Approximate analytical solution for Richards’ equation with finite constant water head Dirichlet boundary conditions

Chen, Xi; He, Dandan; Yang, Guangzhi; Yuan, Yong; Dai, Ying

doi:10.1007/s40314-021-01626-z

Cited by 1 publication

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“…Following the Boltzmann transformation [ 38 , 39 , 42 ], the partial differential Equation ( 7 ) can be transferred into an ordinary differential equation. A Boltzmann variable

is introduced, as a function of the time t and the x -coordinate, i.e.,

The partial derivatives of the Boltzmann variable

with respect to time and x -coordinate read as

Substituting the Boltzmann variable

into the governing Equation ( 7 ) and recalling the implicit differentiation result in

which can be simplified by referring to the partial derivatives of the Boltzmann variable, i.e., Equation ( 11 ), as follows:

Recalling Equation ( 5 ) gives access to the differential,

, which is then substituted into Equation ( 13 ).…”

Section: Establishment Of the Semi-analytical Approachmentioning

confidence: 99%

“…Further developments include the homotopy perturbation method [ 34 ], the homotopy analysis method [ 35 ], the tanh method [ 36 ], and so on. Given the mathematical similarity between the governing equations of the heat conduction problem and the diffusion problem, recent progress on solutions of the linear diffusion equation, utilizing three different Ansätze [ 37 ], and of the nonlinear diffusion equation, utilizing Boltzmann transformation [ 38 , 39 ], are also worth mentioning. The variable transformation technique of the latter was also followed in the present work.…”

Section: Introductionmentioning

confidence: 99%

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Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach

Wang

Chen

Koenders

et al. 2023

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Thermal loading, especially in fire scenarios, challenges the safety and long-term durability of concrete structures. The resulting heat propagation within the structure is governed by the heat conduction equation, which can be difficult to solve analytically because of the nonlinearity related to the thermophysical properties of concrete. A semi-analytical approach for the transient nonlinear heat conduction problem in concrete structures was established in the present work. The nonlinearity related to the temperature-dependent thermal conductivity, mass density, and specific heat capacity of heated concrete was taken into consideration. A Taylor series approximate solution was first established within a small neighborhood, employing the Boltzmann transformation in combination with the mean value theorem. Thereafter, it was extended to the whole domain by utilizing the Bernstein polynomial. The semi-analytical approach was validated by comparing it with the numerical results of two independent Finite Element simulations of nonlinear heat conduction along concrete plates, subjected to either moderate or fierce thermal loading. Absolute values of the relative errors are smaller than 5%. The validated semi-analytical approach was further applied to prediction of the temporal evolution of the temperature field of a scaled model of a subway station, subjected to fire disaster. The nonlinearities, related to the time-dependent surface temperature and the temperature-dependent thermophysical properties of concrete, were taken into consideration. The predictions agree well with the experimental measurements. The established semi-analytical approach exhibits good accuracy and stability, providing insight into the interaction between the thermophysical properties of concrete in the heat conduction process.

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“…Following the Boltzmann transformation [ 38 , 39 , 42 ], the partial differential Equation ( 7 ) can be transferred into an ordinary differential equation. A Boltzmann variable

is introduced, as a function of the time t and the x -coordinate, i.e.,

The partial derivatives of the Boltzmann variable

with respect to time and x -coordinate read as

Substituting the Boltzmann variable

into the governing Equation ( 7 ) and recalling the implicit differentiation result in

which can be simplified by referring to the partial derivatives of the Boltzmann variable, i.e., Equation ( 11 ), as follows:

Recalling Equation ( 5 ) gives access to the differential,

, which is then substituted into Equation ( 13 ).…”

Section: Establishment Of the Semi-analytical Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach

Wang

Chen

Koenders

et al. 2023

Entropy

Self Cite

View full text Add to dashboard Cite

show abstract

Approximate analytical solution for Richards’ equation with finite constant water head Dirichlet boundary conditions

Cited by 1 publication

References 23 publications

Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach

Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach

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