Mass Transfer
10.4 Continuous-Flow Two-Phase Systems
Continuous-flow two-phase systems are ones in which both phases are continuously fed to and removed from the system. They are widely employed for the various unit operations described in Section 10.2 and may be carried out in tank-type or tubular geometries. They are also used in the laboratory to collect experimental data. (Semicontinuous systems, in which one phase is stationary and only one phase flows through the system, are also used, but we will not address them here.) Our
Contactor Phase ll Phase l
Phase l
Phase ll Separator Phase l
Phase ll VI cAIρI
VII cAIIρII
I I I II II II
cAf ρf qf
cAf ρf qf
I I I
cA ρ q
II II II
cA ρ q Figure 10.4. Schematic of a continuous-flow two-phase process.
goal here is to gain an appreciation of the issues in two-phase design and operation by examining an elementary well-stirred continuous-flow tank-type mass transfer system in detail.
A schematic of the continuous-flow system is shown inFigure 10.4.Two pieces of equipment are shown, a tank-type contactor and a device that effects a separation between the phases. We assume that the contents of the contactor are well mixed, so that the two phases are in intimate contact and there is no spatial variation of the concentration of a species in either phase. We also assume that all mass transfer takes place in the contactor and that the sole function of the separator is to separate the two phases that are mixed in the contactor. We do not consider the separator operation in detail; it will often be no more than a holding tank large enough to allow separation by gravity. (In some gas-liquid systems the separation takes place in the contactor, and a distinct separation device is not needed.) The concentrations in the exit stream from the separator are assumed to be the same as those that we would find if we were to sample the phases in the contactor. This is a reasonable assumption since, without agitation in the separator, there is little interfacial area available for mass transfer. The contactor and separator together are frequently referred to as a stage.
Our control volumes will be the same as those designated for the batch two-phase systems, and we can develop the model equations by applying conservation of mass exactly as we did in Section 10.3. The overall mass balance equations are
dρIVI
dt = qIfρIf − qIρI− arA, (10.25I) dρI IVI I
dt = qI If ρI If − qI IρI I+ arA. (10.25II)
Note that we do not consider any mass transfer in the separator, so the streams issuing from the separator have the same compositions as those in the contactor.
This is conceptually the same as if we had assumed that separated streams issue directly from the contactor.
The component equations will be written for the case in which a single species is being transferred. This is a common situation and allows us to develop the important concepts without the algebra becoming too complex. Concentrations everywhere in the tank and at the separator exit are denoted by cIAand cI IA:
dcIAVI
dt = qIfcIAf − qIcIA− arA, (10.26I) dcI IAVI I
dt = qI If cI IAf − qI IcI IA + arA. (10.26II) Transient behavior is important for startup, shutdown, and control of the system, but the basic design is carried out for the steady state, when time derivatives are zero. We thus write, using the rate expression Equation 10.9,
0= qIfρIf − qIρI− Kma[cIA− McI IA], (10.27I) 0= qI If ρI If − qI IρI I+ Kma[cIA− McI IA], (10.27II) 0= qIfcIAf − qIcIA− Kma[cIA− McI IA], (10.28I) 0= qI If cI IAf − qI IcI IA + Kma[cIA− McI IA]. (10.28II) A complete solution to this set of equations will require the constitutive equations between densities and phase compositions.
The general problem with composition-dependent densities is sometimes impor-tant, but we can accomplish our goals by dealing with the more limited case in which the total amount of A that is transferred between phases is insufficient to have an effect on the phase volumes. In that case, qIf = qI, qI If = qI I, and only the com-ponent equations 10.28I and 10.28II are needed to describe the system fully. This approximation leads to negligible error for most applications. Combining Equations 10.28I and 10.28II leads to an alternate equation relating the two concentrations:
cIA= cIAf +qI I qI
cI IAf − cI IA
. (10.29)
Note that Equation 10.29 does not include any terms involving the mass transfer rate; it is simply an overall mass balance that equates the total mass flow rate of A into the system, including both phases, to the total mass flow rate of A out.
Any two of the three equations 10.28I, 10.28II, and 10.29 are independent and can be used to analyze the process. There are eight quantities in these equations:
qI, qI I, cIAf, cI IAf, cIA, cI IA, M, and Kma; six independent quantities must be speci-fied, and the two independent equations can then be used to find the other two.
(Kmand a always appear as a product, so these two experimental quantities cannot
be determined independently as outputs from any experiment that is analyzed by this set of equations.)
10.4.1 Equilibrium Stage
A typical problem is to determine the effluent concentrations. Equations 10.28II and 10.29 can be combined to obtain the following result for cI IA:
cI IA = cIAf
λ + M + (qI I/Kma)+ cI IAf 1+λ+(qI IM/Kma)
. (10.30)
Here,λ = qI I/qI. Note that the volume, or, equivalently, the holdup in the tank, does not appear for either phase. The important transport quantity is the ratio of the flow rate of the dispersed phase to the rate of interfacial mass transfer.
The ratio qI I/Kma can be written qI I
Kma = VI I/Kma
VI I/qI I = VI I/Kma
θI I , (10.31)
whereθI I is the residence time of the dispersed phase. The calculations in Section 10.3.4 show that VI I/Kma is of the order of one minute if the agitation is sufficient to produce droplets with diameters of the order of 1 mm, and M is of order unity. If the system is designed for a holdup of, say, five to ten minutes, so that qI I/Kma 1, then the qI I/Kma terms in Equation 10.31 can be neglected (they can be neglected in the second term only ifλ = qI I/qIis of order unity) and the effluent concentration can be written
cI IA = cIAf + λcI IAf
λ + M . (10.32)
The rate of mass transfer does not appear, and the calculations can be carried out without knowledge of Kma! Clearly this is an extraordinary simplification.
Equation 10.32 is formally equivalent to taking the limit Kma→ ∞ in Equation 10.30. The result is known as an equilibrium stage and can be obtained, with consid-erably less physical insight, in a more direct manner. We simply take Equation 10.29, which is always valid, and assume that we have equilibrium, in which case we write cIA= McI IA, from which Equation 10.32 follows directly. This is the reason for the name equilibrium stage. The concept of the equilibrium stage is used extensively in design calculations for the unit operations touched on in Section 10.2. We illustrate the concept with a few examples here, and we will return to this important topic in Chapter 11.
EXAMPLE 10.1 An aqueous solution containing 200 g/L of acetone is to be purified by continuous extraction with pure trichloroethane. How much acetone can be removed if the flow of both aqueous and organic streams is 10 L/min?
We assume for these calculations that water and trichloroethane are insolu-ble. This is a reasonable assumption for our purposes here; the National Institute
of Standards and Technology (NIST) database*summarizes five studies and con-cludes that 0.0012 grams of trichloroethane or less will dissolve in 1 gram of water in the range 0 – 50◦C. (At a level of over 1,000 ppm the water would probably have to be purified before it could be discharged. More likely, it would be recy-cled for process use, where the very small amount of trichloroethane would be unlikely to be relevant.) We assume that the contactor is an equilibrium stage, the flow rates are constant, and the solvent feed contains pure trichloroethane (cIAf = 0). Because the flow rates are equal, λ = 1. FromFigure 10.1we have M= 2. Hence, from Equation 10.33,
cI IA = λcI IAf
λ + M= 1× 200
1+ 2 = 66.7 g/L, cIA= McI IA = 2 × 66.7 = 133.3 g/L.
That is, the acetone content in the aqueous phase is reduced from 200 to 66.7 g/L in this equilibrium stage.
EXAMPLE 10.2 Suppose now that the flow of organic in Example 10.1 is increased from 10 to 20 L/min and the flow rate of the aqueous phase is maintained at 10 L/min.
We now haveλ = qI I/qI= 0.5, and
cI IA = λcI IAf
λ + M =0.5 × 200
0.5 + 2 = 40 g/L, cIA= McI IA = 2 × 40 = 80 g/L.
By doubling the amount of solvent the acetone content in the aqueous stream is reduced only from 66.7 to 40 g/L.
EXAMPLE 10.3 As an alternative to the increased organic flow in Example 10.2, suppose that the aqueous effluent in Example 10.1 is taken to a second stage and again contacted with a pure trichloroethane stream with a flow rate of 10 L/min.
For the second stage, cI IAf = 66.7 g/L and λ = 1. Thus,
cI IA = λcI IAf
λ + M= 1× 66.7
1+ 2 = 22.2 g/L, cIA= McI IA = 2 × 22.2 = 44.4 g/L.
Thus, by using the same amount of solvent as in Example 10.2, but by dividing the total between two consecutive contacting stages, the residual acetone in the water stream is reduced by nearly a factor of two. We saw a similar result in the treatment of cross-flow dialysis in Section 5.3.
10.4.2 Deviation from Equilibrium
Energy in the form of agitation must be put into two-phase systems to generate adequate interfacial area for efficient mass transfer. Hence, it is evident that there will be situations where equilibrium cannot be attained in a stage. This has led to
* See the Bibliographical Notes at the end of the chapter.
+ +
Figure 10.5. Residual efficiency in a stirred tank as a function of energy input/unit volume.
Data of A. W. Flynn and R. E. Treybal, AIChE J., 1, 324–328 (1955). Reproduced with the permission of the American Institute of Chemical Engineers.
attempts to correlate Kma with quantities like the power input per unit volume in order to compute effluent composition. One quantity that is frequently used in unit operations is the stage efficiency, defined as
Ef = cI IA − cI IAf
cI IAe− cI IAf. (10.33) The stage efficiency defined in this manner is nothing more than the fractional approach to equilibrium. Noting that cI IAe= cIA/M and substituting Equation 10.33 into Equation 10.29, we obtain an equation for the effluent in terms of this additional parameter:
cI IA = Ef
cIAf + λcI IAf
λEf + M +M(1− Ef)cI IAf
λEf + M . (10.34)
When Ef → 1, Equation 10.34 reduces to Equation 10.32 for the equilibrium stage.
As might be expected, the efficiency is simply related to the mass transfer coeffi-cient. Through comparison of Equations 10.31 and 10.35 the relationship can be established as
Ef = M
M+ qI I/Kma. (10.35)
The stage efficiency can be correlated with design variables.Figure 10.5,for example, shows some data of Flynn and Treybal for interfacial transfer of benzoic acid in
toluene-water and kerosene-water systems. The residual efficiency ERplotted in the figure is defined by
ER= Ef − Ef 0
1− Ef 0 . (10.36)
Ef0 is the measured efficiency at zero agitator speed (which is itself high). Clearly, as ER→ 1, Ef→ 1. ε is the energy per unit volume,
ε = P
qI+ qI I, (10.37)
where P is the power supplied to the agitator. The correlation appears to work for each system for the type of mixer used (six-bladed turbine impellors), and it is evident that beyond an energy input of approximately 100 ft lbf/ft3 (or about 5 kJ/m3) an equilibrium stage can be assumed.
Many other correlations for the efficiency and the quantity Kma are available in the published literature. A further discussion of this important practical topic, which is typically covered in a subsequent course in separations or unit operations, is beyond the scope of our introductory treatment, however.