Problem 1: Center-line Vaporization
a) A string of negligible diameter is drawn down the center-line of a pipe containing a Newtonian fluid. At what velocity, w, must the string be drawn in order to vaporize the fluid? Pose the problem in a general fashion but make whatever reasonable simplifying assumptions are necessary to actually come up with a value (or order-of-magnitude) for w. Consider both viscous heating and cavitation, use scaling and discuss whether one or both are dominant in this system given your simplifying assumptions.
b) If the string is accelerated linearly at rate, a(t), to velocity w, what can you say (mathematically) about the rate of acceleration versus heat conduction/convection versus the heat of the string? Obviously this problem will involve some scaling and other innovative approaches to get at an answer.
Problem 2: Icicle Melting/Formation
As an icicle melts a very thin layer of water of thickness, L(z), runs down the surface. Heat transfer from the air at temperature, Ta, to the water is characterized by a constant heat transfer coefficient, h. The air temperature is only slightly above the freezing temperature of water, Tf .
a) What simplifying assumption pertaining to the geometry and dimensions in this system can you make?
b) Determine the m ean water velocity, V (z) in terms of L(z), ρw, µw and gravity. Assume that the velocity in the water film is parabolic in the radial direction.
c) Relate dL dz to the local melting rate, in terms of the surface normal water velocity, vm(z), at the ice-water interface.
d) Relate the local melting rate to the heat transfer coefficient, the reference temperatures, the heat of fusion; assuming the water is nearly isothermal.
e) From parts (b) and (d), determine V (z) and L(z).
f) What must be true for parts (b) and (d) to be valid? Namely, when will the velocity profile be parabolic in r? When can you neglect the temperature gradient in the r-direction? When can you neglect the temperature gradient in the z-direction, even if it is not negligible in the r-direction
Problem 3: Condensate Film Formation
A pure vapor at its saturation temperature, Ts, is brought into contact with a vertical wall that is held at some cooler temperature, Tw. The vapor condenses as a film of thickness, δ(x), which flows down the wall. The mean downward (x-direction) velocity of the condensate is U(x). Far from the wall, the vapor is stagnant and assume steady state has been established.
a) Briefly explain why the condensation rate is nearly independent of the gas density, viscosity and thermal conductivity; ρg, µg and κg, respectively.
b) Develop a relationship between U(x) and δ(x), assuming a parabolic film flow pro-file, vx(x, y).
c) Relate dδ dx to U(x) and the local rate of condensation; which is defined by the liquid velocity normal to the liquid-vapor interface.
d) Apply an energy balance to relate the local rate of condensation to the latent heat and other parameters. Clearly state your simplifying assumptions.
Problem 4: Nusselt and Brinkman
This system has two parallel plates at different temperatures: top plate (y = H) is T2, bottom plate (y = -H) is T1 and T2 > T1. Fluid enters the system (x = 0) at temperature T0. Assume fully developed, Newtonian flow with constant physical properties. DO NOT assume the Brinkman number is small!
a) Determine the temperature profile for large x (i.e. far away from the inlet), including viscous dissipation effects.
b) Evaluate Nu at y = H and y = -H as a c) Under what conditions is Nu < 0 at y = H? Write a brief description as to why this can happen and by sketching T(y)