HEAT AND MASS TRANSFER ANALYSIS OF RADIATION AND CHEMICAL REACTION EFFECTS ON MHD FLOW PAST SEMI-INFINITE POROUS PLATE

Abstract

This study examines unsteady magnetohydrodynamic (MHD) flow past a semi-infinite, vertically permeable, moving plate embedded in a Darcian porous medium, with time-dependent wall suction and a first-order homogeneous chemical reaction. A uniform transverse magnetic field acts on the flow, along with thermal and concentration buoyancy effects. The fluid is assumed to be two-dimensional, laminar, viscous, incompressible, electrically conducting, and heat-absorbing. The plate moves at a constant velocity in the fluid flow direction, while the free stream velocity follows an exponentially increasing small perturbation law. The dimensionless governing equations are analytically solved using a two-term perturbation method, and numerical and graphical results for the velocity, temperature, and concentration profiles within the boundary layer are presented. Findings indicate that increases in radiation and heat absorption reduce both velocity and temperature profiles, while stronger chemical reactions lead to thinner velocity and concentration boundary layers.

Specification

Description:FIELD OF INVENTION
User is interested in analyzing heat and mass transfer in magnetohydrodynamic (MHD) flow over a semi-infinite porous plate, emphasizing radiation and chemical reaction effects. This study explores the interactions between thermal radiation, mass diffusion, and magnetic fields, relevant for industrial applications and energy systems, particularly in enhancing heat transfer efficiency and understanding chemical processes in conductive fluids.
BACKGROUND OF INVENTION
The study of heat and mass transfer in magnetohydrodynamic (MHD) flow over a semi-infinite porous plate has gained significant attention due to its applications in engineering and industrial processes, such as energy generation, cooling of electronic devices, and chemical processing. When an electrically conducting fluid flows over a heated or cooled surface, the simultaneous presence of magnetic fields, radiation, and chemical reactions introduces complex interactions that can alter the fluid's velocity, temperature, and concentration profiles. This setup can model processes in nuclear reactors, cooling systems in fusion devices, or astrophysical phenomena where magnetic fields and heat transfer are critical factors.
In such flows, the magnetic field generates Lorentz forces that impact the fluid's motion, introducing resistance that can control flow rate and enhance stability. Additionally, thermal radiation becomes significant at high temperatures, altering the heat transfer rate and influencing the thermal boundary layer. Chemical reactions, particularly first-order reactions, affect the mass transfer properties by creating a concentration gradient near the surface, further impacting the fluid's thermal and concentration profiles.
The porous nature of the plate adds another layer of complexity, as it allows for fluid suction or injection, which can regulate the boundary layer thickness and enhance the cooling or heating process. Understanding the combined effects of these parameters is essential for optimizing heat and mass transfer in practical systems, especially in environments where efficient control of thermal and chemical properties is vital, such as chemical reactors, heat exchangers, and material processing units.
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SUMMARY
This invention focuses on the analysis of heat and mass transfer in magnetohydrodynamic (MHD) flow over a semi-infinite porous plate, specifically examining the effects of radiation and chemical reactions on the flow characteristics. By integrating magnetic fields, thermal radiation, and chemical reaction kinetics, this model provides a comprehensive framework for studying the thermal and mass diffusion behavior in electrically conducting fluids.
The application of a magnetic field generates Lorentz forces within the flow, which act as a resistive drag, slowing the fluid and affecting the boundary layer dynamics. This control over flow resistance is particularly beneficial for managing fluid behavior in high-temperature environments such as fusion reactors, metallurgical processes, and aerospace engineering. In high-temperature settings, thermal radiation contributes significantly to the heat transfer process, modifying the temperature profile and potentially enhancing the cooling effect. This model accounts for radiative heat flux, providing insights into the thermal boundary layer behavior under radiation-dominant conditions.
Chemical reactions, modeled as first-order reactions, induce concentration gradients that drive mass transfer and influence the fluid's properties. The porous plate enables fluid suction or injection, an essential feature that allows further control over the boundary layer thickness and stability of the flow. This invention's combined approach to studying the MHD flow with these additional effects is vital for designing optimized heat and mass transfer systems. Potential applications range from thermal management in electronic devices to enhancing chemical reactor efficiency, where precise control over temperature, flow, and concentration gradients is critical.
DETAILED DESCRIPTION OF INVENTION
This paper investigates the effects of magnetic body force, radiation, and first-order chemical reactions on unsteady mixed convection boundary layer flow along a vertical, permeable surface embedded in a Darcian porous medium with mass blowing or suction and absorption. The surface is assumed to move uniformly at a constant velocity, while the free stream velocity varies with time. This complex flow scenario, governed by combined buoyancy and magnetic field effects, is highly relevant to many scientific and industrial applications, such as nuclear reactors, chemical processing, and porous media applications.
The interplay between heat and mass transfer with radiation and chemical reactions is significant in various engineering and scientific domains. These processes are especially critical in situations where temperature and concentration gradients occur, influencing fluid flow and convective behavior. Chemical reactions can be categorized as homogeneous if they occur uniformly throughout the fluid phase or heterogeneous if confined to specific regions or boundaries. In cases where reaction rates depend on species concentration, first-order reactions, in particular, directly affect the convective dynamics of the system.
Prior studies have analyzed similar fluid flows under various conditions, such as steady-state flows, oscillating surfaces, and heat absorption effects. Earlier works, including those by Soundalgekar, Cheng, Chamkha, and others, explored convective flows in porous media and the influence of viscous dissipation, magnetic fields, thermal buoyancy, and chemical reactions on such flows. This investigation analytically solves the governing equations and presents the results graphically, offering insights into the velocity, temperature, and concentration profiles within the boundary layer.
The mathematical formulation for this study involves the unsteady two-dimensional flow of a laminar, incompressible, viscous, electrically conducting fluid past a semi-infinite, vertical, permeable moving plate embedded in a uniform porous medium. The fluid is subjected to a uniform transverse magnetic field, and the effects of thermal and concentration buoyancy are considered. It is assumed that there is no applied voltage, which negates any electrical field. With a small magnetic Reynolds number, both the induced magnetic field and Hall effects are negligible, resulting in the decoupling of the Navier-Stokes equations from Maxwell's equations.
The governing equations, derived from the principles of mass, linear momentum, energy, and species concentration balance, are presented in a Cartesian coordinate system as follows:







Continuity Equation

where:
• x and y are distances along and perpendicular to the plate, respectively,
• u and v are the velocity components along the x and y directions,
• g is gravitational acceleration,
• ρ is the fluid density,
• β and β∗ are the thermal and concentration expansion coefficients,
• K is the Darcy permeability,
• B0 is the magnetic induction,
• T and C represent temperature and concentration, respectively,
• σ is electrical conductivity,
• cp is specific heat at constant pressure,
• D is the diffusion coefficient,
• qr is the heat flux,
• Q is the dimensional heat absorption coefficient,
• kr is the chemical reaction parameter.
Boundary Conditions
The boundary conditions for velocity, temperature, and concentration fields are specified as:

where Uw, Tw, and CwCw are the velocity, temperature, and concentration at the wall, U∞ is the free stream velocity, and V0 is a constant representing suction velocity at the plate. The suction velocity normal to the plate is assumed in the form:

where V0 is a positive constant, A and ω are constants with values less than unity. The negative sign indicates suction towards the plate.
Outside the boundary layer, Equation (2) simplifies to:
u=U∞ (7)
Using the Rosseland approximation, the radiative heat flux qrqr is given by:

where σ∗ is the Stefan-Boltzmann constant and kr∗ is the Roseland mean absorption coefficient. Assuming small temperature differences within the flow, T4 can be expressed as a linear function of temperature:

Dimensionless Variables
To non-dimensionalize the governing equations and boundary conditions, the following transformations are introduced:

In terms of these non-dimensional quantities, Equations (2)-(4) become:
Dimensionless Momentum Equation

Dimensionless Energy Equation

Dimensionless Concentration Equation

Dimensionless Boundary Conditions
The boundary conditions transform as:

Solution of the Problem:
Equations (12)-(14) represent a system of coupled, nonlinear partial differential equations, which are challenging to solve in closed form. However, they can be simplified into a system of ordinary differential equations that are analytically solvable. This reduction is achieved by expressing the velocity, temperature, and concentration fields of the fluid near the plate as outlined in Equation (16).
By substituting Equation (16) into Equations (12)-(14), equating the harmonic and non-harmonic terms, and ignoring higher-order terms of small parameters, we obtain the following set of equations:

where primes denote ordinary differentiation with respect to y.
The associated boundary conditions for this system can be represented as:

By solving Equations (17)-(22) with the boundary conditions given in Equation (23), we obtain the velocity, temperature, and concentration distributions within the boundary layer.
Important Physical Parameters:
For this type of boundary layer flow, the skin friction coefficient, Nusselt number, and Sherwood number are key physical quantities.
• Skin Friction Coefficient: Given the velocity field, the skin friction coefficient at the plate can be determined in nondimensional form by:

where τ is the shear stress, ρ is the fluid density, and u is the velocity.
• Nusselt Number: Using the temperature field, the rate of heat transfer at the plate surface is characterized by the Nusselt number in nondimensional form:

where h is the convective heat transfer coefficient, L is the characteristic length, and k is the thermal conductivity of the fluid.
• Sherwood Number: Based on the concentration field, the rate of mass transfer at the plate surface is characterized by the Sherwood number in nondimensional form:

where kc is the mass transfer coefficient, L is the characteristic length, and D is the diffusion coefficient.
These parameters help analyze the momentum, heat, and mass transfer rates in the boundary layer flow near the plate.
Results and Discussion
The effects of various flow parameters-such as the magnetic parameter (M), Grashof number (Gr), modified Grashof number (Gc), heat absorption coefficient (Q), permeability parameter (K), radiation parameter (R), chemical reaction parameter (Kr), Prandtl number (Pr), and Schmidt number (Sc)-on velocity, temperature, concentration, skin friction (τ), Nusselt number (Nu), and Sherwood number (Sh) were analyzed and are presented graphically and in tabular form.
The impact of the magnetic field on velocity profiles in the boundary layer is depicted in Figure 1, where velocity profiles decrease as the magnetic parameter (M) increases. The application of a transverse magnetic field to an electrically conducting fluid induces a resistive force, known as the Lorentz force, which slows down the fluid flow.
Figure 2 shows the effect of the radiation parameter (R) on velocity profiles, revealing a decrease in velocity profiles with an increase in R. Higher values of the radiation parameter reduce the boundary layer thickness, enhancing the heat transfer rate due to thermal and solutal buoyancy forces.
The influence of the permeability parameter (K) on velocity profiles is illustrated in Figure 3, showing that as the porous permeability increases, the drag on the fluid flow reduces, leading to an increase in velocity. Thus, the velocity accelerates as the porosity parameter increases.
Figures 4 and 5 demonstrate the influence of thermal (Gr) and solutal buoyancy (Gc) forces, respectively, on velocity profiles. The results indicate a peak velocity value at Gr = 20 and a minimum at Gr = 1, as the buoyancy force enhances fluid velocity and increases boundary layer thickness with higher values of Gr and Gc.
Figure 6 illustrates the velocity distribution for varying chemical reaction parameters (Kr), showing a decrease in velocity with an increase in Kr. Figure 7 depicts the effect of the heat source parameter (Q) on the boundary layer, where increasing Q leads to a reduced boundary layer as heat absorption decreases buoyancy forces, thereby slowing the flow rate and decreasing velocity profiles.
Temperature profiles for different values of the Prandtl number (Pr) are shown in Figure 8. An increase in Pr results in reduced temperature profiles as smaller Pr values correspond to higher fluid thermal conductivity, allowing heat to diffuse away from the surface more efficiently.
Figure 9 shows the variation in temperature profiles with different heat source parameters (Q), where temperature decreases as Q increases, given that heat absorption reduces buoyancy forces and subsequently lowers temperature profiles. Figure 10 further illustrates the influence of Prandtl number (Pr) on temperature, with higher Pr values leading to a thinner thermal boundary layer and a reduced rate of heat transfer due to lower thermal conductivity.
The effect of Schmidt number (Sc) on concentration profiles is presented in Figure 11. As Sc increases, the concentration decreases, reflecting the reduced molecular diffusivity that leads to a thinner concentration boundary layer. Figure 12 displays the effects of the chemical reaction parameter (Kr) on concentration, showing a decrease in concentration profiles with higher Kr values, resulting in a reduction in fluid velocity along the boundary layer.
The combined effect of Grashof number and magnetic field on skin friction is illustrated in Figure 13. Increased magnetic intensity reduces skin friction, while for a constant magnetic field, increasing the Grashof number raises skin friction, as shown in Figure 14. Figure 15 demonstrates the influence of heat absorption (Q) and radiation parameter (R) on skin friction, where smaller values of Q have a notable effect within the boundary layer, and increasing Q generally contributes to higher skin friction.

M K Gr A Kr Sc Pr Gc R Q SF
1
2
3
4 0.5 5 0.5 0.5 0.78 0.71 5 0.5 0.5 2.1606 1.0443 0.5800 0.3253
2 1
2
3
4 5 0.5 0.5 0.78 0.71 5 0.5 0.5 2.1606 3.6320 4.5605 5.2002
2 0.5 1
2
3
4 0.5 0.5 0.78 0.71 5 0.5 0.5 -0.1942 0.1154 0.4250 0.7347
2 0.5 5 1
2
3
4 0.5 0.78 0.71 5 0.5 0.5 1.0073 0.9332 0.8591 0.7850
2 0.5 5 0.5 1
2
3
4 0.78 0.71 5 0.5 0.5 1.3734 2.0762 3.0921 5.0554
2 0.5 5 0.5 0.5 1
2
3
4 0.71 5 0.5 0.5 1.3928 12.3350
-1.6975
-0.4900
2 0.5 5 0.5 0.5 0.78 1
2
3
4 5 0.5 0.5 5.4049
-2.7019
-1.7394
-1.4102
2 0.5 5 0.5 0.5 0.78 0.71 1
2
3
4 0.5 0.5 1.0083 1.0173 1.0263 1.0353

The governing equations for the unsteady MHD convective heat and mass transfer flow past a semi-infinite, vertically oriented permeable moving plate in a porous medium, with the inclusion of radiation, heat absorption, and a first-order chemical reaction, were formulated. The plate velocity was kept constant, and the flow was influenced by a transverse magnetic field. Through a transformation of the partial differential equations into a set of ordinary differential equations using a two-term series, a closed-form analytical solution was achieved. Numerical evaluations of these solutions were performed, and graphical results for various physical parameters were obtained. The analysis reveals the following findings:
• The presence of the magnetic parameter, radiation, heat absorption coefficient, and chemical reaction leads to a decrease in velocity profiles.
• The permeability parameter, Grashof number, and modified Grashof number contribute to an increase in velocity profiles.
• The Prandtl number, radiation, and heat absorption coefficient contribute to a decrease in temperature profiles.
• The concentration boundary layer thickness reduces with increasing Schmidt number and chemical reaction.
• Chemical reactions decrease both skin friction and the Sherwood number.



DETAILED DESCRIPTION OF DIAGRAM
Fig.1. Velocity profiles for different values of magnetic parameter (M).
Fig.2. Velocity profiles for different values of radiation parameter (R).
Fig.3. Velocity profiles for different values of permeability parameter (K).
Fig.4. Velocity profiles for different values of Grashof number (Gr)
Fig.5. Velocity profiles for different values of modified Grashof number (Gc)
Fig.6. Velocity profiles for different values of chemical reaction parameter (Kr)
Fig.7. Velocity profiles for different values of heat source parameter (Q)
Fig.8. Temperature profiles for different values of radiation parameter (R)
Fig.9. Temperature profiles for different values of heat source parameter (Q)
Fig.10. Temperature profiles for different values of Prandtl number (Pr)
Fig.11. Concentration profiles for different values of Schmidt number (Sc)
Fig.12. Concentration profiles for different values of chemical reaction parameter (Kr).
Fig.13. Skin-Friction for different values of magnetic parameter (M).
Fig.14. Skin-Friction for different values of Grashof number (Gr).
Fig.15. Skin-Friction for different values of heat absorption parameter (Q). , Claims:1. Heat and mass transfer analysis of radiation and chemical reaction effects on mhd flow past semi-infinite porous plate the study investigates the behavior of MHD flow, where the flow of an electrically conducting fluid is influenced by an applied magnetic field.
2. The flow occurs past a semi-infinite porous plate, which is assumed to be extended infinitely in one direction, contributing to boundary layer phenomena.
3. The analysis incorporates the influence of thermal radiation, which can significantly impact the heat transfer rate in high-temperature flows, particularly for optically thick fluids.
4. Chemical reactions, such as first-order reactions, are modeled to evaluate their effects on the concentration and thermal field, influencing the overall flow dynamics.
5. The study emphasizes both heat and mass transfer, analyzing how the temperature and concentration profiles evolve due to the combined effects of thermal radiation, chemical reactions, and MHD forces.
6. The effects of viscous dissipation are taken into account, which play a role in the temperature distribution due to the conversion of kinetic energy into thermal energy in the boundary layer.
7. The flow is subject to convective boundary conditions at the plate surface, with the fluid temperature and concentration gradients influencing the rate of heat and mass transfer.
8. The applied magnetic field modifies the flow characteristics, introducing a Lorentz force that affects the velocity profile, often leading to a decrease in the flow velocity (known as magnetic damping).
9. The analysis considers variations in fluid properties such as thermal conductivity, diffusivity, and viscosity with temperature and concentration, which influence the flow and heat/mass transfer rates.
10. The problem is typically solved using numerical techniques, such as finite difference or finite element methods, to obtain the temperature, velocity, and concentration profiles for different conditions of magnetic field strength, radiation, and chemical reaction rates.

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