Show that for fully developed laminar flow between two flat plates spaced 2a apart, the Nusselt number based on the ‘bulk mean’ temperature and the passage spacing is 4.12 if the temperature of both walls varies linearly with the distance x, i.e., ?T/?x = C. The ‘bulk mean’ temperature is defined as

Question

GIVEN

• Fully developed laminar flow between two flat plates

• Spacing = 2a & ?T/?x = C

• Bulk mean temperature as defined above

FIND

• Show that the Nusselt number based on the bulk mean temperature = 4.12

ASSUMPTION

• Steady state

• Constant and uniform property values

• Fluid temperature varies linearly with x (This corresponds to a constant heat flux boundary)

SKETCH

Answer 1

The solution will progress as follows

1. Derive the temperature distribution in the fluid.

2. Use the temperature distribution to obtain an expression for the bulk mean temperature.

3. Use the bulk mean temperature to derive the Nusselt number.

Beginning with the laminar flow energy equation of



v = component of the velocity in the Y direction = 0



Note that ?T/?x = constant by assumption

The velocity profile u(y) must be substituted into this equation before the equation can be solved for

the temperature distribution. The velocity profile can be derived by considering a differential element

of fluid of width w as shown below



A force balance on this element yields



Since the flow is fully developed



Integrating with respect to y



This is subject to the following boundary conditions



This is subject to the following boundary conditions



Substituting this into the energy equation



Subject to the boundary conditions





Applying the first boundary condition, C1 = 0

Integrating again



Applying the second boundary condition



Therefore, the temperature distribution is



The bulk mean temperature is defined as



Solving the numerator of this expression



The denominator is



The rate of heat transfer is given by