Theory of Open-Tube Distillation J
Columns J. W. WESTHAVER U. S. Patent Office, Washington, D. C.
HE extremely high viscous flow of vapor. The For total reflux and ideal reflux conditions, efficiency required of a descending reflux film is asthe H. E. T. P. of an open-tube distillation rectifying column to sumed ideal-i. e., to have column is found to be (11/48)uari/D negligible surface transfer reeffect even a partial separaD/wa, where w, is the radially averaged vetion of certain close-boiling sistance and uniform composilocity, ro the radius, and D the molecular mixtures, such as isotopes (9) tion throughout its radial thickness, and to wet evenly or isomers (IO), for example, diffusion coefficient, of the vapor stream. the entire tube surface. has raised questions regarding This relation fixes the minimum H. E. T. P. A neglect of surface transfer the limit of efficiency to which at 0.96 ro; the corresponding vapor velocity resistance can be justified by a column can be operated. is below the range of existing experimental kinetic theory considerations Much of the present informadata. At ordinary velocities the theoretical which show that the vapor tion has come from the exbombarding the liquid surface perimental side where the and certain experimental H. E. T. P.’s are is of the order of grams per practice has been to obtain in good agreement; this indicates that square centimeter per second, the performance characterisideal reflux conditions have been obtained. most of this being condensed tics of a column using a The H. E. T. P. can be reduced below that and replaced by other material made-up test mixture; these given by the formula but only by inducing evaporating at the same rate. have then served as a guide This rapid process ensures to anticipate the maximum vapor turbulence. The increase in H. E. equilibrium between the liquid degree of separation of other T. P. due to sampling and temperaturesurface and that portion of mixtures. However, unless gradient instability is discussed. The rethe vapor which moves very one knows the theoretical sults of the theory indicate the possibility slowly in close proximity to it. maximum performance under of obtaining an experimental H. E. T. P. The assumption of a uniform assumed ideal conditions, composition across the film is some doubt still remains as less than 1.0 om. the same as saying that liquid to whether the tested column diffusion, enhanced possibly has given the best possible perby liquid turbulence, is sufficient to mix the film radially, so formance. This doubt can be removed only by making a that liquid diffusion is not a transfer-limiting factor. The comprehensive theoretical analysis of all the factors which justification for this last assumption is to be found in the limit the efficiency of a rectifying column. agreement of the theory with experiment, as will appear later. The purpose of this paper is to outline the equilibrium This paper will be concerned only with the equilibrium theory of the open-tube form of column (6) and to compare operations of the tube and with a binary mixture. We will its theoretical performance with existing experimental data. first consider the equilibrium which is reached under total The open tube was chosen for the analysis in preference t o reflux. Later we will consider a production equilibrium such more complicated structures because the fewer variables as exists in continuous distillation, the pot constituting in efenable a more certain identification of those factors in the fect an unlimited source of fixed composition and the conliquid-vapor transfer process which limit the column effidenser being subjected to continuous sampling. In either ciency. One may then proceed with less hazard to an analysis case there is a dynamic equilibrium of the convection and of other types. diffusion processes such that the composition a t any fixed The following theory for an open tube can serve as the point in the column does not change with time. Hence we basis for a theory of packed tubes by noting certain correlamay write: tions between the two types. We merely state here that the turbulent nature of the flow through a packing appears to produce an effective vapor diffusion coefficient or “eddy diffusion”, reaching as high as ten times that for the laminar vapor stream of a smooth open tube. This in itself confers a (4 (b) (4 tenfold advantage on turbulence-producing structures since, whereX = mole fraction, in the vapor phase, of the more as will be seen, the vapor diffusion coefficient is an important volatile component of binary mixture efficiency-limiting factor. D = molecular vapor diffusion coefficient T, z coordinates measured from axis and base of tube, reTheory of the Open Tube spectively v = vapor velocity at points r, z We take as our model a long, vertical, heat-compensated tube with a pot a t the bottom and a condenser formed as a Equation 1 states that the time rate of change of X in a continuation of the top of the tube (1); and we assume that a static element of volume a t points r , z is due solely to three small, constant pressure gradient produces a laminar or
T
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January, 1942
compensating factors whose total for equilibrium operation is zero. Term (u) is due to the existence of a radial concentration gradient; analogous to this is the radial conduction of heat into a cylinder (3). Term (b) is due to the established vertical concentration gradient. Term (c) represents the time rate of change of X that would occur in the element of volume due to its traversal by the vapor stream, if (a) and (b) were not present. The magnitude of the vapor velocity v is a function of r. We assume that u is independent of z; this prohibits condensation within the tube. TOTAGREFLUX EQUILIBRIUM. For this case we may write a second equilibrium equation stating that there is no net forward flow of the more volatile component. Thus:
Equation 2 means that the amount of the more volatile component carried upward by the vapor stream is compensated by a like amount carried downward in the reflux and by vertical vapor diffusion. The limits of the integration cover only the vapor flow area. This involves the assumption that the liquid mole fraction Y is independent of r-i. e., that the radial composition of the liquid film at any height z is uniform. If this assumption were not permissible, the integration would necessarily extend beyond TO to the tube surface with suitable account taken of the variation of Y and liquid velocity with r. Equation 2 also assumes that vertical diffusion in the liquid is negligible. To obtain the solution of these two equations, we first substitute for v the expression 2va(ri - r2)/r%,where vo is the radially averaged vapor velocity-i. e. (flow in cc./sec.) i mi. This expression follows from Poiseuille’s law for laminar flow through a nonwetted tube of radius ro; it requires v = 0 a t r = ro. I n our model we will assume TO to extend to the downward-moving liquid surface. We thus require that the downward vapor velocity near the liquid surface can be neglected; this is the same as saying that the average liquid velocity is small compared to the average vapor velocity. Equation 1 is now integrated once with respect to r by holding 6X/6z and 62X/6z2independent of r. This is permissible for close-boiling mixtures because the vertical distribution of X along the tube axis is the same as along the liquid surface except for a small shift of X. Introducing the limits dX/dr = O a t r = 0, dX = [(2va/D)(r/2
-
rs/4rE)(6X/6z)
-( r / 2 ) ( P X / 6 ~ 2 ) ] d r (IA)
Equation 2 is next integrated by holding SX/& constant and using the device, J’UdV = UV - J V d U . Letting U = (X- Y ) ,it is seen that dU = d X , since Y may be considered constant for the level z with which we are dealing. This operation gives [(X
- Y)(rEr2/2 - r * / 4 ) ] 2 = Sh(r3-2/2
- r4/4)dX + D(6X/6z)r;/4va
(2A)
The integration of Equation 2A is now completed by substituting for dX its value as expressed in Equation 1A. On collecting terms we find
where X,=,, is the mole fraction of the more volatile component in the vapor a t the liquid surface. The difference [X Y],=,,is therefore due solely to the equilibrium separation factor Cr (i. e., the relative volatility) between the stag-
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127
nant liquid and its stagnant vapor. By definition of a,we have CY = [X/(1 - X)]r-q + IYl(1 - Y)]r-q and, for close-boiling mixtures within the range 1 < a < 1.1, this definition can be written with maximum error less than 5 per cent in the form This last expression is substituted for the left side of Equation 3 which then becomes integrable to give the vertical distribution of X in terms of z and the constants v., ro, and D. Summarizing the operations thus far, i t will be seen that the purpose has been to eliminate the variables u, r, and Y which appear in the basic equations and to arrive, finally, a t an equation giving the vertical distribution of X alongside the liquid surface. Before integrating Equation 3, the term involving S2X/Sz’ will be neglected. This term arises only because of the nonlinearity of the vertical distribution of X. Its magnitude can be estimated by differentiating Equation 3 and setting 6*X/6zaequal to zero. We then find that the term involving SeX/6z2 will be less than 1 per cent of the term involving SX/Sz, in Equation 3, when Va 4.4Dn/~&> (D/~o)[320(lnCY)I 1 - 2x1 (3A)
+
This condition will always be satisfied in the fractionation of close-boiling mixtures. Hence, neglecting the 62X/Sz2 term of Equation 3, substituting for the left side, and rearranging terms, we obtain
which, on integration between the limits z = 1 and z gives
=
0
In S = [(ll/48)varE/D f D / V ~ ] (In - ~ CY)I (5) where S is the separation factor between ends of the column of height I , defined as
s
=
[X/(l
- X)I;-z
f
[X/(l
- X)l;-a
The mole fraction X of the more volatile component may be considered here as being that of either the liquid or the vapor, so far as its vertical distribution is concerned. Equation 5 is of interest in enabling a computation of S or a,whichever is known, without previous calibration of the tube. If the assumed ideal conditions can be realized in practice, it is evident that the open tube, with the aid of Equation 5, can be used to find values for a which are so near unity that a direct measurement is impossible, as is often the case for isotopic mixtures. Considering now a short height h equal to the H. E. T. P., and noting that a column of this height would have a separation factor equal to a,by definition of the H. E. T. P. we see that Equation 5 gives e h = H. E. T. P. = (11/48)v,ri/D D/v. (6)
+
or
N
=
[(11/48)v&/D
+ D / V ~ ] 1- ’ ~
and S = (a)N where N = number of theoretical plates in height 1
(6.4) (6B)
Equation 6 is the principal result sought from the analysis since it predicts theoretically the experimental characteristic of H. E. T. P. us. throughput which has been used in the literature to discuss the performance efficiency of open tubes. On reviewing the operations leading to Equation 6, it will be seen that the term (11/48)u,r;/D results from terms (a) and (c) of Equation 1, together with the integrated transport
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
term v(X-Y) of Equation 2. The term D/va, on the other hand, is present only because of the inclusion of the vertical back-diffusion term, D6X/&,in Equation 2. The relative importance of vertical back diffusion can best be judged by setting the derivative of Equation 6 equal to zero. We then find that a minimum H. E. T. P. exists, equal to 0.96 ro or, roughly, the tube radius; i t occurs when the vapor velocity through our model has been adjusted to and held at 2.1(D/r0) cm./second. Now D , for atmospheric pressure and in the absence of turbulence, will have a value of about 0.05 sq. cm./second, and ro could have a value of, say, 0.2 cm. For this example the vapor velocity for the minimum is only 0.5 cm./second, the theoretical value of h being 0.2 cm. There are reasons, some of which appear later in the treatment of the temperature requirements, why it is doubtful if such low values of Va and h can be realized experimentally. Unless realized, the effect of back diffusion remains unimportant. The vapor velocities so far reported in the literature appear to have been at least ten times that for which a minimum H. E. T. P. due to back diffusion would occur. For these cases, then, Equation 6 reduces to It = H. E.T. P. = (11/48)~,r:/D (7) Equation 7 means that lower H. E. T. P. values are to be sought in practice by reducing the throughput, by reducing the radius of the vapor passage, and by increasing the diffusion coefficient. The latter is possible by reducing the pressure or, in effect, by inducing turbulence in the vapor stream. It appears desirable a t this point to list the assumptions involved in Equation 7 : (1) The mixture is close boiling; i. e., 1.0 < QI < 1.1. (2) The tube is operating at equilibrium under total reflux. (3) The flow of vapor is laminar. (4) The velocity of vapor flow is constant and independent of the height; i. e., the vertical pressure gradient is uniform and small enough so that little condensation occurs within the tube. (5) The average vapor velocity is large compared with the average liquid velocity. (6) The liquid film flows evenly over the entire tube surface and presents neither surface transfer resistance nor radial diffusion resistance. (7) The H. E. T. P. is large compared with the tube radius; otherwise, Equation 6 should be used in preference to Equation 7 . I n addition, we have assumed that D is independent of X. Since there is a slight dependence of D on X,the H. E. T. P. will vary slightly from the base to the top of the tube. For practical purposes, however, the average H, E. T. P. is given by Equation 7 if we use for D its vertically averaged value. APPARATUS SIMILAR TO OPENTUBE.The foregoing analysis was carried through in a similar manner for the case of laminar flow between elongated parallel plates having spacing 2w. For both of the opposed surfaces wetted by the reflux, it is found that h = (17/35) vaW2/D D/v., and when but one D/v,. The first of surface is wetted, h = (52/35) VoW’/D these formulas should hold quite closely for a column of concentric tubes (8) in which the annular spaces have thickness 2w. The second should apply to a recent centrifugal rectifying apparatus of spiral form (6) if h is measured along the spiral. The spacing 2w should be taken as the width of the vapor stream when the film thickness is appreciable. It should be kept in mind that these formulas are for equilibrium at total reflux and for the ideal reflux conditions that have been assumed. EFFECT OF SAMPLINQ ON PRODUCTION EQUILIBRIUM. The preceding formulas were developed with the prerequisite of equilibrium under total reflux. For a production equilibrium under continuous sampling from the condenser, we consider a small continuous forward flow of vapor, in excess of that returned as reflux liquid and designate as Avo, this excess vapor velocity. The new equilibrium is to be calculated from the
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Vol. 34, No. 1
basic equations by noting that Equation 1 remains the same but that Equation 2 must be modified by subtracting within the bracket the term 2A~aY(ri- r 2 ) / $ . Carrying the calculations through as before, letting h’ represent the new H. E. T. P. during sampling and R the reflux ratio, and noting that R = Va/Ava, we obtain, on final neglect of small terms, h’ = h [ l
- [R(1 - X)(In
a)]-1)-1
(8)
where h represents the H. E. T. P. a t total reflux given by Equation 6. Equation 8 shows the necessity for a high reflux ratio when attempting to obtain a continuous sample representing approximate total reflux conditions. For example, if h’ is not to exceed h by more than 5 per cent and a for the mixture is 1.1, Equation 8 shows that R must be at least 200, Equation 8 can be used to compute the total reflux H. E. T. P. when a production H. E. T. P. is known, and is applicable in this respect to packed columns. ECONOMICAL CONDITIONS.The operating conditions for economical output of a concentrated mixture of the more volatile component are seen from Equation 8 t o involve a compromise between high output with low degree of concentration (high H. E. T. P.)and low output with high degree of concentration (low H. E. T. P.). As a working basis i t could be assumed that the effective number of theoretical platea, which is a measure of the degree of concentration, has the same importance as the output rate. Then maximum production economy is obtained when N(va/R) is maximum, or when h‘R is minimum, which occurs in Equation 8 when h’ = 2h; i. e., the column should be operated at a reflux ratio which gives half the number of theoretical plates obtainable at total reflux. The value of R for this condition is 2 [ ( 1 - X ) (In a)]-l, and the corresponding output rate for the case of the open tube is vanri/R cc/second of vapor a t column temperature and pressure. The preliminary time of column operation required to build up to either the total reflux or the production equilibria that have been treated in this paper depends mainly on the holdup and the factor a. This problem was fully treated in a recent paper by Cohen (2).
Comparison with Experimental Data Figure 1 shows the experimental points for an open tube as obtained by converting data reported by Rose (7). The experimental values of va were computed from the reported liquid reflux rates measured a t the base of the tube, by assuming a 50-50 mixture and by neglecting the thicknesses of the downward descending liquid film and vapor sheath. In order for to represent the average velocity throughout the height of the tube, it is necessary that there be suitable insulation and heat compensation; such was the case for the data shown. The experimental H. E, T. P. values are those reported. I n agreement with Equation 7 the experimental points tend to fall on a straight line. The line shown, however, is that predicted by Equation 7 , using for ro the inner radius of the tube and for D the value 0.050 sq. cm./second. The latter was estimated from the viscosities of the pure vapor components as given in International Critical Tables, the “selfdiffusion” of the mixture being found from the formula D = 1.34 v / p as given by Jeans (4),1 and p for the mixture being taken as the arithmetical mean. Had a smaller value of ro been chosen t o allow for film thickness, this would not have altered the relative positions of the points and the theoretical line since the computed experimental values of B. vary as I/& The agreement shown in Figure 1 is surprisingly close, considering the theoretical assumptions involved in the Iine
January, 1942
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INDUSTRIAL AND ENGINEERING CHEMISTRY
and the experimental precautions necessary to obtain the points. However, it is possible that the true value of D may differ from its estimated value by perhaps 40 per cent in an extreme instance, so that the agreement is partly fortuitous. Figure 2 shows three sets of points for open-tube data reported by Bragg (1). These data are for columns that were
FIGURE 1. H. E. T. P. 's OF AN OPENTUBE (ROSE)us. THEORETICAL
apparently not heat-compensated although vacuum jackets were used. The experimental values of ua were computed from the reported reflux rates a t the base, by neglecting film thickness and assuming a 10 per cent ethylene dichloride90 per cent benzene mixture. The lines shown are those predicted by Equation 7, using for ra the inner radius of the tubes and for D the value 0.10 sq. cm./second. The latter is probably too high since that estimated with Jeans' formula is 0.055 sq. cm./sec. This would mean that the experimental efficiency of the tubes is higher than the theoretical. The discrepancy, if real, can be explained by assuming that, because of condensation within the tubes, the experimental values of va computed from the reflux rate a t the base are higher than the average velocity throughout the tubes. I n neither Figure 1nor Figure 2 do the experimental points lie sufficiently above the theoretical lines to show that radial liquid diffusion is a transfer-limiting factor of importance. It would seem, therefore, that our neglect of liquid diffusion in the theory is not serious even though its effect should be greatest under the experimental conditions with which we are dealing-i. e., liquid flow on a smooth wall. A closer scrutiny of this question would require accurate experimental values for D as well as data for open tubes taken under very exacting experimental conditions. The Reynolds number is below 2000 for all the data shown in Figures 1 and 2, and the assumption of laminar vapor flow is therefore justified. However, it is important to note that any means employed to cause vapor turbulence in our model would, in effect, increase the value of D and thus reduce the H. E. T. P. in accordance with Equation 7. The lower H. E. T. P.'s obtainable when the tube is packed (7) can be explained in some cases on this basis alone, although it should also be noted that the packing may reduce the effective value of TO by splitting the vapor stream with reflux liquid. It appears possible to make a direct test of the effect of vapor turbulence by operating an open tube through a range of high Reynolds numbers or by using a mechanical agitator for the vapor. The experimental values of va in Figures 1 and 2 are much too high to give a n indication of the minimum H. E. T. P. of
0.96 TO predicted by Equation 6. An attempt to locate this minimum experimentally would involve the difficulties a t low reflux rates of maintaining a continuous reflux film and of obtaining a sufficient sample for test. I n the case of isotope separation the needed rate of sampling could be minimized by mass-spectrographic analysis, however. There is also the
FIGURE 2. H. E. T.P. ' 8 OF OPEN TUBES (BRAGG) us. THEORETICAL
difficulty of stabilizing a low vapor velocity; this will be discussed next.
Temperature Requirements The velocity of vapor flow through a wetted tube is governed by the longitudinal temperature gradient. When the tube is set vertically, the total temperature gradient is the sum of a flow-producing gradient and a gradient, necessary to sustain the weight of the vapor, which must exist in the presence or absence of Row. When the weight-sustaining gradient is a large percentage of the total gradient, as is often the case for open-tube distillation columns, a small percentage variation in the total temperature gradient causes a large variation in the vapor velocity. The H. E. T. P., as seen from Equation 7, may thus become unstable. Let A p be the total pressure drop and Ap' the drop due to weight of the vapor, both expressed in dynes per sq. cm. Then from Poiseuille's law, uo = (Ap
- ApJ)rz/8ql
Writing for Ap'/Z its value 1.2 X 10-6MP/T, we have Ap/l
5
8qva/rt
+ 1.2 X 10-6 M P / T
Combining this with the approximate Clausius-Clapeyron relation Ap/AT = AHP/RT2
where AT = temperature drop between ends of tube AH = heat of evaporation, calories/mole R = 1.99 cdories/mole/' C . and introducing for v,, its value in terms of h, given by Equation 7,
TO,
and D, as
+
AT/Z = dT/dz = (R/AH)(Ta/P)(35qDh/rd 1.2 x lO-SMMP/T) (9)
Equation 9 fixes the total temperature gradient that must be maintained throughout the height of the tube in order t o
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
130
obtain a stable H. E. T. P, a t total reflux equal to h. To illustrate the magnitude and permissive variation of d T / d z , we choose a 95 per cent benzene-5 per cent ethylene dichloride mixture and assume that the H. E. T. P. is to be held to 1.0 * 0.1 cm. We estimate that AH = 7400 calories/mole and M = 79. At atmospheric pressure 7 = 1.1 x 10-4 poise, D = 0.10 sq. cm./second, and T = 353” K . At 0.1 atmospoise, D = 0.72 sq. cm./ phere pressure, 7 = 0.80 X second, and T = 293” K . These constants are substituted in Equation 9. The results are summarized in Table I for two of the column sizes of Figure 2 and two smaller sizes. TABLEI. TEMPERATURE REQUIREMENTS FOR H. E. T. P. = 1.0 Tube Radius ro,
Pressure, Atm.
0.95 0.47 0.47
1.0 1.0 0.1 0.1 0.1
Cm.
0.20 0.10
EL:
0.1
CM.
dT/dz, 9.0 9.0 8.4
3.C 4.5
C./Cm. x 10-5 x 10-1 x 10-5 X 10-4 x 10-8
Permissive Variation of dT/dz, % .t0.0017 0.027 1.1 7.9 9.8
The last column of Table I shows that extremely close temperature control would be required to extend the atmospheric pressure data of Figure 2 down to an H. E. T. P. of 1.0 cm., although stable values of this magnitude appear obtainable a t 0.1 atmosphere, and even more readily with smaller sizes of tubes. Thus, the weight-sustaining part of the temperature gradient becomes of no consequence a t low pressures and for tubes of small bore. A difficulty arises a t low mass throughput since the longitudinal heat transport is also low. Hence, even though pot and condenser temperatures are fixed, slight variations in the heat balance along the tube may occur, and it is not a permissible practice to depend on condensation or evaporation within the tube to stabilize the temperature gradient. It would seem necessary, therefore, to provide, in addition t o the usual insulation and compensating heater, some means to force the presence of the desired small temperature gradient. Since columns of high throughput have high temperature stability, this suggests that the tube in our model should be surrounded by an auxiliary column sealed from the tube condenser at its top and having an annular flow area sufficient to provide a higher throughput of about tenfold. The auxiliary column would then serve merely as a longitudinal heat conductor but would give to the tube the uniform low temperature gradient necessary for a low H. E. T. P.
Summary 1. Two basic equations are set up to describe the total reflux equilibrium operation of an open-tube distillation column, assuming ideal reflux conditions and laminar vapor flow. 2. The solution of these equations gives H. E, T. P. = (11/48)var8/D D/vo, where vo is the average vapor velocity, T Othe radius of the vapor stream, and D the molecular vapor diffusion coefficient. This result predicts a minimum H. E. T. P . of 0.96 TO which has not been detected experimentally because it occurs a t the very low vapor velocity, Z.l(D/ro). 3. At vapor velocities roughly one hundred times higher than ~ . ~ ( D / Tthe o ) , term D/vo, representing the effect of vertical back diffusion, is negligible, and in this range good agreement is found between the theoretical and several selected experimental performances. 4. The agreement indicates that, for the binary mixtures cited, liquid diffusion is not an important transfer-limiting
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factor in open-tube columns and that practically all of the resistance to transfer resides in the radial vapor diffusion process between the liquid surface and the axis of the vapor stream. 5. The agreement also indicates that in the practical range of vapor velocities the maximum performance for laminar vapor flow has been closely approached experimentally and can be exceeded only by inducing turbulence within the vapor stream. 6. Formulas are also given for the H. E. T. P. a t total reflux between elongated parallel plates, the vapor flow being laminar. 7. A formula is given for the higher H. E. T. P. resulting from continuous sampling a t the condenser, the pot representing an unlimited source. This formula is also applicable to packed tubes. 8. A formula is developed for open tubes with laminar vapor flow to show the pronounced effect on the H. E. T. P. of an unstable temperature gradient. 9. A need is indicated for accurate experimental values of the vapor diffusion coefficient for those binary mixtures commonly used in testing fractionating columns. These would enable a closer scrutiny of such discrepancies as may exist between the theoretical and experimental performances of open tubes.
Acknowledgment The writer is indebted to F. D. Rossini and A. I
