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Closure

# 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)OPEN ACCESS

[+] Author and Article Information
O. D. Makinde

Faculty of Military Science,
Stellenbosch University,
Private Bag X2,
Saldanha 7395, South Africa
e-mail: makinded@gmail.com

A. Aziz

Department of Mechanical Engineering,
School of Engineering and Applied Science,
Gonzaga University,
Spokane, WA 99258
e-mail: aziz@gonzaga.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 1, 2014; final manuscript received July 3, 2014; published online December 2, 2014. Assoc. Editor: Terry Simon.

J. Heat Transfer 137(3), 036001 (Mar 01, 2015) (1 page) Paper No: HT-14-1432; doi: 10.1115/1.4029096 History: Received July 01, 2014; Revised July 03, 2014; Online December 02, 2014

This write-up presents a Closure to “Discussion of ‘Mixed Convection From a Convectively Heated Vertical Plate to a Fluid With Internal Heat Generation,’” Journal of Heat Transfer, 2011, Vol. 33, 122501 and upholds the mathematical correctness of the published article.

The argument of discussion by “Asterios Pantokratoras” with respect to the correctness of the published paper [1] is totally unfounded and misleading. First, it appears that “Asterios Pantokratoras” did not understand the paper or have any idea about the convectively heated boundary layer problem with a prescribed spatially decaying exponential internal heat generation that were discussed in the paper by Makinde and Aziz [1]. The argument of Asterios Pantokratoras is mathematically erroneous, incorrect, and inaccurate with respect to the model problem described in the paper. We emphasize that the paper [1] is correct both in formulation, analysis, solution, and discussion. Below is our response;

The model local similarity equations are Display Formula

(1)$f"'+12ff"+Grxθ=0$
Display Formula
(2)$θ"+12Prfθ'+λxe-η=0$

with Display Formula

(3)$f(0)=f'(0)=0, θ'(0)=-Bi[1-θ(0)]$
Display Formula
(4)$f'(∞)=0, θ(∞)=0$

where the prime symbols indicate derivatives with respect to η. Following Crepeau and Clarksean [2] and Sahin [3], the spatially decaying exponential internal heat generation within the cold fluid in contact with the right surface of the plate as prescribed in the Makinde and Aziz [1] is Display Formula

(5)$q·=λxkRex(Tf-T∞)x2eη⇒λx=q·x2eηkRex(Tf-T∞)$

The above prescribed expression for internal heat generation enables one to obtain the local similarity equation for energy balance in the Makinde and Aziz [1]. Equations (1)–(4) is a local similarity model problem, since some of the parameters still depend on the variable x and are solved numerically using the Runge–Kutta–Fehlberg integration method implemented on MAPLE. The solutions are displayed in the paper for various values of the local similarity parameters for λx = 0, 1, 5, 10 (see Tables 1 and 2 in Ref. [1]). Moreover, we emphasize in Makinde and Aziz [1] that the local similarity parameters Bix, Grx, and λx in the model equations (1)–(4) can become similarity parameters under certain assumptions, for instance if Display Formula

(6)$q·=lx-1⇒λ=lυeηkU∞(Tf-T∞), since Rex=U∞x/υ$

Finally, both the tables and the graphical results highlighted in the Makinde and Aziz [1] satisfy the model equations (1)–(4) as well as the physics of the problem as thoroughly discussed in the paper.

## References

Makinde, O. D., and Aziz, P. O., 2011, “Mixed Convection From a Convectively Heated Vertical Plate to a Fluid With Internal Heat Generation,” ASME J. Heat Transfer, 133(12), p. 122501.
Crepeau, J. C., and Clarksean, R., 1997, “Similarity Solutions of Natural Convection With Internal Heat Generation,” ASME J. Heat Transfer, 119(5), pp. 184–185
Sahin, A. Z., 1992, “Transient Heat Conduction in Semi-Infinite Solid With Spatially Decaying Exponential Heat Generation,” Int. Commun. Heat Mass Transfer, 19(3), pp. 349–358.
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## References

Makinde, O. D., and Aziz, P. O., 2011, “Mixed Convection From a Convectively Heated Vertical Plate to a Fluid With Internal Heat Generation,” ASME J. Heat Transfer, 133(12), p. 122501.
Crepeau, J. C., and Clarksean, R., 1997, “Similarity Solutions of Natural Convection With Internal Heat Generation,” ASME J. Heat Transfer, 119(5), pp. 184–185
Sahin, A. Z., 1992, “Transient Heat Conduction in Semi-Infinite Solid With Spatially Decaying Exponential Heat Generation,” Int. Commun. Heat Mass Transfer, 19(3), pp. 349–358.

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