Discusión:Mecánica de fluidos
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[editar] Transport Phenomena
Chemical Engineering Graduate Student. Teofilo Donaires
Problem 1. Proposed by the teacher
Automobile catalytic converters for treating exhaust gases are typically monolith structures in which the exhaust gases flow through multiple parallel channels in a structure of which the walls comprise a supported catalyst. There are numerous design factors that affect the performance of the converter, but for the purposes of this problem the following simplified picture is adequate. Consider a square channel (edge length B, length L) carrying volumetric flow rate Q of gas. Assume that the velocity profile across the channel is a known function vz(x,y). There is a single reactant that is removed catalytically by reaction assumed to occur at the wall (i.e. neglect the effects of transport into the monolith) and is effectively first-order with rate constant k´´; this seems reasonable because the concentration of the reactants in the gas stream is fairly low. Neglect thermal effects (certainly invalid for a while after start-up).
Write down the differential equation governing transport and reaction of the reactant in the channel at steady-state. Hence determine on what dimensionless independent variables and parameters the average exit concentration of reactant depends. Estimate how long the channel should be for reactant removal to be substantially complete.
Solution:
Problem 2. Deen (3.9). Oswald Ripening.
In a supersaturated solid solution, precipitation may lead to the formation of grains or particles of a new phase. It is found that after an initial stage of this process that involves nucleation, there is a second stage in which the particles grow to much larger sizes. In the second stage the degree of supersaturation is slight, and the rate of particle growth is controlled by diffusion. This coarsening of particles, known as Ostwald ripening, has import effects on the physical properties of various alloys. An extensive analysis of the kinetics of the second stage of this process was given by Lishitz and Slyozov (1961). The objective here is to derive one of their simpler results. Consider a solid solution consisting of components A and B the abundant species. The particles being formed by precipitation are composed of pure A. Assume that at any given time the precipitate consists of widely dispersed spherical particles of radius R(t), which do not interact with one another. Accordingly, it is sufficient to consider one representative particle for which CA=CA(r,t) in the surrounding solution. Far from the particle the concentration of A is constant at CA. A key aspect of this problem is that the concentration of A at the particle surface is influenced by the interfacial energy, and therefore it depends on particle size. The relationship is:
Where the constant is proportional to the interfacial energy. It is the interfacial energy term which provides the driving force for diffusion. a) Derive an expression for the growth rate of the particle, , valid when growth is slow enough that diffusion in the solution is pseudo steady. ( Assume that the densities of the particle and solution are the same, and show that as a consequence there is no convection)
b) Show that eventually R(t)t1/3, which is the result of Lishitz and Slyozov.
c) Use order –of-magnitude reasoning to determine when the pseudosteady assumption is valid.
Solution:
3. Problem 3. Deen (3.15). Velocity of a Solidification Front
Assume that a pure liquid, initially at temperature T, occupies the space x>0. At t = 0 the temperature at x = 0 is suddenly lowered to T0, which is below the freezing temperature TF of the liquid. Solidification will begin, and the solid-liquid interface will advance into the region formerly occupied by liquid. As shown in Fig. P3.15, the interface location at time t is given by x = (t). Assume that the liquid and solid have the same density, but different thermal conductivities (kL and kS, respectively). The latent heat of fusion is . In this situation, (t) may be determined exactly, without invoking pseudosteady or other approximations.
a) State the differential equations and boundary conditions governing the solid and liquid temperatures, TS(x,t) and TL(x,t), respectively.
b) Taking into account the fixed temperatures at x = 0 and x but ignoring for the moment the interfacial conditions at x = (t), show that both the solid and liquid temperatures may be expressed in terms of error functions. The variables involved are , when i = S or L.
c) Use the results of part (b) and the requirement that T = TF at the solid-liquid interface to show that . To define the proportionality constant, use.
Where is a numerical constant which remains to be determined?
d) Use the interfacial energy balance to show that is given implicitly by
Complete the solution for TS(x,t) and TL(x,t) in terms of .
e) Consider the special case where the initial liquid temperature is at the freezing point, T = TF. Show that a pseudosteady analysis involving TS(x,t) leads to
Comparing this with the exact results of parts (c) and (d), show that the pseudosteady and exact results for (t) are in good agreement, provided that . (Hint: as x 0). Use and order –of-magnitude analysis of the time-dependent energy equation and boundary conditions to explain why a pseudosteady approximation is accurate when this requirement is met.
Solution:
