Browsing by Author "Subrahamanyam Upadhyay"

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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 new look in heat balance integral method to a two-dimensional Stefan problem with convection
(Taylor and Francis Ltd., 2022) Vikas Chaurasiya; Subrahamanyam Upadhyay; Kabindra Nath Rai; Jitendra Singh
In the current work, we developed a new approximation function for temperature profile with the help of Legendre wavelet in heat-balance integral method (HBIM) to solve a two-dimensional moving boundary problem with moving phase change material (PCM). It is assumed that PCM moves with induced velocity u along x and y direction. In heat transfer mechanism conduction and convection driven by fluid flow in liquid region is considered. To validate the current approximate method, we compared our numerical results with a previous work and found in strong acceptance. In particular, to show the accuracy of the present approximate method, we compared our numerical results against exact solution by converting present problem into a one-dimensional standard melting problem and found in good acceptance. The effect of Péclet number on temperature profile and moving melting front are analyzed in detail. Furthermore, it is shown that with a moving phase change material (PCM) the liquid/solid interface get accelerated and hence, the melting process becomes fast. This study may be applicable in thermal management and energy storage system. © 2022 Taylor & Francis Group, LLC.
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
Finite difference Legendre wavelet collocation method applied to the study of heat mass transfer during food drying
(John Wiley and Sons Inc., 2019) Subrahamanyam Upadhyay; Vineet K. Singh; K.N. Rai
In this paper, we studied a single-phase-lag model of heat and moisture transfer in a finite slab, cylinder or sphere, undergoing industrial drying of foods. The present model is a boundary value problem of a system of two nonlinear second-order hyperbolic partial differential equations. The solution of the present is model obtained by using finite difference Legendre wavelet collocation (FDLWCM) and Galerkin methods. In case of constant moisture diffusivity, we observe that the finite difference Legendre wavelet Galerkin and collocation solutions are exactly the same. Laplace transform and FDLWCM solutions are approximately the same. The L 2 norm error decreases as Legendre wavelet basis functions increases and strip size decreases. The L 2 norm relative error increases as Luikov number increases. L 2 norm relative error is highest in cylindrical shape and lowest in slab shape while in between spherical shape. The effects of the dimensionless parameters of heat and mass transfer are discussed in detail. © 2019 Wiley Periodicals, Inc.
Galerkin and collocation methods for solution of initial value problem of generalised van der Pol equation
(Inderscience Enterprises Ltd., 2017) Subrahamanyam Upadhyay; K.N. Rai
In this paper, we proposed a van der Pol oscillator with delay feed-back including duffing oscillators and a periodic forcing term model. For solution of proposed model, using collocation and Galerkin method with Legendre wavelet as a basis functions. Convergence and stability analysis of the method are discussed. An algorithm provided for computing numerical data. The solution obtained by both Legendre wavelet collocation method and Legendre wavelet Galerkin method is exactly same as exact solution and those obtained by method of averaging (Atay, 1998), PEM (Kimiaeifar et al., 2010). The Legendre wavelet collocation method for different M and k provides better results in lesser time than Legendre wavelet Galerkin method. It has been observed that the displacement have cyclic behaviour with respect to velocity for different feedback gain and angular frequency of the driving force. The displacement decreases as delay parameter increases in small domain. The displacement also decreases as duffing parameter increases and angular frequency of the driving force decreases. © 2017 Inderscience Enterprises Ltd.
Mathematical modelling and simulation of three phase lag bio-heat transfer model during cancer treatment
(Elsevier Masson s.r.l., 2023) Mukesh Kumar; Harpreet kaur; Subrahamanyam Upadhyay; Surjan Singh; K.N. Rai
We have developed a two-dimensional mathematical model that described the study of bio-heat transfer. Our model is an initial boundary value problem of partial differential equation. The solution consists of the three step procedure- (i) transformation of problem in dimensionless form (ii) by using finite differences, the problem converted into ordinary matrix differential equation (iii) applying Legendre wavelet Galerkin method, the problem is transferred into the generalised system of Sylvester equations which are solved by applying Bartels-Stewart Algorithm of generalised inverse. We have used this method to determine the temperature profile in three different boundary conditions. The consequence of boundary conditions on temperature profile are discussed in detail. The effect of phase lag due to heat flux, phase lag due to temperature gradient and phase lag due to thermal displacement have been observed. And, we have seen that temperature profile increases when the phase lags decreases. We have also observed the effect of blood perfusion rate and metabolic heat generation in specific. Results are validated with exact results in particular case. © 2022 Elsevier Masson SAS
Modelling and Simulation of a Moving Boundary Problem Arising During Immersion Frying of Foods
(Springer, 2019) Subrahamanyam Upadhyay; Sarita Yadav; K.N. Rai
A single phase lagging model of heat and moisture transfer has been developed in an infinite slab undergoing immersion frying of foods. This model is a moving boundary problem of a system of second order hyperbolic partial differential equations. The solution obtained by using finite element Legendre wavelet Galerkin scheme. The stability analysis of present model and method are discussed in detailed and whole analysis presented in dimensionless form. The effect of variability of Luikov number, Kossovich number and Biot number on frying process are discussed in detail. © 2018, The National Academy of Sciences, India.
Taylor–Galerkin–Legendre-wavelet approach to the analysis of a moving fin with size-dependent thermal conductivity and temperature-dependent internal heat generation
(Springer Science and Business Media B.V., 2023) Vikas Chaurasiya; Subrahamanyam Upadhyay; K.N. Rai; Jitendra Singh
In heat transfer, fins are commonly used to enhance the heat transfer rate from surfaces and are widely applicable in heat exchangers and thermal energy storage systems. The material used for fins typically has a high heat conductivity. While the study of temperature-dependent heat conductivity for fins is already available, an insufficient mathematical description is observed in the case of size-dependent heat conductivity and convective heat transfer. In the current work, a heat transfer study is presented to estimate the temperature field in a moving fin that accounts for size-dependent heat conductivity and internal heat generation that depends on temperature under periodic boundary conditions. To observe the temperature field, we have developed a hybrid numerical method based on Taylor–Galerkin and Legendre wavelets. The stability analysis of the developed method is discussed in detail. Our numerical method shows excellent agreement with the analytical solution obtained in a special case. The impact of problem parameters is extensively discussed. This study shows that fin temperature decreases periodically with a space-dependent heat conductivity. In addition, for a problem which accounts for constant heat conductivity and movable fin, have greater temperature response, and standard problem which accounts for constant heat conductivity have weaker temperature response while it is between them for a problem that includes size-dependent heat conductivity and moving fin. It is shown that fin efficiency can be improved by lowering the value of the Knudsen number. Moreover, fin problem with fixed thermal conductivity offer greater efficiency in comparison with size-dependent thermal conductivity. © 2023, Akadémiai Kiadó, Budapest, Hungary.