Question 2.11: Thermal Pipe Flow ( qwall = c ) Consider Poiseuille flow (se......
Thermal Pipe Flow ( \rm{q_{wall} = \cancel c} )
Consider Poiseuille flow (see Example 2.10) where a uniform heat flux, \rm{q_w} , is applied to the wall of a pipe with radius r_o.
(A) Set up the governing equations for the fluid temperature assuming thermally fully-developed flow, i.e.,
{\frac{T_{w}-T}{T_{w}-T_{m}}}\equiv\Theta\!{\left({\frac{r}{r_{0}}}\right)}\qquad\qquad\qquad(\operatorname{E}.2.11.1)
where \rm{T_w(x)} is the wall temperature, T(r, x) is the fluid temperature, and \rm{T_m(x)} is the cross-sectionally averaged temperature, i.e.,
\mathrm{T}_{\mathrm{m}}={\frac{1}{\mathrm{\bar u A}}}\int_A\mathrm{uTdA}~~~~~~~~~~~~~~~~~~~~~~~(E.2.11.2)
Note that Θ = Θ(r) only, describing thermally fully-developed flow.
(A) Solve a reduced form of the heat transfer equation (Eq. 2.44) and develop an expression for the Nusselt number, defined as:
\begin{array}{c c}\rm{{{\frac{\partial T}{\partial t}+(\vec{v}\cdot\nabla){\mathrm{T}}=\alpha\nabla^{2}\mathrm{T}+\frac{\mu}{\rho c_{\mathrm{p}}}\Phi}}}&{{\qquad\qquad}}&{{\qquad(2.44)}}\end{array}
\mathrm{Nu}=\frac{2\mathrm{r_{0}}}{\mathrm{k}}{\frac{\mathrm{q}_{\mathrm{{w}}}}{\mathrm{T_{w}}-\mathrm{T_{m}}}}:={\frac{\mathrm{hD}}{\mathrm{k}}}{\mathrm{~\qquad}\qquad(\mathrm{E}.2.11.3\mathrm{a},\,\mathrm{b})}
where k is the fluid conductivity and D is the pipe diameter.
\mathrm{u}(\mathrm{r})={\frac{1}{4\mu}}({\frac{\Delta\mathrm{p}}{\mathrm{L}}})(\mathrm{r}_{0}^{2}-\mathrm{r}^{2})=\mathrm{u}_{\mathrm{max}}\!\left[1-({\frac{\mathrm{r}}{\mathrm{r}_{0}}})^{2}\right]\qquad\qquad(\mathrm{E}.2.10.3)
| Sketch | Assumptions | Concept |
|
|
As stated, i.e., |
Reduced heat transfer equation (Eq. (2.44)) based on assumptions |
| • u(r) as in Eq. (E.2.10.3) | ||
| • ∂T / ∂t = 0 <steady-state> | ||
| • \rm{(\vec v \cdot \nabla)}T⇒u∂T/∂x | ||
| • \rm{\alpha \nabla T\Rightarrow \frac{\alpha }{r}\frac{\partial}{\partial r} (r\frac{\partial T}{\partial r} )} |
(A) With the result of Example 2.10 and the reduced heat transfer equation in cylindrical coordinates from App. A (see also list of assumptions), we have:
\frac{{\mathrm{u(r)}}}{\alpha}\frac{\partial T}{\partial \rm x}={\frac{1}{r}}{\frac{{\partial}}{\partial r}}(r{\frac{{\partial}T}{\partial r}})\qquad\qquad\qquad({\mathrm{E.2.11.4}})
(B) Employing the dimensionless temperature profile \Theta (\frac{r}{r_0} )\equiv \Theta (\hat r) given as Eq. (E.2.11.1), we can rewrite Eq. (E.2.11.4) as
-\,2\,{\frac{\mathrm{hD}}{{\rm k}}}(1-\hat{\rm r}^{2})={\frac{\mathrm{d}^{2}\Theta}{\mathrm{d}\hat{\rm r}^{2}}}+{\frac{1}{{\hat{\rm r}}}}{\frac{\mathrm{d}\Theta}{\mathrm{d}\hat{\rm r}}}\qquad\qquad\qquad\qquad(\mathrm{E}.2.11.5)
Specifically,
• For \mathrm{q_{w}}=\cancel{c},\,\mathrm{}{\frac{\partial T}{\partial\mathrm{x}}}={\frac{\mathrm{d{T_{w}}}}{\mathrm{d}\mathrm{x}}}={\frac{\mathrm{d{T_{m}}}}{\mathrm{dx}}}={\frac{2}{\mathrm{r_{0}}}}{\frac{\mathrm{q_{w}}}{\mathrm{ρc_{0}\overline{{{u}}}}}}=\cancel{c}\,;\qquad(\mathrm{E}.2.11.6a-c)
• As stated, \mathrm{hD/k}\equiv\mathrm{Nu_{D}}:={\frac{\rm D}{\mathrm{k}}}{\frac{\rm q_{w}}{\mathrm{T_{w}}-\mathrm{T}_{m}}}=\cancel c;\qquad\qquad(\mathrm{E}.2.11.7a,b)
• And with \rm{d\Phi /d\hat r} being finite at \rm{\hat r =0}, we obtain
\rm T-\mathrm{T}_{w}=-(\mathrm{T}_{w}-\mathrm{T}_{\mathrm{m}})\mathrm{Nu}_{\mathrm{D}}\left\lgroup\frac{3}{8}-\frac{\hat{r}^{2}}{2}+\frac{\hat{r}^{4}}{8}\right\rgroup
Now, by definition
\mathrm{T_{w}-T_{m}=\frac{2\pi}{\pi\mathrm{r_{0}^2\bar u}}\int_0^{r_0}{(T_{w}-\mathrm{T})\,u(r)r d r}\,\qquad\qquad\qquad(\mathrm{E}.2.11.8)}
so that, when combining both equations and integrating, we have
1=4\mathrm{N}\mathrm{u}_{\mathrm{D}}\int_0^1{\left\lgroup{\frac{3}{8}}-{\frac{\hat{\mathrm{r}}^{2}}{2}}+{\frac{\hat{\mathrm{r}}^{4}}{8}}\right\rgroup}(1-\hat{\mathrm{r}}^{2}){\hat{\mathrm{r}}}\,\mathrm{d}\hat{\mathrm{r}}
from which we finally obtain:
\mathrm{Nu}_{\mathrm{D}}=\frac{48}{11}=4.36\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\qquad(E.2.11.9)
Comment: It is interesting to note that for hydrodynamically and thermally fully-developed flow in a tube, subject to a constant wall heat flux, the Nusselt number (or the heat transfer coefficient) is constant. The same holds for the isothermal wall condition; however, the Nu-value is lower (see Kleinstreuer 1997; or Bejan 2002).