Discussion: “Performance and Optimization Analysis for Fins of Straight Taper with Simultaneous Heat and Mass Transfer” (Kundu, B., and Das, P. K., 2004, ASME J. Heat Transfer, 126, pp. 862–868)

We begin with the authors’ Nomenclature because it contains several errors, thus making the reading of this manuscript extremely difficult. The Nomenclature is a mirror of the symbols found in the text, which is used to explain their meanings to the reader who frequently needs to consult it.

(ii) I think $Bit=htyb∕k$ should have been $Bit=htyt∕k$⁠, although $ht$ is never defined.

(iii) The authors assert “$f$⁠, $g$ to be functions defined by Eq. (18) and Eq. (19) respectively.” However, Eq. (18) defines the dimensionless volume $U$⁠, whereas $f(Bi,ψ)$ is defined by Eq. (19), and $g(Bi,ψ)$ by Eq. (20).

(iv) The symbol $hm$⁠, which appears in the last term on the left-side of the authors’ Eq. (1), has never been defined. Also, the dimensions of $q$⁠, the actual heat transfer rate, for longitudinal fins should be expressed in watts per meter.

(v) The correct definition of $qe$ is: “the heat transfer from the base of the fin, with the same heat transfer coefficient in its absence.” The assumption for the ideal heat dissipation $qi$ is: “the entire fin is maintained at the base temperature.”

(vi) $Q$ is defined by Eq. (11) not Eq. (10), $Qe$ by Eq. (16) not Eq. (15), $Qi$ by Eq. (14) not Eq. (13), and $U$ by Eq. (18) not Eq. (17).

(vii) The parameter $Z0$ is recognized as the one which used by Gardner (1) to plot his efficiency graphs and is expressed as the product of the aspect number $L∕yb$ and the $Bi$⁠. The quotient $L∕yb$ is always referred in the literature as aspect number, however, the authors’ $ψ$ is the reciprocal, but still defined as the aspect number. As shown in Refs. 2 3 the authors’ parameters $Z02=Bi∕ψ2$ and Bi have physical significances.

(viii) Normally, the solutions should be able to handle any values of the parameter $0⩽λ⩽1$⁠, hence the parameters $Z1$⁠, $Z2$⁠, and $α$ that, are divided by $(1‐λ)$⁠, cannot be used unless they are redefined.

(ix) The authors created a huge aggregate of dimensionless parameters, whereas only a few appearing in their results. The principal parameters that are used by the authors in their result are: $Z0$⁠, Bi, and $λ$⁠; however, the temperature is independent of Bi.

(i) In this section the authors make the following statements: (1) at any point, the temperature over the fin cross section is assumed to be constant and (2) the transport processes can be rendered one-dimensional (1D) as is conventionally done for dry fins. The authors should be reminded that, in mathematical analyses nothing is conventionally done, but it must be based on certain fundamental principles. In this case, there are certain conditions, known as simplified assumptions, described in Ref. 1, that must be met, in order to allow using the 1D conduction model. For example here, the restrictive condition for using the 1D approach is the small values of the Biot number (of the order of 0.01). As shown in Refs. 2 3, when one employs the 1D approach, the simplified assumptions, insulated tip and the length of arc idealization are valid. Thus the expression $1+(dy∕dx)2$ can be eliminated, and the tip boundary condition should be modified accordingly.

(ii) There in no explanation as to why the authors introduce the radius $ri$ in Eq. 1, which for longitudinal fins and spines is zero, and in radial fins the coordinate $x$ varies from 0 to $ri$⁠, instead of $ri⩽x⩽rt$⁠, which is universally used. In this case an important new parameter is introduced namely the ratio parameter $rt∕ri$⁠. For example, what is the value of this parameter in the results plotted in the original Fig. 4?

(iii) Now, in any mathematical equation involving symbols with dimensions, like those in Eq. (1), one should always first check if it is dimensionally correct. For longitudinal fins, the dimensions of the left-side and the first right-side terms are kelvin per meter whereas the dimensions of the last term $hfg$ are watts per square meter, with similar contradictions appearing in the other cases. I believe the authors neglected to divide this term with the thermal conductivity $k$⁠.

(iv) The authors assert that have employed the Chilton Colburn analogy and some algebraic manipulations to derive their Eq. (2). However, the temperature becomes now a function of $Y=y∕yb$ instead of $X=x∕L$⁠. This is indeed a very puzzling operation for mainly two reasons. First, according to the authors’ earlier statement the temperature along this direction is constant. In the other hand, how could the profiles of the fins, which are $y∕yb=f(X)$⁠, possibly influence the results?

At the beginning of this section the authors engage in a long and unnecessary explanation for the self-evident conceptions, of what is meant by dry fins and their performance. Then they present a few graphs, which in the abstract claim to be design curves, while they only refer to one specific case. I am indeed puzzled how these plots were contracted using those formidable equations ((4)–(20)), with all these dimensionless variables, which as was pointed out earlier, they contain several mistakes? What are exactly the differences between efficiencies and effectiveness with the authors’ over-all counterparts? I believe that their results are incorrect despite the agreement with previous work shown in the original Fig. 2. Moreover, the variation of the effectiveness versus $Z0$ in their Fig. 4 physically does not make sense. For example, when the value of $Z0=0$ (absence of the fin), by definition, the effectiveness value is equal to 1 instead of the authors’ 20. To help the authors revise their paper certain additional references are given here.

I regret to say, that due to space limitations of the journal I cannot describe the correct formulation of the problem that the authors should solve.