546
Ind. Eng. Chem. Process Des. Dev. 1980, 79, 546-550
kbwe, C. F., Smh, C. L., Murill, P. W., Instrum. &ntrolSyst., 43(1), 70 (1970). Mosler, H. A., Koppel, L. B., Coughanowr, D. R., AIChf J., 13, 766 (1967). Mvtharasan, R., Coughanowr, D. R., Ind. Eng. Chem. Process Des. Dev., 15, 141 (1976). Mutharasan, R., Coughanowr, D.R., Ind. fng. Chem. Process Des. Dev., 13, 168 (1974). Paraskos, J. A., McAvoy, T. J., AIChEJ., 16(5), 754 (1970).
Seinfeld, J. H., Gavalas, G. R., Hwang, M., Ind. Eng. Chem. Fundam. 9, 651 (1970). Vermeychuck, J. G., Lapidus, L., AIChE J., 19, 123 (1973).
Received for review July 23, 1979 Accepted June 12, 1980
Implementation of a Feedback Direct-Digital Control Algorithm for a Heat Exchanger Gerhard K. Giger, Donald R. Coughanowr,' and Werner Rlcharz Department of Industrial and Engineering Chemistry, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland
The control algorithms developed by Mutharasan and Coughanowr are verified experimentally for a double-pipe heat exchanger. A method to modify these algorithms for inclusion of dynamics and hysteresis of the control valve is presented in this paper. Closed-loop responses of the actual heat exchanger for set point changes in outlet temperature showed these modifications to be important for good transient behavior of the control system. Compared with the traditional proportional-integral control these algorithms have superior transient response characteristics, especially when shifting of set point over a wide range is required.
Background a n d Objectives During the past few years, several authors have studied design and implementation of direct-digital control (DDC) algorithms for distributed-parameter systems. The present work is based on this background which is reported in more detail by Fuhrman et al. (1980). The goal of this work is to verify experimentally the control algorithms, ALGl and ALG2, developed by Mutharasan and Coughanowr (1974) for a double-pipe heat exchanger. For this purpose an experimental system was designed which was similar to that used by Fuhrman et al. (1980). In this case the inner tube of the heat exchanger is filled with Sulzer static mixing elements (Tauscher and Schutz, 1973), for which the mean residence time of the heated water is much shorter than in the experiments mentioned above. The outlet water temperature represents the controlled variable, and the flow velocity is the manipulated variable. The performance of the algorithms was experimentally tested by the closed-loop system response to set point change. Algorithm Implementation by the Computer The experimental studies presented here are based on the same process model and the same control algorithms as described by Fuhrman et al. (1980). The model and algorithms are not presented here. The storage of the past velocities required by the implementation of the algorithm is organized here in a different manner than in the work by Fuhrman et al. (1980). The past 24 values of dimensionless deviation in velocity, m, are stored sequentially in a loop array with a pointer which shifts one position at the end of each sampling period. This kind of storage enables the saving of some computing time. For the same reason, the procedure for the implementation of the control algorithm is organized in a different order and can be summarized as follows: (1)
Sample outlet temperature and convert to dimensionless form. (2) Calculate velocity required to reach the set point in one residence time. (3) Send signal to valve after necessary conversion of units. (4) Store the newest value of velocity and shift the time pointer one position. ( 5 ) Compute the residence time for the following sampling period. By means of this procedure, the time between sampling of the controlled temperature and the control action can be shortened considerably. Inclusion of Static a n d Dynamic Behavior of Final Control Element In the experimental system used in this investigation, sensor dynamics contribute very little to the lag of the entire process and can be neglected. In contrast, the dynamic response of the final control element, which is a significant part of the control system dynamics, can be represented by a first-order delay. Preliminary experiments proved that this model is more accurate than the straight-line approximation used by Fuhrman et al. (1980). In this formulation of the algorithm, the valve response is considered to be slow compared to the process response and the sampling period is of the same order of magnitude as the residence time. As a result of these conditions, there is not sufficient time in one sampling period for the flow rate to reach the value requested by the computer a t the beginning of the sampling period. The variation in flow rate is shown in Figure 1. The flow rate reached at the ith sampling instant is given the symbol mfi-l; ideally, if the valve response were instantaneous rnfi-l = mi-l. As shown in Figure 1for several successive sampling intervals, the variation of flow rate between the value at the beginning of the sampling interval (mfi-2)and the value reached a t the end of the interval (mfi-J varies according to a first-order response. With this background, the flow rate m(0)during the period (i - l)0, I 0 < io, can be expressed by
* Department of Chemical Engineering, Drexel University, Philadelphia, PA 19104. 0196-4305/80/1119-0546$01.00/0
0 1980 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 547
\el
L),
Figure 2. Experimental apparatus.
rl
Figure 1. Characteristic path of a fluid element with first-order valve dynamics.
As a result of this variation in flow rate, the characteristic path of a fluid element will be curved between sampling instants as shown in Figure 1. This is in sharp contrast to the linear variation 'between successive sampling instants which occurs for a valve which has no lag. Equation 1can be integrated to give the average velocity for the sampling period from (i - l)O, to io,
0, -[I - e x ~ ( - 8 , / ~ , ) l ( m ,-- ~mf,-d (2) 0, The final value of the flow velocity can be written as
mfl-l = ml-l - (ml-l- mfL-2)exp(-8,/8,)
(3)
In a control situation, at the time io,, the past values of final velocity, namely, mfl-l, mflT2,...,and the average velocities, ml-l, f i l - 2 , ..., are stored in the computer memory and therefore are available. Along the characteristic path of a fluid element, one may write k
1=
c(i+ ml-,)es+ (1 + m,-,-,)~o, ]=1
(4)
This equation enables! one to calculate the residence time of the exit fluid element a t io, when the dynamics of the control valve are significant compared to the process dynamics. The calculation of the manipulated velocity needed for the outlet temperature to reach the set point in one residence time is given by eq 16 and 17 of Mutharasan and Coughanowr (1974) or eq 1 2 and 13 of Fuhrman et al. (1980); in the second of these equations, the velocity ( m ) is replaced by the average velocity (m) computed from eq 2 ad this paper and the values of k and X are obtained from eq 4. The original control algorithms described earlier are based on linear static flow characteristics of the final control element. However, in the experimental system used in this investigakion, hysteresis of the valve is significant and must be used to further modify the algorithms of Mutharasan and Coughanowr. A reasonable model to describe the control valve characteristics uses a straightline approximation with constant hysteresis. Hence, the calculated manipulated velocity, m,, which will be sent to the control valve, has to be modified. The hysteresis lag ( H / 2 in Figure 3) is added or substracted to the computed value of voltage signal depending on the change of sign of (m,- m1-J. Experimental System A simplified flow sheet of the experimental system is shown in Figure 2. The apparatus consists of a 1-m double-pipe heat exchanger with an inner diameter of 0.0285 m. Water f l o w through the inner tube which is
2
( 4
Coltace at t-e d i - r - a l - a - d q
6
- 5
I(
>
c amcl
Figure 3. Flow characteristics of the final control element.
filled, for better heat transfer rate, with 33 elements of Sulzer static mixer packing and steam condenses on the shell side. By means of a separate control circuit, steam pressure is kept constant at 1.5 X lo5 N/m2, corresponding to a wall temperature of 111.8 "C. Both the time constants of the resistance thermometer and the low-pass filter are small and therefore have been neglected in the modification of the algorithms of Mutharasan and Coughanowr. The digital/analog channel of the computer is interfaced to the control valve through an electro-pneumatic converter. The dynamics of the control valve were determined in preliminary experiments. The computer sends a voltage to the voltage/pressure ( e / p ) converter and the echo velocity is recorded using a flow transducer of the turbine type. A first-order delay having a time constant of 0.9 s proved to be a reasonable model for the final control element. The inherent flow characteristics of the valve shown in Figure 3 were measured with a rotameter. In the flow region between 0.25 m/s and 0.6 m/s where closed-loop experiments were performed, the valve characteristics can be represented with good accuracy by a model having a linear sensitivity with constant hysteresis. Open-Loop Behavior For different flow rates, the outlet temperature of the heat exchanger was determined experimentally and the corresponding heat transfer coefficient was calculated. As illustrated in Figure 4, this coefficient proved to be almost constant for the experimental system used in this investigation. Hence the related parameter, b, in the process model can be set equal zero. The other parameter, p, was calculated to be 0.2. The values of the parameters b and were 0.3 and 1.0, respectively, in the experimental system of Fuhrman et al. (1980). The dynamics of the heat exchanger were determined by measurement of the response of the outlet temperature to flow rate change. The corresponding theoretical response was obtained by solving the process model equation
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
548
.A
2 z
A L G 1 , C= 0 A L G l , -?=-205 ALGl , .4=+203
E
i 0.2
J 0.3
0.4
0.5
velocity t h r o u g h the h e a t e x c h a n g e r , v
C.6
2
0
tm,
A2
A3
15
52 45 64
55 63
16 18
a
6
4
A1
io
55
16
14
12
tlSi
0.7
Figure 6. Response to step change in set point from 30 " C to 25 OC when modeling errors are present; t , = 1.0 s.
m/51
Figure 4. Heat transfer coefficient. 30.0 *2
*3
ALG1
15
52
55
? X 2 , K4.4
14
52
44
A L G 2 , K=2.0
15
46
33
I
I 0
4
2
6
8
tm,
10
14
12
16
t(S)
Figure 7. Response of ALG2 to step change in set point from 30 OC to 25 O C ; t, = 1.0 s.
-
30.0
."
, t =1.0 , tS=2.0 , t =4.0
j
E
Y
A1
A2
4
5
14
50
44
s
17
51
66
s
19
57
92
--
25.0.
y
i
Y
2 0
2
4
8
6
10
12
14
16
tune,
Figure 8. Response of ALG2 to step change in set point: effect of sampling rate; K = 0.4.
0
2
4
6
8
9
A2
A3
14
M
44
10
12
14
16
tm. t!Sl
Figure 9. Response of ALG2 to step change in set point when time delay or hysteresis in final control element is ignored; t, = 1.0 s; K = 0.4.
(7)
These criteria were calculated directly by the computer and printed after each experiment.
Algorithms ALGl and ALGB With algorithm ALG1, the outlet temperature reaches the new set point almost without overshoot for the exact
value of process parameter p as shown in Figure 6. In the presence of modeling error in the parameter 0,the performance of this algorithm is somewhat poorer, but still seems to be stable. The time of the transient of the closed-loop system approaches that of the open-loop system, which is the shortest transient achievable. With the algorithm ALG2, the speed of response can be improved, but overshoot occurs if K is increased. The optimal value of K was found to be about 0.4 as shown in Figure 7. Figure 8 shows a set point response with ALGB for different sampling rates. Even with sampling times twice the process residence time, the control system behaves
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 549
1
35.0
'$
Fmz FLG1 LG 1, mz ,
3C.0,
mz ,
~ 3 . 4 ~ 3 . 4 ~ 4 . 8
A1
A2
x3
4 12 41 44
277 294
1154 53 154 187
277 279
30.0
I
2 5 . 0 v
3
25.0
ir 3
1 2
4
8
6
tune.
-
10
11
12
16
t(S1
30.0
i U 2 , Ka.4,
5 3
ts=l.O s
kpt,,tRopt,t s 3 . 2 5 s PI, k p t , ,tkpt, t ~ 1 . 0s PI,
$
25 0
4
6
8
10
12
14
16
tune, t s s )
Figure 10. Response of ALG2 to step change in set point from 25 " C to 35 " C ; t , = 1.09.
2
2
A?
A2
A3
14
30
44
13
43
44
16
43
67
Figure 12. Response of ALG2 and PI-control to step change in set point from 25 O C to 30 "C when hysteresis is ignored in the PI algorithm.
because of the nonlinearity of the process. Therefore ALGl and ALGB are far better when set point has to be shifted over a wide range of outlet temperature. Conclusions Two feedback direct-digital control algorithms derived by Mutharasan and Coughanowr (1974) were verified experimentally for a double-pipe heat exchanger. Both the algorithms ALGl and ALG2 are easily implemented and give a good transient response even when there is an error in the knowledge of the process parameter. However, it proved to be of essential importance that these algorithms be modified to include the dynamics and hysteresis of the control valve. With this modification, the algorithms showed superior performance compared to the traditional proportional-integral control, which is very sensitive to sampling rate and shifting of set point. It is believed that the type of feedback direct-digital control algorithm used in this investigation is quite versatile and practical in cases where nonlinearities in the controlled system are significant. Acknowledgment The authors wish to thank Dr. R. Mutharasan for his helpful suggestions. Nomenclature A, = cross-sectional area of the heat exchanger A, = surface area of heat transfer A1,2,3= error criteria; see eq 5, 6, and 7 b = exponent in expression for variation in heat transfer coefficient with velocity cp = specific heat of the fluid k = an integer K = constant in the proportional-integral controller; also the tuning parameter in ALG2 m, = computed value of manipulated variable value between ith and (i + 1)st sampling instants mi = mean value of the manipulated variable between ith and (i + 1)st sampling instant m,: = manipulated variable value at the (i + 1)st sampling instant m(0) = manipulated variable, actual value in the process t = time t , = sampling period tR = reset time in the proportional-integral controller T = temperature T , = set point temperature U = heat transfer coefficient u = velocity of fluid Greek Letters (3 = process parameter, defined as UA,/oA,c,p 7 = dimensionless distance along the system 0 = dimensionless time 0, = dimensionless time constant of control valve Or = residence time of fluid element
Ind. Eng. Chem. Process Des. Dev. 1980, 79, 550-555
550
Os = sampling period X = fraction of the sampling period p = density of the fluid
Harriott, P., "Process Control", p 181, McGraw-Hill, New York, 1964. Mutharasan, R., Coughanowr, D. R., Ind. Eng. Chem. Process Des. Dev., 13, 168 (1974). Tauscher, W., Schutz, G., Tech. Rundsch. Sulzer, 2 (1973).
Literature Cited
Received f o r review July 23, 1979 Accepted June 12, 1980
Fuhrman, J. E.,Mutharasan, R., Coughanowr, D. R., Ind. Eng. Chem. Process Des. Dev., preceding article in this issue (1980).
Intrinsic Kinetics of the Reaction between Oxygen and Carbonaceous Residue in Retorted Oil Shale Hong Yong Sohn" and Sun K. Klm Departments of Metallurgy and Metallurgical Engineering and of Mining and Fuels Engineering, University of Utah, Salt Lake City, Utah 84 7 72
The intrinsic kinetics, unaffected by diffusional and mass transfer effects, of the oxidation of oil shale carbonaceous residue have been determined. For samples with undecomposed carbonate minerals, the activation energy is 22.06 kcal/g-mol. The activation energy remains essentially the same when the carbonate minerals in the sample are decomposed and then recarbonated. The oxidation rate shows first-order dependences on b e oxygen partial pressure and the amount of unreacted char. It is shown that a nonisothermal method with linearly increasing temperature provides a useful technique for determining the intrinsic kinetics of gas-solid reactions.
Introduction
During the retorting of oil shale, kerogen is thermally decomposed to produce oil vapor, hydrocarbon gases, and carbonaceous residue. In various above-ground and in situ retorting processes this carbonaceous residue is burned to produce energy for heating the raw shale. Oil shale of 20 gallons per ton (GPT) or higher grade produces a sufficient amount of char to provide all of the energy requirement for retorting if it can be burned completely (Dockter, 1976). Thus, the efficient combustion of the carbonaceous residue is an important factor in increasing the oil yield because oil vapor must otherwise be burned to obtain the required energy. Furthermore, the uniform and rapid combustion of oil shale char would also be important in preventing oxygen from reaching the zone of kerogen decomposition where it can burn the valuable oil product. Kinetics of the combustion of oil shale char have been determined by a number of previous investigators (Dockter, 1976; Dockter and Turner, 1978; Mallon and Braun, 1976; Soni and Thomson, 1978; 1979; Tyler, 1977). Most of the previous work has been concerned with the combustion of oil shale char at high temperatures and in large samples. Under these conditions the overall rate is controlled by the diffusion of oxygen through the ash layer. At temperatures higher than the level at which carbonate minerals begin to decompose (about 700 "C)the reaction between the char and carbon dioxide liberated from carbonate minerals complicate the problem. The rate expression for a diffusion-controlled reaction, however, is not capable of predicting the point at which the controlling mechanism shifts from diffusion to chemical kinetics or vice versa under continuously changing temperature. Thus, in order to determine the total amount of char burned as the combusion front passes an oil shale block, information on the intrinsic kinetics of char oxidation is required. Soni and Thomson (1979) recently reported the results of the reinterpretation of their earlier measurements (1978) of char oxidation kinetics. They used the 0196-4305/80/1119-0550$01.00/0
isothermal method. Their work does not indicate a systematic elimination of diffusional effect, although they make a qualitative mention of the problem. It is the objective of this paper to report the results of an investigation to determine the intrinsic kinetics of oxidation of oil shale char by using an isothermal as well as a nonisothermal technique. Considerable care has been taken to completely eliminate the mass transfer and diffusional effects. The nonisothermal method yields the temperature dependence of the rate constant from a single run whereas the isothermal method requires many runs to obtain the same information. The reactivity of carbonaceous residue contained in retorted shale with its carbonate minerals first decomposed and then recarbonated is also determined. The decomposition and recarbonation of carbonate minerals are expected to take place during the combustion retorting of oil shale. T h e o r e t i c a l Consideration
The intrinsic kinetics for the reaction of a solid with a gas may follow a number of different rate expressions, which in general can be written as
where X is the fractional conversion of the solid reactant, k is the reaction rate constant, f l ( p A )is the dependence of rate on gaseous reactant concentration, and f 2 ( X )is the dependence on the fraction of solid remaining unconverted which is normally a function of geometrical change in solid as the reaction proceeds. Topochemical reaction of nonporous particles and nucleation-and-growth kinetics are examples of the results of such geometrical changes. Equation 1 has been written for the cases where dependences on PA and X can be separated, which is the case for most commonly encountered systems. Integration of eq 1 gives 0 1980 American
Chemical Society
