MrsDrPoe: The First Law of Thermodynamics, Part 6

Hello and welcome to Thesis Tuesday on the blog!  As promised, today we will be finishing our look at the first law of thermodynamics by applying the differential form of the first law to an example problem.

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
Solution: 


Starting with the general form of the differential energy equation for an incompressible Newtonian fluid:

rho*cp*(d/dt(T) + u*d/dx(T) + v*d/dy(T) + w*d/dz(T)) = -(d/dx(qx) + d/dy(qy) + d/dz(qz)) + mu*Phiv

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.

rho*cp*(u*d/dx(T) + v*d/dy(T) + w*d/dz(T)) = -(d/dx(qx) + d/dy(qy) + d/dz(qz))

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:

0 = (d/dx(qx) + d/dy(qy) + d/dz(qz))

The flux terms can be related to the temperature gradient using Fourier's law:

qx = k*d/dx(T), qy = k*d/dy(T), and qz = k*d/dz(T)

Furthermore, since the thermal conductivity (k) is constant, our energy equation becomes:

0 = k*(d/dx(d/dx(T)) + d/dy(d/dy(T)) + d/dz(d/dz(T)))

We can divide both sides by k; the first and third terms cancel since T is not a function of x or z:

0 = d/dy(d/dy(T))

If we separate and integrate this equation twice, we end up with the expression:

T(y) = c1*y + c2

We can solve for c1 and c2 by applying the given boundary conditions at 0 and h:

c2 = T0 and c1 = (Th - T0)/h

So we now know the expression that shows us the temperature distribution in the fluid film:

T(y) = (Th - T0)/h*y + T0

Not so bad, huh?  Next week we'll begin anew with a fresh subject, but until then, happy studying!