MASS TRANSFER OPERATIONS – CH2020
1. What is the general statement and expression of mass balance? Reduce the expression if the fluid has constant density. Mass balance: balance: Total mass is conserved or mass cannot be created or destroyed The general general mass balance expression is given given by:
(For derivation refer summary) summary)
(In Cartesian coordinate system)
If the density of the fluid is constant, the above equation reduces to
1.
Define mass flux, molar flux and find the relation between them.
Mass Flux (kg/m2s (kg/m2s): ): Mass flux of a component is the mass being transported per unit area per unit second. (We use (Density*velocity of component) to keep track of how much of component is getting transported. Molar Flux (mol/m2 mol/m2 s): s): moles being transported per unit area per unit second [N A]. Where NA= Molar flux of A, MA= Molecular mass of A, AUA is Mass flux of A. Mass flux and molar flux can be related rel ated as follows:
2. Prove that the average position of the one dimensional random walker, walking in X-direction will be at origin.
Let’s assume black dot takes n steps and d is the total distance travelled by black dot, it can be either positive or negative. d= a1+a2+a3+a4+…..an Average= But as a1 has an equal probability of +1 or -1, we assume average value is 0. Therefore the average position will be at origin.
3. Show that the mean square of the average distance travelled is related step size and total number of steps and the expression. Even though d can be positive or negative, d2 is always positive. 2
= <(a1 + a2 + a3 + ... + aN)2> = <(a1 + a2 + a3 + ... + aN) (a1 + a2 + a3 + ... + aN)> = ( + 2 2 2 + + ... + ) + 2 ( + + ... + + ... + .) If the step size is l, a1 can either be +l or –l. either way, a12 =l2. Then on average a12 is l2 and 2 2 = l . Since a1a2 is equally likely to be +1 or -1, = 0. Therefore = nl The larger the number of steps, the longer time is taken by the random walker. Distance traveled is proportional to square root of time. 4. Write the expression for component balance and simplify the expression for solution of constant density, no flow and chemical reaction. Analyze the simplified expression (fick’s second law). Using rectangular coordinates, the component balance would be:
For a solution of constant density, this will simplify to (substituting ρA = MAcA):
When there is no flow and chemical reaction, the above equation simplifies to:
This is called Fick’s second law, analogous to Fourier’s second law of heat conduction or
unsteady state diffusion. 5. If air velocity is 1 cm/s in a pipe of diameter 2 cm, what are: overall mass flux, overall molar flux, oxygen mass flux and oxygen molar flux?
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Assumptions: Air contains 21% O 2 and 79% N2, Density of air= 1.184 Kg/m , Mair = 28.84 g/mol Overall mass flux= = air * Uair = 1.184*10^-2 Kg/m2 s. Overall molar flux= total mass flux/ molecular mass of air= 0.410 (mol/m2 s). To find oxygen mass flux and molar flux we need velocity of oxygen, Total mass flux= mass flux of oxygen + mass flux of nitrogen
Total molar flux= molar flux of oxygen + molar flux of nitrogen
Solving these two equations, velocity of oxygen and nitrogen can be found. Uo2 = 2.007*10^-3 m/S, UN2 = 0.01002 m/S. Oxygen mass flux= = 2.87*10^-3 (Kg/m2 S), Oxygen molar flux= mass flux/molecular mass= 8.96*10^-5 (Kg/m2 S).
6. Find the diffusion coefficient and the Sherwood number for diffusion from a sphere where boundry conditions are given by
Answer) Simplified governing equation, l aplacian In spherical coordinates,
Applying the boundary conditions:
Concentration decreases monotonically from cAa to cA ∞.Flux at the sphere surface,
In the above equation, we can also define the flux in terms of a mass transfer coeffcient. Therefore, kc = DAB/a. Mass transfer coefficient is usually represented in the form of Sherwood number:
7.
Define the average velocity and the velocity component in X direction at some point and
find the average of fluctuations term
The velocity at a given point fluctuates even if average velocity with respect to time, at that point, is constant. Therefore, we can write velocity component in x -direction at some point as: ux = ux + u ′x Where the average velocity is defined as,
Where __ is the time window over which averaging is done. This is because, over this time window the fluctuations are random and therefore, their averages are zero:
This window is large enough to average out the fluctuations. 8.
What is the value of Sherwood number according to film theory of mass transfer?
Film Theory: It is assumed that next to the interface, a thin film exists. The thickness of this fictitious film is ZF . • Mass transfer resistance is confined to this film. • It ensures that concentration is uniform out - side the film. Molecular diffusion takes place
within the thin film. Concentration varies linearly within the film. • For better mass transfer, the film should be as thin as possible
The mass transfer and heat transfer coefficients are still called film coefficients. Solution for component balance for steady one-dimensional diffusion across a stationary thin film is:
Therefore, mass transfer coefficient and Sherwood number according to film theory are:
9.
Write the expression for mass flux due to diffusion and hence find the sum of mass flux
due to diffusion for a multicomponent mixture. Mass flux due to diffusion ( jA) : The mass flux due to diffusion will be MiJI. Since the molecular weight of different species will be different, summation of MiJI . However, we could define all the fluxes on mass basis, such that:
Where density and u is component velocity. Based on this we can see that:
10.
What is anomalous diffusion and what are the differences between normal and
anomalous diffusion? The term used to describe a diffusion process with a non-linear relationship to time is called Anomalous diffusion which deviates from normal diffusion. It is either faster or slower compared to normal diffusion
Difference between normal diffusion and anomalous diffusion
Normal diffusion
Anomalous diffusion
1. far away from equilibrium 1) occurs mainly in fluids close to equilibrium 2. ”long flights” of particles, where 2. Particle trajectories are
11.
What are the different boundary conditions used in many realistic applications other
than concentration boundary conditions? Time dependent concentration: When component diffuse in/out of domain, the concentration in the surrounding medium changes
Flux boundary condition : Composite media (1 and 2) boundary condition
Constant flux at the interface: due to reaction, for example rate of production of a species at the electrode based on the current being passed
Impermeable surface: flux at these surfaces is zero.
Moving boundary: Interface changing due to flow or phase change. For example, above a certain concentration diffusivity changes. This will lead to a moving front and boundary condition at that moving front.
Simultaneous heat and mass transfer: Most mass transfer operations involve simultaneous heat and mass transfer, such as evaporation, adsorption etc. In these cases, boundary conditions are linked with heat of evaporation etc
12. Give some of the examples of the equipments used for mass transfer contacting (staged or continues)
Sparged gas, bubble column
Mechanically agitated vessels: one example of mass transfer coefficient correlation for gas-liquid systems:
L refers to liquid. This implies that mass transfer resistance is confined to the liquid phase, and Rayleigh number determines the change in Sherwood number over stagnant bubble case. It also implies that Sh is independent of agitator power. Tray tower Spray tower Wetted wall towers Packed towers
13. Define equilibrium stage, real stage and explain the difference between in staged operations. A stage is a well-mixed unit of the operation. It could be a stirred vessel or chamber, it could be a tray. It is assumed that concentrations in both the phases are constant. It is assumed that the exiting streams from these stages are at equilibrium. Because of this assumption, the stage is called an equilibrium stage, an ideal stage or a theoretical stage. In an ideal stage, therefore, contacting time is infinite or mass transfer rates are sufficient for equilibrium to be reached. If mass transfer operation consists of ideal stages, than the selection/design of the operation can be carried out using the phase equilibrium data. This selection/design would depend on the nature of operating points or lines (depending on what the feeds are) and equilibrium points or lines. Because of the assumption of equilibrium, mass transfer considerations are not required while designing mass transfer operations as equilibrium staged operations. A real stage, on the other hand, has several mass transfer resistances. These are due to finite rates of mass transfer, limited interfacial areas over which contacting occurs and short residence times involved in a stage. Therefore, it is stated that a real stage operates at efficiency less than 100%. This stage efficiency depends on the equipment used for contacting. 14.
Explain the diffusion phenomenon in case of liquids and give the expression for
diffusivity. In liquids the interaction between molecules is very strong, and the average number of collisions is very large (compared to mean free path.). Diffusion in liquids is based on Stokes Einstein relation. Stokes Einstein relation is valid when the diffusing molecule (for example,
solute or colloidal particle) size is much larger than the surrounding molecules (solvent).The shape of the diffusing molecule is also assumed to be spherical. Many molecules, however, are non- spherical. Stokes Einstein relation can be modified for nonspherical shapes. Therefore, Stokes Einstein relation is used as a basis for situations where molecules are comparable to surrounding molecules as well as when molecules are nonspherical. The diffusivity given by this relation is D= KT/6 .where K is Boltzmann’s constant, T is temperature, is viscosity and a is radius of the sphere.