CL: Film Theory

Near a solid boundary it is often assumed that there is a "film" of fluid that is slower than the bulk flow where we might further assume that the velocity varies linearly with position:

For reasons that will be discussed in the next section, this means that the shear stresses on the solid are constant and may be written as:

$\displaystyle{\tau = \frac{F_s}{A} = \mu\frac{U}{\delta}}$

which can be rearranged to give:

$\displaystyle{\delta = \mu\frac{U}{\tau}}$

Recalling that the definition of the friction factor was:

$\displaystyle{f = \frac{\tau}{\frac{1}{2}\rho U^2}}$

means that we can write the film thickness, scaled by some length, L, as:

$\displaystyle{\frac{\delta}{L} = \frac{2}{fRe}}$

As discussed previously, we can write the heat flux due to convection as a product of the heat transfer coefficient and the driving force. For reasons that will be discussed in the next section, if we assume a linear temperture profile within a thermal film (see picture) we can write:

$\displaystyle{\frac{q}{A} = k\frac{\Delta T}{\delta '} = h\Delta T}$

This shows that the "film coefficient" name for the heat transfer coefficient comes from the fact that, using film theory, it can be related to the thermal film thickness and fluid properties by:

$\displaystyle{h = \frac{q/A}{\Delta T}}$

Making an analogy to the friction factor, we can define a dimensionless heat flux as:

$\displaystyle{Nu = \frac{q/A}{k\left(\Delta T/L \right)}}$

which can be combined with Newton's law of cooling to give:

$\displaystyle{Nu = \frac{h\Delta T}{k\left(\Delta T/L \right)} = \frac{hL}{k}}$

Using the film theory definition of h, we can get:

$\displaystyle{Nu = \frac{L}{\delta '} = \frac{L}{\delta}\left (\frac{\delta}{\delta '}\right ) = \frac{fRe}{2}\left ( \frac{\delta}{\delta '}\right )}$

As we will see later in the course, the ratio of film thicknesses can be related to the material properties for heat transfer, through the Prandtl number, Pr, to give:

$\displaystyle{Nu = \frac{fRe}{2}\left ( \frac{\nu}{\alpha}\right )^{1/3} = \frac{f}{2}Re Pr^{1/3}}$

where the factor of 1/3 will be explained later in the course.

This expression is a useful generic formula for the convective heat transfer coefficient that can be used when no better correlation is available (provided you have a decent correlation for f). One further aid to correlations was devised by Colburn, where a "j-factor" has been defined as:

$\displaystyle{j_H = \frac{Nu}{Re Pr^{1/3}}}$

so that a direct relation to the friction factor is given as:

$\displaystyle{j_H = \frac{f}{2}}$

NOTE:

Using the above analysis, one can use correlations for Nu directly, j-factor correlation directly, or film theory along with correlations for f to yield h values!

OUTCOME:

Use film theory and other correlations to obtain h