Nomenclature
reaction components stoichiometric coefficients molar concentration of A, moles/La molar concentration of B, nioles/La true molar density of B, moles/La dimensionless concentration of B, c B = CB/CB,, coefficient defined in Equation 15 coefficient defined in Equation 15 coefficient defined in Equation 16 reaction rate constant per unit volume reaction rate constant per unit surface reaction rate constant per unit mass effective reaction rate constant per unit volume reaction order with respect to solid reactant reaction order with respect to gaseous reactant number of moles reaction rate per unit volume, moles/la8 dimensionless reaction rate, T* = r / r , surface area of B , L z surface area of B per unit volume, L2/L3 surface area per unit mass, LZ/m dimensionless surface area of B , S,* = S,/S,, time, 8 distance, L
SUBSCRIPTS = =
0 W
initial condition per unit mass
GREEKSYMBOLS = exponent in Equation 13 P c = porosity literature Cited
Fujishige, H., Kogyo Kagaku Zasshi 6 6 , 891 (1963). Kasaoka, S., Sakota, Y., Kagaku Kogaku (English ed.) 4, 223 (1966). \_.._,
Lacey, D. T., Bowen, J. H., Basden, K. S., IND.ENG.CHEM. FUNDAM. 4, 275 (1963). Petersen, E. E., A.1.Ch.E. J.3 , 4 4 3 (1957). Wen, C. Y., Ind. Eng. Chem. 60, 34 (1968). ROBERTO E. CUKNINGHAM A L F R E D 0 CALVELO
Universidad Nacional de La Plata La Plata, h g e n t i . n a RECEIVEDfor review August 25, 1969 ACCEPTED February 24, 1970
Degrees of Freedom in a Simple Fractionating Column By computational methods a study was made of possible restrictions on the particular selection of the degrees of freedom available to the designer of fractionating columns. In particular, the choice of only intensive variables, all located at the same point at one extremity of a column comprising an enriching and a stripping section, was considered. Such a choice is permissible and defines the system completely.
FOR
some considerable time, interest has been displayed in determining t,he number of variables that are freely a t the choice of a designer of a fractionating column. For the case of a fract’ionating column with one feed, fixed in composition and condition, with an enriching and a stripping section operating a t a fixed pressure and heat loss per plate, and the products being removed only from the column top and bottom, these tidegrees of freedom’’ were concluded (Gilliland! 1942) to be four. The relationships used by Gilliland in calculat,ing this number were the material and energy balances and the phase relationships that must be satisfied by any column a t steady state. Later work, on a very similar basis (Kwauk, 1956), also calculated that the number of degrees of freedom was four. This argument, though similar to Gilliland’s, is much more highly formalized, and is comparatively difficult to use to obtain an answer which has not’ already been reached intuitively. The “description rule” (Hanson et al., 1962) takes the pragmatic approach that a column, once built and operating steadily, gives fixed products. It concludes t h a t t’he number of factors to be fixed in a calculation is equal in number to that required to define the plant in its physical essentials. This approach gives four for the simple case now discussed. Recent work (Hoffman, 1964) reaches the conclusion that the degrees of freedom for the case described numbers three.
Surprisingly, no comment is made on this answer being different from that of previous investigators. The fact that two different answers can be reached for a case which is almost t,he simplest, possible is disturbing, and reflect,s the considerable difficulty encountered in defining unambiguously which variables, compositions, flows, enthalpies, etc., are fixed when the problem is being specified, and which may be regarded as free variables for the designer t,o manipulate. I n particular, even assuming that four is the correct number of degrees of freedom available to the designer, no guidance is given to whether any restriction exists on the type of variable to be chosen. N a y or may not all the variables be intensive or all extensive; must some specification be made for both the enriching and the stripping column, or can all four be in one column; or is there complete freedom of choice as to which we may regard as the variables capable of being fixed by the designer? T o amplify this argument, it easily can be seen by using the very simplified example of the blcCabe-Thiele construction for a binary distillat,ion that, if one chooses to fix Z Z D , then x l D is fixed; if one then fixes, say, R and n, the enriching column is tot,ally specified, a t least with respect to compositions, and a fourth degree of freedom can be allocated only to a variable either directly associated with the stripping column-for example, zlll., or m-or possibly one indirectly Ind. Eng. Chem. Fundam., Val.
9, NO. 3, 1970
507
TOTAL CONDENSER
P
n EQUILIBRIUM STAGES
XI0
(NOT NECESSARILY INTEGRAL)
= 0,028203
= 0 104619 0 232865 x w = 0 307151 X5D = 0 327163
XZD X30
SPLITTER
z4
‘02 = 0 2 =02 :0 2
25
=02
21 22 21
‘Ii
Z
m EQUILIBRIUM STAGES
F
SATURATED LIQUID
/(NOT
NECESSARILY INTEGRAL1
indication as to whether such a choice of variables is acceptable, even if the number of variables fixed were correct. The present work attempts to find if, by suitably manipulating operating and design factors such as R , D/W, n, and m, identical five-component top products might arise from two different bottom product compositions, both of which satisfied all component material balances. If this proved to be possible, the particular choice of four compositions in the top product would have been shown to be insufficient to define a column uniquely, and arguments previously proposed would have to be re-examined. Procedure
00 d t = I 25 UI
r
TOTAL REBOILER
2 1
d5
SPLITTER
Figure 1 . Schematic layout of column and relevant numerical values
10
8v)
2.5
W I - -
a
&-
J
a
a
-P r 2 a w z tL
0
a
EXACTLY - 3
6-
x 2 W
-
W
5
4-
-2 Y
K
w m
-
2
2-
IO
5
n 4
20
I
10
I
I
12
I
1
1
14
I
16
I
I
18
NUMBER OF STRIPPING PLATES
Figure 2. Relationship between number of enriching plates and number of stripping plates for a material balance of selected group of components
so associated-for example, D/F-which implies a ratio of D / W and therefore x l w . For a ternary mixture, the situation might be that the designer might fix z30and z20, R , and n, and while in this case, it appears superficially true that he has not fixed any variables in the stripping column, he may indeed have done so. I n any case, he has chosen a mixture of intensive properties, such as composition and reflux ratio, and a n extensive property, the number of plates. It is of interest to consider the case where the four degrees of freedom used would all be intensive and all would be associated with the enriching column. For example, for a fivecomponent system, x60, 2 4 0 , 2 3 D , and 220 might be chosen as variables to be allocated to the four degrees of freedom. None of the arguments quoted by Gilliland and Reed (1942)) Hanson and Duffin (1962)) and Hoffman (1964) give any 508
Ind. Eng. Chem. Fundam., Vol. 9, No. 3, 1970
All the results described are from computation. Constant volatilities and molal flows were assumed, and Raoult’s law was assumed to be obeyed exactly. Both enriching and stripping columns were permitted to contain nonintegral numbers of plates, a column being represented within the computation as having a n integral number of perfect plates plus one plate of fractional efficiency (Murphree, vapor). As a preliminary, in order to determine a possible top product composition, a separate calculation was carried out. A column with six enriching plates and 11 stripping plates including the feed plate, and operating a t a reflux ratio of 3 and a reboil ratio of 6, was assumed. The feed was a saturated liquid of a composition indicated in Figure 1. The top product corresponding to these conditions was that used in all the subsequent calculations described. Following this preliminary calculation, the search for two different bottom compositions which might give this fixed top composition was, in essence, simple. After fixing n, m, and R, a value of D was assumed and a calculation made from the fixed top composition to the bottom composition which had to result from these operating conditions. The extent to which component 5 was out of material balance was noted. The assumed value of D was changed and the procedure repeated. From these two values of D and the corresponding unbalances in component 5, an iterative procedure rapidly determined that value of D which resulted in a (nearly) perfect material balance of component 5. At this point, the unbalance of the other four components was calculated and noted. Still retaining the chosen values of n and m, new values of R were selected and the procedure was repeated to find a value of R which, together with the still unchanged values of n and m, gave (nearly) perfect material balances on components 5 and 2. Needless to say, the value of D required to give material balances on these two components did vary as R was changed. Now, by varying n and m, always adjusting R to suit these new conditions in such a way that components 5 and 2 were always in balance, it was possible to find sets of R, n, and m (with D as a self-adjusting variable) such that components 1, 2, and 5, or 3, 2, and 5, or 4, 2, and 5 were simultaneously in balance. The necessary changes in n, m, R, and D were made internally by the computer program. A plot of n us. m for each of the above groups of balanced components is shown in Figure 2. R is shown by means of a n auxiliary axis. The lines cross a t only one point-namely, n = 6, m = 11,and R = 3 (all exactly)-which was the set of operating conditions used for the initial calculation to determine the top product to be used. The curves on Figure 2 have been cut off at m = 16. Points beyond this value become increasingly difficult to determine, but those which have been determined, although somewhat scattered, do not indicate any tendency for the three lines of Figure 2 to reconverge. Discussion
At least for the particular case studied, it was found to be sufficient to fix four compositions a t the top of the column in
order to define the system completely. Fixing three of five compositions and the column top temperature for a fixed operating pressure must also define the system. Therefore, the choice of only four intensive variables related to the top of the column is sufficient. It would appear by the symmetry of the system that the situation would be similar a t the bottom of the column. Although not proved, an extension seems likely, that any four intensive variables fixed anywhere within the column will completely define the operation. The initial choice of the number of plates ( n = 6, m = 11) to be used to obtain a possible top product composition for subsequent calculation was made with the idea that increased values of n might result in smaller values of m being required, and a sought-for second solution might be within this region. It was feared that an initial choice of n = m might result in missing a possible second crossing because of its nearness to the first. Results proved that larger values of n required larger values of m to obtain three components in material balance.
Nomenclature
D = moles of top product per unit time F = moles of feed per unit time n = number of plates in enriching column m = number of plates in stripping column R = reflux ratio W = moles of bottom product per unit time z = mole fraction in liquid phase a = relative volatility SUBSCRIPTS D = in top product 1,2,3,4,5, = component number W = in bottom product Literature Cited
Gilliland, E. R., Reed, C. E., Ind. Eng. Chem. 34, 551 (1942). Hanson, D. N.,Duffin, J. H., Sommerville, G. F., ‘Computation of Multistage Separation Processes,” p. 5, Reinhold, New York, 1962.
Hoffman, E. J., “Azeotropic and Extractive Distillation,” Interscience, New York, 1964. Kwauk, M., A.I.Ch.E. J . 2,240 (1956).
J. S. FORSYTH
Acknowledgment
Facilities and financial support from the National Research Council of Canada made the present work possible.
University of British Columbia Vancouver,B. C., Canada RECEIVED for review September 8, 1969 ACCEPTEDJune 12, 1970
Surface Tension Observation in Distilled and Saline Water Interface In qualitative experiments saline water jets injected into distilled water showed “interface tension” phenomena-i.e., the wavy forms of the jet until its breakup into liquid drops. Interface tension between a small part of a free saline water bubble and a free distilled water surface was observed. More quantitative experiments showed the increase of a saline submerged water bubble in a distilled water medium, These observations proved that a very small interface tension exists between saline and distilled water.
THE
surface tension of a salt solution in a weight concentration of 35% a t 2OoC is approximately 10% higher than t h a t of distilled water. If two immiscible fluids have surface tensions of different magnitudes in air, the surface tension would be expected in the interface of these two liquids. I n the case of saline and distilled water, the liquids can be easily mixed to form one continuous phase; therefore, the interface is not stable and only dynamic surface tension is probable. The existence of surface tension in the interface of distilled and saline water may be useful in desalination processes. I n our experiments we tried to observe surface tension phenomena from which the magnitude of the possible surface tension can be estimated. Saline water jets were injected into distilled water in a transparent vessel. The experimental system is described in Figure 1. The rate of flow was adjusted by three valves (1, 2, and 3) and by changing the position of vessel B. The discharge was measured volumetrically by measuring the water level in vessels A and D. Experiments were conducted with three nozzles, 6 cm long and 0.4, 0.5, and 1.0 mm in diameter. The injected saline water was colored by potassium permanganate, sodium salt fluorescein derivative, and Rhodamine B. The first two are most suitable, as they are soluble in a n ionic solution and a t their extremely dilute concentrations (between 0.025 and O.lyo)they do not affect surface properties. The salt concentration in the saline water was about 150/o.
Some experimental observations are shown in the photographs. It may be concluded that an apparent “surface tension” exists a t least temporarily a t the boundary between the salt solutions and pure water. I n a laminar jet (velocity between 1 and 3 cm per second) phenomena were observed which are usually associated with true interfacial tension (Figures 2, 3, 4, and 5). I n the course of these experiments we observed the instability phenomena described by Levich (1962). According to him, an injected jet has a definite length; then it becomes wavy and is broken into liquid drops owing to interfacial tension. Levich’s theoretical analysis is limited to a free liquid jet; therefore his analysis does not apply precisely to our observations. However, the observed wavy form and the breaking up of the jet are typical of surface tension phenomena, even though no true interface was present. Experiments conducted with colored distilled water jets did not show the phenomena, which indicates that the color itself did not affect “surface tension.” The experimental results are summarized in Table I. I n Figure 6 the distilled water temperature (42OC) was high enough to prevent surface tension phenomena. The experiments with the injected jets proved that surface tension exists a t the boundary between distilled and saline water, but its magnitude apparently is very small. Ind. Eng. Cham. Fundam., Vol. 9, No. 3, 1970 509
