Moisture Transfer Equations Research Articles

Inverse Problem for the Moisture Transfer Equation: Development of a Method for Finding the Unknown Parameter and Proof of the Convergence of the Iterative Process

This study develops a mathematical model for soil moisture diffusion, addressing the inverse problem of determining both the diffusion coefficient and the variation coefficient in a nonlinear moisture transfer equation. The model incorporates specific boundary and initial conditions and utilizes experimentally measured moisture values at a boundary point as input data. An iterative method, based on an explicit gradient scheme, is introduced to estimate the soil parameters. The initial boundary value problem is discretized, leading to a difference analog and the formulation of a conjugate difference problem. Iterative formulas for calculating the unknown parameters are derived, with a priori estimates ensuring the convergence of the iterative process. Additionally, the research establishes the convergence of the numerical model itself, providing a rigorous foundation for the proposed approach. The study also emphasizes symmetry in moisture calculations, ensuring consistency regardless of the calculation direction (from right to left or left to right) and confirming that moisture distribution remains symmetric within specified intervals. This preservation of symmetry enhances the model’s robustness and accuracy in parameter estimation. The numerical simulations were successfully conducted over a 7-day period, demonstrating the model’s reliability. The discrepancy between the numerical predictions and experimental observations remained within the margin of measurement error, confirming the model’s accuracy.

Simultaneous heat, moisture, and salt transfer in porous building materials considering osmosis flow: Part 1: Theoretical modeling based on nonequilibrium thermodynamics

Salt weathering is a common deterioration phenomenon that affects outdoor cultural properties, and it is important to precisely predict the heat, moisture, and salt transfer in porous materials to suppress salt weathering. Osmosis and osmotic pressure were considered in the field of soil research, especially in clay research, but not in the field of outdoor cultural properties and building materials, which are the main target of salt weathering. Osmosis in clay is supposed to be caused by its surface charge. However, it has been suggested that sandstones and bricks that constitute cultural properties and buildings also have surface charge as clay. Thus, osmosis and osmotic pressure can occur in building materials, which may lead to materials degradation. In this study, we derive basic equations, based on nonequilibrium thermodynamics, for the simultaneous heat, dry air, water vapor, liquid water, cation, and anion transfer in building materials by considering osmosis. This equation was compared with existing model for heat and moisture transfer equations as well as models that considered the salt transfer. Based on the previous research for osmosis in clay, we summarized conditions under which osmosis occurs in building materials and presented an outlook for modeling the physical properties of materials related to osmosis.

УЧЕТ ВЛАЖНОСТИ В ПОВЫШЕНИИ ТОЧНОСТИ РАСЧЕТА ТЕПЛОВЫХ ПОТЕРЬ ЗДАНИЯ

A discrete-continuous approach was applied to the moisture transfer equation, which made it possible to obtain an analytical solution for the moisture potential. The new method was applied to study the unsteady-state moisture regime of the facade heat-insulating composite system with expanded polystyrene insulation and aerated concrete base. It was found that the mass moisture content of building materials achieved in the walls during the period of operation is lower than the mass moisture content used in construction regulations, which makes it possible to increase the accuracy of calculating the transmission heat losses of the building. Calculations of heat losses of a two-storey building in the climatic conditions of Moscow (Russia) showed a decrease in the heating system load by 5 %, and the reduction in heat losses for specific premises of the building ranged from 3.6 to 7 %.

К вопросу существования решения первой краевой задачи для уравнения влагопереноса Аллера – Лыкова с оператором дробного дискретно распределенного дифференцирования

The paper investigates the first boundary value problem for the Aller – Lykov moisture transfer equation with the operator of fractional discretely distributed differentiation. Fractional derivatives included in the equation are understood in the Riemann – Liouville sense. The equation in question is a generalization of the classical Aller – Lykov equation. It takes into account the colloidal capillary-porous structure of the soil, including the presence of flows against the moisture potential. The existence of a solution to the first boundary value problem is proved by the Fourier method.

Методика расчета тепловлагопереноса в специальной защитной обуви пожарного-спасателя

Разработана методика расчета тепловлагопереноса в специальной защитной обуви пожарного-спасателя, которая позволяет учесть терморегуляционные механизмы в стопе для оценки комфортности обуви, изготовленной из определенных пакетов материалов и эксплуатируемой в широком диапазоне внешних воздействий, как по тепловым и влажностным факторам, так и в зависимости от уровня активности (тяжести выполняемой работы) пожарного-спасателя. Для определения безопасной работы в специальной защитной обуви необходима оценка степени деструкции кожи (степень ожога). The article describes the solution of the problem of non-stationary heat transfer in the «foot – footwear – environment» system for multilayer flat, cylindrical or spherical materials and non-stationary heat transfer from the outer surface of the special protective footwear to firefighter's foot. The well-known biothermal equation and the generalized heat transfer equation are used as the basis for the model of heat and moisture transfer in boot materials, taking into account sweating on the human skin surface. It is possible to get data on the thermal state of the foot under various conditions by solving these equations with appropriate coefficients and boundary conditions: zero heat flow in the center of the foot and convective environmental conditions on the surface of the last layer of porous shoe material. Modeling for three cases of human activity: resting state, walking with a speed of 4.8 km/h, light work (in the office room) under the same environmental conditions was carried out. The obtained results showed that the model of heat and moisture transfer in protective boots, taking into account thermoregulatory processes in the foot, can be used to predict the functionality of the materials for the bottom and top of the boots in terms of the skin burns likelihood. The method for calculating heat and moisture transfer in special protective boots for a firefighter-rescuer has been developed, based on the simulation, which includes: - construction of a geometric image of a model of protective boots; - determination of the firefighter-rescuer operating mode; - selection of material packages for the details of the boot top and bottom. An indication of the thermophysical, structural, geometric characteristics of the materials allows, using the selected climatic conditions and the firefighter-rescuer operating mode, to establish the initial and boundary conditions for solving the basic system of heat and moisture transfer equations. The solution of the system of equations with the boundary and initial conditions makes it possible to assess the comfort of boots, as well as the rate of destruction of the skin under intense thermal effects (the probability of burns of varying degrees). Based on this analysis, further optimization of the boots design can be carried out by choosing different material packages or evaluating the thermal effect under various external influences.

A boundary problem for the time-fractional Hallaire–Luikov moisture transfer equation with Hilfer derivative

A boundary problem for the time-fractional Hallaire–Luikov moisture transfer equation with Hilfer derivative

Устойчивость и сходимость разностной схемы, аппроксимирующей нелокальную краевую задачу для нагруженного уравнения влагопереноса с производными дробного порядка

We study a boundary value problem for a loaded moisture transfer equation with two Gerasimov - Caputo fractional derivatives of different orders (α, β), variable coefficients, and a nonlocal integral boundary condition. On a uniform grid a difference scheme of order of approximation 𝑂(ℎ2 + 𝜏2) for 𝛼 = 𝛽 and 𝑂(ℎ2 + 𝜏2−𝑚𝑎𝑥⁡{𝛼,𝛽}) for 𝛼 ≠ 𝛽 is constructed. The study is carried out by the method of energy inequalities. For different ratios between the orders of fractional derivatives α and β, for solving the posed nonlocal boundary value problem, a priori estimates in the difference interpretation are obtained, from which the uniqueness of the solution of the posed problem, the continuous and uniform dependence of the solution on the input data, and the conv ergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme.

Boundary-Value Problem for the Aller–Lykov Nonlocal Moisture Transfer Equation

In this paper, a boundary-value problem for the inhomogeneous Aller–Lykov moisture transfer equation with a fractional Riemann–Liouville time derivative is examined. The equation considered is a generalization of the Aller–Lykov equation obtained by introducing the fractal rate of change of humidity, which explains the appearance of flows directed against the potential of humidity. The existence of a solution to the first boundary-value problem is proved by the Fourier method. Using the method of energy inequalities for solutions of the problem, we obtain an a priori estimate in terms of the fractional Riemann–Liouville derivative, which implies the uniqueness of the solution.

DERIVATION OF THE EQUATION OF UNSTEADY-STATE MOISTURE BEHAVIOUR IN THE ENCLOSING STRUCTURES OF BUILDINGS USING A DISCRETE-CONTINUOUS APPROACH

Two differential equations of moisture transfer based on the theory of moisture potential have been considered. The first equation includes the record of moisture transfer mechanisms of vapor and liquid phases and their relationship. The second equation is a simplified form of the first equation which makes it possible to apply a discrete-continuous approach. The peculiar properties of the boundary conditions setting of the outside air for temperature and humidity fields have been presented. It is proved that the use of the discrete-continuous method provides high accuracy of calculations and can be used in engineering practice to assess the unsteady humidity regime of enclosing structures.

The use of sorption and excess sorption isotherm in the mathematical modeling of the unsteady-state heat and humidity regime of the building envelope

In the given article the development of the moisture transfer equation based on the theory of moisture potential is considered. The task of combined heat and moisture transfer is one of the most complicated tasks in the building thermal physics field. The classical equations of moisture transfer by K.F. Fokin representing the transfer of moisture under the action of partial transfer potentials - the gradient of the partial pressure of water vapor and the gradient of humidity F - are listed. The possibility of uniform accounting of the combined water vapor transfer on the basis of the moisture potential F is described. The sorption isotherm for aerated concrete is constructed in accordance with the experiment carried out in a desiccator with an aqueous solution of sulfuric acid. A new equation of moisture transfer which takes into account moistening with vaporous moisture in the sorption zone of moisture and liquid moisture in the excess sorption zone of moisture is derived. In order to simplify the work with the obtained equation a new value of the relative potential capacity is introduced. A graph construction of sorption and excess sorption isotherms which are obtained using an analytical expression for the relative potential capacity is proposed. In the sorption zone of humidification the sorption and excess sorption isotherms coincide with the classical sorption isotherm. Meanwhile, in the excess sorption zone of humidification the sorption and excess sorption isotherms depend on temperature.

The comparison of a discrete-continuous approach and a method of finite differences in solving the problem of unsteady-state heat and moisture transfer in the building envelope

This article is devoted to the development of methods for calculating heat and humidity regime in the building envelope. The equation of steady-state thermal conductivity with boundary conditions of the third kind and the formula for calculating heat losses of a building based on the heat transfer equation have been considered. The equation of unsteady-state thermal conductivity as well as its solution using the discrete-continual approach has also been studied. The solution of the unsteady-state heat conductivity problem with invariable over time boundary conditions using the discrete-continuous approach was proposed by A.B. Zolotov and P.A. Akimov. The subsequent modernization of the solution was conducted by V.N. Sidorov and S.M. Matskevich. The unsteady-state equation of moisture transfer based on Fick’s second law using the theory of moisture potential is derived. The solution of the unsteady-state moisture transfer equation using the finite difference method according to an explicit difference scheme as well as the solution of the unsteady-state moisture transfer equation using the discrete-continuous approach is demonstrated. To prove the effectiveness of using the discrete-continuous approach in the area of the unsteady-state humidity conditions we compared the calculation results of the distribution of moisture in a single-layer enclosing structure made of aerated concrete using two methods of moisture potential theory. It was found that the difference in the results of calculation by the discrete-continual formula and by the method of finite differences does not exceed 3.2%.

On convergence of schemes of finite element method of high accuracy for the equation of heat and moisture transfer

In this paper difference schemes of the finite element method of a high order of accuracy for the non-stationary equation of moisture transfer of Aller are constructed and investigated. The increased order of accuracy is achieved through special sampling of temporal and spatial variables. The stability and convergence of the constructed numerical algorithms are proved, the corresponding a priori estimates are obtained in various norms, which are used later to obtain estimates of the accuracy of the scheme under weak assumptions on the smoothness of solutions to the differential problem.

USING DISCRETE-CONTINUOUS APPROACH FOR THE SOLUTION OF UNSTEADY-STATE MOISTURE TRANSFER EQUATION FOR MULTILAYER BUILDING WALLS

Moisture regime of enclosing structures is one of the most complicated and controversial directions in construction industry. Temporary climate impact on enclosing structures and low moisture inertia of building materials lead to the situation in which it is impossible to calculate the steady-state moisture regime. Numerical methods are usually used to assess the moisture behaviour of the enclosing structures. In the current paper, a differential equation of moisture transfer is formulated. The solution of the unsteady-state equation of moisture transfer was obtained using the discrete-continuous approach. Thus, a formula which allows scientists to calculate unsteady-state moisture transfer in multilayer walls of buildings was obtained. A two-layer building enclosing structure with aerated concrete base and mineral wool insulation was calculated.

MATHEMATICAL MODELS OF DISCRETE IRRIGATION TECHNOLOGIES FOR AGRICULTURAL CROPS IN FURROWS

The article discusses mathematical modeling of discrete technology of irrigation of crops by furrows, based on joint consideration of the system of Saint-Venan equations and equations of moisture and heat transfer in soils. The boundary conditions are described in detail in accordance with the technologies of irrigation by furrows at discrete water supply.

A Boundary Value Problem for a Partial Differential Equation with Fractional Derivative

Abstract In this paper, we investigate a nonlocal boundary value problem for an equation of special type. For y > 0 it is a fractional diffusion equation, which contains the Riemann-Liouville fractional derivative. For y < 0 it is a generalized equation of moisture transfer. A unique solvability of the considered problem is proved.

Modeling heat and moisture transfer of building facades thermally insulated by the panels with ventilated channels

This work deals with a numerical study of the thermal and humidity state of a brick wall of a building insulated by panels with ventilated channels in a long-term operation cycle. A new physical and mathematical model has been developed for calculating heat and moisture transfer processes in facades with such panels. The model is based on the joint solution to a system of non-stationary differential equations of heat and moisture transfer in multilayer porous materials, taking into account adsorption processes and phase transitions. In addition, the model includes equations that describe the processes of heat and vapor transfer in ventilated channels. The results of calculations for a sharply continental climate have shown that two characteristic peaks of relative moisture content are observed in the insulating material of the panel. One of them corresponds to late summer - early autumn; the second peak corresponds to winter - early spring. The seasons corresponding to these peaks are the most critical in terms of wetting the mineral wool panel.

Numerical-analytical method for solving boundary value problem for the generalized moisture transport equation

The paper studies qualitatively new equations of moisture transfer, which generalize the Aller and Aller-Lykov equations. The generalization contributes to revealing in the original equations the specific features of the studied massifs, their structure, physical properties, processes occurring in them through the introduction of the notion of the rates of change of the fractal dimension. We have obtained solutions to the constant coefficient difference equations as a system arising when using the method of lines for the equations with a Riemann-Liouville time fractional derivative with boundary conditions of the first kind. A priori estimates are obtained that imply convergence of the obtained solutions to systems of ordinary differential equations with variable fractional coefficients. Numerical tests have been carried out to confirm theoretical results of the study.

Numerical-analytical method for solving a boundary-value problem for the modified equation of moisture transfer with time-fractional derivative

Numerical-analytical method for solving a boundary-value problem for the modified equation of moisture transfer with time-fractional derivative

АНАЛИТИЧЕСКАЯ ОЦЕНКА ДИНАМИКИ ПОТЕНЦИАЛА ВЛАЖНОСТИ В ЗОНЕ АЭРАЦИИ В ОРОШАЕМЫХ ЗЕМЛЯХ

The issue of forecasting soil moisture capacity using differential equation of moisture transfer in the aeration zone has been considered. An analytical approach for estimating moisture capacity per the depth of active soil layer in the irrigation period has been recommended. The obtained outcomes can be used for different soil types with various physicochemical and hydrophysical properties. To make the recommended approach more demonstrative the results of some numerical computations are introduced at the end of the current work.

Simulation of conjugate heat and moisture transfer in multilayer porous materials with ventilated channels

A physical and mathematical model for calculating the thermal and moisture state of the facade of a building with ventilated channels has been developed. The model is based on the joint solution of a system of non-stationary differential equations of heat and moisture transfer in multilayer porous materials. In addition, equations describing heat and mass transfer in ventilated air channels are included in the model. The calculations of the thermal and moisture state of the brick wall of the building, insulated from the outside with heat-insulating panels with ventilated channels, have shown that the selected geometry of the ventilated channels provides a low level of moisture content in the facade materials.