Created in COMSOL Multiphysics 5.2a

Ev apo r a t i o n i n Po r ou s Med i a w i t h Sma l l E v apo r a t i o n Ra t e s

This model is licensed under the COMSOL Software License Agreement 5.2a.All trademarks are the property of their respective owners. See www.comsol.com/trademarks.

Introduction

Evaporation in porous media is an important process in food and paper industry among others. Many physical effects must be considered: fluid flow, heat transfer and transport of participating fluids and gases. All of these effects are strongly coupled and can be handled by using the predefined interfaces of COMSOL Multiphysics. When having a liquid and a gaseous phase inside the porous domain, some adjustments of the predefined equations are required. In this tutorial, evaporation is assumed to have a negligible impact on the amount of liquid water, making it possible to describe the transport of water vapor with a convection-diffusion equation.

Porous domain

Free flow domain

Air flow

Figure 1: Geometry and principle of the model.

Model Definition

This tutorial describes laminar air flow through a humid porous medium. The air is dry at the inlet and its moisture content increases as it flows through the porous media.

The flow inside the porous medium is described with Brinkman equation. The flow in the surrounding domain is described with the laminar Navier-Stokes equation. The geometry and basic set-up is shown in Figure 1.

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T H E R M O D Y N A M I C P R O P E R T I E S

The thermodynamic properties of air with water vapor can be described using mixture laws, based on the amount of water vapor and dry air. This is done automatically when choosing Moist air as fluid type and the governing equations can be found in the Heat Transfer Module User’s Guide. As input term for water vapor, the concentration c (mol/m³) from the transport equation is used.

F L O W P R O P E R T I E S

To model the fluid flow from the air domain, using the laminar Navier-Stokes equation into the porous domain using Brinkman equation the Laminar Flow Interface is used and extended by enabling porous media domains. This way, the coupling of both flow regimes is done automatically. The resulting velocity field then can be used to model convective heat and species transport.

TR A N S P O R T P R O P E R T I E S

The fraction of water vapor is small and the Transport of Diluted Species interface is used to describe the transport properties in the free flow and porous domain. In the porous domain, the diffusion coefficient for water vapor into air DL (m²/s) needs to be adjusted according to

.

This describes the effective diffusivity inside a porous medium, depending on its structure, characterized by the dimensionless numbers porosity εp and tortuosity τL. Here, the Bruggeman correction is used, which is

(1)

E V A P O R A T I O N

Including the evaporation process is done by adding the mass of water vapor that is evaporated as source term in the transport equation. Evaporation takes place, if the concentration of water vapor is below the equilibrium concentration, which is determined by the saturation concentration csat and water activity aw , with

(2)

with the saturation pressure psat and the ideal gas constant R = 8.314 J/(mol·K). The water activity describes the amount of water that evaporates into air. In general it is a

DeεpτL-----DL=

De εp3 2⁄ DL=

csatpsat T( )

RT--------------------=

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function depending on the water content on dry basis of the surrounding air and the temperature (see for example Ref. 2), but here an approximate value of aw = 0.9 is used because of the moderate variation of water content. Hence the amount of evaporated water is defined as:

(3)

where K (1/s) is the evaporation rate, and c the current concentration. The evaporation rate depends on the material properties and the process which causes the evaporation. It must be chosen so that the solution is not affected if further increased. This corresponds to assuming that vapor is in equilibrium with the liquid or in other words, the time scale for evaporation is much smaller than the smallest time scale of the transport equations. This is true for pore sizes that are not too large (Ref. 1).

The heat of evaporation is then inserted as source term in the heat transfer equation according to:

(4)

where Hvap (J/mol) is the latent heat of evaporation.

mvap K awcsat c–( )⋅=

Q Hvap mvap⋅=

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Results and Discussion

After one minute the temperature field shows strong cooling due to evaporation. As seen below, evaporation mainly occurs at the surface of the porous medium. Inside the domain, the temperature gradients remain moderate.

Figure 2: Temperature distribution after 60 s.

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The concentration distribution is shown in the next plot.

Figure 3: Concentration distribution after 60 s

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The pressure field shows steep gradients at the interface boundaries.

Figure 4: Pressure field combining the pressure from the Laminar Flow interface and the Darcy’s Law interface.

Notes About the COMSOL Implementation

Using a proper mesh size is important to resolve the steep gradients at the interface boundaries. Therefore a customized mesh with boundary layers is used.

To get good convergence of the time dependent behavior, first solve the stationary flow equation only. This solution will then be used as initial value for the time dependent study step providing smooth initial conditions in the modeling domain.

Instead of coupling the Transport of Dilutes Species Interface with the flow interface manually, the Reacting Flow in Porous Media interface provides a predefined coupling of both equations. Then, the coupling to the heat transfer interface is the only coupling that has to be done manually. This interface is available with one of the following modules: Batteries and Fuel Cells Module, CFD Module or Chemical Reaction Engineering Module.

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Reference

1. A. Halder, A. Dhall and A.K. Datta, “Modeling Transport in Porous Media with Phase Change: Applications to Food Processing”, J. Heat Transfer, vol. 133, no. 3, 2011.

2. A.K. Datta, “Porous media approaches to studying simultaneous heat and mass transfer in food processes. II: Property data and representative results, Journal of Food Engineering, Vol. 80, (2007)

Application Library path: Heat_Transfer_Module/Phase_Change/evaporation_porous_media_small_rate

Modeling Instructions

From the File menu, choose New.

N E W

In the New window, click Model Wizard.

M O D E L W I Z A R D

1 In the Model Wizard window, click 2D.

2 In the Select Physics tree, select Heat Transfer>Heat Transfer in Fluids (ht).

3 Click Add.

4 In the Select Physics tree, select Chemical Species Transport>Transport of Diluted Species

(tds).

5 Click Add.

6 In the Select Physics tree, select Fluid Flow>Single-Phase Flow>Laminar Flow (spf).

7 Click Add.

8 Click Study.

9 In the Select Study tree, select Preset Studies for Selected Physics Interfaces>Stationary.

10 Click Done.

G E O M E T R Y 1

1 In the Model Builder window, under Component 1 (comp1) click Geometry 1.

2 In the Settings window for Geometry, locate the Units section.

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3 From the Length unit list, choose cm.

Rectangle 1 (r1)1 On the Geometry toolbar, click Primitives and choose Rectangle.

2 In the Settings window for Rectangle, locate the Size and Shape section.

3 In the Width text field, type 30.

4 In the Height text field, type 10.

Rectangle 2 (r2)1 On the Geometry toolbar, click Primitives and choose Rectangle.

2 In the Settings window for Rectangle, locate the Size and Shape section.

3 In the Width text field, type 8.

4 Locate the Position section. In the x text field, type 6.

Fillet 1 (fil1)1 On the Geometry toolbar, click Fillet.

2 On the object r2, select Points 3 and 4 only.

3 In the Settings window for Fillet, locate the Radius section.

4 In the Radius text field, type 0.25.

5 On the Geometry toolbar, click Build All.

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6 Click the Zoom Extents button on the Graphics toolbar.

Start with defining some parameters.

ParametersOn the Home toolbar, click Parameters.

G L O B A L D E F I N I T I O N S

Parameters1 In the Settings window for Parameters, locate the Parameters section.

2 In the table, enter the following settings:

Name Expression Value Description

p0 1[atm] 1.0133E5 Pa Ambient pressure

T0 20[degC] 293.15 K Ambient temperature

u0 0.1[m/s] 0.1 m/s Free stream velocity

D_wa 2.6e-5[m^2/s] 2.6E-5 m²/s Water-air diffusivity

M_w 0.018[kg/mol] 0.018 kg/mol Molar mass of water

H_vap 2.454e6[J/kg]*M_w 44172 J/mol Heat of vaporization

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Set up the different physics interfaces. Start with the heat transfer interface, where you define the thermodynamic properties of moist air for the free flow and porous domain.

H E A T TR A N S F E R I N F L U I D S ( H T )

1 In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Fluids

(ht).

2 In the Settings window for Heat Transfer in Fluids, locate the Physical Model section.

3 Select the Heat transfer in porous media check box.

Porous Medium 11 On the Physics toolbar, click Domains and choose Porous Medium.

Set the ambient temperature to be used in boundary conditions of the Heat Transfer interface.

2 In the Settings window for Heat Transfer in Fluids, locate the Ambient Settings section.

3 In the Tamb text field, type T0.

Use the velocity from the Laminar Flow interface as input. Later it is customized to take the porous properties into account.

4 In the Model Builder window, click Porous Medium 1.

5 Select Domain 2 only.

6 In the Settings window for Porous Medium, locate the Model Inputs section.

7 From the pA list, choose Absolute pressure (spf).

8 From the u list, choose Velocity field (spf).

Select Moist Air as Fluid type. The moisture content is evaluated using the concentration from the Transport of Dilutes Species interface.

9 Locate the Thermodynamics, Fluid section. From the Fluid type list, choose Moist air.

10 From the Input quantity list, choose Concentration.

11 Locate the Model Inputs section. From the c list, choose Concentration (tds).

12 Locate the Immobile Solids section. In the θp text field, type 0.2.

Similar settings apply for the Fluid node.

K_evap 1000[1/s] 1000 1/s Evaporation rate

a_w 0.9 0.9 Water activity

Name Expression Value Description

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Fluid 11 In the Model Builder window, under Component 1 (comp1)>Heat Transfer in Fluids (ht)

click Fluid 1.

2 In the Settings window for Fluid, locate the Model Inputs section.

3 From the pA list, choose Absolute pressure (spf).

4 From the u list, choose Velocity field (spf).

5 Locate the Thermodynamics, Fluid section. From the Fluid type list, choose Moist air.

6 From the Input quantity list, choose Concentration.

7 Locate the Model Inputs section. From the c list, choose Concentration (tds).

Temperature 11 On the Physics toolbar, click Boundaries and choose Temperature.

2 Select Boundary 1 only.

3 In the Settings window for Temperature, locate the Temperature section.

4 From the T0 list, choose Ambient temperature (ht).

Outflow 11 On the Physics toolbar, click Boundaries and choose Outflow.

2 Select Boundary 9 only.

The next step is to set up the source term for the heat of vaporization according to Equation 4.

Heat Source 11 On the Physics toolbar, click Domains and choose Heat Source.

2 Select Domain 2 only.

3 In the Settings window for Heat Source, locate the Heat Source section.

4 In the Q0 text field, type -H_vap*m_evap.

The variable m_evap is defined according to Equation 3 and depends on the saturation concentration (Equation 2) and water activity.

D E F I N I T I O N S

In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.

Variables 11 In the Settings window for Variables, locate the Variables section.

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2 In the table, enter the following settings:

Continue with the Transport of Diluted Species interface. Water vapor is produced (m_evap) and saturation concentration is reached in the porous domain. For water-air diffusivity the Bruggeman correction (Equation 1) is used to describe the diffusive transport.

TR A N S P O R T O F D I L U T E D S P E C I E S ( T D S )

Transport Properties 11 In the Model Builder window, under Component 1 (comp1)>Transport of Diluted Species

(tds) click Transport Properties 1.

2 In the Settings window for Transport Properties, locate the Model Inputs section.

3 From the u list, choose Velocity field (spf).

4 Locate the Diffusion section. In the Dc text field, type D_wa.

At the inlet for the free air flow the concentration is set to zero which implies dry air.

5 In the Model Builder window, click Transport of Diluted Species (tds).

Concentration 11 On the Physics toolbar, click Boundaries and choose Concentration.

2 Select Boundary 1 only.

3 In the Settings window for Concentration, locate the Concentration section.

4 Select the Species c check box.

Outflow 11 On the Physics toolbar, click Boundaries and choose Outflow.

2 Select Boundary 9 only.

Initial Values 21 On the Physics toolbar, click Domains and choose Initial Values.

2 Select Domain 2 only.

3 In the Settings window for Initial Values, locate the Initial Values section.

4 In the c text field, type ht.porous1.fpsat(T0)/(R_const*T0).

Name Expression Unit Description

c_sat ht.fluid1.fpsat(T)/(R_const*T)

mol/m³ Saturation vapor concentration

m_evap K_evap*(a_w*c_sat-c)

mol/(m³·s) Evaporated mass of water vapor

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Reactions 11 On the Physics toolbar, click Domains and choose Reactions.

2 Select Domain 2 only.

3 In the Settings window for Reactions, locate the Reaction Rates section.

4 In the Rc text field, type m_evap.

Porous Media Transport Properties 11 On the Physics toolbar, click Domains and choose Porous Media Transport Properties.

2 Select Domain 2 only.

3 In the Settings window for Porous Media Transport Properties, locate the Model Inputs section.

4 From the u list, choose Velocity field (spf).

5 Locate the Diffusion section. In the DF, c text field, type D_wa.

6 From the Effective diffusivity model list, choose Bruggeman model.

The remaining task is to set up the flow interface for the free and porous domain.

L A M I N A R F L O W ( S P F )

The next step enables additional features for the flow interface to account for porous domains also.

1 In the Model Builder window, under Component 1 (comp1) click Laminar Flow (spf).

2 In the Settings window for Laminar Flow, locate the Physical Model section.

3 Select the Enable porous media domains check box.

Fluid Properties 11 In the Model Builder window, under Component 1 (comp1)>Laminar Flow (spf) click Fluid

Properties 1.

2 In the Settings window for Fluid Properties, locate the Fluid Properties section.

3 From the ρ list, choose User defined. In the associated text field, type ht.rho.

4 From the μ list, choose Dynamic viscosity (ht).

This step sets the density to the density that is calculated by the moist air feature within the Heat Transfer interface.

Set up the porous domain properties.

5 In the Model Builder window, click Laminar Flow (spf).

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Fluid and Matrix Properties 11 On the Physics toolbar, click Domains and choose Fluid and Matrix Properties.

2 Select Domain 2 only.

3 In the Settings window for Fluid and Matrix Properties, locate the Fluid Properties section.

4 From the ρ list, choose User defined. In the associated text field, type ht.rho.

5 From the μ list, choose Dynamic viscosity (ht).

Inlet 11 On the Physics toolbar, click Boundaries and choose Inlet.

2 Select Boundary 1 only.

3 In the Settings window for Inlet, locate the Boundary Condition section.

4 From the list, choose Laminar inflow.

5 Locate the Laminar Inflow section. In the Uav text field, type u0.

The Entrance length value must be large enough so that the flow can reach a laminar profile. For a laminar flow, Lentr should be significantly greater than 0.06ReD, where Re is the Reynolds number and D is the inlet length scale. In this case, 30 m is an appropriate value.

6 In the Lentr text field, type 30.

Outlet 11 On the Physics toolbar, click Boundaries and choose Outlet.

2 Select Boundary 9 only.

Specify the porous material properties.

M A T E R I A L S

In the Model Builder window, under Component 1 (comp1) right-click Materials and choose Blank Material.

Material 1 (mat1)1 In the Settings window for Material, type Porous Matrix in the Label text field.

2 Select Domain 2 only.

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3 Locate the Material Contents section. In the table, enter the following settings:

Steep gradients are present and the default mesh is adjusted to resolve the flow and transport equations with high accuracy.

M E S H 1

In the Model Builder window, under Component 1 (comp1) right-click Mesh 1 and choose Free Triangular.

Free Triangular 1In the Model Builder window, under Component 1 (comp1)>Mesh 1 right-click Free Triangular 1 and choose Size.

Size 11 In the Settings window for Size, locate the Geometric Entity Selection section.

2 From the Geometric entity level list, choose Domain.

3 Select Domain 1 only.

4 Locate the Element Size section. From the Calibrate for list, choose Fluid dynamics.

5 From the Predefined list, choose Finer.

Free Triangular 1Right-click Free Triangular 1 and choose Size.

Size 21 In the Settings window for Size, locate the Geometric Entity Selection section.

2 From the Geometric entity level list, choose Domain.

3 Select Domain 2 only.

4 Locate the Element Size section. From the Calibrate for list, choose Fluid dynamics.

5 From the Predefined list, choose Extra fine.

Property Name Value Unit Property group

Thermal conductivity k 0.21 W/(m·K) Basic

Density rho 1528 kg/m³ Basic

Heat capacity at constant pressure

Cp 1650 J/(kg·K) Basic

Porosity epsilon 0.4 1 Basic

Permeability kappa 1e-13 m² Basic

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Boundary Layers 11 In the Model Builder window, right-click Mesh 1 and choose Boundary Layers.

2 In the Settings window for Boundary Layers, locate the Domain Selection section.

3 From the Geometric entity level list, choose Domain.

4 From the Selection list, choose All domains.

Boundary Layer Properties1 In the Model Builder window, under Component 1 (comp1)>Mesh 1>Boundary Layers 1

click Boundary Layer Properties.

2 Select Boundaries 2–4, 6–8, 10, and 11 only.

3 In the Settings window for Boundary Layer Properties, locate the Boundary Layer

Properties section.

4 In the Number of boundary layers text field, type 5.

5 In the Thickness adjustment factor text field, type 2.

6 Click Build All.

First, solve the stationary flow equations only to provide a fully developed flow profile for the whole equation system that is solved in a time interval of 1 minute.

S T U D Y 1

Step 1: Stationary1 In the Settings window for Stationary, locate the Physics and Variables Selection section.

2 In the table, clear the Solve for check box for Heat Transfer in Fluids (ht) and Transport

of Diluted Species (tds).

Time DependentOn the Study toolbar, click Study Steps and choose Time Dependent>Time Dependent.

Step 2: Time Dependent1 In the Settings window for Time Dependent, locate the Study Settings section.

2 Click Range.

3 In the Range dialog box, type 0 in the Start text field.

4 In the Step text field, type 1.

5 In the Stop text field, type 60.

6 Click Replace.

7 On the Study toolbar, click Compute.

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R E S U L T S

Temperature (ht)Several default plots are created automatically. The first plot group shows the temperature field as in Figure 2.

1 Click the Zoom Extents button on the Graphics toolbar.

Concentration (tds)The concentration distribution (see Figure 3) is computed from the Transport of Diluted

Species interface.

Pressure (spf)The last plot group shows the pressure field computed from the Laminar Flow interface (Figure 4).

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