Finite Element Analysis of Heat and Mass Transfer past an Impulsively Moving Vertical Plate with Ramped Temperature

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1 Journal of Applied Science and Engineering, Vol. 19, No. 4, pp (2016) DOI: /jase Finite Element Analysis of Heat and Mass Transfer past an Impulsively Moving Vertical Plate with Ramped Temperature Siva Reddy Sheri* and Anjan Kumar Suram Department of Mathematics, GITAM University, Hyderabad Campus, Telangana, India Abstract A numerical investigation is performed to study the finite element analysis of heat and mass transfer past an impulsively moving vertical plate with ramped temperature. The governing equations of the flow are transformed into a non dimensional form using suitable dimensionless quantities. A Finite element method is engaged to solve the dimensionless governing equations. Obtained numerical solution is displayed graphically to illustrate the influence of various controlling parameters of flow. Skin friction and Nusselt number are compared with the earlier results by Seth et al. Finally, numerical solutions are found to be in excellent agreement with previously published results under special cases. Key Words: Heat and Mass Transfer, Vertical Plate, Ramped Temperature, FEM 1. Introduction *Corresponding author. sreddy7@yahoo.co.in Heat transfer is a study of the exchange of thermal energy through a body or between bodies which occurs when there is a temperature difference. Heat always transfers from hot to cold. Whereas mass transfer is the transport of constituent from a region of higher concentration to that of lower concentration. In nature, there exist some flows which are caused not only by the temperature differences but also by concentration differences. These mass transfer differences influence the rate of heat transfer. In industries, many transport processes exist in which heat and mass transfer takes place simultaneously as a result of combined buoyancy affect of thermal diffusion and diffusion through chemical species. The phenomenon of heat and mass transfer frequently exist in chemically processed industries such as food processing and polymer production. Several engineering processes where heat and mass transfer take place simultaneously such as in heat exchanger devices, insulation systems, petroleum reservoirs, filtration, chemical catalytic reactors and processes, nuclear waste repositories, desert coolers, wet bulb thermometers etc. Considerable attention has been given to this area and few investigations have been reported [13]. Furthermore, it is found from the literature that several investigations on free convection flows are available with different thermal conditions at the bounding plate which are continuous and well-defined at the wall. However, most of the practical problems appear with non-uniform or arbitrary conditions at the wall. To study such problems, it is useful to investigate them under step change in wall temperature. The physical implication of this idea can be found in the fabrication of thin-film photovoltaic devices where ramped wall temperatures may be employed to achieve a specific finish of the system [4]. According to [5], periodic temperature step changes are also important in building heat transfer applications such as in air conditioning, where the conventional assumption of periodic outdoor conditions may lead to con-

386 Siva Reddy Sheri and Anjan Kumar Suram siderable errors in the case of a significant temporary deviation of the temperature from periodicity.

However, here we are only highlighting some contributions.

[7] investigated numerical solution of unsteady hydromagnetic natural convection flow of heat absorbing fluid past an impulsively moving vertical plate with ramped temperature. Srinivasacharya et al.

[9] found Transient approach to heat absorption and radiative heat transfer past an impulsively moving plate with ramped temperature.

2 386 Siva Reddy Sheri and Anjan Kumar Suram siderable errors in the case of a significant temporary deviation of the temperature from periodicity. Keeping this in view, several authors have studied free convection flow past a vertical plate with step discontinuities in the surface temperature. However, here we are only highlighting some contributions. Nandkeolyar and Das [6] analyzed unsteady MHD free convection flow of a heat absorbing dusty fluid past a flat plate with ramped wall temperature. Seth et al. [7] investigated numerical solution of unsteady hydromagnetic natural convection flow of heat absorbing fluid past an impulsively moving vertical plate with ramped temperature. Srinivasacharya et al. [8] studied Radiation effect on mixed convection over a vertical wavy surface in Darcy porous medium with variable properties. Siva Reddy et al. [9] found Transient approach to heat absorption and radiative heat transfer past an impulsively moving plate with ramped temperature. The present numerical investigation deals with the finite element analysis of heat and mass transfer past an impulsively moving vertical plate with ramped temperature. The governing equations are solved using finite element method. The numerical result is validated by comparing the values of the Skin friction and Nusselt number obtained through our present scheme with the earlier published results. 2. Mathematical Formulation Consider flow of a viscous incompressible electrically conducting fluid past an infinite vertical plate embedded in a porous medium. The physical model and coordinate system are shown in Figure 1. Choose the Coordinate system in such a way that the x-axis is taken along the plate in the upward direction, y-axis normal to the plane of the plate in the fluid. Under the Boussinesq s approximation, the governing boundary layer equations of the problem are: (3) Initial and boundary conditions for the problem are specified as (4) By using Rosseland approximation for radiative flux is given by (5) T 4 is expanded in Taylor series about a free stream temperaturet and neglecting higher order terms (T T ), we obtain (6) Making use of equations (5) and (6) in equation (2), we Obtain (7) Introducing following non-dimensional quantities and parameters (1) (2) Figure 1. Physical model and coordinate system of the problem.

Finite Element Analysis of Heat and Mass Transfer past an Impulsively Moving Vertical Plate with Ramped Temperature 387 The transformed system of coupled, non-linear and non-homogeneous dimensionless

(12) are solved numerically for momentum, energy and concentration equations by using the extensively-validated and robust method known as finite element method.

assembled equations. An excellent description of these steps presented in the text books Bathe [10] and Reddy [11].

defined as 3.1 Grid Independence Study The grid independent test is carried out by dividing the whole domain into successively sized grids 81x81, 101x101, and 121x121.

3 Finite Element Analysis of Heat and Mass Transfer past an Impulsively Moving Vertical Plate with Ramped Temperature 387 The transformed system of coupled, non-linear and non-homogeneous dimensionless partial differential Eqs (9)(11) under the boundary conditions Eq. (12) are solved numerically for momentum, energy and concentration equations by using the extensively-validated and robust method known as finite element method. This method has five fundamental steps which are discretization of the domain, derivation of element equations, assembly of element equations, imposition of boundary conditions and solution of the assembled equations. An excellent description of these steps presented in the text books Bathe [10] and Reddy [11]. (8) In view of equation (8), equations (1), (7) and (3) reduce to the following dimensionless form: (9) (10) (11) According to the above non-dimensional process the characteristic time t 0 can be defined as 3.1 Grid Independence Study The grid independent test is carried out by dividing the whole domain into successively sized grids 81x81, 101x101, and 121x121. For all the computations 101 intervals of equal length 0.01 is considered. At each node three functions are to be evaluated so that, after assembly of elements a set of 303 non-linear equations are formed, consequently an iterative scheme is adopted. With the imposition of boundary conditions system of equations are solved by using Thomas algorithm. A convergence criterion based on relative difference between two successive iterations was used. When the difference satisfies the desired accuracy 10-6, the solution is assumed to be converged. An excellent convergence for all the results is achieved. Skin-friction (), Nusselt number (Nu) and Sherwood number (Sh) at the plate are given by (13) Using (8) the initial and boundary conditions (4), in non-dimensional form reduces to (14) (15) 3. Method of Solution (12) 3.2 Validation of Numerical Solution The correctness of this numerical scheme and MATLAB code is ensured by comparing the present results with the previous results in the absence of solutal Grashoff number, radiation and Sc = which are shown in Tables 12. Tables 1 and 2 present a comparison of Skin friction and Nusselt number Nu for various values of the magnetic parameter M 2 and Prandtl number Pr respectively with those reported previously by Seth et al. [7]. It is seen from these tables that excellent agreement between the results exists. In addition, both Skin friction

4 388 Siva Reddy Sheri and Anjan Kumar Suram Table 1. Skin friction when R =0,Gc =0andSc = (Gr =4.0,K =0.4,Pr=0.71,Q =2.0andt =0.5) Seth et al. [7] Present results M 2 Ramped Isothermal Ramped Isothermal Table 2. Nusselt number Nu when R =0,Gc =0andSc = (Gr =4.0,K = 0.4, Pr = 0.71, Q =2.0andt = 0.5) Seth et al. [7] Present results Pr Ramped Isothermal Ramped Isothermal and Nusselt number increase on increasing magnetic parameter M 2 and Prandtl number Pr respectively. 4. Results and Discussions To analyze the effects of various parameters on flowfield in the boundary layer region, numerical values of fluid velocity, temperature and concentration are computed from the numerical solutions, are depicted graphically versus boundary layer co-ordinate y in Figures 2 10(c). In the present study we adopted the following default parameter values of finite element computations Gr = 1.0, Gc = 1.0, M = 2.0, K = 0.2, Pr = 0.71, R = 1.0, Q = 1.0, Sc = 2.0 and t = 0.5. All graphs therefore correspond to these values unless specifically indicated on the appropriate graph. Figure 2 represents the effect of thermal Grashof number on fluid velocity for both ramped temperature and isothermal plate. Figure 2 displays that the velocity begins to increase by increasing the values of thermal Grashof number Gr. This is due the reason that the thermal buoyancy force tends to accelerate fluid flow for both ramped temperature and isothermal plate. Figure 3 depicts the effect of solutal Grashoff number Gc on fluid velocity for both ramped temperature and isothermal plate. From Figure 3 it is noticed that there is an increase in fluid velocity in boundary layer region on increasing solutal Grashoff number Gc. This happens due to the reason that concentration buoyancy force has a tendency to accelerate fluid velocity for both ramped temperature and isothermal plate. Figure 4 displays the influence of magnetic field on the fluid velocity for both ramped temperature and isothermal plate. Figure 4 conveys that the velocity begins to decrease by increasing the values of magnetic parameter M 2. This is due to the fact that the application of a transverse magnetic field to an electrically conducting fluid gives rise to a resistive force which is known as Lo- Figure 3. Effect of Gc on velocity profile. Figure 2. Effect of Gr on velocity profile. Figure 4. Effect of M on velocity profile.

5 Finite Element Analysis of Heat and Mass Transfer past an Impulsively Moving Vertical Plate with Ramped Temperature 389 rentz force. This force has a retarding influence on fluid velocity. Figure 5 explains the effect of permeability parameter on the fluid velocity for both ramped temperature and isothermal plate. Figure 5 expresses that the velocity increases by increasing the values of permeability parameter K. Permeability is the measure of the materials ability to permit liquid or gas through its pores or voids. This supports that as permeability increases velocity should Figure 5. Effect of K on velocity profile. also increase. Figures 6(a) and 6(b) display the control of Prandtl number on fluid velocity and temperature for both ramped and isothermal plate. Figures 6(a) and 6(b) reveal that there is a decrease in fluid velocity and temperature on increasing Pr. This implies that thermal diffusion tends to accelerate fluid velocity and temperature throughout boundary layer region. This happens due to the fact that thermal diffusion provides an impetus to the thermal buoyancy force. Since Pr signifies the relative effects of viscosity to thermal conductivity. This implies that, thermal diffusion tends to enhance fluid velocity and temperature. Figures 7(a) and 7(b) demonstrate the influence of heat absorption on fluid velocity and temperature for both ramped temperature and isothermal plate. From Figures 7(a) and 7(b) it is clear that fluid velocity and temperature decrease on increasing Q. This is due to the reason that, heat absorption tends to retard fluid velocity and temperature throughout boundary layer region. This may be attributed to the fact that the tendency of heat Figure 6. (a) Effect of Pr on velocity profile; (b) Effect of Pr on temperature profile. Figure 7. (a) Effect of Q on velocity profile; (b) Effect of Q on temperature profile.

6 390 Siva Reddy Sheri and Anjan Kumar Suram absorption (thermal sink) is to reduce the fluid temperature which causes the strength of thermal buoyancy force to decrease resulting in a net reduction in the fluid velocity. Figures 8(a) and 8(b) depict the effect of radiation parameter on fluid velocity and temperature for both ramped temperature and isothermal plate. From these two figures it is clear that fluid velocity and temperature decrease on increasing radiation parameter in the boundary layer region which implies that thermal radiation tends to reduce fluid velocity and temperature for both ramped temperature and isothermal plate. Figures 9(a) and 9(b) give the details about the control of Schmidt number on velocity and concentration for both ramped temperature and isothermal plate. From 9(a) and 9(b) it is noticed that both velocity and concentration distribution diminishes at all points of the flow field with the increase of the Schmidt number. This due to the reason that the heavier diffusing species have a greater retarding influence on velocity and concentration distribution of the flow field in case of both ramped temperature and isothermal plate. Figures 10(a), 10(b) and 10(c) display the effect of time on fluid velocity, temperature and concentration for both ramped temperature and isothermal plate. From these figures it is noticed that fluid velocity, temperature and concentration increase on increasing time. This implies that, for both ramped temperature and isothermal plate there is improvement in velocity, temperature and concentration with the progress of time. 5. Conclusions A detailed numerical study has been carried out to investigate finite element analysis for heat and mass transfer past an impulsively moving vertical plate with ramped temperature. The conclusions for both ramped temperature and isothermal plates of the study are: The velocity increases with increasing values of thermal buoyancy force, solutal buoyancy force, permeability parameter and time. And for increasing values of magnetic parameter, Prandtl number, heat absorption, radiation pa- Figure 8. (a) Effect of R on velocity profile; (b) Effect of R on temperature profile. Figure 9. (a) Effect of Sc on velocity profile; (b) Effect of Sc on concentration profile.

7 Finite Element Analysis of Heat and Mass Transfer past an Impulsively Moving Vertical Plate with Ramped Temperature 391 Figure 10. (a) Effect of t on velocity profile; (b) Effect of t on temperature profile (c) Effect of t on concentration profile. rameter, and Schmidt number velocity boundary layer decreases. The fluid temperature increases for increasing values of time. And increasing values of Prandtl number, heat absorption and radiation parameter thermal boundary layer decreases. The concentration of fluid increases with progression of time. And increasing values of Schmidt number concentration boundary layer decreases. Acknowledgments The authors are thankful to the University Grants Commission, New Delhi, India for providing financial assistance to carry out this research work under UGC - Major Research Project [F. No /2013 (SR)]. g Nomenclature Acceleration due to gravity Volumetric coefficient of thermal expansion for species concentration * k K C p q r k * * Q 0 D B 0 U 0 t 0 T T C C M K Volumetric coefficient of thermal expansion Kinematic viscosity Fluid density Electrical conductivity Thermal conductivity of the fluid Permeability of the porous medium Specific heat at constant pressure Radiative flux vector Mean absorption co-efficient Stefan-Boltzmann constant Heat absorption coefficient Chemical molecular diffusivity Magnetic induction Characteristic velocity Critical time for rampedness Dimensional temperature Temperature of free stream Dimensional concentration Concentration of free stream Magnetic parameter Permeability parameter

8 392 Siva Reddy Sheri and Anjan Kumar Suram Gr Thermal Grashof number Gc Solutal Grashof number Pr Prandtl number Q Heat absorption R Radiation parameter Sc Schmidt number Skin friction coefficient Nu Nusselt number Sh Sherwood number References [1] Bejan, A. and Khair, K. R., Heat and Mass Transfer by Natural Convection by Porous Medium, International Journal of Heat and Mass Transfer, Vol. 28, No. 5, pp (1985). doi: / (85) [2] Kandasamy, R., Periasamy, K. and Sivagnana Prabhu, K. K., Effects of Chemical Reaction, Heat and Mass Transfer along a Wedge with Heat Source and Concentration in the Presence of Suction or Injection, International Journal of Heat and Mass Transfer, Vol. 48, No. 7, pp (2005). doi: /j.ijheatmas stransfer [3] Yih, K. A., Coupled Heat and Mass Transfer in Mixed Convection Over a VHF/VMF Wedge in Porous Media the Entire Regime, Acta Mechanica, Vol. 137, No. 1, pp. 112 (1999). doi: /BF [4] Chandran, P., Sacheti, N. C. and Singh, A. K., Natural Convection near a Vertical Plate with Ramped Wall Temperature, Heat and Mass Transfer, Vol. 41, pp (2005) doi: /s [5] Narahari, M., Beg, O. A. and Ghosh, S. K., Mathematical Modelling of Mass Transfer and Free Convection Current Effects on Unsteady Viscous Flow with Ramped Wall Temperature, World Journal of Mechanics, Vol. 1, No. 4, pp (2011). doi: /wjm [6] Nandkeolyar, R. and Das, M., Unsteady MHD Free Convection Flow of a Heat Absorbing Dusty Fluid past a Flat Plate with Ramped Wall Temperature, Afrika Mathematika, Vol. 25, No. 3, pp (2014). doi: /s (2013) [7] Seth, G. S., Sarkar, S. and Mahato, G. K., Numerical Solution of Unsteady Hydromagnetic Natural Convection Flow of Heat Absorbing Fluid past an Impulsively Moving Vertical Plate with Ramped Temperature, International Journal of Applied Mathematical Research, Vol. 2, No. 2, pp (2013). doi: / ijamr.v2i2.939 [8] Srinivasacharya, D., Mallikarjuna, B. and Bhuvanavijaya, R., Radiation Effect on Mixed Convection Over a Vertical Wavy Surface in Darcy Porous Medium with Variable Properties, Journal of Applied Science and Engineering, Vol. 18, No. 3, pp (2015). doi: /jase [9] Siva Reddy Sheri, R. S., Raju and S. Anjan Kumar, Transient Approach to Heat Absorption and Radiative Heat Transfer past an Impulsively Moving Plate with Ramped Temperature, Procedia Engineering,Vol. 127, pp (2015). doi: /j.proeng [10] Bathe, K. J., Finite Element Procedures, Prentice- Hall, New Jersey (1996). [11] Reddy, J. N., An Introduction to the Finite Element Method, 3rd Edition, McGraw-Hill Book Company, New York (2006). Manuscript Received: Feb. 22, 2016 Accepted: Aug. 8, 2016

MHD NATURAL CONVECTION FLOW PAST A MOVING VERTICAL PLATE WITH RAMPED TEMPERATURE

VOL. 13, NO., NOVEMBER 18 ISSN 1819-668 6-18 Asian Research Publishing Network (ARPN). All rights reserved. MHD NATURAL CONVECTION FLOW PAST A MOVING VERTICAL PLATE WITH RAMPED TEMPERATURE Ch. Vijaya Bhaskar