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
205
rate of the acid, the local rate of surface reaction, and the geometry of the porous matrix. Despite the deceptive simplicity of the process, there does not now exist a proven design technique for predicting the result of a n acid treatment. This lack of a competent design procedure has resulted in a disinterest on the part of the users of acids in the fundamental mechanism of the chemical reaction, since there is no apparent way of introducing the reaction mechanism into a design. Schechter and Gidley (1969) have proposed a model for the process of matrix acid trectment in which the porous matrix is represented by a collection of randomly distributed capillaries possessing a variety of cross-sectional areas. This so-called capillaric model is not a unique idea for representing a porous matrix, and the success of a variety of applications of this model has been discussed by Scheideger (1960). Schechter and Gidley recognized the basic difficulties inherent in such models and proposed using the model only to determine the evolution of the pore size distribution and to represent the change in the various related phjsical properties rather than to make absolute predictions. This more limited goal seemed attainable but !vas still to be tested experimentally. The basic concept is that the collection of pores enlarges when invaded by the reactive substance as a result of the surface reaction. Thus, the distribution of pore sizes evolves in time. This distribution is also visualized as changing because of pore “collisions” which occur when the walls separating pores dissolve. This collision mechanism is very important when considering the change in permeability of the rock. Schechter and Gidley were able to derive a differential equation describing these processes.
a? + a (W= .E
{LA
*(A’) ? ( A - A’) ?(A’) dA’
-
The quantity ‘k represents the rate of growth of the area of a single pore having a n area A , 7 is the pore size distribution density function, and tis the time. Equation 1 describes the evolution of the pore size with time, and this evolution depends upon the reaction rate through the function \k. The theoretical evaluation of the q function has been discussed by Williams, et al. (1970), and it was even further developed by Guin (1969). The problem is to compute the rate of surface reaction in a single capillary attending the flow of a reactive fluid through that capillary. For the purposes of this paper, we will assume that this function has been defined by careful experimentation, avoiding the pitfalls noted b y Williams, et al. (1970). The system hydrochloric acid-limestone has been studied in a very careful and complete work by Kierode and williams (1971). These investigators have disputed the often-reported first-order nature of that reaction. If the \E function is defined, the problem is a purely mathematical one of solving the evolution equation, given the initial state of the rock matrix. This solution can be approximated using a Monte Carlo simulation scheme as proposed by Guin and Schechter (1971). All theoretical quantities reported in this paper have been obtained using this computational scheme, which avoid? the numerical instabilities associated with other methods of solving the evolution equation (Guin, 1969). An exact solution of eq 1 predicts that acids which react slowly increase the permeability of the porous medium in a 206
Ind. Eng. Chem. Fundam., Vol. 11, No. 2, 1972
way which is uniquely related to the increase in porosity provided the acid is applied under nonflow conditions. If, for example, the acid is introduced into the porous matrix and then permitted to remain stagnant until the reaction goes to completion, then we would classify this as a nonflow reaction. This exact solution for nonflow reactions contains no adjustable parameters and is independent of the reaction rate and the initial pore size distribution. It has been verified experimentally for the reaction of hydrofluoric acid with sintered glass disks (Guip, et al. 1971). HF solutions cooled t o the freezing point to retard the reaction rate were imbibed into sintered glass disks. After the reaction went to completion, the new permeability was measured and found to compare closely with that predicted by the theory. This work was the first experimental verification of the model first proposed by Schechter and Gidley (1969) for the case of the nonflow system. The present paper will provide a second verification of the model by applying it to slowly reacting flow systems. Simulation of Slow Reactions
For the case of fast, diffusion-controlled surface reactions, Guin and Schechter (1971) have shown that the function which determines the rate of pore enlargement varies with the square of the pore area, that is, q 0: A 2 .If this proportionality is used in conjunction with the Aloiite Carlo simulation program, for a given initial pore size distribution a relationship between the permeability ratio and the porosity can be developed. For the case of fast reactions in flow system, no generalized result comparable to the situation found for nonflow reactions can be obtained, and each different initial pore size distribution yields a different relationship between the permeability ratio and the porosity change. B y determining the relationship for a variety of initial pore size distributions, Guin and Schechter (1971) were able to establish that, while a unique relationship betryeen the permeability ratio and the pore size distribution does not exist, the results from four widely differing initial distributions were found to lie in a band which was relatively wide for small changes in the porosity but narrowed as the porosity ratio increased. The conclusion was drawn that the matrix tends to “forget” its initial distribution because the collisions between pores create a few pores of very large size which tend to conduct all of the acid. These large pores, often called “wormholes,” are the only pores then tending to grow, and hence, no matter what the initial distribution, all matrices behave in the same way once wormholes have been established. The central issues such as the number and rate of development of “wormholes” which are so important in design calculations have not yet been completely explored using the model. Before pushing the calculations, we need to know if the model applies to flow systems The fast, diffusion-controlled reactions repregent one end of a possible spectrum of reaction rates with the s l o reactions ~ being a t the other end. For slow reactions it is most convenient to take a boundary layer approach in defining the pore growth function since the alternative is to sum terms of an infinite series if the eigenvalue calculation is made. For slow, first-order reactions with short pore-residence times, Williams, et al. (1970), have shown that for a circular pore
where P is a dimensionless surface reaction rate parameter for a first-order surface reaction, L* is a dimemionless length, and R* is the dimensionless reaction rate. These quantities
are all defined iii the Nomenclature section. For small values of both P and L R* = 4PL*
or R.,
=
kCi
(3)
The error incurred in neglecting all of the terms in the series has an upper bound fixed by the first term in the series since it alternates, The first term can be evaluated using PI = -0.82699 and some of the remaining values of p are given by Guin (1969). T h e average wall reaction rate over the entire reaction area is equal to the product of the rate constant and the acid concentration. Now a material balapce on a single pore may be written -
Ap,L,t+AC -
Ap&t
= crR,,2?rREAt
6
4
2
(4)
Dividing by At on both sides and passing to the limit as At + 0 gives dA q(A,z,t) = dt
Rava2sR
1.75
1.50
2.00
(5) POROSITY RATIO, $/$o
and since r R 2 = A \E(A,z,t) =
1.25
1.00
= ___ PI
Figure 1. Local permeability improvement for acid treatment with a slowly reacting flowing acid
,2 V T k C 1 U A*/2 Pa
Thus, the pore growth rate function is proportional to A'/'. This calculation can be shown to be valid for shapes other than circular and, although the proportionality constant changes with pore geometry, the square-root dependence on area is general. Monte Carlo calculations using the technique reported by Guin and Schechter (1971) were performed with growth function proportional to the square root of the area. Three initial distributions were assumed, the log normal, the 6, and the exponential, defined as 1. log normal: v(A,O) = ( N / A )exp[- ('/Z)(log A
2. delta:
+ 21)21
120
-
100
-
90
-
110
80 70 60
-
50-
40
-
30
-
q(A,O) = N(A -
3. exponential: q(A,O) = AT exp(- lO-9A) for A
q(A,O) = 0 for A
>
cm2
< 10-lo cm2
The constant N was selected so that the initial porosity for all runs was 0.35 with = 0.01 cm. Figure 1 shows the results of the simulation. Each of the curves is the result of averaging five separate simulations, starting each time with a different random sample of 1000 pores obtained from an infinite number having the prescribed initial distribution. The difference between runs for a given initial distribution results from the use of the finite sample rather than an infinite one; 1000 pores is about the largest practical number that can be used in a run. The initial porosity of 0.35 was selected to correspond to experimental conditions, which are reported in the following section. More extensive results for a variety of initial porosities are not given here because it is felt that these have very little field application unless new, slower-acting acids are developed. ,4t the present time, most practical applications are covered by the work of Guin and Schechter (1971) on fast reactions.
0.25
0
0.50
0.75
1.0
CITRIC ACID CONCENTRATION (MOLES/LITER)
Figure 2. Effect of citric acid concentration on dissolution rate for a flow rate = 4 cma/sec Experimental Section
A. The Reaction System. I n order to test the theory, it seemed best to choose a porous matrix constructed of a homogeneous material (homogeneous both chemically and structurally). Therefore, a manufactured matrix was chosen for study instead of a naturally occurring porous structure such as limestone or sandstone. Further requirements were that this material be susceptible to chemical reaction and subsequent solution without the evolution of a gas or the formation of a precipitate. The porous matrix selected was sintered bronze disks 3.5 in. in diameter and 0.5 in. thick. The disks were Type PC-60 made by the Powdercraft Corp. of Spartanburg, S. C. The Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 2, 1972
207
c $, = M
o. 35
Disk No. 1
POROSITY RATIO, $/$o
Figure 3. Experimental results compared with the theoretical local permeability improvement for acid treatment with a slowly reacting flowing acid (disk 1 )
9,=0.35 M
Disk No. 2
POROSITY RATIO, $/$o
Figure 4. Experimental results compared with the theoretical local permeability improvement for acid treatment with a slowly reacting flowing acid (disk 2)
spherical particles from which the disks were made were about 0.02 cm in diameter and were 90.0 wt yo copper and 10.07, tin. A suitable reacting system for bronze is a mixture of ferric citrate and citric acid. This solution reacts slowly enough to allow some of the reactants to flow through the disk u i reacted and, hence, produce an even erosion through the depth of the disk. The ferric citrate-citric acid system pro208
Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 2, 1972
duces both soluble reactants and products in the oxidat'ion of copper and a t the same time possesses a small ionization constant which effectively maintains the concentration of the ionized species at a small value and makes it possible to obtain even erosion of the copper disk. I n the case of ferric citrate, a reduction of the ferric ion concentration by the addition of citric acid increased the reaction rate instead of decreasing it. I t is thought that the presence of the excess citrate ion accelerated the reaction either by participating in it directly or by increasing the solubility of the product, copper citrate. The rate a t which the sintered bronze disks dissolved when contacted by the flowing ferric citrate-citric acid solut'ion is shown in Figure 2. The volumetric flow rate was maintained a t 4 cm3/'sec of reacting solution throughout each 8-hr run. Ferrari (1958) has observed a similar effect in the dissolution of metallic copper in concentrated ferric chloride solutions. He found that for a certain range of HCl concentrations, increasing the HCl concentration resulted in a corresponding increase in the rate of dissolut'ion. There is a maximum rate and further increases in the HC1 concentration do not give any further improvement in the rate of dissolution. Ferrari proposed several mechanisms, but sufficient experimentation to isolate the particular one was not performed. Since the goals of the present research do not require an explicit knowledge of the kinetics, we have not attempted to clarify the mechanism. This seems to be an interesting but open question. Based on Figure 2 and a knowledge of the structure of the sintered bronze (Sines, 1970), it is estimated that P = 6 x IO-* and L* = 3 X l o + for t,he ferric citrate-citric acid system. The leading term in the infinite series of eq 3 can then be shown to be -0.024. Since the series is alternating, the largest error which can be incurred by neglecting the series is represented by t'hat number. Therefore, the reacting system approximates the ideal slowly reacting case with a n error of 2.4% or less. B. Experimental Results. Two bronze disks were dissolved with the citric acid-ferric citrate solutions. For both runs, the citric acid concentration was 0.05 X.Periodically, the flow of reactant to the disk was stopped, and the porosity and permeability of bhe disk were measured. The initial porosity of both disks was 0.35, and it would t'herefore be expected that the observed relationship between the increase in permeability and the change in porosity be represented by the theoretical predictioiis given in Figure 1. This is indeed the case, as the experimental data are seen in Figures 3 and 4 to lie well within the predicted band. The main point to be stressed is therefore that the model correctly predicts the change in permeability which occurs when a porous material is dissolved in a flowing, slowly reacting fluid. This represents the first experimental verification of the model for a flow system. Conclusions
The pore growth rate function is shown to be proportioiial to the square root of the pore area provided the surface reaction is the rate-controlling factor. Experiments dissolving a manufactured porous disk of broiize with a solution which is a mixture of ferric citrate and citric acid were conducted. For this system the reaction is shown to be the limiting step. Predictions based 011 a model first proposed by Schechter and Gidley (1969) are seen to correspond closely t o the experimental observations of the permeability illcrease as a function of the porosity change.
Nomenclature
Literature Cited
A C1 D k
Ferrari, H., Ph.D. dissertation, University of Michigan, Ann Arbor, hlich., 1958. &in, J. A4.,Ph.D. dissertation, The University of Texas at Austin, Aust,in, Tex., 1969; also Report No. UT 69-2, Texas Petroleum Research Commit,tee, Austin, Tex., 1969. Guin, J. A,, Schechter, R. S., SOC.Petrol. Eng. J . 1 1 , 390 (1971). Guin, J. A., Siiberberg, I. H., Schechter, R. S., IND.ENG.CHEM., FUNDAM. 10, 50 (1971). Hurst, R. E., st,atement reported iri Chem. Eng. L\-ews 48 (lo), 20 (1970). Nierode, D. E., Williams, B. B., SOC.Petrol. Eng. J . 11, 406 (1971). Schechter, R. S., Gidley, J. L., A.I.Ch.E. J . 15, 339 (1969). Scheideger, A. E., “The Physics of Flow t’hrough Porous JIedia,” pp 114-124, ~Iacmillan,New York, N. Y., 1960. Sinex. W. E.. Jr.. R1.S. thesis. The Universitv of Texas at Austin. Austin, Tex., 1970; also Report Yo. UT-715-2, Texas Petroleum Research Committee, -4ustin, Tex., 1970. Williams, B. B., Gidley, J. L., Guin, J. il., Schechter, R. S., ISD.ENG.CHEY.,FCXDAM. 9, 589 (1970).
K L*
= = = = = =
-
L P
= =
R R*
=
R,,
=
v t
=
= =
cross-sectional area of a pore acid concentration molecular diffusivity of acid first-order surface reaction rate parameter permeability dimensionless axial length (zD/2?Rz for a circular pore) mean porelength dimensionless surface reaction rate parameter ( k R / D for a circular pore) pore radius dimensionless reaction rate ( 2 R , , ~ / C , ’ ~ for R a circular pore) average surface reaction rate in a pore average fluid velocity in a pore time
GREEKLETTERS 9
=
q
=
pB
=
u
=
=
porosity pore size distribution function solid density mass of solid dissolved per mass of acid expended growth rate for a single pore
RECEIVED for review February 10, 1971 ACCEPTEDSeptember 20, 1971 This work was supported by the Texas Petroleum Research Committee, Austin, Tex.
A Model for the Dissociation of Hydrogen in an Electric Discharge Alexis T. Bell Department of Chemical Engineering, Cniversity of California, Berkeley, Berkeley, Calif. 94720
A model i s developed for predicting the extent of dissociation of hydrogen in an electric discharge. Two cases are discussed. The first i s for a stagnant discharge. The second treats the discharge in the presence of flow. Results are given for the effects of gas pressure and temperature, discharge size, power density, and flow rate. Quantitative comparisons are made with experimental data for a dc discharge and qualitative comparisons are made with data for a microwave discharge.
Interpretation of the kinetics of chemical reactions carried out in electric discharges has been hampered by the lack of a suitable model for relating the observed rates to the experimental variables. d number of earlier studies have indicated the necessary theoretical framework (Bell, 1971; Emeleus, et al., 1936; Lunt, 1969; Lunt and Meek, 1936; Vasil’ev, 1947, 195Oa,b, 1951), but they have not provided a complete model which takes into account reaction kinetics as well a s the mass transport equations necessary to describe the discharge as a chemical reactor. The purpose of the present effort is to illustrate how such a model might be devised to describe the production of atomic hydrogen in either a high-frequency or a dc discharge. This particular process was chosen because of the small number of elementary reactions needed to establish a reaction mechanism, the availability of the necessary physical data, and the reports of several experimental studies against which comparisons could be made. A comprehensive study of hydrogen dissociation in a de glow discharge has been reported b y Poole (1937). Molec-
ular hydrogen was passed through the positive column of the discharge and the atomic hydrogen which formed was detected b y a water-cooled calorimeter. Investigations were made of the effects of power, pressure, and flow rate on the yield of atomic hydrogen. One of the principal objectives of this work was to determine whether the rate of hydrogen dissociation per unit of dissipated power could be uniquely described by E / p as had been predicted earlier by Emeleus, et al. (1936). The experimental results were fouiid to be in good agreement with the theory and indicated that a masimum yield of atoms for a given pomer could be obtained through the adjustment of E / p . More recent efforts have concentrated on the use of high-frequency discharges. An extensive study of hydrogen dissociatioii in a microwave discharge (2450 MHz) was carried out by Shaw (1958). This investigation considered the effects of pressure, gas flow rate, discharge power, discharge tube size, microwave frequency, as well as the merits of continuous wave us. pulsed operation. Atomic concentrations weie determined by isothermal calorimeters as well a5 by esr. The results of the Ind. Eng. Chem. Fundom., Vol. 1 1 , No. 2, 1972
209
