Quick notes on convective mass transfer

Quick notes on convective mass transfer

Cedric J. Gommes

March 26, 2014

Contents

Two portraits 2

Back to basics: convective mass transfer 4

1

Figure 1: Jean Claude Eugene Peclet (1793-1857) was a French Physicist.He was among the first students to study at the Ecole Normale Superieure,and he is one of the founders in 1829 of the Ecole Centrale des Arts etManufactures, a private engineering school in Paris. The aim of the schoolwas to train new types of engineers in the context of the ongoing industrialrevolution. Peclet is notably famous for his Traite de la chaleur considereedans ses applications, in which he introduced a dimensionless number thatnow bears his name (the Peclet number) to quantify the ratio of convectiveand diffusive heat transfer. Since then, that number has been generalisedalso to mass transfer.

C.J. Gommes, March 26, 2014 2

Figure 2: Ludwig Prandtl (1875-1953) was a German engineer who was a pio-neer in the field of aerodynamics. He did most of his career at the Universityof Gottingen. He introduced the concept of boundary layer, which is alsocentral to problems of mass transfer. In his autobiography Theodore VonKarman, who studied under Prandtl, tells the following story: “ Prandtl’slife was marked by overtones of naıvete. When Prandtl was thirty-four, hedecided it was time to marry, so he went to his old professor (Foppl) to askhis daughter’s hand in marriage. But Prandtl didn’t say which daughter. Thecanny professor and his wife had a hurried caucus and prudently decided itshould be the older one. That was fine. The marriage was a long and happyone.”


Back to basics: convective mass transfer

Fick’s law with convection

The general form of mass conservation is the same,

∂tc+∇ · J = r (1)

where c is the concentration of a given molecule rV is the rate at which itappears locally per unit time and per unit volume, and J is its flux. Whenthe fluid is at rest, the only contribution to the flux is diffusive. However, ifthe fluid is in motion, an additional convective contribution has to be added,namely

J = −D∇c+ cv (2)

where v is the fluid velocity.An alternative way to look at this consists in assuming first that there is

no diffusion, so that the flux across a given surface is is simply the volumeof fluid that crosses the surface multiplied by the concentration of moleculesin that volume. This leads to the following expression

J = cv (3)

One has to note, however that all the molecules contained in a small vol-ume element have a different velocity. Therefore, the single velocity v thatis attributed to the small volume element, which you would obtain e.g. bysolving the Navier-Stokes equation, can only be an average value. The devi-ation of the molecules velocities from the average value is responsible for theadditional diffusive contribution. The complete flux is given by Eq. 2.

Inserting the expression of the flux in the mass-conservation law, leads tothe equivalent of Fick’s second law

∂tc+∇ · (cv −D∇c) = rV (4)

Assuming that the system is stationary, that the diffusion coefficient is aconstant, and that the fluid is incompressible, leads to the following simpleequation

v · ∇c = D∆c+ rV (5)

which is sometimes referred to as a convection-diffusion equation. For exam-ple, in Cartesian coordinates that equation would be written as

vx∂xc+ vy∂yc+ vz∂zc = D(∂2xxc+ ∂2yyc+ ∂2zzc

)+ rV (6)


This is the equation that you have to solve to find the space dependence ofthe concentration, using appropriate boundary conditions. Once this is doneyou can, in principle, analyse any mass transfer problem of interest to youby integrating the the flux over the relevant surface.

Figure 3: An object with characteristic size L is immersed in a fluid flow withcharacteristic velocity U (the streamline are sketched). A molecule is presentin the fluid with concentration c0 far away from the surface of the object; achemical reaction (or any other process) occurring in the object maintains itssurface concentration to the value c1. There is no chemical reaction takingplace in the fluid. How do you calculate the flux of molecules from the fluidto the object?

Consider the example of Fig. 3. Let’s see how we would calculate thenet flow of molecules from the fluid to the object. To be explicit, what wewant to calculate is, say N , expressed in moles/s. For that purpose, we firsthave to solve Eq. 5, with rV = 0 (no reaction in the fluid). Let us put it ina dimensionless form by defining the following dimensionless variables

x = x/L v = v/U and c = (c− c1)/(c1 − c0) (7)

With these variables, the equation takes the form

Pe v · ∇c = ∆c (8)

where the dimensionless differential operators are defined as ∇ = L∇ and∆ = L2∆, and

Pe =UL

D(9)

is the Peclet number. To solve Eq. 8 you still need boundary conditions.They are

c = 0 on the surface


c = 1 far away from the surface (10)

Of course, we haven’t solved the problem yet. How could we? We haven’tspecified the shape of the object, its orientation in the flow and so on. Whatwe have learned, however, is that whatever the solution is going to be, wewill be able to put it in the following form

c = f (x, y, z;Pe) (11)

where we have written explicitly the dimensionless Cartesian coordinates.The quantity we are interested in, N , would be calculated in the usual

way, by integrating the flux over the surface of the object

N = −∫surface

dA n · (−D∇c+ cv)surface (12)

where n is the unit vector orthogonal to the surface and pointing towardsthe fluid. In case you preferred defining n as pointing towards the object,you would have to remove the minus sign in front of the integral.

Now, if the object is solid the fluid does not penetrate it, so that n ·v = 0at the surface. The only contribution to N is D∇c. This does not mean thatthe convective transport does not matter. Convection has a dramatic effecton the value taken by ∇c at any point of the surface. Only, at there is noconvective transport orthogonal to the surface. Be sure you understand this.

The total number of molecules arriving at the object can be written as

N

DL(c1 − c0)=

∫surface

dA n · ∇f (13)

where f is the unknown function given by Eq. 11. In the latter equation theright-hand side does only depend on the Peclet number, and the left-handside is nothing but Sherwood’s number.

We have therefore shown that the question of mass-transfer for a situationlike Fig. 3 can be put as

Sh = Function(Pe) (14)

Of course the function depends on the shape of the object, its orientationin the flow, etc. and on the type of flow. The streamlines sketched in Fig.3 are typical of a laminar flow. When the velocity is changed significantly,


you do not expected the streamline to keep the same shape. For example(see Fig. 4) a recirculation zone may appear, which would change the localvalues of the concentration and therefore the mass transfer. Because therelevant number for predicting the type of flow is the Reynolds number, amore general correlation may be written as

Sh = Function(Pe,Re) (15)

The Reynolds-dependence in this type of correlation accounts for a qualita-tive change in the flow patterns. For example, if you have reasons to believethat the flow is laminar in the problem you are trying to solve, check if youcan meaningfully set Re = 0 in that relation. This would bring you to acorrelation of the type of Eq. 14 which is much easier to handle. Remember:when dealing with dimensional analysis, any dimensionless number that youcan get rid of is a huge simplification.

Figure 4: The flow pattern around an object may change when the flowvelocity is changed. Compared to Fig. 3, when increasing the flow one mayexpect to see first a recirculation zone appearing (a), and for larger velocitiesthe flow will generally become turbulent (b). It is to account for the possiblechanges of the flow patterns that the Reynolds number is used, and thatcorrelations of the type of Eq. 15 are used instead of 14.

Boundary layers

The topic of the present section is what old-school chemical engineers wouldcall film theory. Film theory was an empirical way of calculating mass (andheat) transport across what modern engineering refers to as a boundary layer.

You might be familiar with the concept of viscous boundary layers fromlectures you had in fluid mechanics. In that context, the situation is thefollowing. It is a general property of viscous fluids that their velocity vanishes


on solid surfaces. If you have a rapid fluid flow past a solid structure (a wing,a hull, etc.), you may have a situation where the fluid behaves far from thesolid as if it were inviscid, with the effect of viscosity being manifest onlyclose to the solid. In that case, the fluid velocity decreases extremely rapidlyfrom its value far from the solid (say v = U) to v = 0 on the solid surface,within a thin layer surrounding the solid. This thin layer is referred to as theviscous boundary layer.

We repeatedly touched on the fact that momentum, heat and mass aretransported in much the same way. For each of them, the conservation equa-tion has a convective term of the type v · ∇x and a diffusive term of thetype ∆x. The variable x is a concentration in the case of mass transfer,the temperature in the case of heat transfer, and the velocity in the case ofmomentum transfer (see the Navier-Stokes equation). Boundary layers aretherefore not limited to viscous phenomena, they are also expected for heatand mass transfer.

Figure 5: Simple case of diffusive boundary layer: a liquid (in blue) flowsover a plate with a flat velocity profile vx = U . A soluble molecule (in red)is initially on the plate and it diffuses in the liquid as it flows. The thicknessδ(x) of the boundary layer depends on x.

The typical situation is sketched in Fig. 5. Far from the plate the con-centration of molecules is the same as upstream the plate, and the moleculesare only present in a thin boundary layer. We wish to describe that layerand calculate the rate at which the molecules leave the plate.

Because the layer is thin, the space derivatives of the concentration in thehorizontal direction are expected to be much smaller than in the vertical di-rection. This leads to the following simplification of the convection-diffusion


equationU∂xc = D∂2zzc (16)

in the region where the boundary layer exists. The boundary conditions tosolve that equation are

c = 0 for z →∞ ∀x and c = c0 for z = 0 ∀x (17)

If you think of it, Eq. 16 resembles the equation describing the time-dependent evaporation over an infinite surface, which we have solved longago. All you have to do is rename the variable t = x/U and the equationsare indeed identical. It is the contact time of the liquid with the plate thatpays the role of the time.

The exact solution to Eq. 16 with the corresponding boundary conditionsis therefore

c

c0= 1− erf

(z

2√Dx/U

)(18)

which tells you that the boundary layer thickness increases like

δ '√Dx/U (19)

The quantity of interest to us here is the rate at which molecules diffuseinto the liquid. This is calculated by integrating the diffusive flux over thesurface. More specifically, the average flux per unit area of a plate of lengthL is calculated as

JD =1

L

∫ L

0

dx D

[∂c

∂z

]z=0

(20)

Recalling that the error function is defined as

erf (x) =2√π

∫ x

0

e−t2 dt (21)

one finds easily the following result

Sh =2√πPe1/2 (22)

where the Sherwood and Peclet numbers are defined here as

Sh =JD

Dc0/Land Pe =

UL

D(23)


Equation 22 is indeed of the type of Eq. 14.Note that Eq. 22 tells you that the mass transfer depends in very specific

way on the size of the plate, on the fluid velocity, and on the diffusion coef-ficient. If you have a plate of thickness l (its length being L), the number ofmoles that leave it per unit of time is expressed dimensionally as

N =2√πl√U√L√Dc0 (24)

Note, in particular, that N is not proportional to the area of the plate. Thisis because the flux is not homogeneous over it, but it is larger at its upstreampart. Note also the surprising dependences on D and on U .

All this is qualitatively correct. However, the present analysis is notcompletely satisfactory because flat velocity profiles are unrealistic. A morerealistic approach would take into account that, in addition to the diffusiveboundary layer described here, there exists a viscous boundary layer throughwhich the velocity would increase progressively from v = 0 at z = 0 to v = Ufar from the surface. This is the reason why other correlations are often used.The underlying calculations, however, are similar to the one reported here.Only mathematically more involved.


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Quick notes on convective mass transfer