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
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
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
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