A Simplified Model of Low Reynolds Number, immiscible, Gas-Liquid Flow and Heat Transfer in Porous Media (Numerical Solution with Experimental Validation)

: This study investigates the thermal-hydraulic characteristics of immiscible two-phase flow (gas/liquid) and heat transfer through porous media. This research topic is interested among others in trickle bed reactors, the reservoirs production of oil, and the science of the earth. Characteristics of two-phase concurrent flow with heat transfer through a vertical, cylindrical, and homogeneous porous medium were investigated both numerically and experimentally. A generalized Darcy model for each phase is applied to derive the momentum equations of a two-phase mixture by appending some constitutive relations. Gravity force is considered through investigation. To promote the system energy equation, the energy equation of solid matrix for each phase are deemed. The test section is exposed to a constant wall temperature after filled with spherical beads. Numerical solution of the model is achieved by the finite volume method. The numerical procedure is generalized such that it can be reduced and applied to single phase flow model. The numerical results are acquired according to, air/water downward flow, spherical beads, ratio of particle diameter to pipe radius D=0.412, porosity φ=0.396, 0.01≤Re≤500, water to air volume ratio 0≤W/A≤∞ , and saturation ratio 0≤S 1 ≤1. To validate this model an experimental test rig is designed and constructed, and the corresponding numerical results are compared with its results. Also, the numerical results were compared with other available numerical results. The comparisons show good agreement and validate the numerical model. One of the important results reveals that the heat transfer is influenced by two main parameters; saturation ratios of the two fluids; S 1 and S 2 , and the mixture Reynolds number Re. The thermal entry length is directly dependent on Re, S 1 , and the thermofluid properties of the fluids. A modified empirical correlation for the entrance length; X e =0.1 Re.Pr.R m is predicted, where R m =R m (S 1 , S 2 , ρ 1 , ρ 2 , c 1 , c 2 ). The predicted correlation is verified by comparing with the supposed correlation of Poulikakos and Ranken (1987) and El - Kady (1997) for a single-phase flow; X e /Pr=0.1 Re. Immiscible, Concurrent, Darcy model.


Abstract:
This study investigates the thermal-hydraulic characteristics of immiscible two-phase flow (gas/liquid) and heat transfer through porous media. This research topic is interested among others in trickle bed reactors, the reservoirs production of oil, and the science of the earth. Characteristics of two-phase concurrent flow with heat transfer through a vertical, cylindrical, and homogeneous porous medium were investigated both numerically and experimentally. A generalized Darcy model for each phase is applied to derive the momentum equations of a two-phase mixture by appending some constitutive relations. Gravity force is considered through investigation.
To promote the system energy equation, the energy equation of solid matrix for each phase are deemed. The test section is exposed to a constant wall temperature after filled with spherical beads. Numerical solution of the model is achieved by the finite volume method. The numerical procedure is generalized such that it can be reduced and applied to single phase flow model. The numerical results are acquired according to, air/water downward flow, spherical beads, ratio of particle diameter to pipe radius D=0.412, porosity φ=0.396, 0.01≤Re≤500, water to air volume ratio 0≤W/A≤∞, and saturation ratio 0≤S1≤1.
To validate this model an experimental test rig is designed and constructed, and the corresponding numerical results are compared with its results. Also, the numerical results were compared with other available numerical results. The comparisons show good agreement and validate the numerical model. One of the important results reveals that the heat transfer is

I. Introduction
Two phase flows through porous media are known to occur in several industrially relevant branches such as in oil and gas production, trickle bed reactors, geothermal engineering, cooling of debris following accidents in nuclear power plants and many other applications. Fundamental studies related to single-phase flow and thermal convection in  [3] discussed functionally and historically the forms of governing equations that characterize the fluid flow behavior and convective heat transfer through porous media. Single phase forced convection heat flow in a cylindrical packed bed with either a uniform wall heat flux or a constant wall temperature was examined numerically for both the developing and fully developed regions by El Kady [4,5]. Besides the energy equation, the generalized form of the momentum equation including the non-Darcian effects such as the inertia, viscous forces, and the porosity variation was considered. Non-Newtonian effects and fluid flow and heat transfer characteristics were presented by El Kady et al [6].
The literature studies concern the two-phase flow and heat transfer in porous media can be classified into experimental studies and modeling and formulation studies. The first experimental attempt to study this problem was by Larkins [7]. The simultaneous vertical downward flow of liquids and gases over a fixed packed bed was studied experimentally. Rao et al [8], Saez et al [9], and Sai and Varma [10] focused experimentally, the hydrodynamics of concurrent gas-liquid flow in packed beds. A formulation by Wang and Beckerman [11] was approached for a two-phase of constituents of a binary mixture flow model. The Darcy model with the effect of gravity is formulated besides the energy equation while no results were presented. Other macroscopic modeling studies were performed by Rao et al [12], Saez et al [13] and Hilfer [14]. A review by Carbonell [15] performed different theories of multiphase flow models in packed beds of chemical reactor design. Likewise, summaries of the conventional multiphase flow model (MFM) and a more recent multiphase mixture model (MMM) were approached by V.Starikovicius [16]. P. J.
Binning [17] presented a two-dimensional model of unsaturated zone multiphase air water flow and contaminant transport. The incompressible multiphase flow equations were written in two forms: with the two individual phase pressures or using the global pressure saturation approach of petroleum engineering. Coarse scale equations describing two-phase flow in heterogeneous reservoirs are developed by L. J. Durlofsky [18]. The volumeaveraged equations were simplified and applied to the direct solution of a model coarse scale problem. The investigations of both T. W. Patzek and R. Juanes [19] and C. M. Marle [20] intended a three-phase fluid flow of one-dimensional immiscible equations with gravity effect, in the limit of negligible capillarity. Their main study results concluded that a Darcy flow model was insufficient to express all the physics principles of co-current three-phase displacements through reservoirs. As a result of not computing some physical phenomena in Darcy's equation, especially with multi-phase flows in porous media, a multifaceted mathematical approach has been described to derive deterministic conservation equations on the length scale with the realization of thermodynamic constraints by Julia C. Muccino et al. [21]. The study found a set of equilibrium equations that presumed a framework for determining the assumptions inherent in a multiphase flow model. The same factors were investigated numerically by J. Niessner and S. M.
Hassanizadeh [22] and found that the modified model compared to the standard model required an additional physical process such as hysteresis. A 2D numerical model of a porous medium with an incompressible fluid with non-zero Reynolds number was simulated by Sergio Rojas and Joel Koplik [23]. A random array of square cross-section cylinders was utilized. The manifested results found that a transition Darcy from linear flow field at evanescence Re to a cubic transitional regime at low Re, and then a quadratic Forchheimer when Re=0(1). Tore I. Bjørnarå et al. [24] studied 2D numerical model by different types of equations that describe two-phase flow in porous media using the finite element method.
These equations consist of mass balances, partial differential equations for accumulation, transport, and injection/production of the phases. Other auxiliary equations were applied such as hydraulic properties, coupling the phases through the system. The test results predicted that the fractional flow formulation was the fastest and most robust formulation.
A thesis of Jennifer Niessner [25] dealt with thermodynamically consistent modeling of a twostage flow in porous media to optimize the interfaces of the conservation equations for mass, momentum, energy, and entropy of the phases and interfaces through the system. The

II. Definitions:
Porosity: The average porosity of a block in a porous medium is the ratio of the pore space to the bulk volume of the block.

Vt
For two phase flow in porous media, Where V1, V2, and V3 are the volumes occupied by fluid 1, 2 and solid matrix, respectively.
The porosity can be divided into two parts: φ1 and φ2 for fluid 1 and fluid 2. Thus. Saturation: tio of the volume occupied by this ) is defined as the ra where K1 and K2 are the effective permeabilities of fluids 1 and 2 respectively and obviously depend on the nature of the medium under consideration and saturation of the fluid.

Relative permeabilities:
Relative permeability of a fluid is the ratio of its effective permeability to the medium permeability, K.
Kr1=K1/K and Kr2=K2/K Relative permeabilities are functions of saturation of each fluid.
• Square functions; n=2; are suitable for petroleum engineering; oil water system.
• Cubic forms; n=3; are widely used in nuclear safety engineering.
Bejan [2] concluded that satisfactory results have been reported when use has been made of a simple linear relationship namely: Kr1 = S1 and Kr2 = 1-S1.
Mixture density: The mixture density is defined as: Mixture specific heat: The mixture specific heat is defined as a function of individual specific heat of each phase as follows: c = c1 S1+ c2 (1-S1) Mixture velocity: ively (based nd mixture respect , and u are the Darcian velocities of the fluid 1, fluid 2, a 2 , u 1 u on the total cross section area); u = (ρ1u1+ ρ2u2)/ρ Effective thermal conductivity of the medium: The effective thermal conductivity of two phase flow through a porous medium with porosity φ and thermal conductivity k1, k2, and k3 for fluid 1, fluid 2, and solid matrix respectively; is defined as follows: kef = (1-φ) k3 + φ (k1S1+ k2S2)

III. Mathematical Formulation:
A schematic for the physical configuration is shown in No mass sources or heat sources are considered.
The test section is a cylindrical pipe filled with porous medium and is exposed to constant wall temperature Tw on the outer surface. A porous medium consisting of packed spheres is used to illustrate the results. It is also assumed that the two fluids and the solid matrix are in thermal equilibrium.
The thermo physical properties of the solid matrix and the two phases such as the viscosity, thermal conductivity, and effective thermal diffusivity are assumed to be constant. Consider the simultaneous flow of the two phases denoted as liquid (subscripted 1) and gas Where ν is the kinematic viscosity of the mixture; The introduced new mixture density ρk is called the kinematic mixture density. It acquires its name because of its dependence of the relative mobilities of the phases.
The total energy conservation equation for a combined solid matrix and two-phase mixture system is required to determine the temperature field. By assuming that local thermal equilibrium prevails among the solid matrix, gas phase and liquid phase (i.e., T=T1=T2=T3) and for steady conditions without evaporation and with no heat sources.
Assuming constant properties of phases, equation (15) becomes: 1 . W here c1 and c2 are the heat capacities of phase 1 and phase 2, respectively.
In this model, the velocity of each fluid is assumed to be uniform and consequently the mixture velocity. At the outer radius, there is a constant wall temperature Tw. At the inlet of test section, x=0, the fluid has a uniform velocity ui and uniform temperature Ti, i.e., the following boundary conditions are applied:

IV. Fluid Flow and Heat Transfer Characteristics:
The flow through the porous duct experiences in general the boundary frictional drag "fv", a bulk frictional drag induced by the solid matrix (designed as Darcy's pressure drop) "fD", the drag induced by gravity effect "fg", and a flow inertia drag "fi" induced by the solid matrix at high velocities (designed as Forschheimer's form drag) [5,6].
The mean temperature cannot be measured experimentally as calculated in Eq. 29, thus; to compare the numerically calculated Nusselt number with the experimentally measured one, another value of Nusselt number symboled Nu * is introduced. In which the mean temperature is considered as the arithmetic mean value of input and exit temperatures. The thermal entrance length Xe is defined as the distance between the entrance of the pipe and the point at which the mean fluid temperature θm and Nusselt number Nu become independent of the x-location.

V. Numerical Formulation:
Because the present configuration is symmetrical with respect to the centerline, only half of the channel needs to be considered. The finite volume method with variable grid step in X direction is employed. The R domain is discritized into 201 grid points to get an accurate resolution of the important near wall region. A very fine grid size in the X direction near the channel inlet and coarser downstream is used. The grid size at the inlet is 0.0001 and increases gradually in the downstream direction by a multiplier factor 1.01. This is done to capture the steep changes in the temperature field near the entrance. The energy equation was transformed into algebraic finite volume equations by integration following the procedure developed by Patankar [37] and Versteeg [38]. It can be solved considering constant velocity field to get the temperature distribution. A system of tridiagonal algebraic equations for the nodal temperature at any given X position is obtained. Once the temperature profile at each X position is known, the local Nusselt numbed is determined from equation (20). When the mean fluid temperature θm and Nusselt number Nu become independent of the x-location a thermally fully developed flow is assumed and the entrance length is obtained.

VI. Experimental work:
To validate the mathematical and numerical model, an experimental work is done.
Schematic layout of the experimental test rig is shown in Fig. (2), in which the test section (12) is loaded by means of steel stand, which permits to change the porous medium easily.
Water was pumped from a 200 L tank (1) at 5.5 m height to the midpoint of the test section. It flows through a control valve (3) to a rotameter (6), which measures the flow rate of water, J.
W. Dally [39]. Air was drawn from a compressor (15) through a control valve (16) to an air test section (17) of 55 cm length and 3.75 cm diameter. Air velocity and temperature are measured by means of a hot wire anemometer (18), which is located at the end of the air test section. A definite mass flow rate of each air and water were mixed before entering the test section to flow through the porous medium inside the test section. The temperature of outer wall of the test section is maintained constant by means of passing a saturated steam through an outer annulus from an electric boiler. Temperatures were measured by copper-constantan type-J thermocouples (4), which are attached to digital temperature recorder type YOKOGAWA (5) with scale division of 0.1 °C and 24 channels. Four pressure gauges (7) were attached at four levels to measure the pressure at each section along the length of test section. 7-Pressure gauge. 14-Electric super heater.

VII. Results and Discussion:
To validate the numerical model, experimental pressure drop, temperatures and volume flow rates are measured, and average Nusselt number Nua and friction factor ft at different Reynolds number Re are compared with the corresponding numerical model. Figure 4 presents the comparison between the experimental and numerical values of the friction factor at different Reynolds number 0.01≤Re≤500 for spherical sized packed beads of d=7 mm, S1=0.514. The comparison shows good agreement at Reynolds number up to 100. At Reynolds number Re>100, It is expected for friction factor obtained from the numerical model to deviate from the experimental one. Therefore, Darcy model is only valid for low Reynolds number where inertia effect and friction due to the macroscopic shear are ignored.

VIII. Conclusions:
Heat transfer of two-phase concurrent flow through a vertical, cylindrical, homogeneous porous medium is investigated. The system energy equation and a general form of the Darcy flow model for each phase are the points of interest. The test section is exposed to a constant wall temperature after loaded with spherical beads. The numerical solution of the model is achieved by the finite volume method. The numerical procedure is generalized such that it can be reduced and applied to a single-phase flow model. The numerical results are experimentally validated according to air/water downward flow, spherical beads, a ratio of particle diameter to pipe radius D=0.412, porosity φ=0.396, 0.01≤Re≤500, water to air volume ratio 0≤W/A≤∞, and saturation ratio 0≤S1≤1. The main important findings are outlined as follows: • Average Nusselt number is nearely constant up to Re=40 and increases with the increase of Reynolds number.
• With the increase of Re, the average Nusselt number deviates from the experimental one due to the neglection of inertia and friction effects. The error is about 8% at Re=1000. Therefore, Darcy model is only valid for low Reynolds number.
• At Reynolds number Re>100, It is recommended to take inertia and friction effects into account by means of Forschheimer-Brinkman's equation.
• As X increases from 0 to the end of the thermal entrance length; the mean temperature increases up to a constant value for all Reynolds numbers. θm (fully developed) ≈0.67. It also decreases with the increase of Reynolds number at the same section.
• Local Nusselt number has higher values at the entrance section and decreases as X increases until it reaches its fully developed value 4.37 at the end of the thermal entrance length for single phase flow (water or air) and two-phase flow mixture.
• The thermal entrance length Xe is higher for the single-phase flow of the fluid having the higher thermal conductivity "water" and vice versa. For two phase flow mixture, Xe lies between the two previous curves.

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A direct dependence of the thermal entry length on Re, S1, and properties of the flowing fluids exists and gives a modified correlation for the entrance length; Xe w wall.
Compliance with Ethical Standards: No Funding.
Conflict of Interest: The authors declare that they have no conflict of interest. We declare that we have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.