Luikov Equations Research Articles

Numerical analysis of vacuum drying of a porous body in the integrated domain

AbstractIn the present study, the vacuum drying process of an apple slice is numerically modeled based on a control volume method. Transient two‐dimensional Navier–Stokes, energy, moisture, and Luikov equations are solved by numerical coding (Fortran) to simulate the simultaneous heat and mass transfer in the ambient and apple slice, respectively. The privilege of using Luikov's model is that the capillary forces are considered, and a differentiation between air, vapor, liquid, and solid is made. Luikov described the two phenomena associated with the transport of air, vapor, and liquids through the porous media as molecular transport and molar transport. The ambient pressure linearly reduced within a minute until it reached a constant value. One of the intellectually demanding preoccupations among researchers is how to simulate the sample and its surroundings with high accuracy of boundary conditions, which enables to avoid the use of empirical transfer coefficients. This study can be scrutinized from various dimensions, among which nonuse of boundary condition between a porous sample and its surroundings is the most conspicuous novelty. Results showed that although at 50 s, isothermal and iso‐moisture lines inside the sample are symmetric, they are not symmetric at 100, 200, and 400 s. In addition, at first minute, pump operation leads to vary the density of the isothermal and iso‐moisture lines around the sample, but at 100, 200, and 400 s, higher temperature and moisture gradients have been achieved at the right and top of the sample surface.Practical ApplicationsDrying is the main technique of food preservation, so that it reduces the humidity of crops and is the most crucial procedure for safeguarding agricultural crops because it has a considerable impact on the condition of parched goods. In this study, some assumptions of drying including using empirical transfer coefficients between sample and its surrounding and vacuum drying under constant pressure have been eliminated. To achieve this goal, computational fluid dynamics (CFD), plays a crucial role to simulate the parameters inside the sample and its ambient without using boundary condition.

Thermodynamic and mathematical analysis of modified Luikov's equations for simultaneous heat and mass transfer

Interface thermodynamic and mathematical analysis of modified Luikov equations (MLEs) for simultaneous heat and mass transfer in solids were developed. The thermodynamic relations at interface show that air-(wet solid) equilibrium relation may be mathematically approximated by two linear segments. These linear relations conduced to the possibility that MLEs have analytical solution for 1D rectangular, 1D radial cylindrical and 1D radial spherical coordinates system; and, the existence of a topology between solutions. The implications of MLEs thermodynamic and mathematical analysis on lumped equations were discussed. Experimental evolution of moisture and temperature during potato drying were compared with MLEs analytical (for one single linear stage at interface) and numerical (for the 2 linear stages) solutions.

An effect of hygrothermal effects on high velocity impact event for polymer matrix composites

The different loading conditions often occur in polymer matrix composites during their service life. In order to use effectively the materials in high-performance applications, analyzing the effect of hygrothermal effects on a ballistic impact event is necessary. This paper considers a heat sink and humidity sink for the hygrothermal diffusion in the composites and solves the Luikov equations by means of the differential quadrature method (DQM). Then, failure and partial failure of a projectile with high velocity into the composites are studied based on the conservation of energy. Further, as hygrothermal deformation reaches equilibrium, the effect of hygrothermal effects on the ballistic impact event is studied. Finally, based on carbon fiber reinforced composite (CFRP), the effects of the hygrothermal diffusion are discussed on the ballistic impact event.

(Semi-)analytical solution of Luikov equations for time-periodic boundary conditions

The paper addresses the problem of coupled heat and moisture transfer in porous materials with the time-periodic boundary conditions applied. The solution of Luikov equations [1], which describe coupled heat and moisture transfer, is presented. Laplace transform is used, where some terms of the inverse Laplace transform ought to be solved by Gaussian quadrature, meaning that the solution is semi-analytical. The time-periodic boundary conditions are applied to simulate the humidity and temperature oscillations of natural environment. Therefore, the proposed solution is appropriate to evaluate the distribution of moisture and temperature within the porous material exposed to everyday natural cycles. The paper presents convergence tests, validation of semi-analytical solution and application to different building materials are presented in the paper.

Hybrid integral transform solution for the analysis of drying in spherical capillary-porous solids based on Luikov equations with pressure gradient

A hybrid solution based on the Generalized Integral Transform Technique (GITT) is obtained for the analysis of the Luikov equations including pressure gradient effects for drying of capillary-porous solids that have spherical geometry. First, the present results are compared with previous ones in the literature, as well as with those from the NDSolve routine (Mathematica system) for specific situations, in order to validate the numerical codes developed in the present work and to demonstrate the consistency of the final results. Additionally, the results for temperature, moisture and pressure fields are produced with different values of the governing parameters, illustrating the potential of the hybrid numerical–analytical GITT approach in solving drying problems with significant pressure variations, such as in highly intensive and fast drying processes. Therefore, for this purpose, five test-cases are considered in order to make a parametric analysis, which is performed in order to investigate the influence of typical governing parameters (Bulygin and Luikov for filtration numbers) for such a physical situation. The influence of the pressure field in the temperature and moisture distributions is also emphasized by analyzing a typical case of the drying problem of a ceramic material with the Luikov equations with two parameters, namely temperature–moisture model (TM Model) and with three parameters, those in the temperature–moisture–pressure model (TMP Model).

Solution of porous media inverse drying problems using a combination of stochastic and deterministic methods

In the present work the inverse problem of simultaneous heat and mass transfer modeled by Luikov equations is studied using a hybrid combination of the Levenberg-Marquardt (LM), Simulated Annealing (SA) and Artificial Neural Network (ANN) methods. The direct and inverse problems are described, formulated and solved. After the use of an experiment design technique, the hybrid combination ANN-LM-SA yielded good estimates for the heat and mass transfer problem of interest. The proper choice of the set of parameters to be estimated allowed the design of an experiment with higher sensitivity coefficients. One ANN was used to generate the initial guess for the LM, another one to approximate the gradient needed by LM, and, finally, the global minimum was searched using the SA. The experimental data considered in the inverse problem was generated using the solution for the direct problem with the addition of noise.

Numerical Simulation of Vacuum Drying by Luikov's Equations

A two-dimensional mathematical model was developed to simulate coupled heat and mass transfer in apple under vacuum drying. Luikov's equations are the governing equations in analyzing heat and mass diffusion problems for capillary-porous bodies. The model considers temperature- and moisture-dependent material properties. The aim of this study is to analyze the influence of some of the most important operating variables, in particular, pressure and temperature of drying air, on the drying of apple. The resulting system of unsteady-state partial differential equations has been solved by a commercial finite element method (FEM) package called FEMLAB (COMSOL AB, Stockholm, Sweden). Simulations, carried out in different drying conditions, showed that temperature is more effective than air pressure in determining the drying rate. A parametric study was also carried out to determine the effects of heat and mass transfer coefficients on temperature and moisture content distributions inside apple during vacuum drying. A comparison between the theoretical predictions and a set of experimental results reported in the literature showed very good agreement, especially during the first 4,200 s, when experimental data and theoretical predictions overlapped and relative errors never exceeded 2%.

Optimal Heat Input for Estimating Luikov's Parameters in a Heat and Mass Transfer Problem

In this article, the problem of computing an optimal heat input in Luikov's heat and mass transfer problem is detailed and analyzed. The main objective is the establishment of an optimal time-dependent heat flux profile with the goal of maximizing the temperature and moisture sensitivities of some parameters to this excitation in a drying process. Such maximization makes the estimation of the desired parameters possible, easier, and with limited uncertainty intervals. It also helps to reduce the linearity dependence between the parameters of interest and the number of temperature and moisture sensors used. The estimation of the optimal heat input is obtained with Uzawa's algorithm, while the estimation of parameters is performed with Levenberg-Marquardt's method of minimization of the ordinary least-square criterion. The six dimensionless parameters characterizing Luikov's equations are estimated successfully with this optimal heat flux profile, which also helps to reduce the number of both temperature and moisture sensors needed in the estimation procedure. By doing so, the objective of estimating simultaneously the six parameters which appear in the formulation of Luikov's physical problem is reached by using a limited transient temperature and/or moisture measurements taken anywhere in the drying medium.

Conjugate Heat and Mass Transfer from a Drying Rectangular Cylinder in Confined Air Flow

Heat and mass transfer from a porous body subject to convective drying is investigated numerically based on Luikov's equations. The air flow is assumed incompressible, two-dimensional, laminar, confined in a channel, and parallel to the rectangular-shaped solid. The finite-volume method is used and the computed temporal and spatial variations of moisture content, temperature, concentration, and flow parameters for two different flow rates are analyzed. Two flow configurations are studied: with and without a flow divider upstream of the cylinder in an attempt to eliminate the presence of separation zones and study their effect on drying. It was found that such effects may greatly affect the drying process, along with frontal area stagnation and the thickness of the body.

Fire analysis of timber composite beams with interlayer slip

The purpose of this paper is to model the behaviour of timber composite beams with interlayer slip, when simultaneously exposed to static loading and fire. A transient moisture-thermal state of a timber beam is analysed by the Luikov equations, and mechanical behaviour of timber composite beam is modelled by Reissner's kinematic equations. The model can handle layers of different materials. Material properties are functions of temperature. The thermal model is validated against the experimental data presented in the literature. Generally, the model provides excellent agreement with the experimental data. It is shown that the material properties of timber play an important role in the fire resistance analysis of timber structures when exposed to fire.

Mathematical modelling of an industrial furnace for high‐temperature heat treatment of wood

AbstractThere are various high‐temperature treatment methods for wood. In the “Bois Perdure” process, the thermal treatment of wood is carried out in a furnace by contacting it with hot combustion gases over 200°C without the addition of any chemicals in order to improve its dimensional stability and durability. The treatment eliminates free and bound water in the wood and modifies its molecular structure. In this study, a mathematical model describing the industrial furnace has been developed. The overall model consists of a 3‐D unsteady‐state sub‐model which solves for the flow, heat, and mass transfer in the gas coupled with a 1‐D unsteady‐state sub‐model which calculates the heat and mass transfer in the wood. The 3‐D gas sub‐model was developed using the commercial CFD code CFX. The 1‐D wood sub‐model is based on the solution of simultaneous heat and mass transfer equations (Luikov equations) using the implicit finite difference formulation. The model predicts the temperature and moisture distributions in the wood as well as the flow, heat, and moisture profiles in the gas. The model results are compared with the data obtained from the industrial furnace, and a good agreement was found between them.

Study of Drying of a Porous Medium Using Gamma-Ray Attenuation Technique

ABSTRACT In this work the drying of porous media is considered and modeled according to Luikov equations. The experimental apparatus consists of a cylindrical porous medium positioned vertically, with its bottom closed and maintained at a constant temperature. The top surface of this porous medium is in contact with a known flow rate of dry air at room temperature. Under these conditions, the transient axial temperature and moisture concentration profiles are measured. A noninvasive technique has been used to measure the moisture content, which is based on the attenuation of a horizontal beam of gamma-rays penetrating the porous medium. The measurement of this attenuation allows determining the water content of the sample. The mathematical model has been developed using two different approaches. In the first one, the one dimensional Luikov equations are applied considering all the physical parameters as constant. In the second approach, heat and mass balance equations of this process are the simple one dimensional transient diffusion equations, considering both diffusion coefficients as adjustable function of the moisture content. As shown, this second approach is superior to the first one. #In Memoriam.

Estimation of thermophysical properties of moist materials under different drying conditions

This article deals with the solution for the inverse problem of simultaneously estimating moisture content and temperature-dependent moisture diffusivity, together with thermal conductivity, heat capacity, density, and phase conversion factor of a drying body as well as boundary condition coefficients, by using only temperature measurements. Two different physical problems, convective and contact drying, are examined and compared. The present parameter estimation problem is solved with the Levenberg–Marquardt method of minimization of the least-squares norm, by using simulated experimental data. The temperature responses during the drying are obtained with a numerical solution of the non-linear one-dimensional Luikov's equations. As a representative drying body, a mixture of bentonite and quartz sand with known thermophysical properties has been chosen. Analyses of the sensitivity coefficients and of the determinant of the information matrix are presented.

Esquema de discretização Flux-Spline aplicado à secagem, em meio poroso capilar

Este trabalho foi desenvolvido com o objetivo de se apresentar a aplicação de um esquema de discretização mais eficiente para volumes finitos, denominado Flux-Spline utilizando-se, para tal, de dois casos de transporte difusivo de umidade e calor, através de um meio poroso capilar. Os resultados da solução numérica do sistema de equações formado pelas equações de Luikov mostram desempenho adequado do esquema para este tipo de problema, quando comparado ao tradicional esquema de diferença central e ao método da transformada integral.

CORRECTION OF TEMPERATURE AND MOISTURE DISTRIBUTIONS IN WOOD DRYING

A procedure is proposed to improve the numerical solutions of the Luikov equations of heat and mass transfer in capillary-porous bodies based on the measured temperature and moisture data. The main idea of the procedure can be summarized as a modeling technique with online correction ability. The procedure is demonstrated by a one-dimensional problem of heat and mass transfer in spruce sample. Numerical simulation shows that the accuracy of the solutions for the heat and mass transfer can be improved significantly by using the proposed method.

Estimation of dimensionless parameters of Luikov's system for heat and mass transfer in capillary porous media

This work deals with the solution of inverse problems of parameter estimation involving heat and mass transfer in capillary porous media, as described by the linear one-dimensional Luikov's equations. Our main objective is to use the D-optimum criterion to design the experiment with respect to the magnitude of the applied heat flux, heating and final experimental times, as well as the number and locations of sensors. The present parameter estimation problem is solved with Levenberg–Marquardt's method of minimization of the ordinary least-squares norm, by using simulated temperature data containing random errors. Moisture content measured data is not considered available for the inverse analysis in order to avoid quite involved measurement techniques. We show that accurate estimates can be obtained for Luikov, Kossovitch and Biot numbers by using only temperature measurements in the inverse analysis. Also, the experimental time can be reduced if the body is heated during part of the total experimental time.

Complete and satisfactory solutions of luikov equations of heat and moisture transport in a spherical capillary — Porous body

Available results based on the exact solutions of the Luikov system of coupled heat and moisture diffusion problems addressed to the boundary conditions of the third kind for the bodies of simple geometry, have been reconsidered and revised because the transcendental equations have countable number of complex conjugate roots under certain conditions, which were ignored in the original computations. A methodology, quite different from Lobo et al., is presented for evaluating a set of countable number of pairs of complex conjugate roots. The influence of complex eigenvalues on the temperature and moisture distributions in a spherical capillary porous body is demonstrated and benchmark results are obtained for reference purpose.

Periodic solution of Luikov equations for heat and mass transfer in capillary porous bodies

Closed-form solutions for the heat and moisture distribution given by the Luikov system of partial differential equations have been obtained by using periodic solution approach. Boundary conditions of the third type valid for capillary porous structure have been used to find the unknown constants. Calculations have been performed for Spruce wood, the properties of which were known. The results show that moisture reduces the temperature distribution between outside and inside by 2-3°C which is in conformity with the earlier results of Hall.

Periodic solution of Luikov equations for heat and mass transfer in capillary porous bodies

Closed-form solutions for the heat and moisture distribution given by the Luikov system of partial differential equations have been obtained by using periodic solution approach. Boundary conditions of the third type valid for capillary porous structure have been used to find the unknown constants. Calculations have been performed for Spruce wood, the properties of which were known. The results show that moisture reduces the temperature distribution between outside and inside by 2–3°C which is in conformity with the earlier results of Hall. Copyright © 1999 John Wiley & Sons, Ltd.

Solutions of Luikov equations of heat and mass transfer in capillary porous bodies through matrix calculus: a new approach

In this paper an eigen value analysis approach is employed to obtain the solutions of the Luikov system of linear partial differential equations addressed to the most general type of boundary conditions. The Luikov equations provide a well established model for the analysis of various simultaneous heat and mass diffusion problems in capillary porous bodies. However,analytical methods to achieve a complete and satisfactory solution of these equations is still lacking in the literature because of non inclusion of the existence of a countable number of complex roots in almost all the solutions. A specific example on contact drying of a moist porous sheet with uniform initial temperature and moisture distribution is considered. The influence of the complex roots on the dimensionless temperature, moisture content, and the local rate of drying is demonstrated. A set of benchmark results is obtained for reference purposes.