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Coupled Heat and Mass Transfer

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Old   May 29, 2020, 08:23
Default Coupled Heat and Mass Transfer
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Felix Hochwallner
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Hello,

I ultimately want to solve a coupled heat and mass transfer problem of two regions. One region is a liquid desiccant solution and the other region is humid air. The liquid desiccant takes up water content from the air stream. Meanwhile sensible heat is transferred. Furthermore, latent heat is "generated" at the boundary, as the water changes phase from the air (vapor) to the solution (liquid).

I am using the official OpenFoam v7 on an Ubuntu machine.

Assumptions:
  • The humidity/water content in both air and solution is modelled by a passive scalar, which is called cw (mass fraction of water to total mass).
  • Both solutions are incompressible with constant thermophysical properties (besides the vapor pressure, which depends on temperature and water content).
  • Laminar steady state flow in both regions.
  • Ideal gases.
  • Indeformable, flat interface.
  • On the interface between air and solution the temperatures (for heat transfer) and water vapor pressure (for mass transfer) are equal.

In order to solve this problem I started with the chtMultiRegionFoam solver, as it includes multiple regions and already an implementation of the sensible heat transfer. I first solve for the flow field (U,p) and afterwards freeze the field and solve for the coupled heat and mass Transfer (T, cw). For both solvers, the air region is solved first and afterwards the solution region.

The water vapor pressure (ps) in the air region can be obtained directly from the water content and the system pressure ps_air=ps_air(cw_air,p_air). In the solution, the water vapor pressure is a function of the water content and the solution temperature ps_sol = ps_sol(cw_sol,T_sol). This relation is given by a fit of experimental data (fit is conducted as an Antoine equation, with coefficients A,B,C as a function of the water content cw_sol).

The idea behind the region coupling I got from Brent A. Craven and Robert L. Campbell on a presentation given at the 6th OpenFOAM Workshop (13–16 June 2011).

For the temperature interface I created a boundary condition which satisfies a thermodynamic equilibrium (equal temperatures at interface in both regions) and the energy balance. For this reason, the temperature at the interface on the air side is set equal to the temperature at the interface on the solution side (optionally with a relaxation factor). For the solution region I set the gradient in order to fulfill the energy balance. Therefore, the heat transferred to the solution region is equal to the sensible heat flow of the air region plus the latent heat transferred due to the absorbed mass flux.

For the water content interface I created a boundary condition analog to the temperature boundary condition. In the air region, the water vapor pressure at the interface is set equal to the water vapor pressure of the solution at the interface (again optionally with a relaxation factor). For the solution region the gradient of the water content is set to fulfill the mass balance. The gradient is calculated according to the Eckert Schneider relation for equimolar diffusion dotM/A = (rho D snGrad(cw))/(1-cw), where dotM is the absorbed mass flux, A the surface area of the cell, rho the density of the solution/air, D the diffusion coefficient of water in the solution/air and cw the water content.

So far, the simulation is working. However, there are two things I am concerned about:
  1. The residuals of the water content stay somewhat high (1e-2 for the air region, 1e-4 for the solution region)
  2. A step in the gradient for both temperature and water content inside the air region.

Especially the second thing concerns me. Doesn’t this imply, that the energy and mass balance are not fulfilled inside the air region?

This is the geometry which is simulated:

The air region is colored green and the solution region is colored red. Both are very long in z- and x- direction, but fairly thin in y-direction. The height (z) is 0.7m, the length (x) is 0.1m and the width (y) is 6e.4m, 2e-3m (solution/air). The air flows in x direction (with a no slip condition at the interface) and the solution flows in z-direction (with a no slip condition at the wall, y=0).

These are the residuals of the flow problem in the air region (U,p):

As there are no velocities in z-direction the somewhat high residuals for Uz should not be a problem.

These are the residuals of the flow problem in the solution region (U,p):

As there are no velocities in the x-direction the somewhat high residuals for Ux should not to be a problem.

These are the residuals of the heat and mass transfer problem in the air region (T,cw):


These are the residuals of the heat and mass transfer problem in the solution region (T,cw):


This is the temperature distribution in the air region for different heights (z) in respect to the width (y).

A non-steady gradient is visible close to the interface (y=0).

This is the water content distribution in the air region for different heights (z) in respect to the width (y).

A non-steady gradient is visible close to the interface (y=0).

However, the total absorbed mass flux, which the solution takes up from the air stream, does not change anymore over successive iterations:


So my questions are:
  • Do you know what causes the high residuals of the solution of the water content in the heat and mass transfer problem in both regions?
  • What causes the unsteady gradient in the air region for both temperature and water content? Doesn't this imply, that the energy and mass balance are not fulfilled inside the air region?

To me it looks like the fixed gradient boundary condition gives problems to the solver and is actually not fulfilled.
If you need any further information about the solver (solver settings, discretization schemes, mesh, etc.) just leave a reply and I will update the post. The post already got way longer than I wanted, therefore I didn't want to fill it with more content.

I already thank you a lot in advance!

Best,
Felix
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Old   July 12, 2020, 19:24
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Joachim Herb
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I presented something simalar as tutorial at last years OpenFOAM workshop in Duisburg. Have a look at the code and the slides: https://github.com/jmozmoz/OFdevelop...nsationCoupled
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