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Timestamp: 2019-04-24 11:56:01+00:00

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are compared with those of previous writers.
to those considered by t,he phase rule.
of freedom of a system can be applied without this knowledge.
not require a detailed formulation of the syst'em equations.
produced disturbances is the major concern, and every entering disturbance must disturb some of the boundary variables.
variables alone (i.e., iiot involving internal or est,ernal variables).
varying) of those listed in Table 1.
This can arise because the balance equations contain accumulation terms under dynamic conditions. for a given type of process. I n a steady-state design problem. in a system having three degrees of freedom and three material streams crossing the boundary. The formal criterion is that there must be no equation Jvhich cont'aiiis only input variables. all the geometrical paramet'ers are available for specification but would not normally be considered bo be boundary variables. and the more sophisticat'ed control systems have variables of t'his type. if not both. V d . this is not necessarily so. otherwise it would not vary and so would not be a variable.(Balance equations are the only equations Table 1. Since the only differences between dynamic and static equations are the accumulation terms in the balance equations.. . are often stated to be differential equations in time.= 5. where A' is the number of chemical compounds present. v d can be greater than 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. A further consideration. (Flow work is not listed separately. 1 1 . in fact.) Any relationship which exists between a hold-up determining variable and the boundary variables will exist under both dynamic and static conditions." Ind. Eng. If a variable which determines the accumulation term in a balance equatioii is not related to the boundary variables. A given input variable is neutral if it is possible to find a set of independent boundary equations not containing t'his variable. however.. 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. + Rates of Flow across the Boundary Material Heat Shaft work ( L e . which arises in a control problem. (A further point. These "neutral" input variables represent degrees of freedom of the system since they are in fact free t'o be arbit'rarily specified. it can happen that some of the input variables have no effect on these dependent variables. which is required later. the hold-up is subsequently determined only if it was determined a t steady state.) in Table I which are differential in time. This makes no difference to eq 1 and adds no additional types of equation to Table I . 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. There is a further aspect of the iiat'ure of degrees of freedom which must be considered in some cases. Thus. However. 1970). Every variable must be affected by the boundary variables under dynamic or static conditions. for example.. all work except flow work). no matter which boundary variables are of interest. 1972 (4) 199 . but they are not in fact (Dison. their values determine the values of t. No.= v d = Nbv S b v .: that is. then the balance equation will be couiit'ed among A'ber.) or momentum consumption (all opposing forces) (b) Thermal Heat transfer (c) Phase Ilaterial transfer (d) Reaction Reaction (At a given point. the mass flow rates of the three streams cannot be chosen as the input variables a t steady st. because it ib determined by the flow rate and pressure of the material.) b. However. it simply provides additional boundary variables and degrees of freedom. number of plates and plate design in a distillation column) are usually fixed. Thus. I t is quite natural to think of such geometrical variables as boundary variables for a control system. is that the only variables which can be undetermined a t steady state are hold-up variables. Having chosen a set of input variables. 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.ions is always neutral. An input variable which appears in none of the boundary equat. For example. hold-up variables are the only ones which can vary under dynamic conditions while having no unique steady-state values. if the system is imagined to start from a steady state. since some externally operable mechanism must be provided for performing the adjustments. etc. it is oft'en the case that the values of only some of the dependent bouiidary variables are of interest. if the hold-up has a unique value a t steady state for a given set of values of the boundary variables. while this constraint' does not apply in evaluating Vd.ate since they are related by t'he total-mass balance equation. and then lid from v d - v. N b v in eq 1 equals the number of boundary variables plus t'he number of variable geometrical parameters. bnt not both.g.degrees of freedom. Fundam. Chem. is that there are. eq 1 can be replaced by t'wo equations v. I n a control problem. and so are the only ones which contain terms under dynamic conditions which are absent a t steady state. but not among N b e . S h e can be less than Shes. Equilibrium Rote or (a) Mechanical hfechanical energy consumption (friction. two numbers of degrees of freedom which need to be considered: st'at'ic.he internal variables and other boundary variables. Val. from eq 2. Reaction rate equations. V. but containing all the dependent boundary variables of interest and containing a total number of dependent boundary variables equal to the number of equations. Boundary Variables Intensive properties Composition Temperature Pressure Or another set of (W 1) intensive properties. the geometrical parameters of bhe system (e. 2. I n such cases. but not a t steady state. either an equilibrium or a rate equation applies. 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. 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. and dynamic. I n this case. Equations 1.) Hence the simplest approach appears to be to determine V . Usual Boundary Variables and Equations in Chemical Engineering Systems a . that is.
mass balances on nonreacting compounds.d i s c h a r g e Tonk liquid c o n s t a n t . a steady state exists for any set of constant values of the input variables (within operating ianges). I n balance equations. I t must be pointed out here that the possibility of Vd being greater than V.1) compound balances are dependent on the rest. there will be no steady state corresponding to an arbitrary set of values of the input variables. However. there are in fact A.2) additional boundary giving (*V . If the concentrations. K h e n V dexceeds V .ariyes from the way in k\hich the problem has been approached i n termi of boundary variables. Vol. this can also be shown by the formal procedure given above. 1 . I n balance equations in which there is a nonzero consumption term. the total-mass balance. P u m p e d . Such a system nil1 be referred to as sdfregulatory. the reaction rate expression is substituted into the consumption term. Isothermal Reactor Feed product overf lo w perfectly mixed Figure 1. . the number of input variables is usually taken to be the former. which determine the reaction rate are related to the boundary variables. which is what is done b y the above method of analysis. 1972 + . However. does not relate boundary variables in this case. If the rate is determined by the boundary variables then the combination of the balance and rate equations will be a boundary equation. one for each independent undetermined hold-up.. since only the steady-state case is considered. in fact. of course. the appropriate rate equation will be required to determine the consumption... These additional degrees of freedom are easily seen to be neutral.. because of the density and viscosity constraints) provide additional boundary equations. some further remarks can be made about the equations in Table I. the composition and temperature could vary. and the energy balance will always be boundary equations. = 1 is correct..?-be%. However. although there is no guarantee. with appropriate friction loss expressions substituted into the consumption term. but only for certain discrete sets Such a system would not normally be coiisidered satisfactory for contiol purposes. These degrees of freedom can be fixed by a set of Ai. that any such steady qtate is stable.' additional degrees 200 of freedom available a t steady state.2 ) additional degrees of freedom.J 00' Qo(h) ' b . The liquid density is constant and Q o depends only on h. 11. Another point about this example is that it contains a iiumber of irrelevant degrees of freedom. Fundam. i n d e p e n d e n l o f h I Heat Exchanger - constant comp0:ition Liquid A d.3) compound mass balance equatioiis (t1F-o of the (S . Figure 1 shows four simple examples which have been chosen for illustration. and they are not listed in Table I. G r a v i t y . 2. leaving (A' . The friction losses depend on Qi and Q o (boundary variables) but also on h. I n a system for which Vd = V. When exceeds V.. in a mass balance equation for a reacting compound. The analyses are given in Table 11. b u t they cannot be fixed by boundary variables... The energy balance equation plus (S . Thus. and so there is no connection between inlet and outlet mechanical energies. and this determines the variation in Qo. Chem. the best approach is to not count these degrees of freedom. then the material balance-reaction rate combination is a boundary equation.voi. Having decided that only the number of steady-state boundary equations. subject to these two constraints.d i s c h a r g e T o n k Ind. There is no independent equation relating h to the boundary variables. etc. No.V . Simple examples n-here S. They do not affect the relation between entering and leaving material streams and they cannot be directly fixed from outside the system..2 = (. need be directly determined. it is easily seen b y inspection that the result Vd = V . no matter what the value of h. For example... For Figure la. and the mechanical energy balance equation reduces.speed 0. the mechanical energy of the inlet stream is dissipated by impact on the surface in the tank. nT/-f+FL/'o a. f l o w constant pressure cd oe n ssti at y I rquld I I I I ~ L . I n effect.is the number of independent hold-ups which are undetermiiied a t steady state.a. it appears t h a t in no case is any of these degrees of freedom of any use or relevance.1) variables for both inlet and outlet streams. Hence. and not h i b e . if Qo is the only boundary variable of interest. to the relation between Qo and h. Since the input and output rates will always be described by boundary variables. which implies that the viscosity is also constant. Qlcan be varied arbitrarily (within limits imposed by the height of the tank) under both dynamic and static conditions. every balance equation in which the consumption term is zero will be counted among x b e s . the accumulation term will be zero. For a system which is riot selfregulatory. The mechanical energy balance. internal variables. Eng.- syslem boundory 2 .. temperature.
in effect. and would be used to eliminate the volume from each compound equation. which is the same as the exit concentration (boundary variable). etc. the range over which Q i could be varied. for the heat exchanger example.ing variable has a particuInd. flon.. I n the case of a feedback control loop containing integral action (it is assumed that integral action is never used in feedforward loops).pressure and temperature of each inlet stream Temperatures determine energy hold-up. 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. as shown in Table 11) reduces V . inlet flon. by one to zero. h may be arbitrarily specified a t steady state). would be small. which is allowed by the height of the tank.. all are fixed a t steady state Isothermal Reactor Compo. = s = *\. it establishes a relationship between a valve setting or other manipulable variable and another variable (the controller actuating variable). rather than a positive displacement type. but the only input variable contained in it is Ql.. as a preliminary to adding the control loop. the volume of the reactor contents has been assumed constant. and the system is equivalent t o Figure l b ..rate Liquid hold-up is determined by h . compohition. the situation where the in. Chem. However.and out-flows of a tank containing a phase boundary are unaffected by the position of the boundary. xb. it ensures that steady state can only exist when the actuat. before the controller is added. which equal3 product composition adjustable parameters of the controller (set-point. b. and the log mean temperature difference is given by the inlet and outlet temperatures. However.1 = 1 E.V . it will be assumed that the manipulable variable exists in the system before the control loop is added. t. the total rate of reaction for each compound depends only on the composition of the contents. to examine specifically the effect of control loops. then the effect of h on Q o is negligible. If the manipulable variable does not already exist. No. Thus. = I'd . an independent equation for h (k.. If t. is very small compared to the variations in Qlwhich are expected. giving the same result for N b e s .his can be accomplished by providing for the direct variation of some quantity which was previously fixed.. The most common case is the addition of a control valve to a pipeline. Hence. becomes unity. Analysis of Figure 1 Examples Gravity-Discharge Tank a. I n the limit as the effect of h on Qo approaches zero. which is not fixed by the boundary variables C. of course. Fundam.'.o systems containiiig coiitrol loops. since there is no equation relating these. the controller itself does not determine the manipulable variable as a function of the actuating variable a t steady state. This gives the line a variable. in effect..he change in the number of degrees of freedom resulting from their addition will be considered. instead of fixed.T. if one says that the effect of h on Qo. This extra equation (which reduces to Qo = constant. so that Vd = 1. The total rate of heat tiansfer is given by the inlet and outlet temperatures and flow rate of either stream.+ 1 0 Heat Exchanger Flow. which makes v d . = 0 d.g.6 Vd = . the manipulable variable is always a boundary variable since.. Compared to Figure l a . = 1. is negligible. p r e s u r e s determine liquid hold-up. Thus. flow resistance. then this adds one extra degree of freedom. since h is undetermined a t steady state.V. it must be added to the system.that is. the relation between depth and outflow (as in Figure l a ) would determine the contents volume in terms of boundary variables. = A-beS = 12 6 V 3 = 12 . can be arbitrarily varied under dynamic conditions but can only have one value a t steady state. Qo (valve setting does not vary. this is equivalent to assuming that the self-regulatory range of the system is negligibly small.g. the same as for Figure la. 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). = 0 is related to Qo = s b v 2 Qi. the system would be self-regulatory over a small range only. Thus. However. It is easily seen t h a t Q .Firstly. the total heat transfer rate equation can be expressed in terms of boundary variables.. it is found that the total mass balance equation (Q1 = Qo) contains only one dependent variable (Q. T h e heat transfer rate equation is included among Nbe. Sb. the self-regulatory range approaches zero and Vd . because the stem does not cross the boundary) Nbej 1 Material balance V s = 2 . The general method of analysis given above can.ition and flow rate of feed and product. Eng. since there is now.. 11.). Q. v d . and this is the boundary variable of interest.) are fixed. while not zero. while still allowing a steady state. Thus. Figure l b illustrates what appears to be the only common situation where Vd exceeds VB. Val.Vs = 1.his discussion. I n the reactor example.V . be applied direct'ly t. the S compound balance-reaction rate equations are boundary equations.T'. would be affected to a small extent by h and so the system would be equivalent to Figure la. With this choice of input variables.g. that is. a permissible choice of the input variables is Q L plus inlet temperature and ('V . Even for Figure l a the self-regulatory range could be negligible under the conditions being considered. pressure. temperature llaterial balance-reactioii rate for each compound E. 1972 201 .and temperature of the feed Material accumulation is dfterniined by reactor composition. floxv. If the variation of the volume with flow (because of the overflow product-removal system) were taken into account.3) inlet composition variables.V . I\lhv = 2 5 Shes = +1 a\- Vq = 2 5 +1- I'd . For the purposes of t. the mechanical energy balance-friction loss equation now relates boundary variables a t steady state. which Vd . If the range of variation of Qo. it can only be varied from outside the system. However. the other input variables are neutral. If the pump in Figure l b were centrifugal. = 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 . Hence. if the Table II. 2.= 1 = 1 -1=0 V.
the system is equivalent to Figure l b . The system boundary has been drawn to cross the valve stem. Depending on the circumstances. A steady state can exist for a wider range of values of Q>than previously. No. = 1. Simple control systems lar value. V . the same as for Figure la.. and v d = 7. w i t h F l o w C o n t r o l %--- -. 1 1 . and not Q. with finite or infinite controller gain. If any one of these is variable. V 8 = 0. Chem. From the above d i x u w o n . the problem is to determine the effect on the number of degrees of freedom of making the manipulable variable a function of the actuating variable. With close control of the flow. canceling the extra degree of freedom from the control valve.= 1. which indicates that the set-point is fixed. adjustable controller parameters have been assumed fixed. and if Qo is closely controlled Q .. It has. Since the effect of integral action is to give the controller an infinite steadystate gain. However.<. is still a permissible choice as iiiput variable and will have an effect even if h is fixed. with a large gain a t steadystate when integral action is used. 7 c.d i s c h a r g e Tank. simply by reduciiig ‘l’b.) Figure 2b shows another control system for the gravitydischarge tank. adding the controller does not affect V. These effects caii be illustrated for the two liquid-holding tanks of Figure l a and b. 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. Eng. However. This merely determines the value of h required so that Qo has the required value. The above discussion has also been limited to control loops which are added to the system. Since the actuating variable was previously undetermined a t steady state. this adds an extra degree of freedom in all cases. by one. V .. and so V .)does not cross the system boundaries. but reduces Vd by one. Closely controlling h has no additional effect. However. 2. V. Vd and V.is reduced by one. is not changed. but the valve position can be chosen. has only one value for steady state. The addition of the control valve has added an extra degree of freedom.. and the result is Vd = 1. Figure 2a shows the gravitydischarge tank (Figure la) with a level controller. Thus. w i t h L e v e l C o n t r o l Figure 2. The controller removes this new degree of freedom. Q . The filial result is Vd = Ti. Thus. The result depends on whether the actuating variable is an undetermined hold-up variable or iiot 111 the original system. There does not appear to be any specific rule for determining whether degrees of freedom are made neutral by integral action. G r a v i t y .a. (The arrow indicating the controller set-point (sap. are reduced by one. Figure 2c shows a level controller applied to the pumpeddischarge tank (Figure lb). in this case the self-regulatory range has been reduced. 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. and the system is now self-regulatory and equivalent to Figure l a . and this places a constraint on the manipulable variable through the system relationships.d i s c h a r g e T a n k . then the controller fixes the maiiipulable variable as a function of these boundary variables for both dynamic aiid static conditions. = 0. P u m p e d . Fundom. vd is reduced by one. Therefore..d i s c h a r g e T a n k . the valve position then represents an irrelevant degree of freedom since it only affects h . Assuming that the final Eystem is required to be self-regu- v. increased the self-regulatory range of the system. is further reduced. Again the actuating variable (Qo) was previously determined a t steady state and so the controller reduces Vd and V . If the actuating variable is undetermined a t steady state ill the original system. G r a v i t y . Therefore. the manipulable variable is a boundary variable and the actuating variable is related to it in the new system so that Sa. 1972 the actuating variable at steady state this can have a further effect by making some degrees of freedom neutral. the statement (lIurri11. since the actuating variable (h) was previously determined. the simplest approach to an integral controller appears to be to determine its effect without integral action and to then deterniiiie if any further constraint results from letting the steady-state gain become very large (or constraining the steady-state values of the actuating variable to a very small range). . aiid so V. which removes offset in the controlled (actuating) variable. Vol. K h e n Vd exceeds the former is the number of degrees of freedom which applies to this rule. however. When integral action fixes 202 Ind. 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. = 0. adding the control valve and controller has iiot affected the degrees of freedom in this case.p. these degrees of freedom may be irrelevant. In the above discussion. The valve position may be arbitrarily specified a t steady state. which originally had Vd = 1. and each case must be analyzed by the general method given above. If the actuating variable is determined (related to boundary variables) 111 the original system. with close control Q I is no longer a permissible choice for the input variable a t steady state. indicating that the valve setting caii now be varied arbitrarily before addition of the controller. then the manipulable variable after coniiection through the controller is also undetermined. Looking a t this in another way.leading again to the result v d = 1. and Vd are both reduced by one.
one set of iV equations relating top and bottom compositions. The levels in these accumulators are assumed to have negligible effect on product flows ( c j . which agrees n-ith the result of Kwauk (1956) for a biliary system and of Forsyth (19i0) for a quinary system. the present analysis gives V .es. In a design problem. equality of chemical potentials between phases for each compound) are shown. This gives a n extra degree of freedom. temperature.her the accumulators are self-regulatory or not. Thus. discussion of Figures l a and b) so that the friction losses in the accumulators are related to product flows.ion they consider them nonself-regulatory.latory. However. Figure 3 shows a basic two-product distillat’ion column. but this apparent disparity (Forsyth 1970) can be explained. The assumption of equilibrium is only an idealization. The analysis in Table 111can be compared with that of other writers. = 2 Lelels in reboiler and condenser are assumed to h a r e negligible effect on product flow rates cq 5’. 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 . + 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. even if this assumption is not made. which is the present case. If the feed composition. for condenser and reboiler. flon Top composition. other loops may be added which are either external loops holding boundary variables constant or internal loops actuated by previously determined variables. (Xeither of these writers states esplicit. A maximum of V . There is. Basic distillation column tion of uniform column pressure. Vol. in equilibrium with each other (unless the column contains only one ideal stage). or all pressures determined. Fundam. I n Table 111. There is some disagreement between different writers. distillation columns are amoiig the more complex systems and have received considerable attention in connection with the present subject. 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. 2. this assumption cannot be made. instead of the equilibrium equations. Secondly. the number of degrees of freedom is reduced from 4 to 2 . pressuie. . unless a fixed plate design and spacing is used.loops must be actuated b y previously undetermined hold-up variables. temperature. temperature. Showing the heat transfer rat. and the discharge pressures of the two products are related to the reboiler and condenser pressures. no reason why the feed pressure should be specified. S.. Analysis of Distillation Column of Figure 3 Boundary variables S S 2 2 +2 +2 +2 35 Feed composition. this is correct. and hence to each other via the equilibrium relations through the column. presure. pressure. rather than as passing through the water and steam inlet and outlet pipes. of course. K i t h this restriction. and a final number of 3. the number of plates in the column has been taken as fixed. since a variable which is a degree of freedom should be continuously variable over its allowable range. Strictly speaking. The degrees-offreedom analysis of this system is summarized in Table 111. and these are determined independently of the overall mechanical energy balance-friction loss equation by the internal refluxing arrangements. The liquid levels on the trays are assumed sufficiently variable to give adequate regulation. If the accumulator levels have a significant effect.) Hoffman (1964). the numbers of theoretical plates i n the rectifying and stripping sections are not fixed and provide two additional degrees of freedom.ly lvhet. First. 1 1 . I n this case. as has been considered by several writers. since the numbers of plates must be integral. Ind. No. of course.. esplicity by arrows implies that the system boundary is being taken as coincident with the heat transfer surfaces. the only way in which Hoffman’s analysis differs from others is in not counting the numbers of plates as degrees of freedom. which is the usual situation in a control system. The manipulable variables for all iiiternal loops must exist before the rule is applied. of course. However. this is really only a matter of definition. for the design problem where the numbers of theoretical plates are variable. 1972 203 . Chem. Hoffman does not include the pressure in specifying the feed. Eng. four degrees of freedom remain. maintains that there are only three degrees of freedom for this system. Distillation Columns As remarked in t’he introduction. the equilibrium relationships between the phases on each plate through the column give. overall. However. by implicat. the same number of rate equations will be obtained. They make the usual assump- + Table 111. the practical point being that the numbers of plates are variable. This simply leaves some other variable (such as the pressure a t the top of the column) free for specification. Hoffman does not include the two numbers of theoretical plates among the degrees of freedom. A similar comment applies to the mechanical and thermal equilibrium relationships listed.( 2 s 6) = S 4 T’. however. temperature. flow Bottom composition. a i d pressure (or the “column” pressure) are specified. However. flow rate. = N 6. there are numerous additional degrees of freedom in the plate geometrical details. The top and bottom products are not’. S phase equilibrium relations ( L e . I n the above analysis. If real plat’es are considered.
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. This omission also occurs in the analyses of Gillilaiid and Reed (1942) and Howard (1967). Thus. his following analysis actually gives the answer five (as obtained by Smith (1963). Vol. which is a different case. then Vd and V . 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. Also. the controllers. but removed by the control loops. If the top product valve. Hence. 2. Fundam. I n particular. No.5 = N 2. but only to keep them within limits. reduce the static degrees of freedom by 5. be varied arbitrarily. However. = $ 7 and V d = N 9. and flow a t inlet and outlet for the water and steam. for example. and 4 x 3 = 12 variables in the pressure. loops considered. 1964). 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). using Kwauk’s method). 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. resulting from the addition of the extra inlet stream. and so reduces the degrees of freedom by one. five control valves are added. is filed. the material balance and mechanical energy balance-friction loss equations for water and steam. 1972 + + . Eng. together with other control. The reflux control loop has no net effect on the number of degrees of freedom compared to Figure 3. not two./ system b o u n d a r y there are four additional boundary relations. by Murrill and by Howard. The differences in his analysis are: (a) he does not recog- + + Figure 4. Since Vd is greater than V.. The two level controllers are not required to closely control the levels. say. was used in a product flow control system. and pc. Thus. the inlet pressure of the other is determined. and the mater and steam temperatures have been removed. Compared to the column in Figure 3. The other control loops affect V d and V . This is two less than the number of degrees of freedom for the column without control valves and controllers (Figure 3). Figure 4 has 13 . Chem. and so both discharge pressures may be arbitrarily specified. equally. + 204 Ind. The above analysis disagrees entirely iyith that of AIurrill (1965).. if the discharge pressure of the top product is changed. other than the material balance controllers. the first priority usually is to reduce V d so that it equals V. because a degree of freedom has been added via the valve. The inner loop (FRC3) relates the reflux valve setting and the reflux flowv. q. instead of in the material-balance system shown. 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. such a system does not work satisfactorily. A reduction of four is due to the fact that the feed flow and temperature and the steam and water flows are controlled. A similar situation applies for the bottom product discharge pressure. as shown in Figure 4.4 = 9 additional degrees of freedom. by one. As was found in some early installations. will be taken as constant for water and steam. in agreement with Table 111. as has been stated sometimes. The outer loop relates the top temperature to the inner controller set point. as an example. if both flow rates and one inlet pressure are specified. There are then three additional degrees of freedom in Figure 4 (three of the five valves. The answer four is oiily obtained when q. not (h‘ 2). The remaining ( N 2) degrees of freedom could be specified. temperature. Thus. say). degrees of freedom more than an ordinary tray. the complete cascade system reduces the degrees of freedom by one. the inner loop does not affect the degrees of freedom. qr. K h e n two fluids mix (in a T-section.but introduces an additional variable in the set point.. 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 ) . but 17 boundary variables have been added in the fire valve settings. thus achieving “material-balance control” (Buckley. Hence. I t can be seen intuitively that the feed composition and pressure can. in effect). the inlet temperatures and pressures. Thus. the other two replacing q. and discharge pressures. Usually. Howard counts two extra degrees of freedom. in fact. Distillation column with control valves m-I-. leaving VI^ = V . who concludes that Figure 5 has zero degrees of freedom. I --r ~l I ! U s y s t e m boundary L---_/--- 1 * Figure 5. by the feed composition and pressure and the pressures a t the points of discharge of the two products. = A’ 7 . Figure 4 has 13 additional boundary variables. The effect of adding a standard set of control loops will nom be considered. 11. One such set of connections is shown in Figure 5 . a feed tray has (S I ) . each overall loop reducing Vd and Ti.. For dynamic conditions. T o simplify the analysis. g i v i n g v . would both be reduced by 1 and their difference would remain unchanged. their pressures are equal a t the point where they meet.
EKG. 307 (1970). ~ I ~~ From the above discussion.. Schechter* Department of Chemical Engineering. however. Austin. 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.’Cheni.Ch. in some cases. Eng. ---Dison. S.. Chem. 1971 Dissolution of a Porous Matrix a Slowly Reacting Acid by William E. H e states that TRC-2 establishes a relationship between the feed composition and temperature (Ar .551 (1942).. for both steady-state design problems aiid control problems. ‘‘ Design of Equilibrium Stage Processes. 44. G. IKD. Austin. “Techniques of Process Control. W. 2. Sci. formic. _ .” p 84. Y. Howard (1967) obtains AT degrees of freedom. AI. literature Cited Buckley.. by one. P. Sinex. instead of zero. Vol. Texas 7’8712 I.on. E. No. D. Allowing for the extra degree of freedom counted before adding the valves aiid controllers (from omitting the feedplate mechanical equilibrium condition). Interscience. Eng. Znd. (b) adding a control loop reduces V d and V. if the controller actuating variable was previously undetermined a t steady-state.” p 99 ff. Kwauk. then the control loop reduces Vd by 1.. Hoffman. displacing the resident fluids and a t the same time dissolving a portion of the rock. An acid is pumped down the wellbore of an oil well a t rates 1Thich are slow enough to aroid fracturing the rock. The permeability change of the porous bronze disks i s found to agree closely with the theoretical predictions. thereby decreasing the resistance offered by the rock to the flow of oil.CHEY. if the set point is fixed. Box 35000. 34. 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. 240 (1936). 1 1 . 143 (1963).1 degrees of freedom) and the inner loop FRC3 also removes one degree of freedom.FCNDAM. AI. the net result is two degrees of freedom less than the present analysis. Y 1964 .. as it depends to some extent on the initial pore size distribution. The C-niversity of Texas. N . Ne%-York. Ke& York.l and Robert S. Harold Silberberg Texas Petroleum Research Committee. The Vniversity of Texas. many gallons of hydrofluoric. and if the controller actuating variable is not an accumulation variable previously undetermined a t steady state. 6. Eng. RECEIVED for review February 1. Y. o n e method of stimulating oil wells to greater production is to dissolve a portion of the oil-bearing rock with an acid. when the actuating variable !vas originally determined a t steady state... M. l\j. C. acetic.Z. by 2.. E. 1970). 2 5 . aiid other special purpose acids are also used. from eq 2 . but for the initial distributions tested the permeabilities were found to lie in a narrow band. Conclusions trollers: (a) adding a control valve adds a degree of freedom. V. E.. I). Reed. 1963. 1972 205 . Howt. 1971 . The acid invades the oil-bearing formation.E. Forsyth.. FCSDIM. Tesnr. Chem. P.> IND.” pp 10-’15. He counts the seven controllers as removing seven dynamic degrees of freedom. J. C. P. (b) he concludes that the reflux controller plus valve removes -4‘ degrees of freedom. but does not count the three extra degrees of freedom from the reflux valve and the material balance valves. (Hurst.ICCEPTI:D November 27. it is concluded that the following procedure can be used for determining the number of degrees of freedom of a system. 337 (1970). S. Wiley. Texas ?87‘lS The acid treatment of an oil well to increase its productivity i s commonly practiced.. The model i s shcwn to predict a relationship between the increase in porosity and the permeability which i s not precisely unique. (c) integral action can also make a degree of freedom neutral. The process of matris acid treatment is basically a simple one.. in agreement with the present analysis. 2 . 9.J . but leaves V . (b) dynamic degrees of freedom. The distance that the acid penetrates depends on the flow Ind. (I) 1-iicontrolled system: (a) steady-state degrees of freedom. Fluor Corporation. B.O. from eq 4. N . at the present time there i s nc proven method to guide the design of such a process. Fundom.ENG. Gilliland. Howard. About 8 i million gallons of hydrochloric acid are used ailnually to stimulate oil viells in carbonate formations Present address.. These results are independent of any parameters defining the kinetics except that the reaction be slow. AIurrill. R. Hydrocarbon Process. 86 (1967). Vd.CHEX. unchanged.nize that the addition of the two control valves in the material balance loops increases V . 1964. J. Jr. Smith. ( 2 ) Effect of con- McGraw-Hill. I n addition. “Azeotrooic and Extractive Distillation. V P_ Ynrk ~_.

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