Two-dimensional numerical modeling and simulation of heat and moisture transfer during drying

Ishtiaq Ali; Ishtiaq Ali

doi:10.3934/math.2026360

AIMS Mathematics

2026, Volume 11, Issue 3: 8762-8791. doi: 10.3934/math.2026360

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Two-dimensional numerical modeling and simulation of heat and moisture transfer during drying

Ishtiaq Ali ^,

Department of Mathematics and Statistics, College of Science, King Faisal University, P. O. Box 400, Al-Ahsa, 31982, Saudi Arabia

Received: 11 January 2026 Revised: 25 March 2026 Accepted: 27 March 2026 Published: 31 March 2026
MSC : 35K05, 80M20, 65M22

This study explores a two-dimensional model for transient heat and moisture transfer in a rectangular domain subject to convective boundary conditions. The governing equations consist of parabolic partial differential equations, which are derived from Fourier's law of heat conduction and Fick's law of mass diffusion. The model equations are discretized using central finite-difference approximations in space and a Crank–Nicolson (CN) scheme in time. To efficiently handle the multidimensional diffusion operators, an operator splitting strategy based on the Peaceman–Rachford alternating direction implicit (ADI) method is employed. This approach decomposes the two-dimensional diffusion operator into a sequence of one-dimensional subproblems, each associated with a single spatial direction, which significantly reduces the computational complexity of the fully implicit scheme while retaining second-order accuracy in time and space and unconditional stability for linear diffusion operators. The numerical scheme is shown to be unconditionally stable for the diffusion operators and computationally efficient for long-time simulations. The numerical results for the temperature and moisture fields are validated through comparison with available analytical solutions and experimental data from the literature, demonstrating very good agreement in all cases. In addition, a structure-preserving iteration method (SPIM) applied to the unsplit CN formulation is used as a benchmark reference, confirming that the proposed ADI–CN splitting achieves accuracy comparable to the unsplit CN scheme while maintaining competitive computational efficiency.
- heat and moisture transfer,
- parabolic partial differential equations,
- operator splitting strategy,
- numerical simulations
Citation: Ishtiaq Ali. Two-dimensional numerical modeling and simulation of heat and moisture transfer during drying[J]. AIMS Mathematics, 2026, 11(3): 8762-8791. doi: 10.3934/math.2026360

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Abstract

This study explores a two-dimensional model for transient heat and moisture transfer in a rectangular domain subject to convective boundary conditions. The governing equations consist of parabolic partial differential equations, which are derived from Fourier's law of heat conduction and Fick's law of mass diffusion. The model equations are discretized using central finite-difference approximations in space and a Crank–Nicolson (CN) scheme in time. To efficiently handle the multidimensional diffusion operators, an operator splitting strategy based on the Peaceman–Rachford alternating direction implicit (ADI) method is employed. This approach decomposes the two-dimensional diffusion operator into a sequence of one-dimensional subproblems, each associated with a single spatial direction, which significantly reduces the computational complexity of the fully implicit scheme while retaining second-order accuracy in time and space and unconditional stability for linear diffusion operators. The numerical scheme is shown to be unconditionally stable for the diffusion operators and computationally efficient for long-time simulations. The numerical results for the temperature and moisture fields are validated through comparison with available analytical solutions and experimental data from the literature, demonstrating very good agreement in all cases. In addition, a structure-preserving iteration method (SPIM) applied to the unsplit CN formulation is used as a benchmark reference, confirming that the proposed ADI–CN splitting achieves accuracy comparable to the unsplit CN scheme while maintaining competitive computational efficiency.

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