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Reply to “comments on the book review of ‘Diffusional Mass Transfer’”
The Chemical Engineering JournaI, 10 (197.5) 163-164 @ Elsevier Sequoia S.A., Lausanne. Printed in the Netherlands
turn and mass by statistical molecular movement alone, i.e. a function of u/D, the Schmidt number. This dependence is usually represented by I& = (v/D)“. Hence
Reply to “Comments on the book review of ‘Diffusional Mass Transfer’ ” I feel that my reply is best presented by stating my views on the film theory and on the transfer unit concept. In such a way it will be left to the reader to judge the merits of the two approaches. The film theory states that under turbulent flow conditions there is no resistance to mass transfer in the bulk of the fluid. All resistance to transfer is concentrated in a “film” adjoining the interface and is produced by molecular diffusion. Consequently, from the rate equation for the molar flux NA = -(D + E) dCA/dx
K = (D/L)(v/D)” a D’-”
It can thus be shown that there is no direct proportionality between K and D1.O. This result has been obtained without any use of boundary layer assumptions. With reference to the transfer unit approach, it is imperative to start with some fundamental concepts relating to mass transfer equipment. The basic concept here is that of separating power, i.e. the change in composition suitably weighted in terms of the driving force. The separating power of a column, or the separating power which is required to produce the desired composition change, is usually expressed in terms of the number of theoretical stages (plates) or in terms of the number of transfer units. Thus, the unit of separating power is one theoretical stage for stagewise equipment and one transfer unit for differential-contact equipment. The theoretical stage is defined as a piece of equipment of which the two outlet streams are in equilibrium. This statement can also be put in a different form: a mass transfer equipment has the separating power of one theoretical plate if the change in composition yj - yj+ 1 is equal to the driving force referred to the composition of the outlet liquid, i.e. y? - yj+l, where yT= j”(xj) and j denotes an arbitrary plate numbered from the top. Similarly, with some lack of precision, it can be said that a mass transfer equipment has the separating power of one transfer unit if the change in composition is equal to the mean driving force acting over the relevant range of composition change. However, since the latter concept applies to differentialcontact equipment, its precise definition must be given in differential form. Thus, the defining equation for the number of transfer units is, for example,
(1)
it follows that (a) for the bulk, NA = const. but f = 00, and therefore dCA/dX = 0; (b) for the film, E = 0, NA = const. and therefore dCA/dx = const. The film theory can thus be regarded as a mathematical model in which the curvilinear concentration profile is approximated by two straight lines drawn tangential to the profile in the bulk and at the interface respectively. The intersection of these lines defines the thickness 1~ of the film. From the definition of the mass transfer coefficient it can be shown that K = D/ID. By assuming that the film thickness is a function of the Reynolds number only and independent of D, it would in fact follow that K depends on D l.O. In my opinion, however, such an assumption is not part of the film theory and, in addition, it is incorrect. For instance, the film thickness for heat transfer under the same configuration and at the same Reynolds number may be different from that for mass transfer. The film theory thus distinguishes three film thicknesses: ID for mass transfer, IH for heat transfer, and IV for momentum transfer. Of these, only the last can be a function of Reynolds number alone. Consequently, the film theory expression for the mass transfer coefficient becomes K = (D/lv)(l~/l~)
(3)
WOG
(4)
= dylot* -.J')
It should be noted that the definitions of the theoretical plate and of the transfer unit involve only equilibrium and operating relations, but no rate equations. Following the defmition of NOG, it is easy to define the height of a transfer unit, e.g. HOG. Thus,
(2)
Up to this point no elements have been introduced which are not part of the film theory. Thus, for Professor Skelland to be correct, l"/l~ will have to be independent of D. But we know that the ratio of film thicknesses for momentum and mass transfer is a function of the relative ease of transfer of momen-
dh = HOG &VOG 163
(5)
164
LETTERS
where h is the height of packing as a variable. Subsequently, on introducing the additional assumption of plug flow of the phases, one obtains the equation Gdy = KoG~*
- y) a A dh
(6)
from material balance on a differential section of the column. Professor Skelland, in common with a number of other workers, starts with eqn. (6) and uses it to define both NOG and Hoc. This is unsatisfactory for three reasons: (a) a single equation is used to define two fundamental concepts; (b) theoretical plate and transfer unit are analogous concepts and their definitions must follow an analogous pattern; (c) the validity of eqn. (6) is restricted to plug flow. The sole significance of eqn. (6) lies in the relation it gives between HOG, previously defined by eqn. (5), and the parameters KOG, a and G under the assumed flow conditions. If backmixing were incorporated into eqn. (6), then, under certain conditions, it could be shown that Hoc
= (HOG)~IW flow + (HOG)back
mixing
(7)
It has already been mentioned that the analogy between the concepts of theoretical plate and transfer unit requires their analogous treatment. The equivalent concepts are:
separating power: no. of theoretical plates objective parameter: no. of actual plates performance concept: plate efficiency
TO THE
EDITOR
NTU ht. of packing 1/HTU
The use of eqn. (6) to define HOG is therefore equivalent to the inclusion of the plug flow assumption in the definition of plate efficiency. The problem of Chapter 8 is really a matter of opinion. I am well aware of Professor Skelland’s conviction that the subject matter he presented merits inclusion in the book. His view will probably be shared by some readers. Others however, and I include myself in this category, may be of the opinion that a chapter on the “Design of stagewise columns from rate equations” could be utilized in a better way than in presenting a case study of the calculation of a sieve-plate extraction column. H. SAWISTOWSKI
Department of Chemical Engineering and Chemical Technolor], Imperial College, London SW7 ZBY, Gt. Britain
(Received 13 June 1975)
turn and mass by statistical molecular movement alone, i.e. a function of u/D, the Schmidt number. This dependence is usually represented by I& = (v/D)“. Hence
Reply to “Comments on the book review of ‘Diffusional Mass Transfer’ ” I feel that my reply is best presented by stating my views on the film theory and on the transfer unit concept. In such a way it will be left to the reader to judge the merits of the two approaches. The film theory states that under turbulent flow conditions there is no resistance to mass transfer in the bulk of the fluid. All resistance to transfer is concentrated in a “film” adjoining the interface and is produced by molecular diffusion. Consequently, from the rate equation for the molar flux NA = -(D + E) dCA/dx
K = (D/L)(v/D)” a D’-”
It can thus be shown that there is no direct proportionality between K and D1.O. This result has been obtained without any use of boundary layer assumptions. With reference to the transfer unit approach, it is imperative to start with some fundamental concepts relating to mass transfer equipment. The basic concept here is that of separating power, i.e. the change in composition suitably weighted in terms of the driving force. The separating power of a column, or the separating power which is required to produce the desired composition change, is usually expressed in terms of the number of theoretical stages (plates) or in terms of the number of transfer units. Thus, the unit of separating power is one theoretical stage for stagewise equipment and one transfer unit for differential-contact equipment. The theoretical stage is defined as a piece of equipment of which the two outlet streams are in equilibrium. This statement can also be put in a different form: a mass transfer equipment has the separating power of one theoretical plate if the change in composition yj - yj+ 1 is equal to the driving force referred to the composition of the outlet liquid, i.e. y? - yj+l, where yT= j”(xj) and j denotes an arbitrary plate numbered from the top. Similarly, with some lack of precision, it can be said that a mass transfer equipment has the separating power of one transfer unit if the change in composition is equal to the mean driving force acting over the relevant range of composition change. However, since the latter concept applies to differentialcontact equipment, its precise definition must be given in differential form. Thus, the defining equation for the number of transfer units is, for example,
(1)
it follows that (a) for the bulk, NA = const. but f = 00, and therefore dCA/dX = 0; (b) for the film, E = 0, NA = const. and therefore dCA/dx = const. The film theory can thus be regarded as a mathematical model in which the curvilinear concentration profile is approximated by two straight lines drawn tangential to the profile in the bulk and at the interface respectively. The intersection of these lines defines the thickness 1~ of the film. From the definition of the mass transfer coefficient it can be shown that K = D/ID. By assuming that the film thickness is a function of the Reynolds number only and independent of D, it would in fact follow that K depends on D l.O. In my opinion, however, such an assumption is not part of the film theory and, in addition, it is incorrect. For instance, the film thickness for heat transfer under the same configuration and at the same Reynolds number may be different from that for mass transfer. The film theory thus distinguishes three film thicknesses: ID for mass transfer, IH for heat transfer, and IV for momentum transfer. Of these, only the last can be a function of Reynolds number alone. Consequently, the film theory expression for the mass transfer coefficient becomes K = (D/lv)(l~/l~)
(3)
WOG
(4)
= dylot* -.J')
It should be noted that the definitions of the theoretical plate and of the transfer unit involve only equilibrium and operating relations, but no rate equations. Following the defmition of NOG, it is easy to define the height of a transfer unit, e.g. HOG. Thus,
(2)
Up to this point no elements have been introduced which are not part of the film theory. Thus, for Professor Skelland to be correct, l"/l~ will have to be independent of D. But we know that the ratio of film thicknesses for momentum and mass transfer is a function of the relative ease of transfer of momen-
dh = HOG &VOG 163
(5)
164
LETTERS
where h is the height of packing as a variable. Subsequently, on introducing the additional assumption of plug flow of the phases, one obtains the equation Gdy = KoG~*
- y) a A dh
(6)
from material balance on a differential section of the column. Professor Skelland, in common with a number of other workers, starts with eqn. (6) and uses it to define both NOG and Hoc. This is unsatisfactory for three reasons: (a) a single equation is used to define two fundamental concepts; (b) theoretical plate and transfer unit are analogous concepts and their definitions must follow an analogous pattern; (c) the validity of eqn. (6) is restricted to plug flow. The sole significance of eqn. (6) lies in the relation it gives between HOG, previously defined by eqn. (5), and the parameters KOG, a and G under the assumed flow conditions. If backmixing were incorporated into eqn. (6), then, under certain conditions, it could be shown that Hoc
= (HOG)~IW flow + (HOG)back
mixing
(7)
It has already been mentioned that the analogy between the concepts of theoretical plate and transfer unit requires their analogous treatment. The equivalent concepts are:
separating power: no. of theoretical plates objective parameter: no. of actual plates performance concept: plate efficiency
TO THE
EDITOR
NTU ht. of packing 1/HTU
The use of eqn. (6) to define HOG is therefore equivalent to the inclusion of the plug flow assumption in the definition of plate efficiency. The problem of Chapter 8 is really a matter of opinion. I am well aware of Professor Skelland’s conviction that the subject matter he presented merits inclusion in the book. His view will probably be shared by some readers. Others however, and I include myself in this category, may be of the opinion that a chapter on the “Design of stagewise columns from rate equations” could be utilized in a better way than in presenting a case study of the calculation of a sieve-plate extraction column. H. SAWISTOWSKI
Department of Chemical Engineering and Chemical Technolor], Imperial College, London SW7 ZBY, Gt. Britain
(Received 13 June 1975)