Energy Conversion and Management 47 (2006) 3499–3500

www.elsevier.com/locate/enconman

Correspondence

‘‘Effects of viscous dissipation on the heat transfer in forcedpipe flow. Part 1: Both hydrodynamically and thermallyfully developed flow [Energy Conv. Manage. 2005; 46;

757–769] and Part 2: Thermally developing flow [EnergyConv. Manage. 2005; 3091–3102]’’ by Orhan Aydin

Reply to ‘‘Comments on the above referenced articles’’by D.A. Nield, K. Hooman

Orhan Aydin *

Department of Mechanical Engineering, Karadeniz Technical University, 61080 Trabzon, Turkey

Received 13 September 2005; accepted 15 February 2006Available online 18 April 2006

In the following, I reply to the comments by Nield and Hooman on my recently published articles [1,2] on aone to one basis.

First of all, the authors question the need for these two articles [1,2], referencing two papers by Ou andCheng [3,4]. If one closely looks at the references that are cited in my articles [1,2], it will be seen this topichas been studied by others as well, after Ou and Cheng [3,4]. The relation of the topic to many practical fields,especially its importance in heat transfer in micro systems still attracts a great deal of research interest. Atotally different solution methodology than that in [3,4] is used in the study. For the constant wall heat fluxcase, a simpler analytical approach is followed. For the constant wall temperature case, however, an iterativeprocedure is followed by assuming the temperature profile for the case of the constant heat flux at the wall asan initial profile. In addition, in my articles, I focus on the definition of the dimensionless temperature, whichwill lead to different definitions of the Brinkman number. For the constant heat flux case at the wall, two dif-ferent definitions of the dimensionless temperature are introduced, which result in the Brinkman number andthe modified Brinkman number. The thermophysical reasons behind the existence of the singularities at somespecific values of the Brinkman number in some cases are well discussed in terms of the energy balancebetween the wall heat and the viscous dissipation heat.

The authors find the presentation of the CHF case in [1] confusing since Eq. (7) contains a coefficient a thatdoes not appear in the solution. In fact, the coefficient a has been obtained directly from the application of thethermal boundary condition at the wall given in Eq. (9). For the sake of brevity, this simple procedure includ-ing several steps of derivations was not mentioned.

0196-8904/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2006.02.023

* Tel.: +90 462 377 2974; fax: +90 462 325 5526/3205.E-mail address: oaydin@ktu.edu.tr

3500 O. Aydin / Energy Conversion and Management 47 (2006) 3499–3500

The authors claim that the precise relationship between the Brinkman number and the modified Brinkmannumber is not given in [1]. However, a closer look at the study will show that they are separately derived fromthe relevant dimensionless temperature equations, Eqs. (10) and (12).

The authors declare that the Nusselt number should receive the value of 48/5, being independent of Br. Ishould again emphasize the importance of various definitions of the Brinkman number. Actually, this is truefor the definition of the Brinkman number in terms of the temperature difference between the wall and theinlet, see Eq. (6) in Part 2 [2], which is Br ¼ lu2

c

k T w�T eð Þ. As can be seen in Part 2 [2], my analysis yielded the same

asymptotic value, see Fig. 6 [2]. However, in Part 1 [1], I used a different definition for the Brinkman number,

Br ¼ lu2c

k T w�T cð Þ (in terms of the temperature difference between the wall and the centerline), which can not be

expected to lead to the above value of the Nusselt number.In addition, I regret to observe some typing errors in Part 1 [1]. However, it should be noted that they are

correctly treated in the analysis. These are:

1. In Eq. (15), Tm � Tw should be changed as Tw � Tm.2. Eq. (20), NuD ¼ 48

11þBrq, should be corrected as NuD ¼ 48

11þ6Brq.

Finally, I thank the authors very much for their interest and comments on my articles.

References

[1] Aydin O. Effects of viscous dissipation on the heat transfer in forced pipe flow. Part 1: Both hydrodynamically and thermally fullydeveloped flow. Energy Conv Manag 2005;46:757–69.

[2] Aydin O. Effects of viscous dissipation on the heat transfer in a forced pipe flow. Part 2: Thermally developing flow. Energy ConvManag 2005;46:3091–202.

[3] Ou WJ, Cheng KC. Viscous dissipation effects in the entrance region heat transfer in pipes with uniform heat flux. Appl Sci Res1973;28:289–301.

[4] Ou WJ, Cheng KC. Viscous dissipation effects on thermal entrance region heat transfer in laminar and turbulent pipe flows withuniform wall temperature. AIAA Paper No. 74-743 or ASME Paper No. 74-HT-50, 1974, p. 1–7.