Comment on Rooman et al. Entropy Optimization on Axisymmetric Darcy–Forchheimer Powell–Eyring Nanofluid over a Horizontally Stretching Cylinder with Viscous Dissipation Effect. Coatings 2022, 12, 749

Introduction

The concentration shown in Equation (4) in [1] is given as follows:

$$w{\displaystyle \frac{\partial C}{\partial r}}+u{\displaystyle \frac{\partial C}{\partial z}}=D_B\left({\displaystyle \frac{{\partial }^{2}C}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial C}{\partial r}}\right)+{\displaystyle \frac{D_T}{T_{\infty }}}\left({\displaystyle \frac{{\partial }^{2}T}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial T}{\partial r}}\right)$$

(1)

In a physics equation, all terms must have the same units. The above equation in units is given as follows:

$$\begin{array}{l}w(m{s}^{-1}){\displaystyle \frac{\partial C(\dim \mathit{ensionless})}{\partial r(m)}}+u(m{s}^{-1}){\displaystyle \frac{\partial C(\dim \mathit{ensionless})}{\partial z(m)}}=\\ D_B(m^2{s}^{-1})\left({\displaystyle \frac{{\partial }^{2}C(\dim \mathit{ensionless})}{\partial r^2{(m)}^2}}+{\displaystyle \frac{1}{r(m)}}{\displaystyle \frac{\partial C(\dim \mathit{ensionless})}{\partial r(m)}}\right)\\ +{\displaystyle \frac{D_T(m^2{s}^{-1})}{T_{\infty }(K)}}\left({\displaystyle \frac{{\partial }^{2}T(K)}{\partial r^2{(m)}^2}}+{\displaystyle \frac{1}{r(m)}}{\displaystyle \frac{\partial T(K)}{\partial r(m)}}\right)\end{array}$$

(2)

$s^{-1}+s^{-1}=s^{-1}+s^{-1}$

Equation (2) and, consequently, Equation (1) are correct only when the concentration C is dimensionless.

The entropy generation Equation (16) in [1] is as follows:

$$\begin{array}{l}{S}_g={\displaystyle \frac{k_{nf}}{T_{\infty }^2}}\left(1+{\displaystyle \frac{16{\sigma }^{*}{T}^3}{3k^{*}{k}_{nf}}}\right){\left({\displaystyle \frac{\partial T}{\partial r}}\right)}^2+{\displaystyle \frac{1}{T_{\infty }}}\left({\mu }_{nf}{\left({\displaystyle \frac{\partial u}{\partial r}}\right)}^2+{\displaystyle \frac{1}{\beta c}}{\left({\displaystyle \frac{\partial u}{\partial r}}\right)}^2-{\displaystyle \frac{1}{6\beta c^3}}{\left({\displaystyle \frac{\partial u}{\partial r}}\right)}^4\right)\\ +{\displaystyle \frac{RD}{T_{\infty }}}\left({\displaystyle \frac{\partial T}{\partial r}}{\displaystyle \frac{\partial C}{\partial r}}\right)+{\displaystyle \frac{RD}{C_{\infty }}}{\left({\displaystyle \frac{\partial C}{\partial r}}\right)}^2\end{array}$$

(3)

In Equation (3), $R(m)$ is the cylinder radius (Equation (6) in [1]) and $D(m^2{s}^{-1})$ is a diffusion coefficient.

The units of the term ${\displaystyle \frac{k_{nf}}{T_{\infty }^2}}{\left({\displaystyle \frac{\partial T}{\partial r}}\right)}^2$ are given as follows:

$${\displaystyle \frac{k_{nf}(kgm{\mathrm{s}}^{-3}{K}^{-1})}{T_{\infty }^2{(K)}^2}}{\left({\displaystyle \frac{\partial T(K)}{\partial r(m)}}\right)}^2=kg{m}^{-1}{s}^{-3}{K}^{-1}$$

(4)

The units of the term ${\displaystyle \frac{1}{T_{\infty }}}\left({\mu }_{nf}{\left({\displaystyle \frac{\partial u}{\partial r}}\right)}^2\right)$ are as follows:

$${\displaystyle \frac{1}{T_{\infty }(K)}}\left({\mu }_{nf}(kg{m}^{-1}{s}^{-1}){\left({\displaystyle \frac{\partial u(m{s}^{-1})}{\partial r(m)}}\right)}^2\right)=kg{m}^{-1}{s}^{-3}{K}^{-1}$$

(5)

The units of the term ${\displaystyle \frac{RD}{T_{\infty }}}\left({\displaystyle \frac{\partial T}{\partial r}}{\displaystyle \frac{\partial C}{\partial r}}\right)$ are as follows:

$${\displaystyle \frac{R(m)D(m^2{s}^{-1})}{T_{\infty }(K)}}\left({\displaystyle \frac{\partial T(K)}{\partial r(m)}}{\displaystyle \frac{\partial C(\dim \mathit{ensionless})}{\partial r(m)}}\right)=m{s}^{-1}$$

(6)

The units of the term ${\displaystyle \frac{RD}{C_{\infty }}}{\left({\displaystyle \frac{\partial C}{\partial r}}\right)}^2$ are as follows:

$${\displaystyle \frac{R(m)D(m^2{s}^{-1})}{C_{\infty }(\dim \mathit{ensionless})}}{\left({\displaystyle \frac{\partial C(\dim \mathrm{ensionless})}{\partial r(m)}}\right)}^2=m{s}^{-1}$$

(7)

It is clear that the units of the terms in Equation (3) are different, and for that reason, entropy in Equation (3) is wrong, and all results concerning entropy in [1] are incorrect.