Nichols time constant may not vary inversely with either sheet velocity or air flow, as might be expected. In this study time constants were only slightly related to sheet velocity and varied with air velocity in a manner opposite to that expected. The important factor in determining the time constant appears to be the manner in which changes in drying effects are distributed through the dryer.
A factor of primary importance is the distribution of effects produced by changing the heat input. This factor should be considered in estimating the time constants. The dead time in this study was approximately equal to a calculated dead time plus a constant, which is apparently a function of the dryer, rather than process variables.
Conclusions
(1) Campbell, D. P., "Process Dynamics," p. 221, Wiley, New York, 1958. (2) Eckman, D. P., "Automatic Process Control," p. 114, TYiley, New York, 1958. RECEIVED for review October 22, 1962 ACCEPTED October 24, 1963 \.Vork accomplished with the support of the Engineering Experiment Station of Auburn University.
The apparent Ziegler-Nichols time constant of a radiant dryer with countercurrent air flow may be best estimated by considering the incremental deviation in moisture from a n original moisture profile of a n increment as it passes through the dryer.
literature Cited
A MODIFIED MOVING BED GRAIN DRYER Y . K. A H N , H. C. C H E N , L. T. F A N , A N D C. G. W A N DeFartment of Chemical Engineering, Kansas State Uniuersity, Manhattan, Kan.
A method for optimal design of a modified moving bed grain dryer is based on a discrete version of the Maximum Principle. Uniform distribution of the air across the complete length of a conventional moving bed dryer is generally a difficult task. Therefore, it is proposed to modify it by dividing the air supply channel in several equally divided sections so that the air supply rate to each section can b e independently manipulated to achieve the optimal use of the air. First a set of general recurrence equations which relate the optimal operating conditions and control actions i s established. The equations are applied to a specific example of designing a three-stage sorghum grain dryer.
RAIN
drying is a process of considerable importance because
G the amount of moisture contained in grain has definite effects on its storage and milling characteristics. This paper shows how a conventional moving bed grain dryer may be modified and how, with the use of the Maximum Principle, an optimal choice of its operating conditions can be made (5).
where x , must satisfy the initial conditions given in Equation 2 and ti must satisfy the following initial conditions,
(4) The ( a " ) is determined by the minimum or maximum condition S
Formulation According to Maximum Principle
vn;
Suppose that for the general consideration, we have the following system of difference equations in the variables, X I , x2,.
. .x,?: ~ 6 %= F i ( ~ k n - - l ;
where i = 1, 2, n = 1, 2,
I
= ai;
i
1, 2 , 3,
. . ., s
(3)
96
3,
,
= n
v");
Fl(~l"-';
. ., s and n
= 1, 2 , 3,
~ 1 = 0
f ~ l n - 1 - XI^ - XV"
=
(6)
a
F?;
~ 2 "=
. . ., N
l & E C PROCESS D E S I G N A N D DEVELOPMENT
but, we see from Equation? 6 and 7 that
bF2 ~dx1n-1
dF1(x1n-1;
-
0
Letting s = 2 in Equation 3 gives
(2)
1,
= 1, 2,
l
where X is a constant. =
(5)
=1
where H" stands for a Hamiltonian. For a certain class of multistage processes the following special system of difference equations holds:
~ 2 "= ~ Z n - 1
., n!
where superscript n indicates the number of stages. The purpose is to find the sequence of control actions, { v n which makes one of xi.'', say x,h', maximum or minimum (5). In order to find such { o n ) , a new set of variables, zi, is introduced to solve the system of difference equations alongside x's and 2 s :
i
]
~
with the initial conditions, xi0
us) = min. or max.
z,n~, ( x i " - ' ;
=
(1)
v")
. . ., s .
H"
-
0")
bx1n-1
Substitution of Equation 8 b into Equation 8a gives
(7)
From Equation 4, we see that, when
x ~ N is
to be maximized
zZn = 1 = zz"; n = 1, 2, 3, . . ., N
(11)
Substitution of Equation 11 into Equation 9 yields
According to Equation 5, the Hamiltonian for the process is, then
Hn
=
zl*Fl(xln-'; c'") f
+
z~"[xZ~-'
- F(xin-';
0")
-
XU"]
(13)
Application
The use of the method presented in the previous section is illustrated by applying it to a modified moving bed grain dryer of sorghum grains. A conventional moving bed grain dryer is diagrammatically shown in Figure 1 ( 7 ) . In Figure 1, x o and A,Y are the moisture contents of the wet and dried grain on a Xveight basis, respectively, q represents the total mass flow rate of the grain, and G represents the total mass flow rate of air. The decrease of moisture content of the grain during the drying process is related to the operating conditions by the following set of semitheoretical equations given by Hukill ( 7 ) :
Since
dH" ~
dV"
= 21"
and z 2 n
=:
1 from Equation 11, the maximum or minimum of
Hn, is found where
dHm
-dV"
=
2in
-
dK(xl"-'; dV.
V")
-
=
(14)
that is, at
q n
or
-
1
x z
bFl(xin-';
vn)/dvn
Substituting Equation 15 into Equation 12
Letting n = n
Y = t/H
and
dF1(xln-'; P J dP
+ 1, Equation 16 becomes
Uniform distribution of the air across the complete length of the dryer is a difficult task. Therefore, we propose to modify the conventional dryer by dividing the air supply channel in several equally divided sections as shown in Figure 2. This modification would permit us to manipulate the air flow rate to each section independently so that the optimal use of the air can be achieved. The modified dryer can also be schematically represented as a stagewise process as shown in Figure 3
(4). A hot stream of air with a constant humidity h, and drybulb temperature T ois supplied to the nth stage dryer a t the rate of G" and emerges from the dryer with an average humidity of hn. Superscript n stands for the stage number. Material balance with respect to the moisture content a t the nth stage can be written as:
Equating Equation 17 with Equation 15
1 dF1(x 1% ; U " + l ) /bx1" d F l ( ~ l n - ~0;" ) dFl(x1"; Z J ~ + ~ ) / ' ~ V ~ + '
wet grains \I
qlxO(
or
air
-
air
G
air
air
or
air
The optimum values a t the n'th stage can be calculated by solving Equations 6 and 19 simultaneously for v.v and x1Iy--] when the end condition xlN is specified and the value of X is assigned. Using these values of u s and x l x - l , the optimum values a t the (9- 1)th stage can be calculated by solving Equations 6 and 18b simultaneously for v.%'-' and x 1 N V 2 . This is continued to obtain the value of x10. The procedure is repeated until the calculated value agrees with the given x10.
t
91 XN
dried grains
dried grains
Figure 1. Conventional moving bed grain dryer
Figure 2. Modified moving bed grain dryer VOL. 3
NO. 2
APRIL
1964
97
i2c
IOC
Figure 3. Schematic representation of modified dryer as a stagewise process
-t"'
-."g I-"
4c
20
C
I l l l l l l l ( ( j 9 13 17 21 25 29 33 37 41 45 Inlet moisture content X:-'
where G"
on
JG
Figure 4.
= -
as a function of inlet moisture content
4 I n the present work, the following profit function of each stage is considered : Tn
=
xln-l
- xln -
where X is the relative cost of air.
(24)
XL,n
M'e are to choose the air N
rate allocation, { z J ~ ~ } , such that the cumulative profit,
ET",
+ ~ ~ n --1 x l n - ) , p ; x20 = 0
-
__
1
on(T0-
(29)
1
T~n-l
bFl(xp-1;
") _ _ _V ~ -_
bx 1" --I 1
is maximum. I t can be seen that this is equivalent to maximizing x 8 ' if XPis defined as = x2n--l
bxln-'
Substituting Equation 23 into Equation 28
n = l
x2n
ddn __
and
(25) Partial differentiation of Equation 27 with respect to u" gives
The variables in Equations 21, 22, and 23 are related according to Equation 20 a t each stage as
xln =
(
+2u"21'
__--
2n"
)
---- 1
(21"-
- x,)
+
bFl(xln-1;
dun xc
V")
__ -
(26)
and
Dn =
- GpXu(xl"-' ~ _ _ _-_XJC _ (ICpffU"(To T@-l)
-
where C is the cross-sectional area of air flow channel. The value of T ~ n - l in Equation 26 is presented in Figure 4 as a function of ~ l ~ - (- 3l ) . For the sake of simplification, let *1n--l - X' A =G _PX _UC and dn = QCPH ZP(T,. - T0n-l)
Then, Equation 26 becomes
and
bdn _ _ d71"
bFl(xln-1); u")
bV"
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
-
(32)
( T , - Ten-1)
( ~ ~ n -1 x,)* (A In 2) ( 2 y
( P ) 2
98
( P ) 2
- X.
Substitution of Equation 32 into Equation 31 gives
Partial differentiation of Equation 27 with respect to xin--] gives
xIn-l
--
-
1)
( T o - To"-') (2"d")
Substituting Equations 30 and 33 into Equations 18b and 19
Table II.
Specifications of Data Values Used T o = 740"F. Variable 0.21 0.055 0.352 1000 35.2 1140 0.24 140 73 Shown in Fig. 4 0.012 Variable 0.352 1. o 0.5 Variable Variable 2 . 5 x 10-3
I
t II
1st stage
t
'c
0
Gy
(Itqdry air/hc) I
I
(outlet moisture content 1
6
G' G
x
Variable
2nd stage
and A =
(x127-'
- x,)*(rl
In 2)(2y
____._
( v q * ( T o- Tc"-')(2"dA)
-~1) (35)
The optimum values of wV and xls-l can be calculated by solving Equations 35 and 27 for the given values of X and xliv. These values of cS and X S - ~can be substituted into Equations 34 and 27 to solve for optimum values of and x1."-*. This is continued to obtain the value of x10. The procedure is repeated by trial and error for X until the calculated value agrees with the given x10. Each run of trial computations yields the optimal policy corresponding to the computed xlo. Therefore, all the results from computations may be saved to provide information as to the effect of initial value. A similar situation is observed in the use of dynamic programming (1). Suppose that it is desirable to dry the sorghum grain with an initial moisture content of 21% in a bed 30 feet high and 1 foot thick, as shown in Figure 5. The grain is supplied from the top a t the rate of 1000 pounds per hour. Air is supplied a t
Table I. Comparison of Results of Computation by Maximum Principle and by Dynamic Programming Maximum Dynamic Principle Programmin,q 01 x 103 (roo= 8 7 0 F.) 3180 3300 u2x 1 0 3 ( ~ , l = B ~ O F . ) 2940 2970 03 x 103 ( ~ = ~ 9202 F.) 2760 2700 X1h' 0.1527 0.1527 3.51 3.50 Profit X 102/q
dry-bulb temperature of 140' F. and absolute humidity of 0.012 pound of water per pound of dry air. The air supply channel is divided into three equal sections of 10 feet each. The physical properties of air and sorghum grain, and other data for this particular illustration, are tabulated in Table 11. By means of Equations 27, 34. and 35 and the procedure mentioned previously, the optimum sequence of v n was calculated for this illustration using X = 2.5 X 'The results are compared with those obtained by the \vel1 known method of dynamic programming (2) as shown in Table I. The optimal total air rates calculated by both methods are in excellent agreement. However, the use of the method of dynamic programming involved a linear interpolation in the final table look-up procedure, ivhich probably gave rise to some error (2). To assure ourselves further that the optimal policy is truly optimal, the profit from an equivalent case of the same total air rate but Ivith even air distribution was computed. It was found that the use of the optimal policy gives a profit of 1% higher than the case of even distribution. The increase should be much larger than 1% if \ve compare the modified dryer with the conventional dryer operated under any other conditions. In many cases of grain drying, . - however, the end conditions (both initial and final moisture contents of grain) are fixed. The methods developed are still useful, providing that the value of X is not fixed for each trial calculation. Assigning a value to X and using the given value of x1,v, x10 is computed according to the procedure. If the calculated xlo does not match the given value of xlO: the calculation is repeared by assigning a different value to X until the calculated x l 0 agrees with the given value of x1O. In this scheme X is iised as a Lagrange multiplier I
7 ,!
VOL. 3
NO. 2
APRIL 1964
99
Acknowledgment
j N
Assistance given by C. S. Wang is deeply appreciated. Nomenclature
A = GpX,C/qC,H, a = initial condition C, = specific heat of air, B.t.u./lb.-” F. D = dimensionless, see text F = functional notation G = total mass flow rate of drying air at each stage, Ib. dry air/hr. G‘ = mass flow rate of drying air, Ib. dry air/sq. ft., hr. h, = absolute humidity of inlet air, lb. water/sq. Ib. dry air H = time required for fully exposed sorghum to reach a moisture ratio, q
x - x -2 of 0.5, x,, x,
-
hr.
= solid flow rate, dry basis, lb./hr.-sq. ft.
number of state variables time of sorghum grain exposed to drying air, hr. temperature of inlet drying air, F. temperature at which relative humidity of air is in equilibrium with moisture content of sorghum grain, F. 75 = wet-bulb temperature of inlet air, ’ F. = G“ __ sq. ft.
s t To To
= = = =
Q’
x10 =
XI
x,
= =
inlet moisture content, lb. water/lb. dry grain outlet moisture content, Ib. water/lb. dry grain equilibrium moisture content, lb. water/lb. dry grain
= dummy indices in Equations 3 and 5
p
= total number of stages = a particular value at i
.z
=
Y
= see text, dimensionless
a variable defined by Equation 3
GREEK LETTERS = mean distance of grain from air intake, assumed to be half of thickness of bed, ft. p = bulk density of sorghum grain, Ib./cu. ft. A, = unit heat of vaporization of moisture in sorghum grain, B. t. 11. /I b. X = relative cost of drying air, dimensionless 6
literature Cited (1) Anderson, J. A , , Alock, .4.I V . , ed., “Storage of Cereal Grains and Their Products,” Proceedings Am. Assoc. Cereal Chemists, St. Paul, Minn., 1954. (2) Aris, K., “Optimal Design of Chemical Reactors. A Study in Dynamic Programming,” p. 26, Academic Press, New York, 1961. (3) Henderson, S. H., .4,gr. Eng. 33, 29 (1952). (4) Isaacs, G. I V . , Department of Agricultural Engineering, Purdue University, Lafayette, Ind., private communication, 1962. (5) Katz, S., IND.ENG.CIIEM. FUNDAHENTALS 1, 226 (1962)
RECEIVED for review May 20, 1963 ACCEPTED September 26, 1963
NEW METHOD FOR SIMULATION OF M U LTICOM PON ENT D lSTl LLAT IO N E. C. DELAND AND
M. B. WOLF
7’he RA’VII Carp., Sanfa Afonica, C a y .
A new method for the simulation of multicomponent petroleum distillation columns takes advantage of the power of mathematical programming techniques for computing the equilibrium states of physicochemical processes. The procedure is general and is able to incorporate changes of phase, external sources or sinks of mass or energy, and differential equations which describe system dynamics if they are relatively slow with respect to the chemical dynamics. By use of the Gibbs theorem, a chemical equilibrium may b e defined in terms of the thermodynamic free energy of the components, the total free energy being minimized a t equilibrium. A (nonlinear) free energy function is defined and then minimized under the natural physical (linear) restraints of the system. O n the analog computer, the solution method is by steepest descent. A digital solution has also been programmed.
PROCEDURES have been devised for the simulation of particular subsystems in a refinery operation, and practical methods have been developed for modeling mulristage. multicomponent distillation on the computer. Amundsori ( 7 ) . Lyster (9), Greenstadt ( 7 ) , and others have described successful programs on the digital machine; h l a r r ( 7 0 ) , Worley (78), Rijnsdorp and Maarleveld (72), Computer Systems, Inc. ( Z ) , and others discuss simulation on the analog machine. Usually these methods are based upon the equations and techniques developed formerly for hand calculation, but Marr is an exception. in that he proposes a set of partial differential equations for the temperature and composition profiles of the column as a whole. Here, the Gibbs free energy function was used to provide a model for simulation. Several advantages were gained from this method, due principally to its generality. I t is a natural format for representing the subsidiary chemical reactions and classical equilibrium of a complex system, but also it can be such used to model irreversible thermodynamic processes (.I), 100
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
as elution. ion exchange. and forcing functions of various kinds. Thus. the method may be used to simulate other elements of the refinery. Here a n application to multicomponent distillation is illustrated. ‘The procedure originated in a paper by IVhite, Johnson, and Dantzig (75) and has been applied in biological s) stems ( 3 ) , combustion, planet atmospheric studies, and other systems. The present application \\as suggested in a n earlier paper (5); the analog computer results were likeivise obtained from earlier research (77). Fractional Distillation Column
‘The basic idea of fractional distillation is that a homogeneous input mixture of n components (the feed) is to be separated into two principal fractions, a condensed vapor phase (the distillate) and a liquid remainder (the bottoms), \vith reasonable efficiency and control by means of a series of rn staged distillations. Figure 1 illustrates a typical section of a distilla-
