Mass Transfer in Multicomponent Mixtures


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Solution of the linearized equations of multicomponent mass transfer: I

The resulting electrical gradient forces the sodium ions to the left and restricts the amount of hydrogen that can be transferred. There is no such mechanism in conventional mass transfer. It is a plug with fine openings you might think of compressed cotton wool , but otherwise inert. On one side there is helium, on the other argon, both at exactly the same pressure and temperature.

Such a situation can be maintained by having flows of pure helium and pure argon past the ends of the plug, passing out into the open. The concentration gradients of the two gases are equal. A careless application of Figure 2. If you want to understand this you should realise that Figure 2. Figure 2. This movement is balanced by friction between the two gases and the plug. To let the two friction terms cancel, helium must have the higher velocity.

There is no interaction with any plug in the conventional analysis of mass transfer; clearly you should take the plug as a component of the mass transfer system. The viscous flow due to the pressure gradient increases the argon flux and decreases the helium flux. This is analogous to the effect of the electrical gradient in the previous example. The pressure difference required depends on the structure of the plug: the finer the pores, the larger the pressure difference. Even so, rapid diffusion of hydrogen does cause a minute pressure difference.

The transport of nitrogen against its gradient is in small part due to the resulting viscous flow. We have then applied it to a few examples and seen that it does not work well. We will be working these ideas out in the rest of this book. A lively book and a good primer. If you decide not to proceed into the subject of multicomponent mass transfer, Cussler will give you a few excuses. Duncan, J. This describes the two-bulb diffusion experiment. At the top of the Mount Everest their values are about one half of those at sea level.

According to Fick's law, air should be diffusing away into outer space; rough estimates show that the half-life of the atmosphere should be about ten thousand years. This is incorrect; the atmosphere is several hundred times older. Which force counteracts the diffusion and is neglected by Fick's law? It may not seem to be surprising that water is vaporising. Now we put a lid on the glass. This is obviously at variance with Fick's law, although few people seem to notice. What does this theory tell when we put the lid on? There are no concentration differences, so there is no driving force for either water or for the ions.

If we now apply an electrical potential difference across the membrane, sodium will be forced through. This is not surprising: there is a driving force on the sodium, be it one that is not incorporated in Fick's law. What is surprising, is that the electrical force leads to a considerable flux of water. Water is not charged, and there is no obvious driving force. How can this be?

However, transport in media with extremely small pores, such as cellophane, seems to be dominated by diffusion. Why would this be? Here the limitations of conventional approaches to mass transfer are shown using several other examples. We look at the individual species in a mixture, at the forces working on them, and at their resulting motion. There are two kinds of forces: driving forces and friction forces. Their balance is the Maxwell-Stefan equation, which is the basis of this book. We end with a simplified form of the MS-equation and an estimate of the driving force.

Our first concept is that of a potential. Consider the mass in Figure 3. The work you perform is equal to the increase of potential of the mass. If you increase the height by one metre, the potential increases by almost ten Joules. The mass could - potentially - return the same amount of work if its motion were to be reversed. In this book, we will mostly consider not the mass of a species, but the number of moles.

We obtain the potential per mole by mUltiplying our fIrst result by the molar mass. It has a value of about ten Newton per kilogram; for a simple chemical this might be one 3. Driving Forces 35 Newton per mole. We will see that this is a very small force when considering molecular motion. The negative sign shows that it is directed downwards. There are other more important forces, such as those due to electrical fields and pressure fields, those in centrifuges, and the support forces in solid matrices such as membranes. We will come back to these in Chapter 11 electrical forces , Chapter 12 centrifugal and pressure forces and Chapter 14 solid matrices.

Our second concept is that of the chemical potential. Consider a separation of one mole of i from a large amount of a mixture Figure 3. The work required to do this reversibly is the change of the chemical potential. It typically has a value of a few thousand Joules per mole. The chemical potential is related to composition; it usually increases with the concentration of the species.

As you can see, the chemical potential is a logarithmic function of the activity of the species. This activity is the product of the activity coefficient and the mole fraction. The activity is then the same as the mole fraction. This simplifies formulae, and we will often assume ideality in our examples. Note that the chemical potential of a gas can be written in terms of a partial pressure. Our last concept is that of momentum. The momentum of a particle - which is sometimes called its 'amount of motion' - is the product of the mass and velocity of the particle.

The momentum of a system is the sum of that of all parts. The momentum of the system can change due to flows carrying momentum in, out, or by forces working on the system Figure 3. The momentum balance shown is the onedimensional form of a more general relation. The symbol v in this equation stands for 'some velocity'; it does not have the special meaning that it has in the rest of the book. You can regard the momentum balance as a generalisation of the Newton law of mechanics. As a start, we consider a simple experiment with two glass bulbs connected by a capillary Figure 3.

The bulb at the left contains hydrogen species' 1' ; that at the right carbon dioxide species '2'. The whole system is at ambient pressure and temperature. The two components are moving through each other with local velocities u j and uz. The differences in velocity cause friction between the two species.

The velocities that we are talking about are the average 'diffusive' velocities of the species. These diffusive velocities should not be confused with the thermal velocities of individual molecules: they are much lower. Thermal velocities are hundreds of metres per second; the diffusive velocity in a gas might be one centimetre per second. However, the thermal velocities are for the greater part random, and the random part gives neither a contribution to the species velocity, nor to the momentum of the species.

Driving Forces 37 Fig. The amounts are proportional to the partial pressures: friction fo rce oc PIP2 u2 - uI There will also be friction between hydrogen and the wall. However, except in very narrow pores or at low pressures the interactions between the two gases are much larger!. Finally the velocity of hydrogen changes a little across the slice, but the resulting terms are also smalf.

So we neglect these terms. In a steady state the three remaining terms must cancel: We consider this in Chapter See exercise 3. Velocity effects are important in centrifuges Chapter 12 and in shock waves from detonations.

Lec 8: Multicomponent diffusion and diffusivity in solids

The right-hand side is the friction force exerted by carbon dioxide on hydrogen. The gradient of the chemical potential causes an internal force. Internal forces cause motion inside a mixture, but not of the mixture as a whole. The force we saw in the beginning of this chapter - gravity - is an external force. It does have an effect on the whole mixture. Driving Forces for a given T and p in ideal solutions Fig.

It is the basis of the rest of this book. We shall see that it is much more general than the Fick equation. The MS equation does yield the Fick equation as a limiting case for some simple, but important, diffusion problems. However, it can also lead to results that are quite different. Because the Maxwell-Stefan equation is so important, we repeat it once more below - in words Figure 3.

This has forced us to cut a corner. Our choice has been to avoid mathematical complexity as far as we can: 1. The book only considers diffusion in one direction: one-dimensional diffusion. It uses a simple engineering model of the flow near interfaces: the 'film' model. J J J ji'i The left side of the equation contains the ratio of the change of the potential across the film to the film thickness. The right side of the equation uses suitably chosen average values of composition and the velocities.

We come back to these in Chapter 4. The difference form of the equation contains all the main features of the differential equation. Also its accuracy is adequate for most engineering applications. Using it allows us to cover a broad range of mass transfer problems in a short time Figure 3.

Finally: when you have mastered the approximate form, you will be quite far in understanding the behaviour of the complete equations. We will occasionally step outside the limitations of the difference equation. In Chapter 10 we show how to use the difference equation to allow for more realistic flow models than the film theory. In a number of examples we will also look at complete concentration profiles using the differential equation. So we need to solve the Maxwell-Stefan equations simultaneously with a flow model.

The idea is that bulk fluids are turbulent: that they contain convective currents and eddies of all sizes. These cause rapid mixing of the fluid, so that concentration gradients in the bulk cannot develop. Near phase interfaces the eddies die out, and transport is only by 5 To avoid misunderstanding: the film model is not a part of the general theory of mass transfer that we are developing here. It is just a convenient starting point, and a useful model for engineering problems.

The MS-equations can be combined with any flow model. The film theory assumes that the eddies disappear at a defined distance from the interface - at the film thickness. The film is usually very thin; Figure 3. We can of course also use the film model to describe a membrane, and we shall see that it is even approximately applicable to diffusion inside solid particles. The difference form of the composition driving force is given in Figure 3. It also shows that the approximation holds over wide spans of compositions. The approximate difference formula gives better results in the MS-equations than the exact one, so we always use the approximate formula.

Apparently there are compensating errors. You are to calculate the force driving carbon dioxide out of beer or cola, if you prefer into bubbles. Also check the dimensions. The problem should only take you a minute or so to solve. Try it. Every kilogram feels the weight of a supertanker! Figure 3. This illustrates the enormous size of molecular forces in mass transfer. Although the forces are large, their effects are not dramatic. This is because the force has to propel huge numbers of molecules; the force per molecule is not extreme.

You should memorise the approximate form for the potential difference and driving force in Figures 3. R Mass Transfer. McGraw-Hill, New York. A good explanation of the film theory of mass transfer. Wiley, New York.