfer area in cross section and q is the heat flux from solid to liquid. Fig. 10 presents variations along the main flow direction in the porous domain for different inlet pressures. It appears that varies in a limited range at different positions on x-axis. However, its value is mainly determined by the velocity distribution in the pore space. It should be noted that the velocity distribution is mainly influenced by the complex pore structure and the pressure drop across the porous domain. To analytically treat local thermal non-equilibrium phenomenon in porous media, Kuznetsov studied the thermal non-equilibrium effect with the high production of convective heat transfer coefficient and the specific surface area [34]. And several successful applications were performed to analyze the flow and heat transfer in porous media under the local thermal non-equilibrium condition [35-37]. The accurate values of convective heat transfer coefficient and the specific surface area can be obtained from the pore-scale numerical simulation work presented in this paper. So it is available to extend the pore-scale model to analytical model to investigate the transport phenomena in porous media under the local thermal non-equilibrium condition. The three-dimensional temperature profile in the porous media is presented in Fig. 11. It is obvious that the temperature distribution varies intensely in the porous domain. The complex porous structure leads to the flow status varying on the -5Page 5 of 9
different location in the pore space, the heat transfer process will vary accordingly. In order to obtain a better understanding of whole heat transfer characteristics, a relation between Nusselt number and Reynolds number can be derived based on the numerical prediction in this work. The characteristic length used to calculate the Nusselt number (
) and Reynolds number (
) is based on the permeability in this paper. The relation between heat
transfer and flow is presented in Eq. (22), for which . It should be noted that this relation is suitable for the specific porous media adopted in this numerical work. For different porous media, the same procedure can be carried out to obtain the unique correlation between flow and heat transfer, which can be extended for the continuum modeling of the transport process in porous media. (22)
4. Conclusion The flow and heat transfer were successfully predicted in a complex porous medium at the pore-scale by doublepopulation thermal LB method. The porous geometry was reconstructed based on the micro CT scan images of Berea sandstone sample. The thermal LB model was numerically tested before simulation and a good agreement was achieved by compared with the existing CFD and analytical results. The numerical approach established in this work is capable of predicting transport phenomena in porous media at pore-scale by importing 3D reconstructed porous domain into the numerical model. The detailed velocity and temperature distributions in complex pore spaces were obtained from the pore-scale numerical simulation. The correlations for flow and heat transfer in the specific Berea sandstone porous media were derived based on the numerical results and the procedure developed in this paper can be applicable for porous media with different geometrical characteristics.
Nomenclature A c cs Cf f g h H K Nu p Pr q Re t T u U x,y,z x
heat transfer area, m2 velocity, m/s pseudo sound speed, m/s inertia coefficient, particle velocity, m/s particle distribution function of velocity, particle distribution function of temperature, heat transfer coefficient, W/(m2·oC) height, m permeability, md Nusselt number, pressure, N/m2 Prandtl number, heat flow, W Reynolds number, time, s Temperature, K x-velocity, m/s dimensionless velocity, - or superficial velocity, m/s velocity vector, m/s Cartesian coordinates position, (x,y,z)
Greek
Ω
velocity, m/s dimensionless temperature, dynamic viscosity, kg/(m·s) density, kg/m3 dimensionless relaxation time, viscosity, Pa·s collision operator, -
Subscripts b f
boundary fluid
μ ρ
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in lf s T
inlet local fluid solid temperature
Acknowledgements This paper is supported by the National Natural Science Foundation of China through grant no. 51306119 & 51536005, the National Basic Research Program of China (973 Program) through grant no. 2012CB720404 and Key Basic Research Projects of Science and Technology Commission of Shanghai through grant no. 12JC1405100.
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Figure Captions Figure 1 Lattice velocity directions (a) D3G15 (b) D3Q6 Figure 2 Validation for the numerical model (a) Velocity profile (b) Nu variation Figure 3 Reconstruction of porous media (a) 2D scan images (b) Reconstructed porous media (5mm×5mm×5mm) Figure 4 View of geometry model and boundary condition Figure 5 Averaged velocities along the flow direction for different Pin Figure 6 Velocity distribution in the plane of z=2.5 mm (a) Pin=20Pa (b) Pin=40Pa (c) Pin=60Pa (d) Pin=80Pa Figure 7 Velocity vectors in the longitudinal and cross sections (a)z=2.5mm (b)x=2.5mm Figure 8 Temperature distribution at z=2.5 mm (a) Pin=20Pa (b) Pin=40Pa (c) Pin=60Pa (d) Pin=80Pa Figure 9 The dimensionless temperature distribution along the main flow direction for different inlet pressures Figure 10 The local convective heat transfer coefficient along the flow direction for different inlet conditions Figure 11 3D temperature profile in reconstructed porous media (P in=80 Pa)
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