Week 11 Online lecture notes
Convective mass transfer – two fluids in contact
In Week 11 we examine mass transfer process between two fluids in contact from an alternative angle – The two-resistance theory.
We have studied the lecture recording and notes, and have already got the basic idea about the theoretical treatment of mass transfer from one fluid phase to the other (contacting) fluid phase. The theoretical treatment in the lecture recording is based on a scenario of NH3 transfer from an oil phase to water phase; it considers the phase equilibria and the division of the interface region into two films.
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In this lecture, we shall consider a scenario where mass transfer occurs from a gas
phase to a liquid phase, and we shall use the concept of mass transfer resistance to solve the mass transfer problem.
Gas absorption and Liquid stripping
Transfer of a solute A between contacting gas phase and a liquid phase. The two situations are Gas absorption and Liquid stripping
Figure 1. Situation of gas absorption: Concentration gradients between contacting gas and liquid phases where “A” is transferred from the bulk gas to the bulk liquid.
Figure 2. Situation of liquid stripping: Concentration gradients between contacting gas and liquid phases where “A” is transferred from the bulk liquid to the bulk gas.
The interface transfer involves the following three processes (taking gas absorption as an example, Figure 1):
1) Mass transfer from the bulk of phase 1 (gas) to the interfacial;
2) (Diffusive) transfer across the interface into the 2nd phase (liquid);
3) Mass transfer from the interfacial surface into the bulk of phase 2 (liquid). These processes are shown in Figure 1.
Assumptions
Steady state: – concentrations at any position in the mass transfer equipment (e.g. tower) do not change with time.
Sharp boundary: – interface between the gas and the liquid phase is clear and sharp (no mixing zone or a third phase)
Laminar flow in two-film region: – mass transfer inside the two-film region is through molecular diffusion.
Equilibrium: – at the interface (not the whole film), concentrations on the two contacting phases are in equilibrium;
Negligible diffusive MT resistance: – (diffusive) mass transfer resistance across the interface is negligible, equilibrium concentration relations can be applied.
No chemical reaction: – Rate of diffusion across the gas- phase film must be equal to the rate of diffusion into the liquid-phase film.
In Lecture 11 recording, we have seen that the strategy of calculating the mass transfer flux from the oil to the water fluid phase must be linked to the bulk concentration of the transferring component A, so as to avoid the difficulties in determining interfacial concentrations pA,I and cA,i. (the rationale of taking this strategy is that the bulk concentrations can be determined relatively easily, but the
interface concentrations are much more difficult to determine.)
Now we investigate the above strategy from a new angle – the mass transfer resistance
When “A” transfers through the bulk gas phase to the interface, it has a mass transfer resistance RA,G (G is gas); similarly, “A” has a mass transfer resistance when transferring from the interface to the bulk liquid phase, RA,L. These resistances are individual resistances. In the two-resistance theory we assume that the diffusive resistance in the two-film region is far smaller than the convective resistances, therefore negligible. We can also consider that when A transfers from bulk gas to bulk liquid phase, it has an overall resistance, Rtotal. From Figure 1, we know that the overall convective mass transfer resistance is equal to the sum of RA,G, and RA,L :
Rtotal = RA,G, + RA,L
The resistance in mass transfer has the same mathematical form as the heat transfer resistance in heat transfer. In heat transfer, the overall heat transfer coefficient is U, the corresponding overall heat transfer resistance is 1/U. Below is a quick recap of the relationship between the heat transfer coefficients of a process and the corresponding heat transfer resistances.
In heat transfer through a wall formed by two materials of different thicknesses, l1 and l2, we need to consider the two different thermal conductivities, k1 and k2, and two convective heat transfer coefficients of the two outer surfaces of the wall, h(inf),0 and h(inf),3:
Rewrite the eq. We have:
𝑞̇ = ∆”!,# and 𝑅 = ‘
$ #$#%& ( %
The overall heat transfer resistance is the reciprocal of the heat transfer coefficient. Their relationship can be summarised by the following eqns:
Now, let’s return to the two-resistance theory of Mass Transfer; first, let’s look at the math relationships between the individual mass transfer resistances and the overall mass transfer resistance.
Individual convective mass transfer flux equations across the gas and liquid films for the gas and liquid phases:
Two points need to be pointed out:
(1) in the above eqs. pA and cA cannot be compared
directly, because of different units. However, they can be compared if we consider the equilibrium distribution of A between the gas and liquid phases, i.e. the Henry’s law.
(2) Mass transfer resistance of individual step can be written in the following manner:
From this point onward, we will work on the overall connective mass transfer flux. The Overall convective mass transfer flux (NA) equation can be written using the overall mass transfer coefficient (KG or KL) and the driving force.
Next, we find out the equilibrium conditions to specify pA* and CA*. But first let’s read Figure 3 to understand our system; then in Figure 3 we read out all the driving forces and do the final derivation of the mass transfer resistance equation.
When reading Figure 3, consider the following eqns. Our objective is to understand why the driving forces can be drawn in such a way as shown in Figure 3.
Assuming all P ~ C relations in Figure 4 are governed by the same equilibrium relation by the Henry’s law, we have the simplified Figure 4:
What we want to do is to write Into
Now we can do this:
Rearranging the mass transfer equations following the pictorial analysis of Figure 4 and 5:
In an experimental study of the absorption of NH3 by water in a wetted-wall column, the overall mass transfer coefficient KG was found to be 2.74 x 10-9 kmol/(m2.s.Pa). At one point in the column, the gas phase contained 8 mol/m3 of NH3 and the liquid phase concentration was 0.064 kmol/m3 solution. The tower operated at 293 K and 1.013 x 10-5 Pa. At this temperature, the Henry’s law constant H = 1.358 x 103 (Pa.m3)/kmol. If 85% of the total mass transfer resistance is encountered in the gas phase, determine the individual film mass transfer coefficients and interfacial compositions.
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