Problem: A liquid is flowing downward along an inclined plane surface, as shown in the figure:
The free liquid surface (y = h) is maintained at temperature Th, and the solid surface (y = 0) is maintained at To. Determine an expression for the temperature distribution in the film, recognizing that the viscous heating effects can be ignored.
Given: non-isothermal film flow, T(0) = T0 and T(h) = Th
Find: An expression for the temperature distribution in the fluid film
Assumptions: steady, laminar, incompressible, Newtonian fluid, ignore viscous heating effects, constant thermal conductivity, assume T = f(y), constant properties
Starting with the general form of the differential energy equation for an incompressible Newtonian fluid:
Steady flow eliminates the first term on the left side; the last term on the right side is eliminated because we are ignoring viscous heating effects.
Our assumption of laminar flow means that v = w = 0; the first term on the left side is also zero since T is in not a function of x:
The flux terms can be related to the temperature gradient using Fourier's law:
Furthermore, since the thermal conductivity (k) is constant, our energy equation becomes:
We can divide both sides by k; the first and third terms cancel since T is not a function of x or z:
If we separate and integrate this equation twice, we end up with the expression:
We can solve for c1 and c2 by applying the given boundary conditions at 0 and h:
So we now know the expression that shows us the temperature distribution in the fluid film:
Not so bad, huh? Next week we'll begin anew with a fresh subject, but until then, happy studying!