Advanced Heat Transfer---Mid-Exam 2 05/13/2009
1. Consider a liquid metal (Pr<<1), with constant freestream conditions u∞ and T∞, in
parallel flow over a flat plate. Assume the flow to be steady, two-dimensional,
laminar, and incompressible, and the fluid properties to be constant. The plate
temperature varies in the flow direction according to
Tw  T  Ax n
where A is a constant and n is a positive index ( n  1) and x is the axial coordinate
measured from the leading edge of the plate.
(a) Assuming that u= u ∞ throughout the thermal boundary layer, write the
boundary layer energy equation and the appropriate initial and boundary
conditions describing the temperature field in the flow.
(5%)
(b) Show that the dimensionless temperature, θ, is a function of the similarity
variable η only, where
T  T

and   y u / x
Tw  T
In the above expression, y is the vertical coordinate measured upward from the
surface of the plate and α the thermal diffusivity of the fluid.
(18%)
(c) Show that the local Nusselt number is given by
Nu x   (0) Re 1x/ 2 Pr1 / 2
where
 d 
(8%)
 (0)   
d

   0
(d)
(e)
(f)
(g)
(h)
Is  (0) a positive or negative quantity?
(2%)
Does  (0) depend on A?
(3%)
Does local convective heat transfer coefficient, hx, depend on A?
(3%)
"
Is the local surface heat flux, q w , dependent on A?
(6%)
Perform an order-of-magnitude analysis to show that the local thermal
boundary layer thickness would vary according to
 t ~ x Re x1/ 2 Pr 1 / 2
where
u x
Re x  

is the local Reynolds number.
(10%)
1
2. Consider steady, incompressible, parallel, laminar flow of a viscous fluid falling
between two infinite vertical walls, as shown in Fig. 1. The distance between the
walls is h, and the gravity acts in the negative z-direction. There is no applied
pressure driving the flow; that is, the fluid falls by gravity alone. The pressure is
constant everywhere in the flow field. The left wall is kept at constant temperature,
T1 and the right wall is maintained at constant temperature, T2. Assume that
viscous dissipation cannot be neglected.
(1) Calculate the velocity field.
(15%)
(2) Please find the temperature distribution between the walls.
(12%)
(3) Please find the heat flux to each of the walls.
(8%)
(4) Please find the location which maximum temperature occurs.
(10%)
Fig. 1
2