Browsing by Author "K.N. Rai"

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A computational solution of a phase-change material in the presence of convection under the most generalized boundary condition
(Elsevier Ltd, 2020) Vikas Chaurasiya; Dinesh Kumar; K.N. Rai; Jitendra Singh
This article presents a mathematical model describing inward melting process of a phase change material in the presence of convection under the most generalized boundary condition. It is assumed that the material within a container have different geometrical configuration like circular cylinder or sphere. All thermophysical properties of the solid and liquid regions are assumed to be homogeneous. Initially, we convert the mathematical model into an initial value problem in the form of vector matrix representation using a finite difference technique. Two numerical methods; the operational matrix of integration for Bessel functions and finite element Legendre wavelet Galerkin method, are applied to solve the initial value problem. Thus, the obtained results from both methods analyzed for constant (or time depending) temperature or constant (or time depending) heat flux. The whole study is presented in dimensionless form. The effect of Stefan number, Peclet number, Kirpichev number and Biot number on dimensionless temperature profile and dimensionless moving front are illustrated graphically. © 2020 Elsevier Ltd
A Mathematical Model for Hyperbolic Space-fractional Bioheat Transfer during Thermal Therapy
(Elsevier Ltd, 2015) P. Kumar; Dinesh Kumar; K.N. Rai
In this paper, we have developed a hyperbolic space-fractional bioheat transfer model based on the single-phase-lagging constitutive relation. The numerical solution of the present problem has been done using fractional backward finite difference scheme and Legendre wavelet Galerkin approach. The effect of fractional parabolic and hyperbolic bioheat transfer model on temperature profile within living biological tissues has been studied and compared with their respective standard cases. Numerical results are presented graphically in both standard and anomalous case for different values of parameters. © 2015 The Authors.
A mathematical model for solidification of binary eutectic system including relaxation time
(Begell House Inc., 2016) S. Yadav; S. Upadhyay; K.N. Rai
In this paper we have developed the time relaxation model for solidification of a binary eutectic system. In this model, we have considered the melt of a binary eutectic composite filled in a container; the flat probe is kept inside the container. The surface temperature of the flat probe decreases linearly with time. The solidification process occurs in three stages and, whole region is divided into solid, mushy, and liquid regions. The heat released in the mushy region is considered as discontinuous heat generation. The solid fraction present in the mushy region is characterized in two different ways: (i) when the solid fraction depends on distance and (ii) when the solid fraction depends on temperature. To solve this model we have developed the Legendre wavelets spectral Galerkin method. The whole analysis is presented in a dimensionless form and the results thus obtained are discussed in detail. © 2016 by Begell House, Inc.
A new iterative least square Chebyshev wavelet Galerkin FEM applied to dual phase lag model on microwave drying of foods
(Elsevier Masson SAS, 2019) Subrahamanyam Upadhyay; K.N. Rai
In this paper, we developed a dual-phase-lag (DPL) model of heat and mass transfer in presence of a source term for microwave drying foods of different geometrical configuration like slab, cylinder or sphere under the most generalized boundary conditions. The particular cases of present model are the diffusion model, Luikov model, Weng-Chang model and Andarwa-Tabrizi model. The proposed model is the most generalized boundary value problem of a coupled system of two hyperbolic second order partial differential equations. Stability analysis of present model is provided. A new iterative least square Chebyshev wavelet Galerkin finite element method provided for solution. The discretization in space and then application of Chebyshev wavelet Galerkin method converts our problem into a coupled system of two most generalized Sylvester equations. The iterative least square method provide solution of coupled system of Sylvester equations. Convergence and stability analysis of present method is discussed in detail. In a particular case of DPL model, the solution obtained by present method is compared with exact solution (Laplace transform technique) and are approximately same. Effect of relaxation time, Fourier number, phase change coefficient, thermo-gradient coefficient, heat generation, radius of the food and diffusion coefficient on heat and mass transfer are discussed in detail. This generalized model and its solution play important role in the study of microwave food drying. © 2019 Elsevier Masson SAS
A numerical study on dual-phase-lag model of bio-heat transfer during hyperthermia treatment
(Elsevier Ltd, 2015) P. Kumar; Dinesh Kumar; K.N. Rai
The success of hyperthermia in the treatment of cancer depends on the precise prediction and control of temperature. It was absolutely a necessity for hyperthermia treatment planning to understand the temperature distribution within living biological tissues. In this paper, dual-phase-lag model of bio-heat transfer has been studied using Gaussian distribution source term under most generalized boundary condition during hyperthermia treatment. An approximate analytical solution of the present problem has been done by Finite element wavelet Galerkin method which uses Legendre wavelet as a basis function. Multi-resolution analysis of Legendre wavelet in the present case localizes small scale variations of solution and fast switching of functional bases. The whole analysis is presented in dimensionless form. The dual-phase-lag model of bio-heat transfer has compared with Pennes and Thermal wave model of bio-heat transfer and it has been found that large differences in the temperature at the hyperthermia position and time to achieve the hyperthermia temperature exist, when we increase the value of τT. Particular cases when surface subjected to boundary condition of 1st, 2nd and 3rd kind are discussed in detail. The use of dual-phase-lag model of bio-heat transfer and finite element wavelet Galerkin method as a solution method helps in precise prediction of temperature. Gaussian distribution source term helps in control of temperature during hyperthermia treatment. So, it makes this study more useful for clinical applications. © 2015 Elsevier Ltd.
A numerical study on non-Fourier heat conduction model of phase change problem with variable internal heat generation
(Springer Science and Business Media B.V., 2021) Jitendra; K.N. Rai; Jitendra Singh
In this study, the authors proposed one-dimensional non-Fourier heat conduction model applied to phase change problem in the presence of variable internal heat generation and this has been performed by finite element Legendre wavelet Galerkin method (FELWGM). We derived the stability analysis of the non-Fourier heat conduction model in our present case. The finite difference technique has been used to change the non-Fourier heat conduction model into an initial value problem of vector-matrix form and then we applied Legendre wavelet Galerkin method for the numerical solution of the present problem. The location of moving interface is analytically obtained under the steady-state condition. The effectiveness of the proposed numerical technique is verified through the experimental value of parameters which indicate promising results. In addition, the effect of Stefan numbers, internal heat generation, and its linear coefficient on the location of moving interface are discussed in detail and represented graphically. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.
A study of cryosurgery of lung cancer using Modified Legendre wavelet Galerkin method
(Elsevier Ltd, 2018) Mukesh Kumar; Subrahamanyam Upadhyay; K.N. Rai
In this paper, we have developed a new mathematical model describing bio-heat transfer during cryosurgery of lung cancer. The lung tissue cooled by a flat probe whose temperature decreases linearly with time. The freezing process occurs in three stages and the whole region is divided into solid, mushy and liquid region. The heat released in the mushy region is considered as discontinuous heat generation. The model is an initial-boundary value problem of the hyperbolic partial differential equation in stage 1 and moving boundary value problem of parabolic partial differential equations in stage 2 and 3. The method of the solution consists of four-step procedure as transformation of problem in dimensionless form, the problem of hyperbolic partial differential equation converted into ordinary matrix differential equation and the moving boundary problem of parabolic partial differential equations converted into moving boundary problem of ordinary matrix differential equations by using finite differences in space, transferring the problem into the generalized system of Sylvester equations by using Legendre wavelet Galerkin method and the solution of the generalized system of Sylvester equation are solved by using Bartels-Stewart algorithm of generalized inverse. The whole analysis is presented in dimensionless form. The effect of cryoprobe rate on temperature distribution and the effect of Stefan number on moving layer thickness is discussed in detail. © 2018 Elsevier Ltd
A study of heat transfer during cryosurgery of lung cancer
(Elsevier Ltd, 2019) Mukesh Kumar; Subrahamanyam Upadhyay; K.N. Rai
In this study, a mathematical model describing two-dimensional bio-heat transfer during cryosurgery of lung cancer is developed. The lung tissue is cooled by a cryoprobe by imposing its surface at a constant temperature or a constant heat flux or a constant heat transfer coefficient. The freezing starts and the domain is distributed into three stages namely: unfrozen, mushy and frozen regions. In stage I where the only unfrozen region is formed, our problem is an initial-boundary value problem of the hyperbolic partial differential equation. In stage II where mushy and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations and in stage III where frozen, mushy, and unfrozen regions are formed, our problem is a moving boundary value problem of parabolic partial differential equations. The solution consists of the three-step procedure: (i) transformation of problem in non-dimensional form, (ii) by using finite differences, the problem converted into ordinary matrix differential equation and moving boundary problem of ordinary matrix differential equations, (iii) applying Legendre wavelet Galerkin method the problem is transferred into the generalized system of Sylvester equations which are solved by applying Bartels-Stewart algorithm of generalized inverse. The complete analysis is presented in the non-dimensional form. The consequence of the imposition of boundary conditions on moving layer thickness and temperature distribution are studied in detail. The consequence of Stefan number, Kirchoff number and Biot number on moving layer thickness are also studied in specific. © 2019 Elsevier Ltd
A study of solidification on binary eutectic system with moving phase change material
(Elsevier Ltd, 2021) Vikas Chaurasiya; K.N. Rai; Jitendra Singh
A one-dimensional moving boundary problem describing solidification of a eutectic system under imposed material movement occupying a semi-infinite medium is solved for two different cases of solid fraction distribution within the mushy zone. In the first case it is assumed that the solid fraction distribution has a linear relationship with temperature and in the second case solid fraction distribution is varying linearly with distance within the mushy zone. An exact solution of the problem is obtained with the help of a similarity technique. To demonstrate the current study experimental data of Al-Cu solidification are presented. All the thermal-physical properties of each part are discussed in detail for both models. The temperature profile in each region and moving interfaces are calculated for different Peclet number Pe. In the present study it is shown that the moving interfaces are enhanced, growing relatively faster and assisting in the process of phase-transition when material moves in the direction of freeze but transition is delayed when material moves in the reverse direction. It is also shown that mushy zone becomes thinner when surface temperature is lower than the solidus temperature for different Peclet numbers. In addition, the heat removal Q at the surface ξ=0 is shown with respect to time for different Peclet numbers. To validate our study, we compare our results with a previous published work and they are found to be close. © 2021 Elsevier Ltd
A study on DPL model of heat transfer in bi-layer tissues during MFH treatment
(Elsevier Ltd, 2016) Dinesh Kumar; P. Kumar; K.N. Rai
In this paper, dual-phase-lag bioheat transfer model subjected to Fourier and non-Fourier boundary conditions for bi-layer tissues has been solved using finite element Legendre wavelet Galerkin method (FELWGM) during magnetic fluid hyperthermia. FELWGM localizes small scale variation of solution and fast switching of functional bases. It has been observed that moderate hyperthermia temperature range (41-46 °C) can be better achieved in spherical symmetric coordinate system and treatment method will be independent of the Fourier and non-Fourier boundary conditions used. The effect of phase-lag times has been observed only in tumor region. FCC FePt magnetic nano-particle produces more effective treatment with respect to other magnetic nano-particles. The effect of variability of magnetic heat source parameters (magnetic induction, frequency, diameter of magnetic nano-particles, volume fractional of magnetic nano-particles and ligand layer thickness) has been investigated. The physical property of these parameters has been described in detail during magnetic fluid hyperthermia (MFH) treatment and also discussed the clinical application of MFH in Oncology. © 2016 Elsevier Ltd.
A study on thermal damage during hyperthermia treatment based on DPL model for multilayer tissues using finite element Legendre wavelet Galerkin approach
(Elsevier Ltd, 2016) Dinesh Kumar; K.N. Rai
Hyperthermia is a process that uses heat from the spatial heat source to kill cancerous cells without damaging the surrounding healthy tissues. Efficacy of hyperthermia technique is related to achieve temperature at the infected cells during the treatment process. A mathematical model on heat transfer in multilayer tissues in finite domain is proposed to predict the control temperature profile at hyperthermia position. The treatment technique uses dual-phase-lag model of heat transfer in multilayer tissues with modified Gaussian distribution heat source subjected to the most generalized boundary condition and interface at the adjacent layers. The complete dual-phase-lag model of bioheat transfer is solved using finite element Legendre wavelet Galerkin approach. The present solution has been verified with exact solution in a specific case and provides a good accuracy. The effect of the variability of different parameters such as lagging times, external heat source, metabolic heat source and the most generalized boundary condition on temperature profile in multilayer tissues is analyzed and also discussed the effective approach of hyperthermia treatment. Furthermore, we studied the modified thermal damage model with regeneration of healthy tissues as well. For viewpoint of thermal damage, the least thermal damage has been observed in boundary condition of second kind. The article concludes with a discussion of better opportunities for future clinical application of hyperthermia treatment. © 2016 Elsevier Ltd
An analytical study on sublimation process in the presence of convection effect with heat and mass transfer in porous medium
(Elsevier Ltd, 2022) Jitendra; K.N. Rai; Jitendra Singh
The mathematical study of one dimensional heat and mass transfer phenomena is investigated. A Stefan like phenomena is defined as a heat and mass transfer problem with presence of convection in vapour region for heat and moisture flow and convective term due to mass transfer of the water vapour in frozen region. The sublimation process considered in two region: (i) In frozen region, s(t)< x < ∞, there is no effect of moisture in this region. (ii) In vapour region, 0 < x < s(t), there is an effect of heat and moisture flows. The thermo-physical properties of each region are constant, but may be differ for different region. The analytical solutions are obtained for the temperature and moisture profile and moving sublimation interface by using the similarity transformation procedure. The effect of some parameters included in the sublimation process with various enclosures, i.e. dimensionless quantity l0, Peclet number Pe, Luikov number Lu, KT, a21 and other constant parameters are comprehensively investigated and represented graphically. © 2021 Elsevier Ltd
An exact Analysis of Melting Phenomena Based on Non-Classical Heat Equation with Moving Boundary Problem: Application of Melting-Freezing
(Pleiades Publishing, 2024) Jitendra; K.N. Rai; Jitendra Singh
Abstract: Purpose: The purpose of this study is to introduce a melting phenomena based non-classical heat equation with latent heat as a function of the moving interface and its velocity, which has not often been taken into consideration previously in the literature available due to the non-linearity of the interface condition. In view of these problems, three mathematical models are proposed for non-classical moving boundary problems including both conduction and convection effects. The applied heat flux on the surface is subjected to a control function at x = 0. In a certain situation, latent heat varies with moving interface and its velocity, and in another, latent heat may be treated as constant. Design/methodology: In the context of non-linear variable latent heat, we provided an analytical analysis for single-phase and double-phase moving boundary problems. The similarity transformation approach has been used to obtain analytical results. The impact of associated problem parameters are discussed in detail. Findings: From this study, it is observed that in the case of variable latent heat, moving interface get accelerated more in comparison to constant latent heat. Furthermore, when the Peclet number and the value of the coefficient of control function increase then the melting process become enhanced. In the present study, convection plays a key role during the melting process. This study may improve the theoretical and mathematical understanding of a shoreline problem and is applicable in geology and thermal management systems. © Pleiades Publishing, Ltd. 2024.
Analysis of classical Fourier, SPL and DPL heat transfer model in biological tissues in presence of metabolic and external heat source
(Springer Verlag, 2016) Dinesh Kumar; Surjan Singh; K.N. Rai
In this paper, the temperature distribution in a finite biological tissue in presence of metabolic and external heat source when the surface subjected to different type of boundary conditions is studied. Classical Fourier, single-phase-lag (SPL) and dual-phase-lag (DPL) models were developed for bio-heat transfer in biological tissues. The analytical solution obtained for all the three models using Laplace transform technique and results are compared. The effect of the variability of different parameters such as relaxation time, metabolic heat source, spatial heat source, different type boundary conditions on temperature distribution in different type of the tissues like muscle, tumor, fat, dermis and subcutaneous based on three models are analyzed and discussed in detail. The result obtained in three models is compared with experimental observation of Stolwijk and Hardy (Pflug Arch 291:129–162, 1966). It has been observe that the DPL bio-heat transfer model provides better result in comparison of other two models. The value of metabolic and spatial heat source in boundary condition of first, second and third kind for different type of thermal therapies are evaluated. © 2015, Springer-Verlag Berlin Heidelberg.
Analytic approximation for one phase space-time fractional moving boundary problem with time varying temperature on surface
(Begell House Inc., 2015) Jitendra Singh; K.N. Rai
A mathematical model of space and time fractional derivative for melting of solid fluid in a finite slab under time varying temperature on fixed boundary are presented. The approximate analytical solution of this problem is obtained by the Homotopy analysis method. The problem has been studied in detail by considering different order space and time fractional derivatives and different time varying temperature on fixed boundary. In this paper we discussed the melting process under different order fractional derivative with time varying temperature on surface. The nondimensional temperature and the moving interface for different order fractional derivative are shown graphically. The model and the solution are the generalization of the previous work. © 2021, Begell House Inc. All rights reserved.
Analytical solution of Fourier and non-Fourier heat transfer in longitudinal fin with internal heat generation and periodic boundary condition
(Elsevier Masson SAS, 2018) Surjan Singh; Dinesh Kumar; K.N. Rai
In this paper, the analytical solution of Fourier and non-Fourier model of heat transfer in longitudinal fin in presence of internal heat generation has been studied under periodic boundary condition. The whole analysis is given in dimensionless form. These two mathematical models have been solved analytically, using Laplace transform technique. Temperature distribution in longitudinal fin is measured using residual theorem in complex plane for the inverse Laplace transform technique. Thermal wave nature is appeared for small value of Fo. The longitudinal fin temperature is evaluated for different value of parameters with respect to space coordinate. The effect of variability of different parameters on temperature distribution in fin is studied in detailed. It has been observed that the cooling process is faster in non-Fourier model in comparison to Fourier model. © 2017 Elsevier Masson SAS
Application of He's homotopy perturbation method for multi-dimensional fractional Helmholtz equation
(2012) Praveen Kumar Gupta; A. Yildirim; K.N. Rai
Purpose - This purpose of this paper is to find the approximate analytical solutions of a multidimensional partial differential equation such as Helmholtz equation with space fractional derivatives α, β, γ(1≤α ,β,γ<2). The fractional derivatives are described in the Caputo sense. Design/methodology/approach - By using initial values, the explicit solutions of the equation are solved with powerful mathematical tools such as He's homotopy perturbation method (HPM). Findings - This result reveals that the HPM demonstrates the effectiveness, validity, potentiality and reliability of the method in reality and gives the exact solution. Originality/value - The most important part of this method is to introduce a homotopy parameter (p), which takes values from [0,1]. When p =0, the equation usually reduces to a sufficiently initial form, which normally admits a rather simple solution. When p→1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. Here, we also discuss the approximate analytical solution of multidimensional fractional Helmholtz equation. Copyright © 2012 Emerald Group Publishing Limited. All rights reserved.
APPROXIMATE CLOSED FORM ANALYTICAL SOLUTION OF HEAT CONDUCTION IN BODIES OF SIMPLE CONFIGURATION.
(1987) K.N. Rai; S.N. Sinha
Heat conduction, in bodies of simple configuration such as an infinite plate, an infinite cylinder and an infinite sphere, is solved. This is accomplished by transforming the boundary value problem of partial differential equation into a boundary value problem of ordinary differential equation and using the Galerkin's method to solve the transferred equation. The results obtained are in good agreement with the exact solutions.
Convective-radiative fin with temperature dependent thermal conductivity, heat transfer coefficient and wavelength dependent surface emissivity
(Elsevier Ltd, 2014) Surjan Singh; Dinesh Kumar; K.N. Rai
In this paper, we have studied heat transfer process in a continuously moving fin whose thermal conductivity, heat transfer coefficient varies with temperature and surface emissivity varies with temperature and wavelength. Heat transfer coefficient is assumed to be a power law type form where exponent represent different types of convection, nucleate boiling, condensation, radiation etc. The thermal conductivity is assumed to be a linear and quadratic function of temperature. Exact solution obtained in case of temperature independent thermal conductivity and in absence of radiation conduction parameter is compared with those obtained by present method and is same up to ten decimal places. The whole analysis is presented in dimensionless form and the effect of variability of several parameters namely convection-conduction, radiation-conduction, thermal conductivity, emissivity, convection sink temperature, radiation sink temperature and exponent on the temperature distribution in fin and surface heat loss are studied and discussed in detail. © 2014 National Laboratory for Aeronautics and Astronautics
Correction to: A numerical study on non-Fourier heat conduction model of phase change problem with variable internal heat generation (Journal of Engineering Mathematics, (2021), 129, 1, (7), 10.1007/s10665-021-10143-1)
(Springer Science and Business Media B.V., 2023) Jitendra; K.N. Rai; Jitendra Singh
A typographical mistake of writing the different quantities (Formula presented.) by a single (Formula presented.) has been made in this published article. This creates confusion, especially with regard to physical units. The authors apologize for the typographical mistake and the corrections are given below. On page numbers 6 and 14 of [1], the mathematical inequality for stability in theorem 1 as well as in the conclusion section should read as On page 6 of [1], the characteristic polynomial associated with Eq. (28) should read as On page numbers 6, 7 and 14 of [1], the mathematical inequality for stability in theorem 2 as well as in the conclusion section should read as On page 6 of [1], the characteristic polynomial associated with Eq. (31) should be written as where, On page number 8 in Eq. (42) of [1], the matrix (Formula presented.) , the first fraction of the last row should read as (Formula presented.) instead of (Formula presented.). On page number 10, Eq. (57) of [1], should read as On page numbers 6 and 7 of [1], In Eq. (30) and Eq. (33), the characteristic polynomial should involve (Formula presented.) instead of (Formula presented.). Revised versions are given below of Figs. 2, 3, 4 and 6 in reference [1]. (Figure presented.) (Figure presented.) (Figure presented.) (Figure presented.) Effect on moving interface in steady state with coefficient of internal heat generation (Formula presented.) and (Formula presented.) Effect on moving interface in steady state with internal heat generation (Formula presented.) and (Formula presented.) = 1 The effect on moving interface for different Stefan numbers with (Formula presented.) (Formula presented.) (Formula presented.) (Formula presented.) (Formula presented.) (Formula presented.) The effect on moving interface for coefficient of internal heat generation with (Formula presented.) (Formula presented.) (Formula presented.) (Formula presented.) (Formula presented.) (Formula presented.). © 2023, Springer Nature B.V.