Free Convection in Porous Media

Solved with COMSOL Multiphysics 4.3a

F r e e C o nv e c ti o n in Porou s Med i a Introduction This example describes subsurface flow in porous media driven by density variations that result from temperature changes. The model comes from Hossain and Wilson (Ref. 1) who use a specialized in-house code to solve this free-convection problem. This COMSOL Multiphysics example reproduces their work using the Brinkman Equations interface and the Heat Transfer in Porous Media interface. The results of this model match those of the published study.

Model Definition The following figure gives the model geometry. Water in a porous medium layer can move within the layer but not exit from it. Temperatures vary from high to low along the outer edges. Initially the water is stagnant, but temperature gradients alter the fluid density to the degree that buoyant flow occurs. The problem statement specifies that the flow is steady state.

Figure 1: Domain geometry and boundary conditions for the heat balance in the free-convection problem. Th is a higher temperature than Tc, while s is a variable that represents the relative length of a boundary segment and goes from 0 to 1 along the segment.

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FREE CONVECTION IN POROUS MEDIA


You model this free-convection problem by introducing a Boussinesq buoyancy term to the Brinkman’s momentum equation, and then linking the resulting fluid velocities to the Heat Transfer in Porous Media interface. The Boussinesq buoyancy term that appears on the right-hand-side of the momentum equation accounts for the lifting force due to thermal expansion T μ --- u + ∇p – ∇ ⋅ μ --- ( ∇u + ( ∇u ) ) = ρgβ ( T – T c ) κ ε

(1)

∇⋅u = 0. In these expressions, T represents temperature while Tc is a reference temperature, g denotes the gravity acceleration, ρ gives the fluid density at the reference temperature, ε is the porosity, and β is the fluid’s coefficient of volumetric thermal expansion. The heat balance comes from the heat transfer equation ρC L u ⋅ ∇T – ∇ ⋅ ( k eq ∇T ) = 0

(2)

where keq denotes the effective thermal conductivity of the fluid-solid mixture, and CL is the fluid’s heat capacity at constant pressure. The boundary conditions for the Brinkman equations are all no-slip conditions. Using only velocity boundaries gives no information on the pressure within the domain, which means that the model produces estimates of the pressure change instead of the pressure field. However, without any seed information on pressure, the problem is unlikely to converge. The remedy is to arbitrarily fix the pressure at a point in the model using point settings. The boundary conditions for the Heat Transfer interface are the series of fixed temperatures shown in Figure 1.

Implementation: Initial Conditions for Boussinesq Approximation The simple statements in Equation 1 and Equation 2 produce a strong nonlinear problem that represents a difficult convergence task for most nonlinear solvers. Even without the Boussinesq term, the Brinkman equations are nonlinear alone. To ease the numerical difficulties, you let the coefficient of volumetric thermal expansion β increase gradually, rasing the Rayleigh number of the experiment. When β = 0, the momentum and temperature equations are uncoupled, so the model converges easily. Then you increase β, using the previous solution as the initial guess for the next parametric step, and so on, until you reach a Rayleigh number of 105. The iteration protocol is an easy process with COMSOL Multiphysics’ parametric solver.

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Results This example reproduces a model reported by Hossain and Wilson (Ref. 1). After extracting the input data from the paper, the author constructed the model in less than an hour, including all the steps from geometry input to postprocessing of the results. Figure 2 shows the temperature distribution throughout the porous slice.

Figure 2: Temperature in a porous structure subjected to temperature gradients and subsequent free convection. The COMSOL Multiphysics simulation is in excellent agreement with published results from Ref. 1. Figure 3 gives the COMSOL Multiphysics solution for the flow field.

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Figure 3: Temperature contours and velocity arrows for a Rayleigh number of Ra=105.

Reference 1. M. Anwar Hossain and M. Wilson, “Natural Convection Flow in a Fluid-saturated Porous Medium Enclosed by Non-isothermal Walls with Heat Generation,” Int. J. Therm. Sci., vol. 41, pp. 447–454, 2002.

Model Library path: Subsurface_Flow_Module/Heat_Transfer/ convection_porous_medium

Modeling Instructions MODEL WIZARD

1 Go to the Model Wizard window. 2 Click the 2D button.

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3 Click Next. 4 In the Add physics tree, select Fluid Flow>Porous Media and Subsurface Flow>Brinkman Equations (br). 5 Click Add Selected. 6 In the Add physics tree, select Heat Transfer>Heat Transfer in Porous Media (ht). 7 Click Add Selected. 8 Click Next. 9 Find the Studies subsection. In the tree, select Preset Studies for Selected Physics>Stationary. 10 Click Finish. GLOBAL DEFINITIONS

Parameters 1 In the Model Builder window, right-click Global Definitions and choose Parameters. 2 In the Parameters settings window, locate the Parameters section. 3 In the table, enter the following settings: Name

Expression

Description

rho

1000[kg/m^3]

Fluid density

mu

0.001[Pa*s]

Fluid viscosity

beta

1e-6[1/K]

Fluid volumetric thermal expansion

k

6[W/(m*K)]

Fluid thermal conductivity

gamma

1

Fluid ratio of specific heat

Cp

4200[J/(kg*K)]

Fluid heat capacity at constant pressure

epsilon

0.4

Porosity

kappa

1e-3[m^2]

Permeability

p0

1[atm]

Reference pressure

Tc

20[degC]

Reference temperature

Th

42[degC]

High temperature

L

0.1[m]

Length scale

Pr

mu*Cp/k

Prandtl number

Ra

Cp*rho^2*g_const*beta* (Th-Tc)*L^3/(k*mu)

Rayleigh number

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GEOMETRY 1

Square 1 1 In the Model Builder window, under Model 1 right-click Geometry 1 and choose Square. 2 In the Square settings window, locate the Size section. 3 In the Side length edit field, type L. 4 Click the Build Selected button.

Point 1 1 In the Model Builder window, right-click Geometry 1 and choose Point. 2 In the Point settings window, locate the Point section. 3 In the x edit field, type L. 4 In the y edit field, type 0.01. 5 Click the Build All button. BRINKMAN EQUATIONS

Fluid and Matrix Properties 1 1 In the Model Builder window, under Model 1>Brinkman Equations click Fluid and Matrix Properties 1. 2 In the Fluid and Matrix Properties settings window, locate the Fluid Properties section. 3 From the ρ list, choose User defined. In the associated edit field, type rho. 4 From the μ list, choose User defined. In the associated edit field, type mu. 5 Locate the Porous Matrix Properties section. From the εp list, choose User defined. In

the associated edit field, type epsilon. 6 From the κbr list, choose User defined. In the associated edit field, type kappa.

Initial Values 1 1 In the Model Builder window, under Model 1>Brinkman Equations click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the p edit field, type p0.

Volume Force 1 1 In the Model Builder window, right-click Brinkman Equations and choose Volume Force. 2 In the Volume Force settings window, locate the Domain Selection section.

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3 From the Selection list, choose All domains. 4 Locate the Volume Force section. In the F table, enter the following settings: 0

x

rho*g_const*beta*(T-Tc)

y

Pressure Point Constraint 1 1 Right-click Brinkman Equations and choose Points>Pressure Point Constraint. 2 Select Point 4 only. 3 In the Pressure Point Constraint settings window, locate the Pressure Constraint

section. 4 In the p0 edit field, type p0. H E A T TR A N S F E R I N PO RO U S M E D I A

1 In the Model Builder window’s toolbar, click the Show button and select Discretization

in the menu. 2 In the Model Builder window, under Model 1 click Heat Transfer in Porous Media. 3 In the Heat Transfer in Porous Media settings window, click to expand the Discretization section. 4 From the Temperature list, choose Quadratic. 5 Click to collapse the Discretization section.

Porous Matrix 1 1 In the Model Builder window, under Model 1>Heat Transfer in Porous Media click Porous Matrix 1. 2 In the Porous Matrix settings window, locate the Immobile Solids section. 3 In the θp edit field, type 1-epsilon. 4 Locate the Heat Conduction section. From the kp list, choose User defined. Locate the Thermodynamics section. From the ρp list, choose User defined. From the Cp,p list,

choose User defined.

Heat Transfer in Fluids 1 1 In the Model Builder window, under Model 1>Heat Transfer in Porous Media click Heat Transfer in Fluids 1. 2 In the Heat Transfer in Fluids settings window, locate the Model Inputs section. 3 In the pA edit field, type p0.

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4 From the u list, choose Velocity field (br/fmp1). 5 Locate the Heat Conduction section. From the k list, choose User defined. In the

associated edit field, type k. 6 Locate the Thermodynamics section. From the ρ list, choose User defined. In the

associated edit field, type rho. 7 From the Cp list, choose User defined. In the associated edit field, type Cp. 8 From the γ list, choose User defined. In the associated edit field, type gamma.

Initial Values 1 1 In the Model Builder window, under Model 1>Heat Transfer in Porous Media click Initial Values 1. 2 In the Initial Values settings window, locate the Initial Values section. 3 In the T edit field, type Tc.

Temperature 1 1 In the Model Builder window, right-click Heat Transfer in Porous Media and choose Temperature. 2 Select Boundary 2 only. 3 In the Temperature settings window, locate the Temperature section. 4 In the T0 edit field, type Th.

Temperature 2 1 Right-click Heat Transfer in Porous Media and choose Temperature. 2 Select Boundaries 3 and 5 only. 3 In the Temperature settings window, locate the Temperature section. 4 In the T0 edit field, type Tc.

Temperature 3 1 Right-click Heat Transfer in Porous Media and choose Temperature. 2 Select Boundary 1 only. 3 In the Temperature settings window, locate the Temperature section. 4 In the T0 edit field, type Th-(Th-Tc)*s.

Temperature 4 1 Right-click Heat Transfer in Porous Media and choose Temperature. 2 Select Boundary 4 only. 3 In the Temperature settings window, locate the Temperature section.

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4 In the T0 edit field, type Tc-(Tc-Th)*s. STUDY 1

Parametric Sweep 1 In the Model Builder window, right-click Study 1 and choose Parametric Sweep. 2 In the Parametric Sweep settings window, locate the Study Settings section. 3 Click Add. 4 In the table, enter the following settings: Parameter names

Parameter value list

beta

0 1e-12 1e-11 1e-10 1e-9 1e-8 1e-7 1e-6

Solver 1 1 Right-click Study 1 and choose Show Default Solver. 2 In the Dependent Variables settings window, locate the Scaling section. 3 From the Method list, choose Initial value based. 4 In the Model Builder window, under Study 1>Solver Configurations>Solver 1>Stationary Solver 1 click Fully Coupled 1. 5 In the Fully Coupled settings window, locate the Method and Termination section. 6 From the Nonlinear method list, choose Double dogleg. 7 In the Model Builder window, right-click Study 1 and choose Compute. RESULTS

Velocity (br) The first default plot group shows the velocity magnitude. Add an arrow plot to see the flow direction.

Pressure (br) 1 In the Model Builder window, expand the Velocity (br) node.

Because the cavity is closed, the pressure distribution is solely due to gravity.

Temperature 1 In the Model Builder window, expand the Pressure (br) node.

The third default plot group shows the temperature field as a surface plot. 2 Click the Plot button.

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