Problem 1: Microscopic Momentum Balance In sweep flow filtration (see Figure below), a pressure drop forces fluid containing particles between two porous plates and there is a transverse (cross) flow that forces the particles to collect on one of the plates. A key design question is how one determines the length of the filter. Due to the nature of the problem, the fluid inside the channel will have nonzero x and y components of velocities v and v. For the purposes of this problem you may assume that both vx and vy x y can at most be only a function of y. Suppose that the distance between the two plates is d, the fluid has viscosity µ and density ρ. Further, the transverse velocity ( vy ) at the upper porous plate is fixed at a value = V. (a) Using continuity equation, prove that v y (y) = V everywhere. (b) Using Navier-Stokes equations, derive an expression v ( ) x y. See hint below. (c) What is the average velocity of the flow in the x direction. (d) How would one determine the length of the filter? Derive an explicit expression for the same. d y dy Hint: The solution of a differential equation of the form: + α + βy = γ is given as: dx dx mx 1 mx γ 4 4 (For β 0) y= Ce 1 + Ce +,with m α + α β 1 ; m α α = = β β x γ x (For β = 0) y= C1+ Ce α + α where C1 and Care constants to be determined. V Fluid p 0 d p L (Cross Flow)
Problem : Macroscopic Mass Balance Calculate the Sherwood number for a sphere sublimating into a stagnant film. Sh = Dh/D where D is the sphere diameter, D is the diffusivity in air of the sphere material, and h is an overall mass transfer coefficient, defined as h = [mass loss rate per unit area]/([concentration at the surface] [bulk concentration]) (a) The differential mass balance in the stagnant film can be expressed as: 1 d dc = 0 r r dr dr Solve for the concentration profile. (b) Solve for the mass flux at the surface. (c) Solve for the Sherwood number.
Problem 3: Microscopic Heat Balance As depicted in the figure below, a laminated semi-infinite slab generates heat and is cooled by a convective flow. From x = 0 to x = x h, in the heated region, thermal energy is generated uniformly with a volumetric heat generation rate of S o. To the left of the heat generating region is perfect insulation with a thermal conductivity k pi = 0. On the right of the heat generating region is some normal insulation with thermal conductivity k I and just outside of this region, beyond x I, a heat convecting fluid is flowing with heat transfer coefficient h. The normal insulation cannot operate above a temperature T fail otherwise it fails and does not operate properly. The device is operated at steady-state. What is the maximum fluid temperature T f that the flowing fluid can have in order for the device to continue to operate properly and for the normal insulation to not fail?
Problem 4: Microscopic Mass Balance A sphere of iodine, initially having a radius, r 1, of 0.5 cm, is placed in still air at 40 C and 747 mm Hg total pressure. At this temperature, the vapor pressure of iodine is 1.03 mm Hg. Derive an expression for the pseudo-steady state rate of sublimation of the sphere into the air, and use it to estimate the rate of sublimation of the sphere in mols/hr when the sphere has a radius of 0.5 cm. The diffusion coefficient of iodine in air is 0.0888 cm /s, and the gas constant, R, is 8.06 cm 3 atm/(g-mol K). One atmosphere is 760 mm Hg.
Problem 5: Macroscopic Heat Balance A hypodermic needle with an external diameter of d 1 inches is to be used to transfer a reactant preheated to 00 F to a laboratory reactor. In an attempt to reduce the heat loss from the transfer line, the hypodermic needle is threaded through the center of a solid rubber insulating tube d inches in diameter. Calculate the rate of heat loss from the hypodermic needle with and without rubber installation. Comment on your answer. Data: The rubber installation has a thermal conductivity of 0.1 Btu/(hr Fft); the ambient temperature is 70 F; the heat transfer coefficient from the outside surface of the transfer line is primarily due to natural convection and radiation and is approximately equal to.0 Btu/(hr Fft ) independent of radius. Take d 1 = 0.01 inch Take d = 0.5 inch, 1 inch, inches