Generalised Analysis of Heat and Mass Transfer through Boundary Layers

The particular results obtained for the laminar boundary layer over a flat plate at zero incidence may be written

\begin{displaymath}Nu_{x} = \frac{h_{x} x}{k} = Pr^{\frac{1}{3}}.f\left( Re_{x} \right)
\end{displaymath} (1)

and

\begin{displaymath}Sh_{x} = \frac{k_{c x} x}{D_{AB}} = Sc^{\frac{1}{3}}.f\left( Re_{x} \right)
\end{displaymath} (2)

These results, in this generalised form, are found to apply to a very wide range of heat and mass transfer situations, for forced convection transfer to or from surfaces curved or flat, parallel or inclined, to turbulent or laminar flows, where x is a characteristic dimension of the system.

The following examples are given for heat transfer but would apply equally well to mass transfer

1.
For laminar flow through long smooth tubes, having diameter d as the characteristic dimension, h far from the entrance, is found semi-empirically to be given by:

\begin{displaymath}Nu_{d} = \frac{h d}{k} = 0.040Re_{d}^{0.75}Pr^{\frac{1}{3}}
\end{displaymath}

2.
For turbulent flow through long smooth tubes, having diameter d as the characteristic dimension, h far from the entrance, is found empirically to be given by:

\begin{displaymath}Nu_{d} = \frac{h d}{k} = 0.023Re_{d}^{0.8}Pr^{\frac{1}{3}}
\end{displaymath}

3.
For flow through packed beds of spheres, diameter d, by experiment

\begin{displaymath}Nu_{d} = \frac{h d}{k} = 1.82Re_{d}^{0.49}Pr^{\frac{1}{3}} {\rm\ \ \
for \ \ \ } Re_{d} < 350
\end{displaymath}


\begin{displaymath}Nu_{d} = \frac{h d}{k} = 0.989Re_{d}^{0.59}Pr^{\frac{1}{3}} {\rm\ \ \
for \ \ \ } Re_{d} > 350
\end{displaymath}

4.
For turbulent flow over a flat plate

\begin{displaymath}Nu_{x} = \frac{h x}{k} = 0.037Re_{x}^{0.8}Pr^{\frac{1}{3}}
\end{displaymath}

Dividing equation (1) by RexPr

\begin{displaymath}\frac{Nu_{x}}{Re_{x}Pr} = \frac{\frac{h x}{k}}{\frac{\rho u_{...
..._{p}} =
Pr^{-\frac{2}{3}}\frac{f\left( Re_{x} \right)}{Re_{x}}
\end{displaymath}

and

\begin{displaymath}\frac{h}{\rho u_{o} c_{p}} Pr^{\frac{2}{3}} = F\left( Re_{x} \right)
\end{displaymath}

ie

\begin{displaymath}j_{H} = F\left( Re_{x} \right)
\end{displaymath}

the Chilton-Colburn j-factor for heat transfer which is a function of Rex only.

\begin{displaymath}j_{H} \equiv \frac{h}{\rho u_{o} c_{p}} Pr^{\frac{2}{3}}
\end{displaymath}

- a dimensionless heat transfer coefficient, a function only of Re, the form of the function depending on shape and kind of flow system.

Dividing equation (2) by RexSc

\begin{displaymath}\frac{Sh_{x}}{Re_{x}Sc} = \frac{\frac{k_{c x} x}{D_{AB}}}{\fr...
..._{o}} =
Sc^{-\frac{2}{3}}\frac{f\left( Re_{x} \right)}{Re_{x}}
\end{displaymath}

and

\begin{displaymath}\frac{k_{c x}}{u_{o}} Sc^{\frac{2}{3}} = F\left( Re_{x} \right)
\end{displaymath}

ie

\begin{displaymath}j_{D} = F\left( Re_{x} \right)
\end{displaymath}

the Chilton-Colburn j-factor for mass transfer which is a function of Rex only.

\begin{displaymath}j_{D} \equiv \frac{k_{c x}}{u_{o}} Sc^{\frac{2}{3}}
\end{displaymath}

- a dimensionless mass transfer coefficient, a function only of Re, the form of the function depending on shape and kind of flow system.

The exact solution for the boundary layer over flat plates gave

\begin{displaymath}Nu_{x} = 0.332 Re_{x}^{\frac{1}{2}}Pr^{\frac{1}{3}}
\end{displaymath}


\begin{displaymath}Sh_{x} = 0.332 Re_{x}^{\frac{1}{2}}Sc^{\frac{1}{3}}
\end{displaymath}

and

\begin{displaymath}\frac{\tau_{o}}{\rho u_{o}^{2}} = \frac{c_{f}}{2} =
0.332Re_{x}^{-\frac{1}{2}}
\end{displaymath}

Thus for the laminar boundary layer over a flat plate

\begin{displaymath}j_{H} \equiv \frac{h}{\rho u_{o} c_{p}} Pr^{\frac{2}{3}}
= j_...
...c^{\frac{2}{3}} =
0.332Re_{x}^{-\frac{1}{2}} = \frac{c_{f}}{2}
\end{displaymath}

This is the complete form of the Chilton-Colburn analogy, relating all three forms of transport in one expression. The equation is exact for laminar boundary layers over flat plates and is satisfactory for systems of other geometries provided no form drag is present. For systems with form drag it has been found that

\begin{displaymath}j_{H} = j_{D} \neq \frac{c_{f}}{2}
\end{displaymath}

ie

\begin{displaymath}\frac{h}{\rho u_{o} c_{p}} Pr^{\frac{2}{3}} = \frac{k_{c x}}{u_{o}}
Sc^{\frac{2}{3}}
\end{displaymath}

This equation, relating convective heat and mass transfer, is valid for gases and liquids within the ranges, 0.6 < Sc < 2500 and 0.6 < Pr < 100


Next : Dimensional Analysis


David Balmer
Last modified: Wed Dec 2 16:19:47 GMT