1.1 Thermal
Steady-state one-dimensional heat transfer equation
\[\begin{split}v\frac{dT}{dx} = \frac{1}{c} \biggl(
\underbrace{\dot{q}}_{\substack{\text{through}\\\text{wall}}} +
\underbrace{\frac{\lambda}{d} \frac{v^2 |v|}{2}}_{\substack{\text{Dissipation}\\\approx\,0}} \biggr)\end{split}\]
\(x\) flow direction
\(v(r)\) velocity
\[\begin{split}\begin{split}
v\frac{dT}{dx}
&= \frac{\dot{q}}{c}
= \frac{1}{c} \frac{\dot{Q}}{m} \\
&= \frac{1}{c} \frac{\left(\lambda_w \, 2\pi rL \frac{\partial T}{\partial r}\right)_a -
\left(\lambda_w \, 2\pi rL \frac{\partial T}{\partial r}\right)_i}{\rho\pi(r_a^2-r_i^2)L} \\
&= \frac{1}{\rho \, c}\frac{2}{r_a+r_i}
\frac{\left(\lambda_w \, r \frac{\partial T}{\partial r}\right)_a -
\left(\lambda_w \, r \frac{\partial T}{\partial r}\right)_i}{r_a-r_i} \\
&= \lim_{\substack{r_a \to r^+ \\ r_i \to r^-}} (\ldots) \\
&= \frac{1}{\rho \, c}\frac{1}{r}
\frac{\partial}{\partial r} \left(\lambda_w \, r \frac{\partial T}{\partial r}\right) \\
&= \frac{1}{r}
\frac{\partial}{\partial r} \left(\frac{\nu_w \, r}{\mathrm{Pr}} \frac{\partial T}{\partial r}\right)
\quad\text{with}\quad
\mathrm{Pr} = \frac{\nu \, \rho \, c}{\lambda}
\end{split}\end{split}\]
with Hagen-Poiseuille law of an incompressible and Newtonian fluid in laminar
flow flowing through a long cylindrical pipe of constant cross section
\[\begin{split}\begin{split}
v(r) &= \frac{1}{4\mu}\frac{\Delta P}{\Delta x}(R^2-r^2) \\
v(r) &= \frac{2\dot{V}}{\pi R^4} (R^2 - r^2)
\quad\text{with}\quad
\Delta P=\frac{8\mu L \dot{V}}{\pi R^4}
\end{split}\end{split}\]
\(\dot{V}\) volumetric flow rate
\(\Delta P\) pressure difference between the two ends of the pipe
\(\Delta x = L\) length of the pipe
\(R\) radius of the wall, the inside pipe radius
\(\mu\) dynamic viscosity
\(v(r=R) = 0\) “no-slip” boundary condition at the wall
\(\left.\frac{dv}{dr}\right|_{r=0} = 0\) axial symmetry
\[\begin{split}
\frac{2\dot{V}}{\pi R^4} (R^2 - r^2)\frac{dT}{dx}
&= \frac{1}{r}
\frac{\partial}{\partial r} \left(\frac{\nu_w \, r}{\mathrm{Pr}} \frac{\partial T}{\partial r}\right)
\end{split}\]
twice integration plus boundary conditions
\[T(r) = T_0 - \frac{2 \dot{V} \mathrm{Pr}}{\pi R^2 \nu}
\left( \frac{1}{4}r^2 - \frac{1}{16}\frac{r^4}{R^2} \right) \frac{dT}{dx}\]
Reynolds number
\[\mathrm{Re} = \frac{\rho \bar{v}d}{\mu} = \frac{\bar{v}d}{\nu}
= \frac{2\dot{V}}{\pi R \nu}\]
\(d = 2R\) characteristic linear dimension. For a circular pipe it is the inside pipe diameter
\(\bar{v} = v_{max}/2\) mean flow velocity, which is half the maximal flow velocity in the case of laminar flow
\[T(r) = T_0 - \mathrm{Re}\,\mathrm{Pr}\,R
\left( \frac{1}{4}\frac{r^2}{R^2} - \frac{1}{16}\frac{r^4}{R^4} \right) \frac{dT}{dx}\]