INDUSTRIAL AND ENGINEERING CHEMISTRY
940
On the large construction, packed with barium carbonate, a mixture of 25 liters of oleic acid and 60 liters of ricinoleic acid in 20 liters of ether were similarly separated, both compounds being obtained in highly purified form by elution with moist ether. Recovery of all constituents except the ether was nearly quantitative. Also un the large construction 100 liters of a t h y alculiolic solution containing 25 liters of octan-2-01 and 18 liters of methyl hexyl ketone m r e passed through activated charcoal and e l u t d with moist alcohol under rotation, when the two constituents could be collected in separate fractions. These contained 90c; of the higher alcohol and 85Yc of the ketone. h mixed interniedia r y fraction containing the remainder of both constituents \vas also obtained and would, in production, h a r e to hc rt;tui,ned to the separator after drying. On the laboratory scale a disk was packed \vitli a mixture ol 75% alumina and 25% 8-hydroxyquinoline. .4 dilute sulfuric acid solution of vanadium, iron, nickel, and zinc was pasbed through under rotation, Clear zones with blank interspaces were obtained for vanadium, iron, and nickel, whereas the ziti? zone could be made visible under t h r mcrcury vapor lamp.
Vol. 39, No. 8
A packing of activated alumina 1r-a~ u . d fur the purification of castor oil on tlie works scale. A dark oil (82yo glycerol triricinoleate) containing ricinolvic acid, dihydroxystearic acid, and colloidal impurities was thoroughly dried and passed through the chromatofuge at a rate of 500 liters in 1.5 hours. A little dry ether \vas added to reduce the viscosity and assist passage. The effluent was found to consist of ether and pure, colorless, neutral castor oil in what could be regarded as quantitative yield. On examining the opened container under ultraviolet light, and tiy taking analyticalsamples along a given diameter, clear zones for cach c-onstituent were identified. The colloidal impurities \vc>readsorbed or, more likely, filtered off in a small zonc round the crnti,al tube, the other constituents being separated by blank z o n r s of c l i w dcfinition. Castor oil was found loosely adsorbed in thr outnniost zone bordering on the perimeter. Elution with (+her cwntaining 5"; methanol under rotation allowed the zones to bc ciillected individually. It is hoped that these expc~i~iments and tlirir publication v,ill \videti the scope and application in industry of chromatography, and provide a means for using the wealth of experimental results of wsearch norliers in this field in works practice.
ECONOMIC PROCESS OPERATION Method for Determination of Optimum Operating Conditions
A
mathematical method is yreseriled for the poaiti\e clrtermination of the process conditions that giTe maxim~rrn profit for a plant operation. The complete interrelations of the process variables and the costs of additional unitc of each of the variables are necessary for the calculatiou. T w o illustrative examples are given, and the epecnial requirements of the method are discussed.
0
YE: of thr most important arid, at tht. smie tiin(,, m o ~ t
difficult problems of a process engineer is the determiriatioil of the optimum operating conditions of a plant. The decisions as t o what temperatures, pressures, flows, etc., lead to inaxiriiuin plant monetary return can easily mean the difference betwwn profit and loss for an entire organization. The operating conditions chosen are usually based on the engineer's judgment of the economic and engineering relations involved and ordinarily have little or no mathematical background. In the present article a mathematical systthni is tlevc.lopcd for organizing engineering and economic data to make possible a complete, positive solution for the optimum operating condition, of a plant on the single basis of maximum monetary return. The mathematics of the method is exact, and the rrsults obtaiiiod r a n be made as accurate as the data ujed. RIATHER14TICS OF METHOD
A simple Hiid familiar example of economic design i,< tlit, calculation of the optimum thickness of insulation or lagging for a steam line. Figure I gives a typical set of data for avcbragetl weather conditions. The cost of the steam lost, the co>t of the lagging, and the total cost, all ill dollars per year, arc plotted against the t'hickness of lagging i n inches. The steam loss in M
p o u u d ~ih tit35igiitlttd as u, and tiit, thickness of lagging in inches is t l t 4 g n a t d as b. Also the cost of an additional M pounds of ,steam is -1, and the cost per year of an additional inch of lagging i,s H. The total cost per year of maintaining the steam flow is 7'. \\-it11 these definitions B is the slope of the cost-of-lagging vurvt, of Figure 1. Also d a / d b is the change in the steam loss with change in thickness of lagging. Then A(&/&) is the change in the cost of the steam loss with change in thickness of laggingthat is, A ( d a / d b ) is the slope of the cost-of-steam-loss curve of the are given by t,he figure. Plots of these slopes, B and -A(&/&), broken curves of Figure 1. At the optimum design the total cost curve goes through a rnimimum cost point. But an additional property of the optimum is that the slopes of the cost lines, A ( d u / d b ) and B, are equal and opposite in sign. This is shown graphically in Figure 1 n-here the broken line plots of - 9 ( d a / d b ) and B cross at the minimum cost point. This same property can be stated algebraically as follows: The slope of the total cost curve, d T / d b , is the sum of the slopes of the cost-of-steam loss and cost-of-lagging curves, and a t the minimum cost point the slope of the total cost curve, d?'/dh, is xei'o. Then the equation A(da/db)
+u
=
dT,flb
=
0
(1)
is true a t the minimum cost point. This equation can also be B d b = 0. This simple relation can be used ivritten as A da to define the economic optimum for any system of two variables. I t states that a t the economic optimum an additional dollar outlay yields an additional dollar return. If the return is more than a dollar, then the outlay should he increased, and if the return is less than a dollar t1ic.n the outlay should be decreased. This assumes that there arc no discontinuities in the immediate vivinity of the optimum values.
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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
August 1947
94 1
Minimum cost points as in Figure 1 c-an.in gcneral, be used to obtain the optimum values of only two variables. But the relation of Equation 1 can bc applitd to any two variables of x number of variables and can he used to dcfincx thc optimum values for systems of any number of variables. This i- the basis of the present method. The example of Figure 1 just discussed is for the original design of a steam line. Holyever, Equation 1 ran bc applied c,qually well to types of engineering variables other than equipment design. The remainder of this article will deal only with the process variables, the temperatures, pressures, flows, etc., of continuously operating plants. Equation 1 can be used to define the economic optimum for any operation or design that can be reduced to two variables. In the extension of this relation to systems of more than two process variables the following nomenclature will be used : I
b, c , . . e = process variables measured as units A , B , C . . , E = cost or value in dollars of an increment’al unit of the corresponding process variable
a,
In general, the desired gains of a process will be taken as dependent variables, and the operating conditions by which they are accomplished will be taken as independent’. Thus in the example of Figure 1, the steam loss can be considered to be a dependent function of the thickness of lagging. The signs of the cost terms A , B , C. . . E are taken as positive for product values and other plant gains, and negative for necessary outlays, such as compressing power or heat. For a system of n process variables the relations for defining the optimum conditions corresponding to Equation 1 for t’wo variables, can be written as follows:
A(aa/dh),. A(bU/dC)b.
,
+B . ., + c
=
O*
@ab)
=
0
(2ac) (2be)
*A
discussion of the extreme values of functions of several variables and a rigorous derivation of this system of equations as a property of such valueare given in Courant’s “Differential and Integral Calculur,” Tol. 2 , pp 183-208. S e w T o r k , Sordeniann Pub. C o . , Inc , 1936.
These equations are, of course, simply Equation 1 for t x o vari.abies viith other variables present, but considered constant. Equation 2 is true for any combination of two variables a t the optimum conditions and can be used to define completely the optimum operation. For a system of n process variables only n- 1 of the equations of Equation 2 can be considered to be independent. The remaining relation necessary to solve for the n unknowns consists in the interrelations of the process variables of the plant. In the use of Equation 2 to solve for the optimum conditions for a given operation, three types of data are required: First’, the complete interrelations of t,he process variables must be known in the vicinity of the optimum conditions; second, the values of the partial terms-(da/db); . (daldc) b. . .
