University of Illinois, Urbana Champaign

CHBE 421

ChBE 421 The FINAL Exam
1 (70 pts) Answer briefly.
(a) The material derivative D /Dt is defined as ∂ /∂t + u ∇ .
What is the physical significance of this derivative?
(b) A solid object with length scale L and density ρs falls under the action of gravity.
How does its velocity change if the size L is doubled while the density remains the same?
Assume the drag coefficient doesn’t change.
(c) Sketch the velocity profile for an irrotational vortex. If a vertical cylinder is placed in a large tank
of water and rotates with angular velocity Ω, sketch the shape of the free surface.
(d) For a laminar boundary layer on a flat plate, the boundary layer thickness is given by
δ ∼ (ν x/U )1/2. How does the force on the plate change if the velocity is doubled? Give a
precise answer and include an equation or two to justify your answer?
If experiments showed that the force increased by more than this amount, how would you explain?
(e) A metal pipe wrapped with a layer of insulation is carrying condensing steam at temperature T0 in
a room at temperature T∞. If the thickness of the insulation is increased causing q to decrease, how
does the temperatures change (i) at the outside metal surface of the pipe,
(ii) at the outside surface of the insulation. You may assume hi and ho are constant.
(f) In heat transfer by conduction, the heat transfer rate is directly proportional to the thermal
conductivity k. In forced convection, the heat transfer rate increases with increasing k, but not as
fast. Explain why.
(g) In natural convection, correlations are usually found in the form . Nu ∼ Pr1/4 Gr1/4
For a vertical square plate of side L, how does the heat transfer rate (e.g. kcal/hr) change if the size
of the plate is doubled (L → 2L) ?
2 (30 pts) A large bronze winged figure acts as the nozzle for the center jet of a public fountain.
Unfortunately, the engineer for the department of public monuments forgot to bolt the figure down, and
the force of the water blew it off its pedestal. If the figure weighs 150 lbf, calculate the height of the
fountain jet just before the figure took off.
Nozzle dimensions: entrance diameter 4 in., exit diameter 2 in.
Neglect all frictional losses. Neglect changes in potential energy over the length of the nozzle.
Hint: you may need to consider two separate control volumes for this problem.
ρH20 = 62.4 lbm/ft3 1 lbf = 32 lbm ft/sec