Department of Chemical Engineering Transport Phenomena Midterm Exam
Problem 1 - The Finger Thermometer
Human skin was the world's first thermometer, however, it can be deceptive. For example a a metal wall and an OSB wall in thermal equilibrium in the same room will feel as though they are drastically different in temperatures. By touching which material do you get the best idea of the true temperature of the room?
To receive full credit on this problem you must (1) identify what is being conserved in this problem, (2) write out a non-steady state shell balance for this system approximating your skin as a infinite plate and assume that the metal/OSB is an infinite thermal mass, (3) non-dimensionalize the system, identify any characteristic variables and explain what they represent in a physical sense, (4) make reasonable simplifying scaling assumptions to solve or at least describe the short-time behavior of the system in order to answer the primary question at hand.
Problem 2 - The Orange Farmer
Why would an orange farmer worry about his crops on a cloudless, dead-calm night with lows predicted to be 35 F but sleep easy if it's overcast and windy? Explain your answer from the perspective of transport phenomenon to receive full credit. (The answer is not "It may get colder than 35 F.")
Problem 3 - Pigging
Pigging is a method by which a "pig" (inspection/cleaning gauge) is pushed through a pipe to perform certain maintenance without stopping the production, or flow, of product within pipeline. Some have postulated that a pig, moving counter to the fluid motion and stabilized/transported by a magnetic field, could be used to induce a transverse pressure wave within the pipe. The pressure wave would be used to detect stress, cracks and seal defects within the pipeline.
Your job is to investigate the pig design with consideration to minimal drag and maximum transverse pressure. Let the pig have length l, a radius of zero at the tip and radius = Ro at the back end, and some arbitrary shape, f(r), between the two endpoints. Assuming the pig is moving against fully developed pipe-flow within a pipeline of radius Rp >> Ro, do the following:
(a) Draw the shell balance you would use to model the region of fluid defined by the pipe and pig surfaces. Define the surfaces of the differential shell which experience non-symmetric momentum transport. Write out the differential equation which governs the momentum flux in the system.
(b) Assuming Newtonian, laminar, incompressible and irrotational flow; write out each respective momentum flux and each corresponding differential equation governing fluid velocity in this region.
(c) Write out the boundary conditions for this system.
(d) Non-dimensionalize the DE's and BC's. What do the non-dimensional scaling constants say about the physics and importance of each term in the problem?
(e) Under your choice of sensible limiting cases, what can you say about the shape, f(r), that (i) maximizes transverse pressure or (ii) minimizes drag on the pig? Does the same shape that maximizes transverse pressure also minimize drag? What limiting cases do you think preserves the dominant physics of the problem? Can you come up with a functional form for f(r)?
Problem 4 - Internal Boundary Layers
The boundary layer approximation is not restricted to heterogeneous interfaces. It also can be applied to interfaces where two fluids of different velocities are brought into contact. Imagine that you have a small slit where fluid is flowing into a very large volume of stagnant fluid. Let the initial momentum of the jet be known (i.e. the slit is infinitely thin compared to the stagnant volume). Given the pressure of the outer region (stagnant fluid) is imposed on the boundary layer (jet), P is constant within the jet.
(a) Focus on a planar (x-y) geometry and write down the x-component of the NavierStokes equation, eliminating terms to come up with the N.S. equation that governs the boundary layer flow.
(b) Following our analysis for stream functions & boundary layers, convert the N.S. equation (x-component) and obtain the following equation:
(∂ψ/∂y) (∂2ψ/∂x∂y) - (∂ψ/∂x)(∂2ψ/∂y2) = ν(∂3ψ/∂y3) (1)
where ν is the kinematic viscosity.
(c) Assume a solution of the form ψ(x, y) = xpf(η), where the change of variable is defined by η = y/g(x), and obtain 1 differential equation in f(η) where you are allowed to assume the two ODE coefficients are constants (say, C1 and C2); giving you 1 algebraic equation for g(x) and 1 first order ODE in terms of g(x) and g'(x) as well.
(d) Write down the boundary conditions for f.
(e) Amazingly, this non-linear ODE is analytically solvable. Assume the constant in front of the ff'' and (f')2 term is 2 and 1, respectively. Use the fact that (ff')' = f'' + (f')2 and solve the f-ODE to obtain:
f(η) = btanh(bη) (2)
where b is an integration constant.
(e) Use the boundary condition at f(∞) to show what kind of values (i.e. real, complex, etc) the solution is valid for.
Problem 5 - Nuclear Meltdown
A cylindrical nuclear fuel element of radius R is placed in a liquid cooling media of temperature Ta. Heat is generated at a temperature-dependent volumetric rate Sv = a + bT where a and b are known positive constants. Heat transfer within the fuel is by conduction with thermal conductivity, k. Heat is transferred to the cooling medium at a flux given by h(T(R) - Ta), where h is a known heat transfer coefficient and T(R) is the unknown fuel temperature at radius R.
Determine the steady-state temperature profile T(r) and the total heat transfer rate Q(J/s) to the coolant. List all assumptions necessary to arrive at your results.
Problem 6 - Fluid Mechanicist in a Diner
You're having lunch in a diner with Claude-Louis Navier and George Stokes. As they are talking about the stress-strain behavior of fluids, you are bored and are passively fiddling with the drinking straw in your glass of water. Sealing the top of your straw with your finger, plunging it into the cup and then removing your finger - at which point you've drawn the attention of Navier and Stokes.
(a) What fluid phenomena did your fiddling cause to draw their attention? And, once they start to discuss this phenomena, what perspective would you guess each would stage their arguments from when discussing the phenomena?
(b) Either starting from a shell balance or the Navier-Stokes equations (your choice), derive the fundamental equation(s) that govern the phenomena observed - eliminating obvious unnecessary terms (please state your assumptions that allow you to eliminate EACH AND EVERY term that you cancel).
(c) Assuming the oscillatory restoring force is proportional to the displacement (i.e. harmonic), the net force on the fluid is given by the balance of potential (kz(t)) and kinetic energy (kv(t)):
F = - kz(t)/V - (c/V)(∂z/∂t) = ρ(∂2z/∂t2) (3)
where k is a "spring" constant, V is the volume of the fluid, ρ is the fluid density, c is the viscous damping coefficient and z(t) is the position of the leading edge of the fluid/air interface.
Re-constitute this equation to show that the force balance can be written in the form:
∂2z/∂t2 + 2ξωo ∂z/∂t + ωo2z = 0 (4)
where ξ (dimensionless) and ω (Hz) are the damping ratio and un-damped angular frequency, respectively. How to these two quantities depend on k, m, c and V?
(d) Assuming you know (or specify) the initial conditions for position and velocity in the above equation, what values/ranges of ξ give you different kinds of damping? How many are there? What kind of damping do you observe in your system? Does this depend on how far you submerge the straw or if you allow some fluid to be in it prior to removing your finger?
(e) Coupling the harmonic oscillator force balance with your momentum conservation equation(s), non-dimensionalize this equation and obtain a "Strouhal number", a Reynolds Number and a dimensionless pressure. Based on your assumptions for this system, what parameters will you choose to scale pressure? Can you come up with a scaling parameter that relates the damping phenomena with the other properties associated with the motion of the fluid? If so, what do they tell you? Based on your system parameters, can you estimate the time it takes (or order of magnitude) for the system to reach equilibrium?
(Feel free to conduct "experiments" to collect data for this part!)
(f) Manipulating your equation of motion, under what limiting cases can you solve the system? What are these limiting cases (in concise statement(s)) and what does the resulting closed form solution tell you?
(g) Outline the problem you thought I was going to ask for this question.
Problem 7 - The Dynamic Telescope Lens
A common way to make a telescope lens is to rotate a container of mercury at a constant angular speed, ω.
(a) In terms of this speed, container dimensions (R & h) and fluid properties (ρ & µ), come up with a relationship between the lens' focal point and the speed of rotation.
(b) A problem with this lens is that aberrations can occur on the edges of the image due to the meniscus formed at the Hg-container interface due to the surface tension of Hg. One possible way to eliminate this is to suspend the Hg in an immiscible fluid and rotate the heterogeneous suspension. In this case, the no-slip boundary condition is no longer valid at the Hg boundary. What boundary condition would you use in this case? Would you have one single system or two coupled systems to solve in this case? Determine the governing equations for this system, solve them analytically if possible, or use scaling analysis and state the dominant physics in this system.
(c) Model the non-steady behavior of this system and discuss it's properties (i.e. change in focal point with time, fluid wall height with time, etc).
Problem 8 - Flow in a Contraction
Steady laminar flow of a fluid with constant viscosity and density is driven by a known pressure drop in a constant radii portion of a pipe. At some point (z = 0), the pipe is contracted linearly from radius Ri to Rf. For convenience, assume the pipe is horizontal and gravity is pointing downward.
(a) Draw the differential shell you would use for the linear contraction region, conserve mass and momentum to obtain the differential equations governing the contraction region. What are the boundary conditions for this region?
(b) Simplify the Navier-Stokes equations to check and see if you came to the correct answer for part (a).
(c) Non-dimensionalize the equations in such a way that the Reynolds number appears.
(d) Solve the differential equations. If it's a PDE write down the limiting case in which it collapses to an ODE and then solve the ODE!