PLANT MATERIAL BALANCE CONSTR UCTlO N AND INTERPRETATION In-process losses cost money. Statistical analysis can show how much and what corrective course to take
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1
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1
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I
FRED H. TINCEY
A decision must be made whethn or not a process is to be shut down. Basis for the decision is a suspected loss of material above thnt which can be considered “reasonable.” Suspected i s the referred wmd here since some materials in process, particular& in a chemicnl reuctm, m e not m e p t i b l e to direct meamremmt. These materials must be accounted for in process malerials balame by engineering estimates. The decision i s most generally based on the doUar loss to be incurred through loss of product by actively interrupting processing against the dollm value of thc material being lost by continuing to process. Since cngiwcrtng estimates only are available for a large segment of thz material balance and measurement m m s at best enter into thc calculation, certain risks me involved in making the decision. The accompanying article considers thcse risks and dewlops a procedure by which an “optimum” decision can be made on the basis of the data at hnnd. Thc technique seems most aPprc@iate to processes chmacteriud by high unit dollar value and/or sign$cant strakgic value. Also, those processes in which a potentially hazardous sihiation might result from an undetected acnunulatron of an appreciable quantity of material, euen thrmgh the dollar value may be small, would be especially amenable to this technique.
36
INDUSTRIAL AND ENGINEERING CHEMISTRY
riodlc material balances must be made around an operating unit in the interest of process and cost control. This applies to chemical processing in general and nuclear-fuel processing in particular. But the construction and proper interpretation in either field often presents a fundamentalproblem. Primary concern is directed toward the accuracy of data used for accounting of withdrawals from process, either intentional or unintentional. Proper understanding of the elements going into a material balance is the key to the solution. Usually the material constituting the basis for the balance can be broken into two broad classifications:
P ’.
-Material capable of direct measurement-a total inventory can be made or at least samples can be taken and corresponding analyses made -Material in process not amenable to direct measurement-only engineering estimates are available An example of the former in, say, a liquid-liquid extraction system would be that material held up in column run tanks. The material in the column itself at any given time would not be capable of measurement and would belong to the latter classification. Interpretation of material balance data usually depends heavily on measurement uncertainties inherent in the calculation. These uncertainties in turn depend in large measure on the distribution pattern of the material between the two classifications. A common practice is to reduce the uncertainty hy changing the shape or form of the material to makr it easier to meas-
Any given computation of P, has associated with it a qrtain error which: -Reflects
the error in the measured variableeA,, B,, C,
-Necessarily is functionallydependent on total throughput at time t
t
A-
And, if P,were measured without error, a certain amount of inherent variation in process holdup would cause Pt to vary with time over and above that anticipated from changes in processing rates. It is assumed here that no ultimate res0lu)ion of an apparent imbalance caq be made until P, is free of process inventory, nonmeaswable quantities. A certain risk that the apparent “withdrawal” will fail to materialize is associated with any givem sweep down (elimination of inventory) when a rigorous “dean plant” balance is constructed. It is also conceivable that due to a canbination of circumstances, a given Pt could appear to be “very reasonable” even while material was beiig lost. A certain economic background can be associated with each course of action. A certain financial loss, primarily the cost of sweeping down the plant plus loss of production, is incurred in the event the sweep down is unnecessary. And a ccrtain financial loss, primarily the value of the material, develops in the event sweep down of the plant is not carried out and material is in fact withdrawn. D&ning Ihe Risks I n v o l d
ure. This action takes the form of transferring as much m a t d as possible from the classification identified with engineering estimates to that associated with direct measurements. In many cases this can be done with little extra effort and is almost automatic More any material balance is constructed. However, when successive cleanouts are needed to verify the data, it becomes expensive. There is a natural reluctance to proeeed along this path unless some assurance can be given that the end justifies the means. Certain risks are also involved if an apparent imbalance is ignored and processing continued. It is conceivable that by 80 doing quantities of material withdrawn from inventory which would be detected thmugh a reduction of the nonmeasurable inventory would be irretrievably lost. M n i n g the Balance by Mahemolical Formulas
The usual chemical process material balance is characterized at any given time, t, by:
A, B, C,
= = =
cumulated mcasured feed charged to process cumulated measured product cumulated measured railinate
Apart from measurement error in the absence of unauthorized withdrawal, process imbalance plus inventory is defined :
P, = A , - B, - C,
(1)
It is logical to define the risk m c i a t e d with sweep down criterion as:
Risk = (loss due to wrong decision) X (probability of wrong decision) When this dehition is applied to the two alternative courses of action, risks R1 and RI are: R1 = (loss resulting from sweeping down the plant) X (probability that the criterion will lead to unnecessary sweep down) R, = (loss resulting from undetected withdrawal of 6 units of material) X (probability S units will go undetected) Note that the risk of the criterion failing to detect withdrawal is a function of the unknown amount of material (a) withdrawn, among other things. The procedure proposed here determines a critical value for PI which minimizes the maximum risk to be d a t e d with the incorrect course of action. A criterion is established for Pt which minimizes with respect to Pt the maximum risk, maximized with respect to the unknown amount S withdrawn.
In the mathematical formulation, assume: -True process inventory, as defined by Equation 1 in the absence of a withdrawal, is randomly distributed in time. This assumption may not be rigorously satisfied for extremely short time intervals. But t h i s is very likely true for any practicable interval V O L 5 4 NO. 4 A P R I L 1 9 6 2
S?
-Distribution of the in-process inventory with time is essentially Gaussian with parameter mean p, assumed to be known for a given processing rate, and variance 2 up,also assumed to be known -Errors of measurement associated with each component in Equation 1 are also Gaussian-distributed with corresponding known parameters -Process variation and measurement errors are uncorrelated -Loss associated with undetected withdrawal is directly related to the amount withdrawn -Loss associated with sweeping the plant unnecessarily is fixed by the cost of sweep down and not related to how necmsary (how far the computed P , exceeded the critical P,) the sweep down appeared to be Mechanics and Derivatives
There is no loss in generality if we assume p = 0, because for any throughput rate and corresponding mean holdup the observation Y can be so adjusted. If we make the substitution = 0 and the transformation
-
t=--
in Equations 2 and 3 we get
0
and
where F ( t ) is the cumulativc distribution function for f ( t ) , the normalized normal distribution. Since F ( t ) is a monotonically increasing function oft,and since L and 0 are nonnegative, Rl is a maximum with respect to 6
5
0 when 6 = 0 . Thus
For a given processing rate and corresponding to a given time t :
6
= amount withdrawn P = mean value for P , 2 UD = variance of P, Y , = the “apparent” value for P , reflecting measurement errors 2 u,n, = composite measurement uncertainty in P, reflecting the uncertainties in the terms comprising the right side of Equation 1
+
02
= u;
Y
= w+6
c,
2
R2
=
5 = 86
= cost in dollars of sweeping down the plant = dollar value per unit of material processed
C26F
(-8) L-6
where 6 >_ 0
This expression is to be maximized with respect to 6 in the region defined by -6 2 0 where L and Bare both nonnegative. The derivative of Equation 7 with respect to 6 results in
Uml
including loss of production
cz
Consider
CZ
[- 68 1L (- --6 ~ )+ F (o-)] L--6
(8)
By equating this to zero and making the substitution
With these definitions note that the risk R1 is: we have
where y
I p.
Negative values for 6 are allowed for qenerality. Similarly, the risk RZ is
where b = -L
8
where y
2
p.
Subscript t has been dropped as a matter of convenience. L is defined as the critical value for Y such that if any time, Y exceeds L, the plant will be swept down. But as long as Y is less than L , no sweep down will be made.
The solution of this conditional equality for t along with Equation 9 results in the desired maximizing relationship between 6, L, and 8. T o effect an approximate explicit solution to Equation 10, expand ( t - b ) F ( t ) into a Taylor series about zero, ignorinp. j(t) all terms but the first two. If we define
+
go) = (t - b)i(O
+ F(t)
then
g(O) AUTHOR Fred H . Tingey is Head of the Operations Analysis Branch, Atomic Energy Diu., Phillips Petroleum Co. Hzs interests lie in the application of mathematzcal Jtatzstics to reactor technology and nuclear fuels reprocessing.
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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
and
=
-bj(O)
+
1 -
2
=
-
b
__
l/%+2
1
which when equated to zero gives
TABLE I
4 ) 0.0556')
The accuracy of this approximation is demonstrated by an examination of the data of Table I. This table lists explicit solutions of Equation 10 and corresponding i n c e values for Equation 11 in t for various values of b. S b = L/O, in any given application, is not expected to differ greatly from unity, thisapproximation is sufficiently accurate for most cases. In the event a more accurate relationship is needed, a second degree polynominal has been derived from the data of Table I and corresponding values determined for purpow of comparison. The polynomial can he used to provide the ultimate solution in a manner analogous to that given below.
0.0
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00
That, in fact, Equation 10 defines a maximum rather than a minimum is established by noting
is negative when
- b) + 2 > 0
(-t)(t
or
-0.75 -0.58 -0.42 -0.27 -0.13 0.00 0.12 0.23 0.33 0.43 0.51 0.59 0.66 0.73 0.80 0.85 0.91 0.96 1.OO 1.05 1.10
-0.63 -0.50 -0.38 -0.25 -0.13 0.00
-0.72 -0.57 -0.42 -0.28 -0.15 -0.02 0.10 0.21 0.32 0.42 0.51 0.60 0.68 0.75 0.81 0.87 0.92 0.97 1.01 1.04 1.06
0.12
0.25 0.37 0.50 0.62 0.75 0.87 1.00 1.12 1.25 1.37 1.50 1.62 1.75 1.87
f2-fb-2
