Closure to “Discussion of ‘Mixed Convection From a Convectively Heated Vertical Plate to a Fluid With Internal Heat Generation’” (2011, ASME J. Heat Transfer, 33, p. 122501)

A numerical approach has been adopted to study steady mixed convection from the right face of a vertical plate of finite thickness. Cold fluid flowing over the right face of the plate contains a heat generation that decays exponentially with a dimensionless distance from the wall. The left face of the plate is in contact with a hot flowing fluid. The heating process on that side is characterized by a convective boundary condition that takes into account the conduction resistance of the plate as well as a possible contact resistance between the hot fluid and the left face of the plate. Using a pseudo similarity approach, the continuity, momentum, and energy equations for mixed convective flow over the right face of the plate are transformed into a set of coupled ordinary differential equations. It is found that for a true similarity solution, the convective heat transfer coefficient associated with the hot fluid must be proportional to x−1/2, and both the thermal expansion coefficient and the internal heat generation rate for the cold fluid must be proportional to x−1, where x is the upward distance along the plate. The equations give local similarity solutions. The effects of local Grashof number (defined to represent a mixed convection parameter), Prandtl number, Biot number, and the internal heat generation parameter on the velocity and temperature profiles are illustrated and interpreted in physical terms. The present results agree closely with the existing results for the special cases of the problem. This close agreement lends support to the validity of the present analysis and the accuracy of the numerical computations. The paper also contains a table in which the data for the local skin friction and local Nusselt number are provided for various combination values of the parameters that govern the momentum and energy transport in the mixed boundary layer.

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The conjugate conduction–natural convection heat transfer along a thin vertical plate with non-uniform internal heat generation

International Journal of Heat and Mass Transfer ◽

10.1016/s0017-9310(99)00331-2 ◽

2000 ◽

Vol 43 (15) ◽

pp. 2739-2748 ◽

Cited By ~ 31

Author(s):

F Méndez ◽

C Treviño

Keyword(s):

Heat Transfer ◽

Natural Convection ◽

Heat Generation ◽

Vertical Plate ◽

Convection Heat Transfer ◽

Internal Heat ◽

Internal Heat Generation ◽

Natural Convection Heat Transfer ◽

Convection Heat ◽

Conjugate Conduction

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Non-similar solutions for natural convection from a moving vertical plate with a convective thermal boundary condition

Thermal Science ◽

10.2298/tsci130530050p ◽

2016 ◽

Vol 20 (6) ◽

pp. 1847-1853

Author(s):

Asterios Pantokratoras

Keyword(s):

Heat Transfer ◽

Boundary Condition ◽

Thermal Expansion ◽

Heat Generation ◽

Wall Temperature ◽

Vertical Plate ◽

Internal Heat ◽

Thermal Boundary Condition ◽

Internal Heat Generation ◽

Wall Heat Transfer

In a recent paper by Makinde (Thermal Science, 2011, Vol. 15, Suppl. 1, pp. S137-S143.) the effect of thermal buoyancy along a moving vertical plate with internal heat generation was considered. The plate thermal boundary condition was a convective condition with a heat transfer coefficient proportional to x-1/2 . The fluid thermal expansion coefficient was proportional to 1-x and the internal heat generation was assumed to decay exponentially across the boundary layer and proportional to x-1 in order that the problem accepts a similarity solution. In the present work, the same problem without heat generation is considered, with constant heat transfer coefficient and constant thermal expansion coefficient which is more realistic and has much more practical applications. The present problem is non-similar and results are obtained with the direct numerical solution of the governing equations. The problem is governed by the Prandtl number, the non-dimensional distance along the plate and a convective Grashof number, which is introduced for the first time. It is found that the wall shear stress, the wall heat transfer and the wall temperature, all increase with increasing distance and the wall temperature tends to 1. The influence of the convective Grashof number is to increase the wall shear stress and the wall heat transfer and to reduce the wall temperature.

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