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 (40 pts) 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 mean 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 (40 pts) 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 (30 pts) 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 function of the Brinkman number (Br).
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). Problem 5 (50 pts) Beer Bubbles After this exam, you may likely find yourself at the local bar determined to not think of the subject again. Looking at the CO2 bubbles rising in your glass you regrettably are now stuck pondering the mass and momentum transport dynamics occurring in your beer!
a) Using stream and potential functions, show that the liquid velocity is given by: vr = Ucosθ
vθ = −Usinθ
where R and U are the bubble radius and constant bubble velocity in the stagnant beer, respectively. Assume Stoke’s equation is valid (creeping flow). State what conditions must be satisfied with respect to mass transfer of CO2 to and from the gas/liquid phasesuch that the Strouhal number gives the condition of psuedo-steady state behavior (i.e. R is more-or-less constant). Assume that CO2 in equilibrium with the gas bubble and far from the bubble are C0 and C∞, respectively. Assume that Re â‰ª 1 in both phases and that the beer viscosity is much greater than that of the CO2 gas in the bubble. Assume that Sc is large so that P e = ReSc â‰« 1.
b) Derive the local average Sherwood number for the beer, for P e −→ ∞ and show thatSh¯ ' P e1/2
This result was shown by Levich, See Levich, V.G. ”Physiochemical Hydrodynamics.” Prentice-Hall, Englewood Cliffs, NJ, 1962 c) Unrelated to previous Let’s say you’re an English Draught drinker. Explain in 1 or 2 concise sentences why the bubbles fall downward in your glass. Problem 6, (50 pts) Expansion Perturbation for Flow in an Ill-Aligned Annular Region Going back to flow in an annular region defined by an off-center rod of radius Ro inside of a pipe of radius κRo. Where the rod is off-center in x by , that is the pipe has origin x = 0, y = 0 but the rod is centered about x = , y = 0. The surface of the pipe should be obvious, referenced to that origin, the surface of the rod is given by: r = R(θ, ) (4) where any point on the surface of the rod can be given by: [x − ] 2 + y 2 = R 2 o (5) a) Upon substituting in for r, show that the following equation is obtained R 2 − 2Rcos(θ) = R 2 o − 2 (6) b) Upon defining your mapping as θ = θo (7) and R = R(θ, ) = Ro + R1(θo) + 1 2 2R2(θo)+..... (8) What are R1 and R2? c) Finally, use the mapping for Q = Qo +Q1 + 1 2 2Q2+......, where Q is the volumetric flow rate (defined as an integral) and incorporate the appropriate mapping for vz to show whether Q1 and Q2 increase, decrease or are zero and what this means. Justify this result in terms of momentum transfer. You should already know what vz,o is as we have solved this and it’s solved in BSL!