## Q. 3.19

Thermal Pipe Flow with Entropy Generation
Deriving the irreversibility profiles for Hagen–Poiseuille (H–P) flow through a smooth tube of radius $\rm r_0$ with uniform wall heat flux q [W/m²] at the wall, the velocity and temperature for fullydeveloped regime are given by:

$\rm \mathrm{u}=2\mathrm{U}[1-({\frac{r}{r_{0}}})^{2}]$                                    (E.3.19.1)

and

$\mathrm{T}-\mathrm{T}_{\mathrm{s}}=-{\frac{\mathrm{qr}_{0}}{\mathrm{k}}}[{\frac{3}{4}}-({\frac{\mathrm{r}}{\mathrm{r}_{0}}})^{2}+{\frac{1}{4}}({\frac{\mathrm{r}}{\mathrm{r}_{0}}})^{4}]$                                    (E.3.19.2)

${\rm S_{gen}\equiv S_G=\frac{k}{T^2}\left[(\frac{\partial T}{\partial x} )^2+(\frac{\partial T}{\partial y} )^2+(\frac{\partial T}{\partial z} )^2\right]} \\\qquad\qquad{\rm +\frac{\mu}{T}\left\lgroup2\left[(\frac{\partial u}{\partial x} )^2+(\frac{\partial v}{\partial y} )^2+(\frac{\partial w}{\partial z} )^2\right] \right. +(\frac{\partial u}{\partial y} +\frac{\partial v}{\partial x} )^2} \\ \qquad\qquad{\rm\left. +(\frac{\partial u}{\partial z} +\frac{\partial w}{\partial x} )^2+(\frac{\partial v}{\partial z} +\frac{\partial w}{\partial y} )^2 \right\}}$                             (3.95)
 Conceptions Assumptions Sketch • Volumetric entropy generation rate Eq. (3.95) • Fullydeveloped HP flow • Thermal entropy generation • Constant properties and parameters • Frictional entropy generation

## Verified Solution

The wall temperature $\rm T_s = T(r = r_0 )$ can be obtained from the condition

$\rm \frac{\partial T}{\partial x} =\frac{\partial T_s}{\partial x} =\frac{2q}{\rho c_pUr_0}$ = constant                                       (E.3.19.3)

$\rm\frac{\partial T}{\partial r} =\frac{q}{k} [2-\frac{r}{r_0} -(\frac{r}{r_0} )^3]$                                      (E.3.19.4)

$\rm\frac{\partial u}{\partial r} =\frac{-4Ur}{r_0^2}$                                          (E.3.19.5)

Hense, the dimensionless entropy generation for fully-developed tubular H–P flow can be expressed as:

${\rm \mathrm{S}_{\mathrm{gen}}{\frac{\mathrm{kT_{0}}^{2}}{\mathrm{q^{2}}}}={\frac{4\mathrm{k}^{2}}{(\mathrm{pc_{p}}\overline{{{\mathrm{U}}}}r_{0})^{2}}}{\frac{\mathrm{T_{0}}^{2}}{\mathrm{T^{2}}}}+(\mathrm{2R-\mathrm{R}^{3})^{2}}{\frac{\mathrm{T_{0}}^{2}}{\mathrm{T^{2}}}}+{\frac{\mathrm{16kT_{0}}^{2}\mathrm{\mu}\overline{{{\mathrm{U}}}}^{2}}{\mathrm{q}^{2}Tr_0^2}}^{}R^2}\\\qquad\qquad{\underbrace{=\frac{16}{P e^{2}}(\frac{T_{0}}{T})^{2}+(2R-R^{3})^{2}\frac{T_{0}^2}{T^{2}}}_{\text{heat transfer}}+\underbrace{\phi\frac{T_{0}}{T}R^{2}}_{\text{fluid friction}}}$                (E.3.19.6)

with

$\mathrm{Pe}=\mathrm{Re}\cdot\mathrm{Pr}={\frac{2\mathrm{r_{0}}\mathrm{\rho c_p}\mathrm{U}}{{\mathrm{k}}}}$                                      (E.3.19.7)

and

$\rm R=\frac{r}{r_{0}},\,\mathrm{and}\,\,\phi=\frac{16{k}\,\mathrm{T_{0}\mu}{U_{}}^{2}}{{ q}^{2}\,{ r_{0}}^{2}}$                              ( E.3.19.8a, b)

Here, R is the dimensionless radius, $\rm T_0$ is the inlet temperature which was selected as the reference temperature. On the right side of the Eq. (E.3.19.6), the first term represents the entropy generation by axial conduction, the second term is the entropy generated by heat transfer in radial direction, and the last term is the fluid friction contribution. Parameter $\phi$ , Eq. (3.19.8b), is the irreversibility distribution ratio $(\frac{S_{\rm gen}(fluid~friction)}{S_{\rm gen}(heat ~transfer)} ).$

Graph:

Comments: As expected, according to Eq. (E.3.19.6), at the center point, i.e., R = 0, only the first term in the right side contributed to the dimensionless entropy generation rate; however, for Pe >> 1, the irreversibility due to axial conduction is negligible in the fully developed range. In contrast, in the wall region both thermal and frictional effects produce entropy with a maximum at R ≈ 0.8 generated by dominant heat transfer induced entropy generation. 