Degrees of Freedom in Dynamic and Static Systems D. Christopher Dixon School of Chemical Engineering, University of N e w South I17ales, Kensington, S.S.IT7. 2033, Australia
Provided that the physical relationships governing a system are understood, the number of degrees of freedom can be determined by counting the boundary and geometrical variables and subtracting the number of independent relationships among the boundary variables. The dynamic number of degrees of freedom can exceed the static, the difference being equal to the number of independent hold-ups which are undetermined at steady state. Addition of control loops can make dynamic and static numbers of degrees of fteedom equal, if they were not previously, but not every control system changes the number of degrees of freedom. A distillaticn column i s used as a special example, and the results of applying the present analysis are compared with those of previous writers.
T h e r e has been some discussion in recent years of the determination of the number of degrees of freedom available in steady-state design problems (Forsyth, 1970; Hoffman, 1964) and in cont'rol syst'em design problems (Howard, 1967; Yurrill, 1965). The accent has been on distillation columns, aiid other multistage separation processes, since these are among the more complex of commoii processes. The following discussion will concentrate 011 the control syst,em design case, which has some aspects which are not involved in the st'eadystate case and which does iiot appear to have been properly treated in the literature. However, the approach to be presented is also applicable to steady-state design problems and may be of interest as an alternative to previous methods in this contest. The number of degrees of freedom in a given problem is equal to the number of variables involved minus the number of independent equations (constraints) relating them. This is the number of variables whose values may be arbitrarily specified (within certain ranges aiid as functions of time in an unsteady-state situation), thereby determining the values of the other variables. The phase rule gives the number of degrees of freedom required to fix the intensive state of a tem a t (mechanical, thermal, aiid chemical) equilibrium. However, in the more general cases to be considered here, open systems are involved and several types of constraint other than equilibrium. Thus, although the analysis could be based on the phase rule with additional const'raints (Gillilaiid aiid Reed, 1942), it seems simpler t'o start' from first principles, lisbing all constraints without special attention to those considered by t,he phase rule. The determination of the iiumber of equations relating the variables of a system requires a thorough understanding of the physical relationships involved. It does not seem likely that any method for determining the iiumber of degrees of freedom of a system can be applied without this knowledge. Hence, the best t,hat' can be expected is a method n-hich does not require a detailed formulation of the syst'em equations. Basic Analysis
The number of degrees of freedom equals the number of variables minus the number of independent equations relating them. The number of variables involved in a system can, however, be very large, and in a control problem one usually restricts attention to boundary variables. The reason for this 198 Ind. Eng. Chem. Fundam., Vol. 1 1, No. 2, 1972
is, of course, that the response of the system to externally produced disturbances is the major concern, and every entering disturbance must disturb some of the boundary variables. Thuq, an independent set of variables, equal in number to the number of degrees of freedom, is chosen from among the boundary variables, aiid these are commonly referred to as the input variables. Using this approach, the number of degrees of freedom, or variance, V , is given by
where l y b a is the number of boundary variables and h-be is t'he number of independent equations containing boundary variables alone (i.e., iiot involving internal or est,ernal variables). d boundary variable is a quaiit,ity which has a value a t a point on the system boundary and which can vary under the conditions being considered. d boundary quantity which is physically capable of being varied (such as the temperature of the feed to a reactor), but which is iiot varying under the specified conditions, is not counted among Xbv.Thus, - Y b v is not solely a property of the system but depends also on the conditions being considered. If this constraint lvere iiot placed on . v b v it' would be very large even for a simple chemical engineering systems, the variables a t a point on the bouiidary are usually some or all (depending on which are varying) of those listed in Table 1. : v b e does iiot include equations which involve internal variables. These place no constraint8 on the boundary variables, since it has been decided not, to choose the degrees of freedom from among the internal variables. External variables, also, are not included among the degrees of freedom, since the system is t,he only portion of space which is beiiig considered, by definit,ion. The t,ypes of equation which commonly describe chemical engineering pyst,ems are also listed in Tahle I. Relationships established by controllers will be excluded for the present. Table I is iiot intended to imply that a given equation, which is one of t,he t'ypes listed, necessarily relates boundary variables alone. It often happens that two equations have to be combined, thereby eliminating: an internal variable, to obtain one boundary equation. This is discussed furt,her belox. The input variables are defined above as a set of iadependent boundary variables equal in number to the number of
degrees of freedom. The word "independent" is included in this definition because it is not always possible to choose any set of V boundary variables as the input variables. The formal criterion is that there must be no equation Jvhich cont'aiiis only input variables. For example, in a system having three degrees of freedom and three material streams crossing the boundary, the mass flow rates of the three streams cannot be chosen as the input variables a t steady st,ate since they are related by t'he total-mass balance equation. There is a further aspect of the iiat'ure of degrees of freedom which must be considered in some cases. Having chosen a set of input variables, their values determine the values of t,he internal variables and other boundary variables. However, it is oft'en the case that the values of only some of the dependent bouiidary variables are of interest. I n such cases, it can happen that some of the input variables have no effect on these dependent variables. These "neutral" input variables represent degrees of freedom of the system since they are in fact free t'o be arbit'rarily specified, but if one is only interested in the number of ways in which the result of the process can be influenced t'hey are not' relevant and would not be counted. A given input variable is neutral if it is possible to find a set of independent boundary equations not containing t'his variable, but containing all the dependent boundary variables of interest and containing a total number of dependent boundary variables equal to the number of equations. An input variable which appears in none of the boundary equat,ions is always neutral, no matter which boundary variables are of interest. I n a control problem, the geometrical parameters of bhe system (e.g., number of plates and plate design in a distillation column) are usually fixed. However, this is not necessarily so, and the more sophisticat'ed control systems have variables of t'his type. This makes no difference to eq 1 and adds no additional types of equation to Table I ; it simply provides additional boundary variables and degrees of freedom. I t is quite natural to think of such geometrical variables as boundary variables for a control system, since some externally operable mechanism must be provided for performing the adjustments. I n a steady-state design problem, however, all the geometrical paramet'ers are available for specification but would not normally be considered bo be boundary variables. I n this case, N b v in eq 1 equals the number of boundary variables plus t'he number of variable geometrical parameters. A further consideration, which arises in a control problem, is that there are, in fact, two numbers of degrees of freedom which need to be considered: st'at'ic, V,, and dynamic, V d . The static (or steady-state) number of degrees of freedom is the value of V under the constraint' that the system is a t steady state, while this constraint' does not apply in evaluating Vd. Thus, eq 1 can be replaced by t'wo equations
v,= v d
=
Nbv S b v
- Arbes
(2)
-
(3)
S b e
where N b e s is the value of Aibe when the syst'em is a t steadystate. v d can be greater than V,: that is, S h e can be less than Shes. This can arise because the balance equations contain accumulation terms under dynamic conditions, but not a t steady state. If a variable which determines the accumulation term in a balance equatioii is not related to the boundary variables, then the balance equation will be couiit'ed among A'ber, but not among N b e .(Balance equations are the only equations
Table 1. Usual Boundary Variables and Equations in Chemical Engineering Systems a . Boundary Variables Intensive properties
Composition Temperature Pressure Or another set of (W 1) intensive properties, where A' is the number of chemical compounds present.
+
Rates of Flow across the Boundary
Material Heat Shaft work ( L e , , all work except flow work). (Flow work is not listed separately, because it ib determined by the flow rate and pressure of the material.) b. Equations 1.
Balance
input output consumption accumulation rate rate rate rate nhere t = time 2 = amount in the system = hold-up (a) Material (b) Energy (total) (c) llechanical energy or momentum 2.
Equilibrium
Rote
or
(a) Mechanical
hfechanical energy consumption (friction, etc.) or momentum consumption (all opposing forces) (b) Thermal Heat transfer (c) Phase Ilaterial transfer (d) Reaction Reaction (At a given point, for a given type of process, either an equilibrium or a rate equation applies, bnt not both.)
in Table I which are differential in time, and so are the only ones which contain terms under dynamic conditions which are absent a t steady state. Reaction rate equations, for example, are often stated to be differential equations in time, but they are not in fact (Dison, 1970).) Any relationship which exists between a hold-up determining variable and the boundary variables will exist under both dynamic and static conditions. The accumulation term in a dynamic balance equation gives the change in the hold-up (of whatever entity the balance is accounting for) from some initial value. Thus, if the system is imagined to start from a steady state, the hold-up is subsequently determined only if it was determined a t steady state, that is, if the hold-up has a unique value a t steady state for a given set of values of the boundary variables. (A further point, which is required later, is that the only variables which can be undetermined a t steady state are hold-up variables. Every variable must be affected by the boundary variables under dynamic or static conditions, if not both; otherwise it would not vary and so would not be a variable. Since the only differences between dynamic and static equations are the accumulation terms in the balance equations, hold-up variables are the only ones which can vary under dynamic conditions while having no unique steady-state values.) Hence the simplest approach appears to be to determine V , from eq 2, and then lid from v d
-
v,= 5,"
Ind. Eng. Chem. Fundam., Val. 1 1 , No. 2, 1972
(4) 199
a;.voi. f l o w
constant pressure
cd oe n ssti at y I rquld
I
I I I
~
L - - - -
syslem
boundory 2 - - - - J
00' Qo(h)
' b . P u m p e d - d i s c h a r g e Tonk
liquid
c o n s t a n t - speed
0, i n d e p e n d e n l o f h
I
Heat Exchanger
-
constant comp0:ition
Liquid A
d. Isothermal Reactor Feed
product overf lo w
perfectly mixed
Figure 1.
Simple examples
n-here S,,is the number of independent hold-ups which are undetermiiied a t steady state. K h e n V dexceeds V , the number of input variables is usually taken to be the former. I n a system for which Vd = V,, a steady state exists for any set of constant values of the input variables (within operating ianges). Such a system nil1 be referred to as sdfregulatory, although there is no guarantee, of course, that any such steady qtate is stable. For a system which is riot selfregulatory, there will be no steady state corresponding to an arbitrary set of values of the input variables, but only for certain discrete sets Such a system would not normally be coiisidered satisfactory for contiol purposes. I t must be pointed out here that the possibility of Vd being greater than V,ariyes from the way in k\hich the problem has been approached i n termi of boundary variables. When exceeds V,, there are in fact A,' additional degrees 200
of freedom available a t steady state, b u t they cannot be fixed by boundary variables. These degrees of freedom can be fixed by a set of Ai,, internal variables, one for each independent undetermined hold-up. However, it appears t h a t in no case is any of these degrees of freedom of any use or relevance. They do not affect the relation between entering and leaving material streams and they cannot be directly fixed from outside the system. Hence, the best approach is to not count these degrees of freedom, which is what is done b y the above method of analysis. Having decided that only the number of steady-state boundary equations, .?-be%, need be directly determined, and not h i b e , some further remarks can be made about the equations in Table I. I n balance equations, the accumulation term will be zero, since only the steady-state case is considered. Since the input and output rates will always be described by boundary variables, every balance equation in which the consumption term is zero will be counted among x b e s . Thus, the total-mass balance, mass balances on nonreacting compounds, and the energy balance will always be boundary equations. I n balance equations in which there is a nonzero consumption term, the appropriate rate equation will be required to determine the consumption. If the rate is determined by the boundary variables then the combination of the balance and rate equations will be a boundary equation. For example, in a mass balance equation for a reacting compound, the reaction rate expression is substituted into the consumption term. If the concentrations, temperature, etc., which determine the reaction rate are related to the boundary variables, then the material balance-reaction rate combination is a boundary equation. Figure 1 shows four simple examples which have been chosen for illustration. The analyses are given in Table 11. For Figure la, it is easily seen b y inspection that the result Vd = V , = 1 is correct. Qlcan be varied arbitrarily (within limits imposed by the height of the tank) under both dynamic and static conditions, and this determines the variation in Qo. The mechanical energy balance, with appropriate friction loss expressions substituted into the consumption term, does not relate boundary variables in this case. The friction losses depend on Qi and Q o (boundary variables) but also on h. There is no independent equation relating h to the boundary variables, and the mechanical energy balance equation reduces, in fact, to the relation between Qo and h. I n effect, the mechanical energy of the inlet stream is dissipated by impact on the surface in the tank, no matter what the value of h, and so there is no connection between inlet and outlet mechanical energies. Another point about this example is that it contains a iiumber of irrelevant degrees of freedom. The liquid density is constant and Q o depends only on h, which implies that the viscosity is also constant. However, subject to these two constraints, the composition and temperature could vary, 1 - 2 = (.V - 2) additional boundary giving (*V - 1) variables for both inlet and outlet streams. The energy balance equation plus (S - 3) compound mass balance equatioiis (t1F-o of the (S - 1) compound balances are dependent on the rest, because of the density and viscosity constraints) provide additional boundary equations, leaving (A' - 2 ) additional degrees of freedom. These additional degrees of freedom are easily seen to be neutral, if Qo is the only boundary variable of interest, and they are not listed in Table I. However, this can also be shown by the formal procedure given above.
nT/-f+FL/'o
a. G r a v i t y - d i s c h a r g e T o n k
Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
+
Firstly, a permissible choice of the input variables is Q L plus inlet temperature and ('V - 3) inlet composition variables, since there is no equation relating these. With this choice of input variables, it is found that the total mass balance equation (Q1 = Qo) contains only one dependent variable (Q,,), and this is the boundary variable of interest, but the only input variable contained in it is Ql. Hence, the other input variables are neutral. Figure l b illustrates what appears to be the only common situation where Vd exceeds VB;that is, the situation where the in- and out-flows of a tank containing a phase boundary are unaffected by the position of the boundary. Compared to Figure l a , the mechanical energy balance-friction loss equation now relates boundary variables a t steady state, since there is now, in effect, an independent equation for h (k., h may be arbitrarily specified a t steady state). This extra equation (which reduces to Qo = constant, as shown in Table 11) reduces V , by one to zero. However, since h is undetermined a t steady state, v d - V , = 1, so that Vd = 1, the same as for Figure la. It is easily seen t h a t Q , can be arbitrarily varied under dynamic conditions but can only have one value a t steady state. If the pump in Figure l b were centrifugal, rather than a positive displacement type, Q,, would be affected to a small extent by h and so the system would be equivalent to Figure la. However, the range over which Q i could be varied, while still allowing a steady state, would be small; that is, the system would be self-regulatory over a small range only. I n the limit as the effect of h on Qo approaches zero, the self-regulatory range approaches zero and Vd - V , becomes unity. Thus, if one says that the effect of h on Qo, while not zero, is negligible, which makes v d - Vs = 1, this is equivalent to assuming that the self-regulatory range of the system is negligibly small. Even for Figure l a the self-regulatory range could be negligible under the conditions being considered. If the range of variation of Qo, which is allowed by the height of the tank, is very small compared to the variations in Qlwhich are expected, then the effect of h on Q o is negligible, in effect, and the system is equivalent t o Figure l b . T h e heat transfer rate equation is included among Nbe. for the heat exchanger example. The total rate of heat tiansfer is given by the inlet and outlet temperatures and flow rate of either stream, and the log mean temperature difference is given by the inlet and outlet temperatures. Thus, the total heat transfer rate equation can be expressed in terms of boundary variables. I n the reactor example, the volume of the reactor contents has been assumed constant. Hence, the total rate of reaction for each compound depends only on the composition of the contents, which is the same as the exit concentration (boundary variable). Thus, the S compound balance-reaction rate equations are boundary equations. If the variation of the volume with flow (because of the overflow product-removal system) were taken into account, the relation between depth and outflow (as in Figure l a ) would determine the contents volume in terms of boundary variables, and would be used to eliminate the volume from each compound equation, giving the same result for N b e s . Effect of Control loops
When a control loop (feedback or feedforward) is added to a system (which means that it is placed inside the defined system boundary), it establishes a relationship between a valve setting or other manipulable variable and another variable (the controller actuating variable), if the
Table II. Analysis of Figure 1 Examples Gravity-Discharge Tank
a.
Qo (valve setting does not vary, because the stem does not cross the boundary) Nbej 1 Material balance V s = 2 - 1 = 1 E.g., inlet flon- rate Liquid hold-up is determined by h , which Vd - V , = 0 is related to Qo =
s b v
2
Qi,
b.
Sb,.=
1
=
1
-1=0
V,
=
I'd
- V, = 1
1
Pumped-Discharge Tank
Q i (Qo is constant) Material balance Q i must equal Qo a t steady state Liquid accuniuhtioii is determined by h , which is not fixed by the boundary variables C.
xb,. =
A-beS
=
12 6
V 3 = 12 - 6 Vd
=
- T.', = 0
d. I\lhv
=
2 5
Shes =
+1
a\-
Vq = 2 5
+1-
I'd - T',
=
s = *\- + 1 0
Heat Exchanger
Flow, pressure, and temperature for each entering and leaving stream 2 x material balance 1 X energy balance 2 X mechanical energy balance-friction loss 1 X heat transfer rate 6 E.g., floxv,pressure and temperature of each inlet stream Temperatures determine energy hold-up, p r e s u r e s determine liquid hold-up; all are fixed a t steady state Isothermal Reactor
Compo..ition and flow rate of feed and product; temperature llaterial balance-reactioii rate for each compound E.g., compohition, flon- and temperature of the feed Material accumulation is dfterniined by reactor composition, which equal3 product composition
adjustable parameters of the controller (set-point, etc.) are fixed. The general method of analysis given above can, of course, be applied direct'ly t,o systems containiiig coiitrol loops. However, to examine specifically the effect of control loops, t,he change in the number of degrees of freedom resulting from their addition will be considered. For the purposes of t,his discussion, it will be assumed that the manipulable variable exists in the system before the control loop is added. Thus, the manipulable variable is always a boundary variable since, before the controller is added, it can only be varied from outside the system. If the manipulable variable does not already exist, it must be added to the system, as a preliminary to adding the control loop. If t,his can be accomplished by providing for the direct variation of some quantity which was previously fixed, then this adds one extra degree of freedom. The most common case is the addition of a control valve to a pipeline. This gives the line a variable, instead of fixed, flow resistance. I n the case of a feedback control loop containing integral action (it is assumed that integral action is never used in feedforward loops), the controller itself does not determine the manipulable variable as a function of the actuating variable a t steady state. However, it ensures that steady state can only exist when the actuat,ing variable has a particuInd. Eng. Chem. Fundam., Val. 11, No. 2, 1972
201
a. G r a v i t y - d i s c h a r g e T a n k , w i t h L e v e l C o n t r o l
oi
L
_ _ _ _ _ _ _ _ _ _ _ _1
system b o u n d a r y /f
b. G r a v i t y - d i s c h a r g e T a n k , w i t h F l o w C o n t r o l
%---
-,than previously. (The arrow indicating the controller set-point (sap.)does not cross the system boundaries, which indicates that the set-point is fixed.) Figure 2b shows another control system for the gravitydischarge tank. Again the actuating variable (Qo) was previously determined a t steady state and so the controller reduces Vd and V , by one, canceling the extra degree of freedom from the control valve. However, in this case the self-regulatory range has been reduced, and if Qo is closely controlled Q , has only one value for steady state, and the result is Vd = 1, V , = 0. Looking a t this in another way, with close control Q I is no longer a permissible choice for the input variable a t steady state, but the valve position can be chosen. The valve position may be arbitrarily specified a t steady state. This merely determines the value of h required so that Qo has the required value. However, the valve position then represents an irrelevant degree of freedom since it only affects h , and not Q,,,leading again to the result v d = 1, V 8 = 0. With close control of the flow, the system is equivalent to Figure l b . Figure 2c shows a level controller applied to the pumpeddischarge tank (Figure lb), which originally had Vd = 1, V, = 0. The addition of the control valve has added an extra degree of freedom. Since the actuating variable was previously undetermined a t steady state, adding the controller does not affect V. but reduces Vd by one. The filial result is Vd = Ti, = 1, and the system is now self-regulatory and equivalent to Figure l a . In the above discussion, adjustable controller parameters have been assumed fixed. If any one of these is variable, this adds an extra degree of freedom in all cases. The above discussion has also been limited to control loops which are added to the system. A control loop which is external to the system (because of the defiiiition of the system bouiidary) can reduce the number of degreeg of freedom by removing (nearly) the variatioii in a boundary variable. Vd and V. are reduced by one, simply by reduciiig ‘l’b,. From the above d i x u w o n , the statement (lIurri11, 1965) that “the number of iiideloendeiitly-actiug controllers which may be added to any system cannot eweed the number of degrees of freedom which are iiiherent in the system” requires some qualification. K h e n Vd exceeds the former is the number of degrees of freedom which applies to this rule. Assuming that the final Eystem is required to be self-regu-
v.,
latory, S,,loops must be actuated b y previously undetermined hold-up variables. A maximum of V , other loops may be added which are either external loops holding boundary variables constant or internal loops actuated by previously determined variables. The manipulable variables for all iiiternal loops must exist before the rule is applied. Distillation Columns
As remarked in t’he introduction, distillation columns are amoiig the more complex systems and have received considerable attention in connection with the present subject. There is some disagreement between different writers. Figure 3 shows a basic two-product distillat’ion column, as has been considered by several writers. Showing the heat transfer rat.es, for condenser and reboiler, esplicity by arrows implies that the system boundary is being taken as coincident with the heat transfer surfaces, rather than as passing through the water and steam inlet and outlet pipes. The degrees-offreedom analysis of this system is summarized in Table 111. I n Table 111, S phase equilibrium relations ( L e . , equality of chemical potentials between phases for each compound) are shown. The top and bottom products are not’, of course, in equilibrium with each other (unless the column contains only one ideal stage). However, the equilibrium relationships between the phases on each plate through the column give, overall, one set of iV equations relating top and bottom compositions. A similar comment applies to the mechanical and thermal equilibrium relationships listed. The assumption of equilibrium is only an idealization, of course. However, even if this assumption is not made, the same number of rate equations will be obtained, instead of the equilibrium equations. The inclusion of the mechanical equilibrium and mechanical energy balance-friction loss equations is only permissible because of the assumed poor self-regulation associated with the two accumulators. The levels in these accumulators are assumed to have negligible effect on product flows ( c j . discussion of Figures l a and b) so that the friction losses in the accumulators are related to product flows, and the discharge pressures of the two products are related to the reboiler and condenser pressures, and hence to each other via the equilibrium relations through the column. The liquid levels on the trays are assumed sufficiently variable to give adequate regulation, and these are determined independently of the overall mechanical energy balance-friction loss equation by the internal refluxing arrangements. I n the above analysis, the number of plates in the column has been taken as fixed, which is the usual situation in a control system. In a design problem, the numbers of theoretical plates i n the rectifying and stripping sections are not fixed and provide two additional degrees of freedom. If real plat’es are considered, there are numerous additional degrees of freedom in the plate geometrical details, unless a fixed plate design and spacing is used. The analysis in Table 111can be compared with that of other writers, for the design problem where the numbers of theoretical plates are variable. I n this case, the present analysis gives V , = N 6. If the feed composition, temperature, flow rate, a i d pressure (or the “column” pressure) are specified, four degrees of freedom remain, which agrees n-ith the result of Kwauk (1956) for a biliary system and of Forsyth (19i0) for a quinary system. (Xeither of these writers states esplicit,ly lvhet,her the accumulators are self-regulatory or not. However, by implicat,ion they consider them nonself-regulatory, which is the present case. They make the usual assump-
+
Table 111. Analysis of Distillation Column of Figure 3 Boundary variables
S S 2 2
+2 +2 +2
35
Feed composition, temperature, presure, flon Top composition, temperature, pressure, flow Bottom composition, temperature, pressuie, f l o ~ Reboiler heat flow and steam temperature Condenser heat flow and steam temperature Total
+ 10
Steady-state boundary relations
s 1 1 1 1
s 2
Xaterial balance Energy balance l\Iechanical energy balance-friction loXeciianical equilibrium Thermal equilibrium Phase equilibrium Heat transfer rate-reboiler and condenser Total 3-Y 10 - ( 2 s 6) = S 4 T’, = 2 Lelels in reboiler and condenser are assumed to h a r e negligible effect on product flow rates
cq 5’. + TVd
= =
+
+
s y s t e m boundary
condenser
I
I I I
reflux
I
(
I
t o p product
I
feed I N compounds
negligible heat loss
>
I
I
I
I I I
Figure 3.
Basic distillation column
tion of uniform column pressure, or all pressures determined. If the accumulator levels have a significant effect, this assumption cannot be made.) Hoffman (1964), however, maintains that there are only three degrees of freedom for this system, but this apparent disparity (Forsyth 1970) can be explained. First, Hoffman does not include the two numbers of theoretical plates among the degrees of freedom, since the numbers of plates must be integral. Strictly speaking, this is correct, since a variable which is a degree of freedom should be continuously variable over its allowable range. However, this is really only a matter of definition, the practical point being that the numbers of plates are variable. K i t h this restriction, the number of degrees of freedom is reduced from 4 to 2 . Secondly, Hoffman does not include the pressure in specifying the feed. This gives a n extra degree of freedom, and a final number of 3. There is, of course, no reason why the feed pressure should be specified. This simply leaves some other variable (such as the pressure a t the top of the column) free for specification. Thus, the only way in which Hoffman’s analysis differs from others is in not counting the numbers of plates as degrees of freedom. Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 2, 1972
203
/
system b o u n d a r y
there are four additional boundary relations; the material balance and mechanical energy balance-friction loss equations for water and steam. Thus, Figure 4 has 13 - 4 = 9 additional degrees of freedom. T o simplify the analysis, the inlet temperatures and pressures, and discharge pressures, will be taken as constant for water and steam. There are then three additional degrees of freedom in Figure 4 (three of the five valves, the other two replacing q. and pc, in effect), g i v i n g v , = $ 7 and V d = N 9. Since Vd is greater than V., the first priority usually is to reduce V d so that it equals V,. From the previous discussion it is seen immediately that this can be done by actuating two of the valves from the levels in the condenser and reboiler. One such set of connections is shown in Figure 5 , together with other control. loops considered, as an example, by Murrill and by Howard. The two level controllers are not required to closely control the levels, but only to keep them within limits, thus achieving “material-balance control” (Buckley, 1964). If the top product valve, say, was used in a product flow control system, instead of in the material-balance system shown, then Vd and V , would both be reduced by 1 and their difference would remain unchanged. As was found in some early installations, such a system does not work satisfactorily. The other control loops affect V d and V , equally, each overall loop reducing Vd and Ti, by one. I n particular, the cascade loop (TRC2 and FRC3) which adjusts the reflus valve to control the top temperature reduces the number of degrees of freedom by one only, not two, as has been stated sometimes. The inner loop (FRC3) relates the reflux valve setting and the reflux flowv,but introduces an additional variable in the set point. Hence, the inner loop does not affect the degrees of freedom. The outer loop relates the top temperature to the inner controller set point, and so reduces the degrees of freedom by one. Thus, the complete cascade system reduces the degrees of freedom by one. Thus, the controllers, other than the material balance controllers, reduce the static degrees of freedom by 5, leaving VI^ = V . = A’ 7 - 5 = N 2. This is two less than the number of degrees of freedom for the column without control valves and controllers (Figure 3). A reduction of four is due to the fact that the feed flow and temperature and the steam and water flows are controlled, and an increase of t x o is due to the addition of the two valves in the material balance loops (these loops do not affect X b e r ) . The reflux control loop has no net effect on the number of degrees of freedom compared to Figure 3, because a degree of freedom has been added via the valve, but removed by the control loops. The remaining ( N 2) degrees of freedom could be specified, for example, by the feed composition and pressure and the pressures a t the points of discharge of the two products. I t can be seen intuitively that the feed composition and pressure can, in fact, be varied arbitrarily. Also, if the discharge pressure of the top product is changed, the material-balance control will adjust the valve to restore the product flow, and nothing else in the system vi111 be affected (except an adjustment of the reflus valve to compensate for the effect of a change in accumulator level). A similar situation applies for the bottom product discharge pressure, and so both discharge pressures may be arbitrarily specified. The above analysis disagrees entirely iyith that of AIurrill (1965), who concludes that Figure 5 has zero degrees of freedom. The differences in his analysis are: (a) he does not recog-
+
+
Figure
4.
Distillation column with control valves
m-I-,
I
--r
~l I !
U s y s t e m boundary
L---_/---
1
*
Figure
5. Distillation column with control system
+
+
Although Kwauk states in the introduction to his paper that there are four degrees of freedom for the present case, his following analysis actually gives the answer five (as obtained by Smith (1963), using Kwauk’s method). The answer four is oiily obtained when q, is filed, which is a different case. The extra degree of freedom results from overlooking a mechanical equilibrium condition which must be satisfied a t the point TT here the feed enters the feedplate. K h e n two fluids mix (in a T-section, say), their pressures are equal a t the point where they meet. Thus, if both flow rates and one inlet pressure are specified, the inlet pressure of the other is determined. Hence, a feed tray has (S I ) , not (h‘ 2), degrees of freedom more than an ordinary tray, resulting from the addition of the extra inlet stream. This omission also occurs in the analyses of Gillilaiid and Reed (1942) and Howard (1967). For dynamic conditions, Howard counts two extra degrees of freedom, in agreement with Table 111. The effect of adding a standard set of control loops will nom be considered. Usually, five control valves are added, as shown in Figure 4. Compared to the column in Figure 3, Figure 4 has 13 additional boundary variables. q., qr, and the mater and steam temperatures have been removed, but 17 boundary variables have been added in the fire valve settings, and 4 x 3 = 12 variables in the pressure, temperature, and flow a t inlet and outlet for the water and steam. However.
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204 Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
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+
nize that the addition of the two control valves in the material balance loops increases V , by 2; (b) he concludes that the reflux controller plus valve removes -4‘ degrees of freedom, instead of zero. H e states that TRC-2 establishes a relationship between the feed composition and temperature (Ar - 1 degrees of freedom) and the inner loop FRC3 also removes one degree of freedom. Howard (1967) obtains AT degrees of freedom. He counts the seven controllers as removing seven dynamic degrees of freedom, in agreement with the present analysis, but does not count the three extra degrees of freedom from the reflux valve and the material balance valves. Allowing for the extra degree of freedom counted before adding the valves aiid controllers (from omitting the feedplate mechanical equilibrium condition), the net result is two degrees of freedom less than the present analysis. Conclusions
trollers: (a) adding a control valve adds a degree of freedom; (b) adding a control loop reduces V d and V, by one, if the set point is fixed, and if the controller actuating variable is not an accumulation variable previously undetermined a t steady state; if the controller actuating variable was previously undetermined a t steady-state, then the control loop reduces Vd by 1, but leaves V , unchanged; (c) integral action can also make a degree of freedom neutral, in some cases, when the actuating variable !vas originally determined a t steady state. literature Cited
Buckley, P. S., “Techniques of Process Control,” p 99 ff, Wiley,
V P_ Ynrk ~_. l\j. Y 1964 . _ . ---Dison, D. C.’Cheni. Eng. Sci. 2 5 , 337 (1970). Forsyth, J. S.> IND. EKG.CHEX, FCSDIM. 9, 307 (1970). Gilliland, E. R., Reed, C. E., Znd. Eng. Chem. 34,551 (1942). Hoffman. E. J.. “Azeotrooic and Extractive Distillation.” pp 10-’15, Interscience, Ke& York, N . Y., 1964. Howard, G. AI., IKD.ENG.CHEY.,FCNDAM. 6, 86 (1967). Kwauk, M., AI.Z.Ch.E.J . 2 , 240 (1936). AIurrill, P. W., Hydrocarbon Process. 44, 143 (1963). Smith, B. I)., ‘‘ Design of Equilibrium Stage Processes,” p 84, ~
I
~~
From the above discussion, it is concluded that the following procedure can be used for determining the number of degrees of freedom of a system, for both steady-state design problems aiid control problems. (I) 1-iicontrolled system: (a) steady-state degrees of freedom, V,, from eq 2 ; (b) dynamic degrees of freedom, Vd, from eq 4. ( 2 ) Effect of con-
McGraw-Hill, Ne%-York, N . Y., 1963.
RECEIVED for review February 1, 1971 ;ICCEPTI:D November 27, 1971
Dissolution of a Porous Matrix a Slowly Reacting Acid
by
William E. Sinex, Jr.,l and Robert S. Schechter* Department of Chemical Engineering, The C-niversity of Texas, Austin, Texas 7’8712
I. Harold Silberberg Texas Petroleum Research Committee, The Vniversity of Texas, Austin, Texas ?87‘lS
The acid treatment of an oil well to increase its productivity i s commonly practiced; however, at the present time there i s nc proven method to guide the design of such a process. This research examines the ability of a previously proposed model to predict the changes in a porous matrix when invaded by a slowly reactive fluid which dissolves a portion of the solid. The model i s shcwn to predict a relationship between the increase in porosity and the permeability which i s not precisely unique, as it depends to some extent on the initial pore size distribution, but for the initial distributions tested the permeabilities were found to lie in a narrow band. These results are independent of any parameters defining the kinetics except that the reaction be slow. It i s shown experimentally that the reaction of ferric citrate in the presence of citric acid with porous bronze disks satisfies the condition of being a slow reaction. The permeability change of the porous bronze disks i s found to agree closely with the theoretical predictions.
o n e method of stimulating oil wells to greater production is to dissolve a portion of the oil-bearing rock with an acid, thereby decreasing the resistance offered by the rock to the flow of oil. About 8 i million gallons of hydrochloric acid are used ailnually to stimulate oil viells in carbonate formations Present address, Fluor Corporation, P.O. Box 35000, Howt,on, Tesnr.
(Hurst, 1970). I n addition, many gallons of hydrofluoric, acetic, formic, aiid other special purpose acids are also used. The process of matris acid treatment is basically a simple one. An acid is pumped down the wellbore of an oil well a t rates 1Thich are slow enough to aroid fracturing the rock. The acid invades the oil-bearing formation, displacing the resident fluids and a t the same time dissolving a portion of the rock. The distance that the acid penetrates depends on the flow Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 2, 1972
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