Consider the flow and heat transfer in a planar channel with height 2h. The plates have a constant..

Consider
the flow and heat transfer in a planar channel with height 2h. The plates have
a constant wall temperature T0 for x<= 0="" and="">W for
x>0. The velocity profile of the laminar flow in the planar duct can be
simplified to a slug flow profile with u = u_ = const: The velocity component
in the ydirection v = 0. All fluid properties can be considered to be constant.
Axial heat conduction in the flow cannot be neglected, because of small values
of the Peclet number. The temperature distribution in the fluid can be
calculated from the energy equation

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Consider
the flow and heat transfer in a planar channel with height 2h. The plates have
a constant wall temperature T0 for x<= 0="" and="">W for
x>0. The velocity profile of the laminar flow in the planar duct can be
simplified to a slug flow profile with u = u_ = const: The velocity component
in the ydirection v = 0. All fluid properties can be considered to be constant.
Axial heat conduction in the flow cannot be neglected, because of small values
of the Peclet number. The temperature distribution in the fluid can be
calculated from the energy equation

What
is the resulting differential equation and the boundary conditions in dimensionless
form?

(b)
Show that the above considered problem reduces in case of negligible axial heat
conduction in the fluid .

(c)
Use the method of separation of variables to solve the given problem

(H =
f(~x)g(~y)). What equations are obtained for the functions f and g?

(d)
Predict the complete solution for the temperature field.

(e)
Compare the obtained temperature distribution with the one of Problem 3.3 for
three different Peclet numbers (Pe = 0.1, 1, 2) for x/h = 1. What influence has
the axial heat conduction on the temperature profile? (f) Was it physically
correct to prescribe a constant temperature distribution for x = 0? Explain
your answer.

Problem
3.3

Consider
the flow and heat transfer is a planar channel with height 2h. The plates have
a constant wall temperature T0 for x < 0="" and="">W for x
0. The velocity profile of the laminar flow in the planar duct can be
simplified to be a slug flow profile with u = u_ = const: The velocity
component in the ydirection v = 0. All fluid properties can be considered to be
constant. The temperature distribution in the fluid can be calculated from the
energy equation

(a)
Introduce the following dimensionless quantities

What
are the resulting equation and the boundary condition in dimensionless form?

(b)
Use the method of separation of variables to solve the given problem

(H =
f(~x)g(~y)). What equations are obtained for the functions f and g?

(c)
Calculate the eigenfunctions and display the first and the third eigenfunction
as a function of ~y in the interval [0, 1].

(d)
Predict the complete solution for the temperature field.

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