D Y N A M I C S OF A R A D I A N T DRYER WITH C O U N T E R C U R R E N T AIR FLOW M . E. FINDLEY AND M . W .
MALONEY
Chemical Enginpering, Auburn 17nirersity, Auburn. Ala.
Dynamic response to heat input changes in an infrared dryer with countercurrent air flow was studied.
A semiempirical relationship between the empirical Ziegler-Nichols constants and the drying variables was obtained. In responding to heat changes in any one of three sections, dead time corresponded to the transport time from rhe section exit to the measuring point plus a constant.
The experimental time constants were
related best with two pseudo-time constants calculated for edch run, one for single sections and another for all sections possibly affected. Both were based on deviations with time from the original moistureposition profile. A “weighted” combination of these values gave the best estimates obtained.
increasing use of automatic process control. better knowledge of the dynamics of various processes is needed to determine the behavior of control systems applied to these processes. T h e more complex processes require more complex instrumentation and more knowledge about the process dynamics involved. Drying operations are common in the chemical industry, and are frequently directly related to the profit obtained in a process. ,4lthough it is often necessary to control drying operations. little information is available which relates the dynamic characteristics of more complex drying systems to drying conditions. This study attempted to determine a relationship bet\+een drying variables and the dynamic response of a radiant sheet dryer with countercurrent air flobv (Figure 1). Of particular interest was the response of exit sheet moisture to changes in heat radiated. Such changes \vould normally be utilized to provide moisture control. 1TH
In addition to these direct effects, many variables are interrelated. In particular, the temperature of the sheet depends upon the over-all heat transfer and the mass transfer from the surface, and, in turn: affects the rates of both transfers. Because of the number and complexity of the relationships involved, any’ study of either steady-state or dynamic conditions must be partly empirical or involve a number of simplifying assumptions. It was desired to obtain a semiempirical relationship based on the major drying factors, Lvhich would approximate the dynamics of a particular radiant dryer. A successful relationship \ \ o d d indicate that similar methods might be useful in the stud!- of other dynamically complex systems. The Ziegler-Nichols method of approximating the dynamics of complex systems u s e one dead time function and one time constant function to approximate a response curve (2).
Theoretical Considerations
I n general, drying of solids involves three major rate processes: mass transfer from the wet surface to the surrounding gases, heat transfer to the solid from the sources of heat, and moisture diffusion through the solid to the surface. In radiant drying of a sheet in air, the following factors would be important: Factors Affecting Mass Transfer from Sheet
Partial pressure of water at sheet surface Partial pressure of water in the air Temperature of air Temperature of sheet Velocity and mass flow of air Factors Affecting Heat Transfer to Sheet
Heat supplied in radiation Temperature of sheet Temperature of dryer walls Temperature of air Velocity of air Geometry of dryer Radiation absorptivity of sheet, walls, and air Humidity of air Moisture content of sheet Factors Affecting Moisture Diffusion in Sheet
Moisture of sheet Temperature Moisture gradient Thickness of sheet Present address, Textile Fibers Department, E. I. du Pont de Nemours & Co., Inc., Richmond 12, Va.
Figure 1. Apparatus and wiring diagram of radiant dryer under study A. 6.
E. L. M.
P.
Anemometer Blower Sheet exit and sample point Lamps Motor and variable-speed drive Paper roll
VOL.
3
R.
Rubber rolls
S. Paper sheet SW. Switch T. Thermometers VT. Variable transformer W. Wattmeter WT. W a t e r tank
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1964
89
A response curve is obtained following a step change in the desired variable and the constants are obtained in a specified manner. Such constants are related to the dynamics of the system, but are ordinarily measured and used in an empirical manner. I n this study it was desired to find a correlation between experimental, empirical. Ziegler-Nichols dynamic constants and any possible theoretical or other values calculated from drying variables. Dead time, L , normally due to transfer lag: can be estimated by Equation 1 : L = D/v
(1)
where D is the distance between the points where a change occurs and \\.here the change is measured, and LI is the velocity of transport bet\veen the t\vo points. In the case of complex dynamics. an apparent lag might also be due to a number of dynamic functions operating in series. For certain systems, including many drying operations, the effect of a step change is distributed over a considerable distance on the same order of magnitude as D . Thus the empirical dead time of the Ziegler-Sichols approximation should vary approximately as in Equation 1. but the value of D necessary is unknown and other factors may be involved. Campbell (7), in discussing a roll dryer, assumes that D is the distance from the most distant point where an effect occurs to the measuring point. This would give dead time for a step change in incoming moisture, but not a suitable value for response to a change in heat input. I t was desired to determine the proper distance to use in Equation 1 for a radiant dryer, and lvhether any empirical corrections are necessary. In seeking a relationship between the empirically determined time constant and the drying variables, simplifying assumptions Lvere made to allo\v calculation of theoretical time constants. A number of different procedures were used to caland moisture content ( T.,I) time constants culate thermal (T,) under various assumed conditions. The calculated time constants could then be compared graphically with those experimentally determined. Two derived time constants were used unsuccessfully in attempts to find correlations with experimental time constants.
Equation 2 gives a thermal time constant derived from a heat balance around the sheet, assuming constant moisture contents at given positions, linear moisture-position and temperature-position relationships. and constant inlet temperatures.
bV
M, H
= sheet width, ft. = section length, ft. = sheet density, lb./cu. ft. = sheet thickness, ft. = heat capacity of dry sheet = heat capacity of water = average moisture content = sheet velocity, ft./min. = final moisture content = enthalpy of water vapor leaving sheet with reference
r
= water evaporation rate per unit area, Ib./(min. sq.
X p
b
C,, C,, Mu u
Mass transfer time constants based on similar assumptions were derived and calculated. These LZ ere approximately inversely proportional to sheet velocity. Equation 3 \\.as based on assuming that the sheet in a section bchaved as a mixed quantitv, and that the heat input change changed the rate of Lvater removal:
A more successful approach considered the effect of a change in heat input on an increment of sheet as it passes through the dryer. This effect may be expressed as differences, A M and AT. from the previously existing steady-state moisture and/or temperature at a given point. Thus moisture and temperature of an increment of sheet lrithin a section will gradually change \vhen that section ischanged and A M and A T from the original moisture-temperature-position relationship will change until the increment leaves the section where the input was varied. After this time A M and AT are assumed to remain constant. Actually they probably change, but the effect on the final moisture content may be assumed fixed once the increment is out of the altered conditions. Considering changes from original conditions only, a materiaf balance gives
where t = time A M = change in moisture from original moisture position profile AT = similar temperature changes The change in rate of drying per unit area, AY. may be assumed to be a function of sheet temperature and moisture. Even in sections \\here drying rate is constant. changes in moisture content cause moisture changes near the exit where drying rate is falling. and thus affect the over-all drying rate. Thus, it is assumed that
Ir
=
Insofar as over-all effects are concerned,
In an incremental thermal balance per unit area. change in heat accumulation = -latent heat lost due to AT - latent change in heat lost due to A M - heat transfer loss due to A T heat radiated to sheet, or
+
pb(Cp,
-+ IMC,,)
dA T bY = -HAT dt bT
ft.)
T , = average sheet temperature = film heat transfer coefficient h
90
I&EC PROCESS DESIGN A N D DEVELOPMENT
bY AM bM hAT AqR
- H-
+
(5)
where qR is the heat radiated to the sheet per unit area. Then in operational notation according to Eckman ( 2 ) : Equation 5 becomes
to sheet inlet temperatures
Since the calculated T , is very small compared to experimental values, steady-state temperatures can be assumed.
bY dr - AT+ - AM bT bM
where s is the operator.
In these runs, the coefficient of s is small, and it may be assumed that
I
-P
1
17
Substituting Equation 6 in Equation 4 0
4
8 TIME
br - AM bM
(7)
br bM
br Hbr bM bT br H E + h
_-
and using operator notation, Pb
[(1
br
S
H
I
+1
ff-+h b 1‘
___.__
br (HE
AM=
3r _ -bT Ah __
+ h)
- bT
(8)
br H-+h
bT
‘Thus AM from the original moisture profile may vary approximately as a first-order time constant sj-stem, as a n increment passes through the dryer with time. If the apparent time constant of a n increment passing through the changed section of the dryer is, 1;. which is equal to ob
the change in moisture of the sheet leaving the changed section of the d q e r will be the solution to
(TIS
+ 1)A.M
=
KJq,
(9)
a t a time equal to the length of exposure of the particular increment to the change in heat input. K5 is the coefficient of AqR in Equation 8. Solving Equation 9,
AMExlt= K5AqR (1 -
16
20
Figure 2. Determination of experimental dynamic constants from typical curve of moisture vs. time Procedure
Dividing by the negative coefficient of AM on the right,
-.
12 (MINI
(10)
where t, is the residence time in the section of the dryer in which the heat input has been changed. After a time determined by the length of the changed section divided by the velocity of the sheet, the residence time becomes constant for all increments leaving the section, and the exit moisture content should remain approxirnately constant with increasing time. Even though some of the assumptions in Equation 10 may not be correct, it might be possible to relate the above type dbnamics to the empirical time constants obtained by the ZieglerNichols approximation.
A dryer was constructed as shown in Figure 1, utilizing countercurrent air flow and industrial infrared lamps as a source of radiation. The dryer was in three sections, which could be adjusted and the power input measured separately. Rolls of industrial paper toweling were used as the material to be dried. The sheet was fed continuously through a water bath to saturate i t with water and then through the dryer. T h e moisture content after drying was measured by sampling at a point 10 inches past the sheet exit from the dryer. In making a run, the lamps, blower, and drive were turned on, adjusted, and maintained constant long enough for steadystate operation (30 minutes in most cases). About four samples were then taken a minute apart to determine the steady-state moisture under initial conditions. A step change was made a t a given time, and samples were taken until the change of moisture content was essentially complete. All samples were stored in stoppered test tubes until they could be weighed, then oven-dried and weighed again to determine moisture. The results of a more or less typical run are shown as points in Figure 2. The dead time, L. and apparent time constant, T‘: were determined as described by Eckman ( Z ) , and as illustrated in Figure 2, by extending the maximum slope to the final moisture content. In most of the runs a time constant response calculated using T’ approximated the actual response only for the initial part of the moisture change. ‘4better fit to the over-all actual response was obtained by determining the time required after the “dead” period for a change of 63.2% of the steady-state change, A.W,?. This value, 7“‘, was then used as an empirical time constant. In Figure 2, curve M‘ is the calculated response using 2 ’ ; and curve MI‘ is calculated using T”. Since T” could be obtained more accurately, it was felt to be the most appropriate experimental and empirical time constant to use in the Ziegler-Nichols approximation. Most of the attempts at relating these constants to system variables \\-ere made lvith I, and 7’” experimental values. In some runs it was apparent that changes kvere too small to be used with confidence, other changes were occurring simultaneously, or initial and final conditions were not at the steady-state value. Such data were discarded. In a number of runs the moisture tended to oscillate as in Figure 2. apparently because of changes in tension caused by the unwinding roll. If the average steady-state value could be ascertained reasonably. the data were not discarded. I n most control systems on similar processes it would be desirable to manipulate the input where the dead time would be least (Section I in Figure 1). Because the dynamic response to a manipulation in this section would be of most interest, most runs were made changing the heat input to Section I . Runs were also obtained with changes in heat VOL. 3
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Table 1.
Experimental Data
Sheet Velocity, In./Min.
Exit Air Inlet Velocity, Rel. Hum. MPH STEPCHANGES IN SECTION I
Run JVO.
Min
L,
T', Min.
T", Min.
15 18 30 33 34 38 42 58 60 62 64 68 70 72 75 79D
2 1.4 1.6 1.7 2 1.9 2.2 3.1 1.6 1.6 1.3 2.9 1.2 1 .o 1. o 2.0
2.2 1.5 4.4 2.3 4.1 3.1 2.6 2.2 2.4 1.5 3.5 3.9 3.4 2.1 2.2 3.0
2.0 0.9 3.1 1.3 3.1 1.9 1.8 1.6 2.0
2.7 2.3 2.8 1.5 1.8 1.8
7.9 8.0 7.5 7.5 7.5 7 9 6.9 6.25 10.75 10.25 10.25 3.25 15.00 14.25 15.25 7.25
12 14 26 73 80D 82D 85 90 92 93
4.1 4.0 3.0 2.1 4.0 4.2 3.0 2.8 3.5 2.9
1.3 2.3 2.0 1.3 4.0 4.8 1, I 1 .o 1.4 1.6
0.9 1.4 1.8 0.8 2.4 3.5 0.9 0.8 0.9 1.2
STEPCHANGES I N SECTION I1 6.0 0.81 11 7.9 0.54 11 10.0 0.51 11 15.25 0.46 11 7.75 0.28 6 7.75 0.34 11 12.5 0.45 11 11 . o 0.75 11 9.0 0.72 3 11 . o 0.31 3
11 13 16 61 65 78D
5.9 6.0 4.6 4.8 4.9 6.4
1.9 2.8 3.3 1.7 1.3 4.8
1.1 1.9 2.1 1.1 1.2 3.1
1. o
0.63 0.61 0.40
n
36
0.37 0.40 0.42 0.38 0.61 0.39 0.40 0.40 0.39 0.64 0.40 0.37
11 11 11 6 6 11 6 3 3 6 11 3 11 11 6 6
STEPCIIASGESIN SECTION I11 7.5 0.57 11 7.9 0.80 11
input to Sections I1 and 111. Four runs were made with two sheets passing through the dryer together, and runs were made a t various sheet velocities, air velocities, power input, and relative humidities (Table I). Table I1 gives pertinent dimensions of the dryer. Data on inlet, outlet, and intermediate temperatures were close to ambient temperatures and did not vary appreciably. Outlet relative humidities were close to inlet values and were not utilized. The data do not completely cover the ranges of all variables, because a number of runs produced nonmeasurable results.
Step Change in Lamb _ Power, Watts _ From To
~
Steady-State Exit Moisture, G./G. Before After ~ change change
250 250 250 250 250 250 250 250 250 300 300 0 400 400 450 400
100 350 100 100 0 450 100 0 0 150 0 200 150 150 200 0
0,115 0.153 0.067 0.058 0.078 0.080 0.060 0.062 0.790 0.350 0.071 0,116 0.079 0.228 0,100 0.119
0.235 0.113 0,072 0,064 0.170 0.072 0,066 0.073 1.210 0.487 0,105 0.068 0.118 0,425 0,250 0.224
250 250 250 450 440 440 250 250 150 150
150 in0 150 150 250 200 450 450 400 400
0,240 0.120 0.180 0.150 0.098 0,067 0.290 0.300 0.275 n ,800
0.450 0.280 0.248 0.325 0.325 0.134 0.097 0.090 0.064 0.360
250 250 250 300 250 400
100 100 350 150 150 0
0.145 0.158 0.283 0.232 0.163 0.066
0.320 0.336 0.140 0,452 0.250 0.210
particularly at low values of L. The distance indicated by such a plot was from a point between the center of a section and the exit, to the sample point. On subtracting from L a calculated dead time. L,, based on the distance from the exit of a section to the sample point, a difference was obtained which \vas not related to sheet velocity or to other factors tried. In fact: it appeared to be relatively constant at an average of about 0.4 minute for all sections. Thus the best calculated value of dead time was obtained by the equation:
Analysis of Results
The most obvious way of relating dead time to system variables would be to plot the L obtained by experiment against the reciprocal of sheet velocity and determine the equation L = D / v was applicable a t any particular distance. Such a plot indicated a reasonable correlation, with certain discrepancies,
Table II. Dryer Dimensions Distance from Exit Sec. I to sample pt., inches Exit Sec. I1 to exit Sec. I, inches Exit Sec. I11 to exit Sec. 11, inches Sheet inlet to exit Sec. 111, inches Area of air exit or sheet inlet, sq. inches Sheet width, inches Width dryer, inches Area of opening between sections, sq. inches Area of sheet outlet, sq. inches Height of dryer, inches
92
10 18 18 18 47 91/4
19
40.5 231~ 23'/2
l&EC PROCESS DESIGN A N D DEVELOPMENT
where D is the distance from the changed section exit to the sample point, and K6 is a constant, in this case 0.4 minute. I t was felt this constant was related to the dynamic response of the dryer walls, the bulbs? and/or the techniques used in carrying out the run. A plot of experimental dead time, L , us. estimated dead time, L , 0.4, is shown in Figure 3. In attempting to relate an apparent time constant to system variables, considerable difficulty was experienced: possibly due to unknown changes taking place during the run, or the number of variables affecting the drying of the sheet. One of the variables, which normally is related closely to the time constant, would be the throughput rate, in this case proportional to the velocity of the sheet. The experimental time constants, T", however, shoJved little relationship to velocity. This may be seen from Figure 4 , in which the abscissa is approximately proportional to the reciprocal of velocity. In attempting to relate the experimental time constants, T ' a n d T"?with time constants assumed in Equation 3 and similarly
+
From run 68 under initial steady-state conditions with zero po\ver, both IC3 and / I were calculated, assuming the sheet was a t the wet-bulb temperature. The bP,!bZ \vas assumed to be the change in water vapor pressure per degree a t a normal run wet-bulb temperature of about 20’ C. To obtain 3 r ; d M a relationship was assumed and plotted, giving Ar:AP as a function of average exit moisture content for all runs. AP is change in lamp poiver per unit area of sheet. This curve was differentiated graphically to give
( D t V 1 - k 0.4 (MIN) ESTIMATED DEAD TIME
Figure 3. Experimental vs. estimated dead time
derived equations, it \vas found that the calculated time constants ‘rvere either very small or very dependent on velocity. Thus by using such methods, little if any correlation could be obtained. Figure 2 and other data suggested that an equation such as 10 might be the best way to describe the dynamics. In attempting to use Equation l o ? it \vas necessary to evaluate b r , b l ’ , br/d.&f, and /i> from experimental data and certain assumptions. If I = R, (P,7- Pa),Lvhere P, and Pa are sheet and air partial pressures of \vater, K , is the mass transfer film coefficient, and K , and P, are constant, then for all runs,
The product of this times the average poiver in a changed section \vas used as an estimate of br/bM. This method gives a rough value of exit moisture. ‘I‘he br,’bM as obtained \vas a function of the exit moisture from the dryer as \vel1 as poiver, and these variables \ M i n .
2.0 1.4 1.6 1.7 2.0 1.9 2.2 3.1 1.6 1.6 1.3 2.9 1.2 1. o 1. o 2.0 4.1 4.0 3.0 2.1 4.0 4.2 3.0 2.8 3.5 2.9 5.9 6.0 4.6 4.8 4.9 6.4
Time Constant, TI’, .Idin, 2.0 0.9 3.1 1.3 3.1 1.9 1.8 1.6 2.0 1.o 2.7 2.3 2.8 1.5 1.8 1.8 0.9 1.4 1.8 0 8 2.4 3.5 0.9 0.8 0.9 1.2 1. I 1.9 2.1 1.1 1.2 3.1
VOL. 3
NO. 2
Estd. T”, diin. 2.1 1.4 2.6 2.0 2 6 1.6 2.0 I .i 2.2 1.4 2.2 0.9 1.2 1.o 0.6 2.1 1.7 1.6 1.3 0.7 1.3 1.4 0.8 0.9 0.9 0.8 1.5 1.4 1.4 1.1 1.2 1.5
APRIL
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Combining Equations 11 and 12, 1
Knowing TI and u, T p can be calculated. If T I is large compared to 18/v, which it was in most cases, T p is almost proportional to l/u, and again little correlation of Tpwith 1'" exists. T pis given in Table I11 and 7'"is plotted against T p in Figure 4. The above calculations assume that all the response to manipulation of the radiation occurs in the section manipulated. O n observation of Figure 1, it is apparent that changing the power input to Section I produces some heating of the air, which affects the drying rate throughout the dryer. Similarly a change in Section I1 might affect both Sections I1 and 111. For this reason, a second predicted time constant \\as calculated, T,, based on the response to changes in Section I taking place in all sections, responses for Section I1 taking place in I1 and 111, and Section I11 calculations remaining unchanged. Thus for Section I changes, 1
__-
1I
-
1
- e-TQ
I
0.632
0
2
I
3
4
Tp ( M IN.I
J",
Figure 4. Experimental time constant, vs. JP, calculated for single section
o-54,'t1'12
C
(1 3 4
and for Section I1 changes,
T I for these calculations was changpd somewhat because the average power per unit area of sheet was slightly different for the complete dryer than for the section changed. These values are given as T,z in 'Table 111; values of T, are also presented. In Figure 5, T"is plotted us. TQ. While Figure 5 indicates some relationship betbyeen experimental and calculated values of the time constants? it is obviously unsatisfactory. Comparing Figures 4 and 5, it appears that most of the experimental time constants are between those calculated for a single section, 1,: and those calculated for the maximum dryer length which could be affected, TQ. Thus a method of "weighting" the t\vo calculated time constants to obtain a compromise time constant might provide a better correlation. The deviations from both curves were related to lamp power, air velocity, and relative humidities, but ivith considerable scattering of points. Drying effects due to the air flow would be expected to be approximately proportional to air velocity, a , and to the unsaturation of the air, and to be distributed throughout the dryer. Drying effects due to radiation would be roughly proportional to power supplied, and would be distributed to one section only from a given set of lamps. Since these factors appeared to be related to the experimental time constants, it was assumed that the effect of the change in power to one section could be partly due to change in direct radiation to the sheet, and partly due to heating the air, \vhich would affect all sections downstream with air. If the first part were proportional to power and the second proportional to [Q (1 - R ) ] ,the following relationship among time constants might be reasonable: I .I/ =
P
I
P
+ K7a(l - R )
T,
- n) + P $-K@(l Kyu(1 - R )
-0
I
5
4
3
b
6
IMINI
Figure 5. Experimental time constant, J" vs. for maximum dryer length affected
JQ,
calculated
I- V A L U E S )
A
.3
.I
0 7 ' Q
2
0
(I4)
.o I
D2
03
.04
.05
d(I - R ) P
where the coefficient of 7', is assumed to be the fraction of the effect due to radiation, and the coefficient of l', is the fraction assumed due to air effects. R refers to relative humidity and R7is a n equivalence factor. 94
I h E C PROCESS DESIGN A N D DEVELOPMENT
Figure 6.
Relation of
~ (l R)/P
JQ, J",
and v to distribution factor,
- - - - - - Assumed relationship for Section I1 - - - Approximation of values for Section I
8
d
kW a E 6 W,
25 W E
7-4
-+
J
2 I
2
ESTIMATED
4
3 T" (MINI
Figure 7.
Experimental time constant, T", vs. T" estimated from T Q a n d dashed lines in Figure 6
0
0
2 ( D N ) +0.4
4 i- T " FROM
ESTIMATED
6 FIG. 6
7'" =
---I' +
P + ICja(1 P - R) 3 TQ
[I
-P + IC7a(l - R )]7Q
or,
P
0.632 (54) U
Multiplying by LN and then inverting,
1
_____
-
u(T" - TQ)-
3
-
[12(0.632) ( 5 7 1
1 ~~
U(
TQ - TI')
P
[
K@(l - 0.044 f
+ Kja(1 - R) P
--I
IO TQ
L iT" ( M I N j
Figure 8. Experimental values of I mated values
Since T , is approximately equal to '/a TQ>and TQ approaches 0.632 (54)/u for Section I ? Equation 14 might be arranged approximately as folloivs:
8 AND
3. T"
vs. esti-
were obtained for all runs. and in Figure 7 experimental T" is plotted against 7"' estimated for all data. \t'hile there are a number of points out of line? this appears to be the best relationship obtainable from the data. 'Threr of the points most in error are among the four points from two-sheet runs. Another point is from the lowest sheet velocity run? \rhicli was about one half the next loivrst velocity used. 'I'hese points might be expected to be sornewhat diffeterit, since the various factors might be altered in these cases. The values of 7'" estimated ate also given in 'I'able 111. In obtaining the data, the most accurate experimental time value obtained !vas from the step change to the point Xvhere 63.2% of the final moisture change occurred. l'his corresponds to the dead time plus the time constant, Ti'. In some runs small errors might have caused increases in L at the 7'" would be a expense of T " , or vice versa. Because L more reliable method of evaluating the relationship obtained, TI' is plotted against estimated L T" experimental L in Figure 8.
+
-+
+
- R)
P
To check this relationship, Figure 6 was plotted with I / u ( TQ - T " ) as a function of a ( l - R ) / P . Because of the nature of the ordinate, two negative values were plotted a t the t o p of the graph. With the exception of two points, the data on Section I appear to follow reasonably a curve with about the right intercept but not a constant slope. Thus Ks appears to be a function of a ( 1 - R ) / P in Equation 15. A straight line \+as d r a w n to obtain a n approximation of the values for Section I (the lower dashed line in Figure 6). For Section I1 a similar equation can be obtained.
For Section 111, TQ= Tp and a similar equation has no significance. The upper clashed line in Figure 6 shows the relationship assumed for Section 11. Using the straight lines in Figure 6, estimated values of T"
Discussion of Results
I n considering the dynamics of a control system. in many cases a very accurate value of the dynamic constants is not necessary, because a control system under rrlativrly stable conditions is not particularly sensitive to changes in time constants within certain limits. Therefore even though the relationships obtained in this study were not very consistent, they might be useful in obtaining approximate values for time constants and dead times, for estimating the stability of proposed drying operations. In the more complex systems. \vhere more accurate dynamic data may be required, tlie relationship presented here might be the basis of a more refined and reliable niethod of obtaining the drsired data. If better estimates of 3r/dh1' and 3r;'aT' under various dryilig conditions and of the distribution of heat effects Xvithin the dryrr could be determined, more accurate equations might be developed. One way in which these relationships might be used fairly reliably would be to use T p and Equation 13 to predict the miiiimum time constant, and T, from Equation 13A to predict maximum values. As indicated by Figures 4 and 5. this procedure \vould be scriously in error i i i only a very few cases. One important result was to indicate that the ZieglerVOL. 3
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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)
