Settling of Solid Suspensions under and between Inclined Surfaces Eli Zahavi and Eliezer Rubin* Department of Chemical Engineering, Technion-Israel
Institute of Technology. Haifa, Israel
A n experimental and theoretical study was conducted on t h e effect of inclined planes on batch settling of solid-liquid suspensions. T h e experiments were conducted with clay suspensions in vertical, rightangled trapezoid a n d parallelogram shaped vessels. Within t h e range of variables studied (suspension concentration, 2-10% by volume; angles of inclination, 35-65'; initial suspension height, 20-40 cm;
base length, 9.5-98 c m ) it was found that initial settling rates in the presence of inclined planes can be higher by a factor of u p to 3 compared to settling rates in regular vertical vessels u n d e r t h e same conditions. O n t h e basis of experimental observations with and without dye injection, t h e phenomena causing this increase in settling rate are described qualitatively. A quantitative theoretical model is presented capable of predicting settling rates in t h e presence of inclined planes over wide ranges of independent variables, using settling rate vs. concentration data for regular vertical vessels, and t h e settling rate obtained from a single experiment conducted in a trapezoid shaped vessel. Experimental data are presented and discussed. Very good agreement between theory and experimental data was obtained. T h e present work, though conducted in batch systems, indicates that introduction of inclined planes may increase considerably t h e capacity of semicontinuous and continuous settlers.
Introduction It was first reported by Boycott (1920) that solid suspensions in liquids settle faster in inclined test tubes than in vertical tubes under the same conditions. Since then relatively little has been published on this and related phenomena, i.e., settling of suspensions in the presence of inclined surfaces. Boycott, who observed the enhanced settling in experiments with blood, attributed it to Brownian motion effects. He also observed that initial settling rates increased with decrease of angle of inclination (to the horizontal) of the test tubes. Following additional reports of experimental results with mostly descriptive and qualitative observations and interpretations (Bergezeller and Wastel, 1923; Linzenmeir, 1925; Lundgren, 1927, 1928), Nakamura and Kuroda (1937) were the first to present a quantitative theoretical model. They considered settling between two parallel inclined planes. In their model they assumed that the clear liquid region at the top above the suspension originates from two sources (Figure 1): vertical settling of particles in the upper part (from plane AB to plane E F in Figure 1) and settling of particles under the upper inclined plane (from plane AC to plane ED in Figure 1). They assumed that the clear liquid from the latter source ascends a t infinite velocity to join the clear liquid region in the upper part. The combined effect is a faster settling rate (plane A'B' instead of plane E F in Figure 1). Assuming also uniform suspension concentration, the following equation, applicable to the beginning of settling. was obtained b It is important to emphasize that this model takes into account only the geometry of the vessel and neglects effects of solids concentration changes and actual ascending liquid velocities under the inclined plane. Kakamura and Kuroda conducted settling experiments with blood suspensions in 61 mm long square test tubes inclined at 45". Fairly good agreement was obtained between the experimental results and their model. Following Nakamura and Kuroda's work, results of several investigations with various solid-liquid suspensions have been reported (Kinosita, 1949a,b; Inoge, et al , 1954; 34
Ind. Eng. Chem., Process Des. Develop.,Vol. 14, No. 1, 1975
Pearce, 1962). These investigations. conducted mostly in inclined test tubes of various shapes and sizes, indicated usually considerable deviations from the Nakamura and Kuroda theory. Attempts at correcting the Nakamura and Kuroda model by introducing corrections to eq 1 have also been reported. Graham and Lama (1963) introduced a correction factor multiplying the right-hand side of eq 1. whereas Ghosh (1963) and Vohra and Ghosh (1971) introduced correction factors only to the term resulting from the contribution of the ascending clear liquid under the inclined surface (the right-hand term in eq 1). In all these attempts it was found that the correction factors were far from constant and depended on various parameters. They seem, therefore, to be of little practical value. According to a different model, proposed by Oliver and Jenson (1964) the ascending clear liquid under the upper inclined surface forms a triangular channel. However, they solved the problem by assuming a model which is based on countercurrent steady-state flow of two liquids of similar densities between parallel plates at equal fluxes. They obtained two simultaneous differential equations which were solved with an analog computer. Only fair agreement was obtained between their theory and their experimental results. It is clear from the available literature that not only has little been published on the subject, but that most of the published data are restricted to test tubes of various sizes and shapes. In addition, the few published theories are usually only in fair to poor agreement with experimental data. The phenomenon of the enhanced settling rate of suspensions observed in inclined tubes can be used to advantage in industrial settling equipment. The design as well as the understanding of the phenomena involved in semicontinuous and continuous sedimentation equipment is based on observations and data obtained from batch experiments. The limited available literature on the subject, interesting and important as it may be, gives rise to many questions stemming. for example, from interactions due to the proximity of the walls in test tubes. In addition. the limitations of available theories reduce their usefulness even for predicting batch experimental data and practically preclude their utilization in the design of continuous sedimentation equipment.
D
Figure 1. Theoretical model of S a k a m u r a and Kuroda.
1 4 /1 8,
nation of these two effects (parallelogram). Special car? was taken to avoid any leakage of suspension between the three compartments. In addition to the two vessels mentioned a b o v e . a third vessel was used for dye injection experiments and carefiil qualitative observation. This Perspex fixed-dimension vessel had the form of a right-angled trapezoid. Its dimensions were: height 24 cm. base length 29 cm. and angle of inclination 51". Measurements of suspension height as a function of time were made under various conditions after preparing the desirable suspension with special care to mix it until homogeneous before starting the experiments. The rar, ge of variables investigated was: base length. 9.5-49.5 cm in the parallelogram and 68-98 cm in the right angled trapezoid: angle of inclination, 35-65" to the horizontal; S J S pension concentrations, 2-20% by volume in the vertical vessel and 2-1070 by volume in the trapezoid: 1ni:ial height of suspensions, 20-40 cm. Qualitative Observations
2,
Figure 2. Schematic diagram of experimental system with inclined planes: (1) right-angled inverted trapezoid compartment; ( 2 ) parallelogram compartment: ( 3 ) right-angled trapezoid compartment; (4)supports for inclined plates; (5) inclined plates: (6) drain valves.
In the present work, undertaken in order to study batch settling of suspensions under and between inclined planes. we try to overcome some of the questions and develop a more useful theoretical approach. First, experimental techniques are described which enable one to obtain separately the contributions of phenomena under and above inclined surfaces without the disturbing wall effects which are unavoidable in test tubes. Following are a qualitative description of observed phenomena and development of a new theoretical model based on these observations, and finally experimental results are presented and discussed. It is hoped that this work and its conclusions will throw a new light on the subject and present a step in the possible utilization of the enhanced settling rate in the presence of inclined planes in industrial settling equipment. Experimental Section Clay (specific gravity 2.71 g/cm3 and mean particle diameter of 0.022 m m ) suspensions in water were used in all the settling experiments. A vertical rectangular vessel made of Perspex, 20 x 20.7 cm base and 45 cm height, was used for simple vertical settling experiments. Another Perspex vessel, 20 cm wide, 130 cm long, and 60 cm high, with two inclined Perspex plates in it (20 X 60 cm) was used for settling experiments in a right angled trapezoid, parallelogram and inverted right angle trapezoid (Figure 2). With an appropriate arrangement it was possible t o change the distance between the plates and their angle of inclination. Experiments with this equipment enabled us to examine separately the behavior and settling rates of the suspensions under an inclined plane (right-angled trapezoid), above an inclined plane (inverted right-angled trapezoid), and between inclined planes, i e . , the combi-
Before developing an appropriate theoretical model which can be used for prediction of settling rates and presenting quantitative results, a qualitative description of phenomena is required. Careful observations with and without dye injection (potassium permanganate) *ere made in the right-angled trapezoid. The first and most pronounced observation was the immediate formation of a thin (about 1-2 mm thick) layer of clear liquid ascending relatively very fast under the inclined plane of the trapezoid. This layer of clear liquid separating between the suspension and the inclined plane, and observable with and without dye injection, see.ns to be of practically constant thickness. In order to observe the phenomena under the inclined plane and the origin of the clear liquid layer. dye n'as injected during the settling process at different pla-es. It was possible to follow the ascending liquid path st arting from the suspension. At the beginning, liquid rises very slowly between the particles in the suspension up to the inclined plane (AB in Figure 3R). Afterward, the clear liquid flows very rapidly under the inclined plane up to the upper liquid-suspension interface (BC in Figure 3E). This shows that in contrast to the Nakamura and Kumda assumptions most of the clear liquid laver originates from the suspension itself and not from settling of solids under the inclined plane. Dye injection near the inclined plane indicated that the dye spreads upward and also horizontally in the direction of the inclined plane (Figure 3R). As the settling process continued. it was noted that waves formed on the suspension surface just under the inclined plane (Figure 3A). Thus many small particles moved upward (Figure 3C). It was also noticed, in dye injection experiments, that the clear liquid ascendi-ig under the inclined plane caused some local turbulence TNhen entering the upper liquid-suspension interface. This phenomenon may be of particular importance with higher initial suspension heights and small liquid-suspension interfacial areas. Increasing solids concentration did .not affect the patterns and behavior of the ascending liquid under the inclined plane but caused, as expected, a decrease in the liquid velocity through the suspension (AB in Figure 3B). Based on the qualitative observations, the niechanism of the formation and flow of clear liquid under the inclined plane may be described as follows. Consider a vessel in the form of a right-angled trapezoid filled with a homogeneous suspension. Just in the beginning of the batch settling process a thin layer of clear liquid forms Ind. Eng. Chern., Process Des. Develop., Vol. 14, No. 1 , 1975
35
a
+-
’ I
Figure 4. Schematic presentation of settling in a trapezoid according to the proposed model.
75
97
In)
IC1
Figme 3.lSchematic presentation of some phenomena in a rightangled trapezoid.
under the inclined plane. Once this thin layer of clear liquid forms, additional clear liquid is supplied primarily from two sources: ( a ) liquid moving upward through the suspension, because of increase in solids concentration at the bottom, and reaching the inclined plane region. For later reference we shall call this contribution of clear liquid ,:he “settling effect;” ( b ) at a given horizontal plane there exists a local difference in pressure between the liquid in the suspension of the higher density and the clear liquid layer under the inclined plane. This pressure difference causes liquid to flow and penetrate from the suspension into the clear liquid layer through a thin solids layer forming the upper boundary of the suspension under the inclined plane. The thin solids layer serves as a kind of porosive plate. For later reference we shall call the contribution of clear liquid by this phenomenon the “filtration effect .” The formation of clear liquid because of settling under the inclined plane seems to be of minor importance. The fast upward flow of the clear liquid layer causes considerable disturbance of the suspension and seems to preclude particli? settling under the inclined plane. The fast upward motion of the thin layer of clear liquid is caused by the pressure exerted by the higher density suspension. Theoretical Approach The quantitative macroscopic theoretical model and derivations presented herein for batch settling are based on the physical picture described qualitatively before. The main feature of the observations is that the increase in settling :rate results from the addition of clear liquid originating from the flow under the inclined plane, to the clear liquid originating from settling of solids a t the liquid-suspension interface a t the upper part of the vessel. Designating L’\ as the contribution of settling rate a t the liquid-swspension interface. and u,, as the contribution of clear liquid from under the inclined plane, the actual settling rater. a t a given moment in the presence of an inclined surface, u i n , is 2’in
=
2‘”
+
L3D
(2)
uv, which depends on suspension concentration, can be obtained from settling experiments in vertical vessels in the regular manner, i.e., from the slope of h us. t. In order to develop appropriate expressions for u,,, let us designate by Q as the average flow rate of clear liquid per unit area ,of inclined plane. This definition of 4 enables to introduce geometrical factors through S , the surface area of 36
Ind. E.rig. Chern., Process Des. Develop., Vol. 14,
No. 1 , 1975
inclined plane up to the upper boundary of clear liquidsuspension. S can be calculated easily. Note that by definition 9 vanishes when there is no inclined plane. Thus, a t a given suspension height, the instantaneous total flow rate of clear liquid under the inclined plane is Qp = q * S
(3)
and
(4)
For the beginning period of settling, i e , when the changes in suspension height are relativelv small, may be assumed essentially constant regardless of suspension concentration. This assumption is supported primarily by the quantitative experimental result presented later. In addition, this assumption can be partly explained on the basis of the two main reasons for clear liquid supply under the inclined plane which affect q in the opposite direction with change in concentration. Increasing initial suspension concentration decreases the “settling effect .” On the other hand, the “filtration effect” increases because of the increase in suspension density and hence the hydrostatic pressure difference between suspension and clear liquid region under the inclined plane. The net result is a fairly constant 4 . q is expected, naturally, to depend strongly on the types of materials composing the suspension. However, for a given material, the experimental supported assumption of constant Q is very helpful and useful for calculations capable of reproducing experimental data. Thus, whereas q has to be determined experimentally, its being a constant indicates that only one experiment with an inclined plane suffices for its determination and utilization for wide changes of independent variables for a given type of suspension. Knowing g enables us to predict the initial settling rate, u l n , for every batch settling experiment in vessels of different geometries and solids concentrations. Derivations for right-angled trapezoid and for a parallelogram are presented below. Right-Angle Trapezoid. Let us examine the upper layer of the suspension in a batch settling process in a right angle trapezoid of unit width (Figure 4 ) . At the beginning the suspension interface is at height ho from the bottom. The concentration of solids in the upper layer is co The settling velocity of the suspension at that level is
(uI,)o is obtained from regular data of vertical settling experiments a t concentration CO.
but and
Substituting in eq 5 we obtain
The total settling velocity a t x cm from the top is
As the settling process continues, each suspension layer descends to a larger cross section. Assuming equal and uniform concentration a t any horizontal cross section, the suspension concentration decreases to different values at every height. The upper suspension layer descends to a larger cross section area and the effective inclined surface area S becomes smaller. Solids concentration in the upper laver will be after a descent of x cm
(10) However
A, = h
(Ii, -
-
X) cot
Q
(11)
and therefore c,
=
cot
11 - 11,
c”?;
- (11, -
e
x) c o t s
Since c I < co. the vertical settling velocity of the particles rises from ( u v ) to ( o \ . ) ~which is the vertical settling velocity of the suspension at concentration c x . The effective inclined surface area is now S, = ( 1 ~ ~x)/ s i n
e
(13)
and
The total settling velocity at x cm from the top is
(22) By comparing ( u i n ) x with ( c i n ) O it can be seen that > ( L ’ \ ) o and ( u , , ) < ~ (uIl)o. Some Conclusions from the Theoretical Derivations. It is instructive t o analyze some of the equations obtained with respect to the effect of several independent variables on the settling rate, Decreasing the angle of inclination does not affect u,, which depends only on solids concentration. However. it increases the inclined surface area, S, and therefore Q,) and uI,. The actual settling rate ( u i n ) 0 , will increase according to eq 9 and 18 for a trapezoid and parallelogram, respectively. Decreasing base length does not change uv and S, and therefore Q1, will remain constant. On the other hand. the cross section area of the vessel becomes smaller and therefore u,) will increase accordingly (eq 4 ) . The actual settling rate, ( u i n ) o , will increase according to eq 9 and 18 for a trapezoid and a parallelcgram, respectively. Increasing the initial height of the suspension causes an increase in S, and u l n will increase because of the consequent increase in ~ 1 ~ ) Increasing . solids concentration will cause only a decrease in u, and a consequent decrease in u i n accordingly. T h e Improvement Factor. It is convenient to define an improvement factor as the ratio between initial settling velocities in a vessel with an inclined plane and a vertical vessel under the same initial conditions ~
1
,
~
.
(15) When comparing eq 9 and 15 it should be noted that (uL)-( > (u,10 and (urj).,- < ( L ’ , , ) o . Parallelogram. Let us examine the upper layer of the suspension in a batch settling process in a parallelogram of unit width (Figure 5 ) . At the beginning the upper layer of the suspension is a t ho cm from the bottom and its concentration is c g . The settling velocity of the suspension is (‘.in)n
=
(2sv)o
+ ({‘p)o
(16)
and S , = Iz,/sin 8:A, = b
therefore
-
(17)
y.h,/sin (18) b When the settling process continues, part of the particles of the suspension settle on the lower plane of the parallelogram and stay there, or slide down. In any case, they are “out of the game” in batch settling. After settling x cm from the top, the portion of the suspension that settled on the lower plane is x cot H . The rest, b - x cot 8 , forms a new liquid suspension interface equally distributed in the cross section b. As a result. the solids concentration drops and the vertical settling velocity rises from ( u \ ) o to ( u \ ) ~ , which is the settling velocitv of particles a t cx. (1,inh
=
(7,v)o
+
It is worth noting that according to the theoretical approach increasing solids concentration causes a decrease in ( u l n ) o but an increase in the improvement factor, since only uv in eq 23 depends on solids concentration and its value decreases. N a k a m u r a a n d Kuroda Model for a Right-Angled Trapezoid. Most of the published papers since Nakamura and Kuroda proposed that their model compare experimental results with this model or variations based thereon. Therefore, in the present work experimental data were compared with the present model and the Xakamura and Kuroda model. The Nakamura and Kuroda model was derived for settling between two parallel planes. In order to be able to compare this model with the present work, it is necessary to extend their derivations to a right-angle trapezoid. Nakamura and Kuroda assumed that the clear liquid which formed under the inclined plane ascends immediately to loin the upper clear liquid. Referring to Figure 6, according to their assumption the volumes ACDE + ABEF = ABB’A’, where A’B’ is the new liquid suspension interface after a time period o f t + dt Calculations of the above volumes and rearrangement of the equations yields for the beginning of the settling process
The effective inclined surface area is smaller Ind. Eng. Chem., Process Des. Develop., Vol. 14, No. 1. 1975
37
--
'.o'b---l
I
0
1 5
I
1
20
10
',"Ol"me
Figure .5. Schematic presentation of settling in a parallelogram according to the proposed model.
1
- T 1 - 7 - - T - - 7
"T----t \
'1.
Figure 7 . Effect of solids concentration on settling in vertical vessel: ho = 35 em.
r
I
Figure 6. Theoretical model of h a k a m u r a and Kuroda for a trapezoid.
Results and Discussion Settling in a Vertical Vessel. The batch settling experiments in vertical vesse1.s were conducted primarily for comparison of settling rates with those obtained in vessels containing inclined planes, and in order to utilize the data for the theoretical equations. The two independent variables examined were solids concentration and initial height of the suspension. Initial settling rates as a function of concentration are shown in Figure 7 . The initial settling rates were obtained from the slopes of the straight lines of plots of suspension height us. time during the constant rate period. The descent of settling velocity with increase in concentration is exponential, within our experimental range; i e . , a straight line is obtained when the experimental points in Figure 7 are plotted on semilogarithmic paper. The results of settling experiments with different initial suspension heights indicate that the initial height has very little effect on settling rate. Analysis of batch settling processes in vertical vessels was made by Kynch (1952). Since initial suspension height hardly affected settling rates, our clay suspensions were practically "ideal" according to Kynch assumptions; L e . , the settling rate depends only on solids concentration. If a suspension is not "ideal" its sludge is compressible and the settling rate rises with initial height (Dick et al., 1967). Settling in the Presence of Inclined Planes. The parameters which were examined in these experiments, conducted with right-angled trapezoids and parallelograms, are base length, angle of inclination, solids concentration, and initial height of the suspension. Typical experimental results for suspension height us. time are shown Figure 8 for a right-angled trapezoid and in Figure 9 for a parallelogram. Appropriate curves obtained in the vertical vessel are also shown for compari38
Ind. Eng. Chem., Process Des. Develop., Vol. 14, No. 1, 1975
0
I
I
4
i
I
1
a
I2
16
1, mi"
Figure 8. Change of suspension interface height as a function of time: c = 5% by volume; ho = 30 em: 7 , vertical vessel; i c , ) ~ = 1.1 cm/min; A, right-angled trapezoid: 0 = 65': b = 68 cm; (cIII)u = 1.5 cm/min: 0 , right-angled trapezoid: H = 35": b = 68 cm: ( u , , , ) ~= 2.25 cm/min.
1 , mi"
Figure 9. Change of suspension interface height as a function of time: c = 5% by volume; ho = cm: 7 , vertical vessel: i t ' , ) o = 1.1 cm/min; 0 , parallelogram: f ) = 35"; b = 1 9 . j cm: (vlli)o = 3 cm/ min; A , parallelogram: H = 65": b = 19.5 cm, (cill)0 = 1 . i cm/ min .
son. The considerable improvement in initial settling rate in the presence of inclined planes is self-evident. From both figures it is clear that a substantial part of the curves obtained during the initial settling period is straight, indicating a constant settling rate for a pronounced period of time for our suspensions. It should be pointed out that eventually, after a long enough settling time, when the settling of suspension terminates, all the curves will coincide. The effects of base length and angle of inclination on the improvement factor are shown in Figures 10 and 11 for
I
c 70
1
90
80
b . tm
Figure I O . Effect of trapezoid base length and angle of' inclination on t h e improvement factor: c = ,5% by volume: ho = 30 cm: i i . i ) o = l.lcm/min: . . H = 3 5 " : 0 , H = 4 . i " ; ~ . H = R 5 " : ~ , H = 6 5 " .
I
1
I
I
20
25
30
35
I
bo, . I (, > I
1 s t
i a
1
I 80
,
, 90
I
,
I
100
b,\ \
I
I
I
I
i
L
I i
\\
'\
i
L
\ - I
i -I
I
I,'
1
1
1 4
1
I
r.ro1um.
F i g u r e 14. Effect of parallelogram base length on the improvement factor: c = 5% by volume: ho = 30 cm; (vV)o = 1.1 cm/min; B = 35"; 0 , experimental; --, presently proposed model; - - - - - -, Nakamura and Kuroda model.
ently proposed theoretical model, and the Nakamura and Kuroda model. The striking disagreement of the Nakamura and Kuroda model with the experimental points is evident. This disagreement stems from the fact that the Nakamura and Kuroda model does not consider changes in solids concentration during the settling process. Finally, experiments were conducted in the inverted right-angled trapezoid. In this case solids settle on the inclined plane. The solids may slip down or just stay on the plane. In all cases these two possibilities hardly affect settling rate in batch experiments; however, they are of utmost importance in semicontinuous or continuous settlers with inclined planes. In continuous operation, if the solids accumulate on top of inclined planes the equipment will be plugged after a short time of operation. Experiments conducted with the inverted right-angled trapezoid indicated that a t angles of inclination larger than about 43" the solids slip down along the inclined plane.
Conclusions The conclusions from the present work may be somewhat restricted to the experimental system used, as far as quantitative data are concerned. However, the conclusions are quite general with respect to observations and approach. Within the range of variables studied it was found that initial batch settling rates in the presence of inclined planes can be higher by a factor of up to 3 compared to settling in a regular vertical vessel under the same conditions. The increase in settling rate in the presence of inclined planes originates from the formation of a thin clear liquid laver under the planes ascending relatively very fast to join the clear liquid region above the suspension. Clear liquid is supplied constantly into this thin layer from the suspension. Initial settling rate in the presence of inclined planes, relative to settling rate in the absence of the planes (improvement factor), increases with decrease of angle of inclination or distance between planes and with increase of initial suspension height or suspension concentration. The theoretical model proposed is proved to be a useful, experimentally motivated method which gives good agreement between calculated settling rates and experimental data for vessels with inclined planes. Using this model, the experimental data necessary for prediction of settling rates, 40
Ind. Eng. Chem.,
Process Des. Develop., Vol. 14, No. 1 , 1975
I
I
a
b
I
I 10
I
F i g u r e 15. Effect of initial solids concentration in a trapezoid on settling rate: ho = 30 cm; 0 = 45': b = 68 cm; 0 , experimental; _ _ , presently proposed model: - - - - - -, Xakamura and Kuroda model.
in the presence of inclined planes, over wide ranges of independent variables are: settling rate as a function of concentration in a regular vertical vessel and settling rate obtained from one settling experiment in the presence of an inclined plane; for example one experiment in a rightangled trapezoid. The theoretical model may fail because of mixing and turbulence effects when the distance between planes becomes relatively small. For the clay suspensions studied, solids settling on a plane inclined above about 43" will slip down. Such angles of slip are unimportant as far as batch settling rate is concerned; however, they are very important for the design of continuous settlers with inclined planes. The present work, though conducted in batch system, indicates that the introduction of inclined planes may increase considerably the capacity of semicontinuous and continuous settlers. Experiments in this direction are presently in progress.
Nomenclature A = horizontal cross section area, cm2 b = horizontal distance between planes, cm c = concentration of solids in suspension, g/cm3 F = correction factor, dimensionless h = height of suspension-liquid interface from the bottom, cm Q,, = flow rate of clear liquid under inclined plane, cm3/min q = average flow rate of clear liquid per unit area of inclined plane, cm3/cm2 min S = area of inclined plane from the bottom to suspension-liquid interface t = time u = settling rate of suspension,cm/min u p = settling rate of suspension-liquid interface because of supply of clear liquid from under the inclined plane, cm/min x = distance of suspension-liquid interface from liquidgas interface, cm
Subscripts in = in a vessel with inclined planes 0 = initial p = because of inclined plane v = vertical, without inclined plane x = a t distance x from liquid-gas interface Literature Cited Bergezeller. L.. Wastel, F. H..Biochem. Z., 142, 524 (1923) Boycott. A. E., Nature. 104, 532 (1920). Dick, R. I . , Ewing, 6. B., J . Sanit. Eng. D P . .Amer. SOC. Civil Eng.. 93, SA49 (1967).
Ghosh, B . . lndian Chem. Eng. Trans., 2, (1963). Graham. W . . Lama, R., Can. J. Chem. Eng., 41, 31 (1963). inoge, K.. Uchibori, T., Katsuai. T., KolloidZ., 139, 167 (1954) Kinosita. K.. Mem. Fac. Eng. Nagoya, 1, 15 (1949a). Kinosita. K , J. Colloid Sci.. 4, 525 (1949b), Kynch, G. K.. Trans. Faraday Soc., 48, 166 (1952). Linzenmeier. G., Munchen. Med. Wochenschr,. 5 (1925). Lundgren, R.. Acta. Med. Scand.. 67, 63 (1927). Lundgren. R . . Acta Med. Scand.. 69, 405 (1928)
Nakamura H., Kuroda, K., Keiio J. Med.. 8, 256 (1937). Oliver, D. R., Jenson. V . G., Can. J. Chem. Eng., 42, 191 (1964) Pearce, K. W., Third Congress of European Federation of Chemical Engineering London, A30 (1962). Vohra. 0. K., Ghosh, B., lndianchem. Eng.. 13, 32 (1971).
Received for reuieu: .January 17, 1974 Accepted August 27, 1974
Design of a Catalytic Reactor-Separator System with Uncertainty in Catalyst Activity and Selectivity Gary J. Powers* and Jerald F. Mayer' Department of Chemical Engineering. Carnegie-Meilon University. Pittsburgh, Pennsylvania 75273
Uncertainty in the design of a piece of process equipment often resides in a number of design parameters such as physical property data and design correlations. The uncertainty in these parameters affects both the equipment's cost and performance and the cost of other processing units to which it is connected. A method based on the expected value criterion is used to obtain the optimum overdesign factor for a catalytic reactor-separator system. In t h e example the catalyst activity and selectivity are uncertain
Introduction The design of chemical processes requires knowledge of thermodynamic properties, rate constants, transport coefficients, etc. Quite often, however, exact values for these parameters are not available, and uncertainty must be taken into account in the design. Kittrell and Watson (1966) considered the design of a chemical reactor where the value of the reaction rate constant was uncertain. They used the expected value criterion and minimized expected cost. Wen and Chang (1968) applied the expected relative sensitivity criterion to the same problem. Watanabe, et al. (1973), analyzed the same problem using a more sophisticated technique based on statistical decision theory. All these studies indicated that the reactor should be overdesigned to compensate for the uncertainty in the rate constant. The optimum amount of overdesign varied with the relative uncertainty and also with the objective function used by the various authors. Villadsen (1967) has studied the overdesign of distillation columns. These studies considered uncertainty in only one parameter and determined the effect of this uncertainty on the design of a single item of processing equipment. In the design of a complete process it is important to consider how uncertainty in the design of one part of the process will affect other parts of the system. In certain instances propagation of uncertainty may make it necessary to add or remove complete parts of a process. These design changes can be very costly both in terms of equipment and installation costs and the losses due to business interruption. Hence, it is important to consider more than one part of a process when analyzing the effects of uncertainty on the design. In the design of a process uncertainty is very seldom confined to just one variable in one piece of equipment. The design of even a simple piece of equipment often in-
volves considering several uncertain parameters whose effects may be in opposition. For example, consider the design of a vacuum distillation column separating a temperature sensitive mixture. Uncertainty in tray efficiency leads to overdesign through the use of more than the optimum number of trays computed assuming complete knowledge. These trays are added to avoid penalties associated with using reflux ratios higher than the optimum. However, in the separation of temperature sensitive mixtures the temperature in the reboiler is often limiting. The uncertainty in the temperature a t which decomposition occurs often suggests using a temperature lower than the maximum allowed. The temperature in the reboiler of vacuum distillation columns is fixed via the phase rule by the pressure in the reboiler. The higher the pressure the higher the temperature. The pressure in the reboiler depends on the number of trays in the column. The more trays the higher the pressure in the reboiler. Hence, for this situation, uncertainty in tray efficiency indicates that more trays should be used while uncertainty in the decomposition temperature requires fewer trays. The relative costs of trays and utilities us. product decomposition will determine the overdesign required. The purpose of this paper is to illustrate the effect of uncertainty in two parameters on the optimum size for a single item of equipment which has an effect on the other pieces of equipment in the process. An example is presented for a catalytic reactor in which both the activity and selectivity of the catalyst are uncertain. The problem discussed is greatly simplified, but it demonstrates how uncertainty in a second variable can significantly affect design decisions. The interaction of the reactor with a separator in the process illustrates how uncertainty in one piece of equipment can propagate through a complete process. Problem Statement
' Present address, Chevron Research, Richmond, Calif
Consider a reaction in which a catalyst is used to conInd. Eng. Chem., P r o c e s s Des. Develop., Vol. 14, No. i , 1975
41
