Simultaneous Heat and Moisture Transfer in a Porous Cylinder

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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.


Contributed by: Clay Gruesbeck (January 2018)
Open content licensed under CC BY-NC-SA



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.


moisture capacity

heat capacity

convective heat transfer coefficient

convective mass transfer coefficient

moisture conductivity

thermal conductivity


initial cylinder moisture

moisture of the air

cylinder radius

space coordinate



initial cylinder temperature

temperature of the air

moisture potential


equivalent moisture diffusion coefficient

thermo-gradient coefficient

ratio of vapor diffusion coefficient to coefficient of total diffusion of moisture

equivalent temperature diffusion coefficient

heat of vaporization


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:



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 .


[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.

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