11514

Simultaneous Heat and Moisture Transfer in a Porous Cylinder

This Demonstration illustrates a model of heat and moisture transfer accompanied by phase change in a porous cylinder. The porous cylinder is initially at a constant temperature and moisture. It is suddenly placed in contact with a stream of hot air that exchanges heat and moisture by diffusion and convection. The moisture movement and the phase change occurring within the cylinder generate a coupled relationship between mass and heat transfer.
The governing equations [1] for this model are the Luikov equations:
,
.
Here and are temperature and moisture potential, respectively, is the space coordinate, is time, and are positive coupling coefficients determined by moisture and heat migration, respectively, and and represent the temperature and moisture diffusion coefficients.

DETAILS

The Luikov equation coefficients are:
.
The moisture potential is related to moisture content by . The cylinder is assumed to have an infinite length to radius ratio, thus only radial transport is considered; material properties are assumed to be constant.
Nomenclature
moisture capacity
heat capacity
convective heat transfer coefficient
convective mass transfer coefficient
moisture conductivity
thermal conductivity
moisture
initial cylinder moisture
moisture of the air
space coordinate
time
temperature
initial cylinder temperature
temperature of the air
moisture potential
Symbols
equivalent moisture diffusion coefficient
ratio of vapor diffusion coefficient to coefficient of total diffusion of moisture
equivalent temperature diffusion coefficient
heat of vaporization
density
coupling coefficient due to moisture migration
coupling coefficient due to heat conduction
At the boundary of the cylinder, the latent heat of vaporization becomes part of the energy balance, and the mass diffusion caused by the temperature and moisture gradients affects the overall mass balance; the boundary conditions at are:
and
.
The right-hand side of the first boundary condition represents the heat flux due to convection plus the energy transfer due to phase change, and the right-hand side of the second condition represents the mass flux due to convection plus the mass flux due to the temperature gradient.
At because of symmetry we have:
.
The initial conditions are
.
The system of Luikov equations is solved using the finite element method as implemented in the built-in Mathematica function NDSolve, and the results are presented in space and time plots for various values of the temperature and moisture diffusion coefficients and .
Reference
[1] H. R. Thomas, R. W. Lewis and K. Morgan, "An Application of the Finite Element Method to the Drying of Timber," Wood and Fiber Science: Journal of the Society of Wood Science and Technology, 11(4), 1980 pp. 237–243. wfs.swst.org/index.php/wfs/article/download/534/534.

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