Ind. Eng. Chem. Fundam. 1981,20, 141-147
141
Mass Transfer and Kinetic Effects in an Electrode-Driven Homogeneous Reaction Roland Bauer, Davld K. Friday, and Donald J. Klrwan' Department of Chemical Englneerlng, Unlversw of Vlrglnk, Charbttesvll/e, Vlrgink 2290 7
The homogeneous, enzyme-catalyzed oxidation of ethanol to acetaldehyde, which requires the oxldlzed form of nicotinamide adenine dinucleotide (NAD') as a co-reactant, is employed as a model system to demonstrate the
control of a homogeneous reaction via a coupled electrochemical reaction. Active NAD is continuously produced by oxidation of NADH at an anode. The surface kinetics and selectivity of the electrochemical oxidation of NADH on graphite and platinum electrodes at pH 9 and +700 mV (SCE) were determined in a batch reactor. A film model accounting for mass transfer accompanied by homogeneous and heterogeneous reactions predicts batch production rates of acetaldehyde in the coupled system which compare favorably with experimental observations. The selectivity of the electrode reaction limits the production of acetaldehyde in the batch system.
Introduction This paper describes experimental results and a mathematical model for a chemical reaction system which produces a desired product by the simultaneous use of heterogeneous electrodic catalysis and homogeneous catalysis in solution. Although the specific chemical system studied is one involving a well-known, enzyme-catalyzed reaction, it can represent a model system for other chemistries. The basic concept is illustrated by the reaction scheme cat.,
A+B-R+P cat2
R-B
The desired reaction is the conversion of the reactant A to product P via a reversible, homogeneous, catalytic reaction (I). This reaction requires a co-reactant or cosubstrate B in stoichiometric amounts. The co-reactant can be regenerated by the irreversible heterogeneous, catalytic reaction of R (11). If the second reaction is an electrochemical one, then its rate and reversibility can be influenced by the applied potential thereby providing a means of driving and controlling reaction I. The specific example we have explored is the oxidation of ethanol to acetaldehyde catalyzed by the enzyme, alcohol dehydrogenase (Dickinson and Monger, 1973; Dickenson and Dickinson, 1975). C2H50H+ NAD+ NADH
5 C2H40+ NADH + H+
electrode
NAD+ + 2e-
+ H+
(111)
(IV) This enzymatic reaction requires the cofactor (co-reactant) nicotinamide adenine dinucleotide in its oxidized form (NAD+). The oxidized and reduced forms of NAD represent the most common cofactor system for oxidation/ reduction reactions in living cells. The oxidized form can be obtained from the reduced form by oxidation on a solid electrode such as platinum or carbon (Janik and Elving, 1968; Blaedel and Jenkins, 1975; aBraun et al., 1975). Because the equilibrium of reaction IV can be made to lie far to the right, reaction I11 can be effectively forced to continually produce acetaldehyde from ethanol. This basic concept of reaction coupling is very well known in biochemical reaction networks in cells where all of the reactions are catalyzed by enzymes whether free in solution or immobilized on membrane-like structures (Mahler and
Cordes, 1971; Bailey and Ollis, 1977). There have been a few studies of enzyme-catalyzed reactions in which an electrocatalyst has been employed to regenerate a cofactor (Kelly and K h a n , 1977;Alexander and Coughlin, 1975; Aizawa et al., 1976). A key aspect of these investigations is the determination of a reuse number for the cofactor-the average number of reuses of the cofactor molecule as measured by product concentration divided by the original cofactor concentration. The effective cost of cofactor is simply the purchase cost (for NAD+ $15000/g-mol) divided by the turnover number. The reuse number is limited by the selectivity of the electrochemicalreaction but can also be influenced by mass transfer effects and the kinetics of the homogeneous reaction. There are nonbiological examples of an electrode reaction being one step in a reaction sequence (Adams, 1969; Eberson and Weinberg, 1971). Often, such studies are confined to elucidating mechanisms of electroorganic reactions although the Baizer-Monsanto process for producing adiponitrile is in industrial use (Prescott, 1965). Our work is specifically devoted to an analysis of the use of an electrode reaction to drive a homogeneous reaction toward a desired product. The analysis and experiments to be described consist of first examining the electrochemical oxidation kinetics of NADH including possible side reactions to inactive forms of the cofactor. The second part consists of operating the complete system to examine the reuse numbers achieved as influenced by NADH oxidation kinetics and selectivity,mass transfer to and from the electrode surface, and the enzyme reaction kinetics. Experimental Section Both the oxidation and turnover number experiments were conducted in the magnetically stirred, two-compartment cell shown in Figure 1 (Bauer, 1980). The stopcock connecting the compartments could be adjusted to prevent flow of solution but to allow current flow. The batch experiments were conducted at a constant potential maintained by an Elron potentiostat, Model CHP-1. All experiments were conducted in pH 9, 0.05 M sodium tetrapyrophosphate buffer a t an anode potential of +700 mV relative to the saturated calomel electrode. Earlier work (Kelly and K h a n , 1977; Searby, 1977) had indicated that these conditions were near optimum for high reuse numbers. Stirring rates in the cells could be controlled and measured.
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OI96-4313f81II02O-0141$01.25fO 0 1981 American Chemical Society
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Ind. Eng. Chem. Fundam., Vol. 20, No. 2, 1981
4
ELECTRODE
1 % NADH
A
2-0
2-
L
la!
1 Figure 1. Schematic of electrochemical cell.
I
NAD' b
f4 2-
, (ENZYME REACT CLI
fEhZYME R X I
I
I= L
0 f bi
Figure 3. Film model: (a) oxidation experiment; (b) reuse number experiment.
i WE, V,h
Figure 2. First-order plot for absorbance change during oxidation of NADH on graphite.
The working electrode (anode) was either platinum or anode grade carbon filled with linseed oil (Union Carbide) while the cathode was anode grade carbon in all experiments. The platinum electrode was a thin sheet that curved about the inside wall of the cell. The geometric surface area per volume of solution, a, was 0.2 cm-'. The graphite electrodes were rods of rectangular cross section either 1 cm X 0.5 cm X 11 cm long (thick electrodes) or 2.5 cm X 0.08 cm X 12.5 cm long (thin electrodes) immersed 4 cm into the solution. The interfacial area could be changed by immersing different numbers of the electrodes into the solution. The experiments were conducted with either 4 or 8 thick electrodes (a = 0.48 or 0.96 cm-l) or with 2 thin electrodes (a = 0.40 cm-I). The reference electrode was saturated calomel. The rate expression for the electrochemical oxidation kinetics of NADH was determined by monitoring the ultraviolet absorption (340 nm) of samples taken from the reactor operating in a batch mode. The extinction coefficients of NADH and NAD+ at this wavelength, pH 9 and 23 "C are 6.22 X lo4 and 0.05 X lo4 cm-' M-I, respectively (Bauer, 1980). Usually, buffer solution containing 0.65 M ethanol was made up to 1.28 X 10" M NADH for the start of the experiment. Experiments were conducted at stirring speeds ranging from 200 to 900 rpm. A first-order process was generally observed for the disappearance of NADH (Figure 2), but the rates were mass transfer influenced. The selectivity of the reaction was determined at the conclusion of a run by adding alcohol dehydrogenase to a sample from the reactor allowing the active NAD+ formed to react forming NADH which could be monitored
spectrophotometrically. The equilibrium constant for the enzyme-catalyzed reaction was employed to calculate the total amount of active NAD+ formed during oxidation although this correction was quite small as the forward direction for reaction I11 is highly favored under these conditions. Some experiments on the influence of electrode pretreatment on oxidation kinetics were conducted and will be discussed below. Proper analysis of these experiments yielded the rate constants for the formation of NAD+,the oxidation of NADH to inactive species, and the mass transfer coefficient from solution to the electrode surface at various stirring speeds. The reuse number experiments were conducted in the same apparatus at pH 9,700 mV (SCE),and 900 rpm with initial solution concentrations of 0.652 M ethanol, 1.28 X 10" M NADH and lo4 M alcohol dehydrogenase (ADH). The cofactor was supplied as NADH rather than NAD+ so that the reaction could not commence until the potential was applied to the anode. Both platinum and graphite electrodes were used. Because of the volatility of acetaldehyde the reactor was sealed with a rubber gasket through which a hyperdermic needle could be inserted to withdraw a sample. The acetaldehyde produced was measured by an assay procedure described by Aizawa and Coughlin (1977). A sample from the reactor was diluted approximately 1:lOO into a pH 7.5 buffered solution containing 1.28 X M NADH and about lo4 M ADH. Under these conditions the acetaldehyde will react to ethanol using a stoichiometric amount of NADH which is readily measured spectrophotometrically. A control using buffer solution rather than a reactor sample is also used and corrections are made for the equilibrium of reaction 111. Different acetaldehyde conditions can be assayed by adjustment of the degree of dilution of the sample and the reagent solutions. The production of acetaldehyde during the course of this batch reaction is therefore readily measured and the reuse number (acetaldehyde concentration/initial NADH concentration) as a function of time is determined. Model As depicted in Figure 3 we chose to employ a film theory approach to analyze the influence of mass transfer on the combined heterogeneous and homogeneous reactions occurring in this system. The film thickness is presumed to be a function of the stirring speed in the vessel. Although the experimental reactor was a batch system and bulk concentrations change with time, we assume that a pseu-
Ind. Eng. Chem. Fundam., Vol. 20, No. 2, 1981
do-steady-state approximation may be employed to calculate the profiles in the film at any given time. As long as the diffusion time, L2/DN (see Nomenclature for meaning of symbols) is short with respect to the experimental time, the pseudo-state approximation should be valid. Electrochemical Oxidation of NADH. Material balances for NADH (N) and active NAD' (P) within the film are
d2Cp Dp?=O dz Because the solutions are very dilute, bulk flow contributions to the flux are neglected. The appropriate boundary conditions are
In boundary condition 2 it has been assumed that the oxidation of NADH and the production of active NAD+ both follow first-order kinetics at a given potential with rate constants, k3 + k4 and k3, respectively. This is supported by the experimental results described below. Material balances on NADH and NAD' in the bulk phase are
with initial conditions, bN = bNo,bp = 0. These equations are conveniently put in dimensionless form using the definitions: Y N E CN/bNo, Yp = Cp/bN', WN = bN/bNo, W p = bp/bN', T = tDN/L2, = Z/L. d2YN -- - 0 dZ2 d2Yp
z
-=o
dZ2
T=o:
W N = ~ (94 =0 (9b) The solution to this set of equations is readily obtained to yield the bulk compositions as a function of time.
wp
The observed selectivity is a constant given by bp(t) WP =- k3 1 - WN bN' - bN(t) k3 + k4
--
-
143
(11)
As expected, eq 10 does suggest that NADH disappears by a first-order process dependent upon the mass transfer coefficient and the sum of the electrode reaction rate constants (k3 + k4) for NADH oxidation. The surface constant, k3 + k4, can be determined from experimental data at a series of stirrer speeds in order to account for the mass transfer term. If one measures the amount of active NAD+ formed after oxidation of a given amount of NADH then eq 6 permits calculation of k3 once k3 k4 has been obtained. Reuse Number. When enzyme is present in the solution, the situation is rather more complex as the homogeneous reaction occurs both in the film and in the bulk of the solution. Let rl - r2 represent the net (forwardreverse) enzyme reaction rate per unit volume at any point in the film and (rl - r2)b the enzyme reaction rate at the bulk conditions. If we again employ the film theory model and the pseudo-steady-state approximation in the film,the dimensionless material balances are film:
+
where YA CA/bNois the dimensionless acetaldehyde concentration in the film.
B.C.:
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Ind. Eng. Chem. Fundam., Vol. 20, No. 2, 1981
Note that the dimensionless bulk concentration of acetaldehyde, WA = bA/bNo, is the reuse number. The homogeneous reaction kinetics for the alcohol dehydrogenase catalysis of ethanol to acetaldehyde have been extensively studied by Dickinson and Monger (1973) and Dickenson and Dickinson (1975), who established a mechanism containing some 18 reaction rate constants. Simplification of the mechanism is possible so that a highly nonlinear rate expression containing some 13 parameters obtainable from literature data can be formulated (Searby, 1977). The above model could be solved numerically employing this rate mechanism. Alternatively, as long as the enzyme is present in significant amounts (-lo-’ M), it is reasonable to assume that the enzyme reaction is at equilibrium everywhere in the film. Friday (1980), who conducted a numerical solution for this reaction system and a porous carbon electrode, included the enzyme reaction rate expression but found that the compositions do, indeed, stay near their equilibrium values for the enzyme reaction. The system of equations is readily transformed using a technique originally described by Olander (1960) by assuming equal diffusivities for NAD’ and NADH and that the concentrations everywhere correspond to equilibrium for reaction 111. The equality of diffusion coefficients for NAD+ and NADH is justified because: (1) their molecular weights differ by only 2 out of 780 and (2) the high concentration of supporting electrolyte (-0.05 M) M) reduces the electrical relative to that of NAD+ potential contribution to the transport of NAD+ to a negligible value (Chapman, 1969). film: (12a) + (12b)
i.e., the flux of NAD+ (124 - (12a)
The solution to this set of equations may be written in the form
where YN(0), the dimensionless NADH concentration at 2 = 0, is calculated from
(28) In order to obtain an approximate solution, the magnitudes of the terms in eq 28 can be examined. wN and wp are -1; wA, the reuse number, is > l ; D 1,and K for pH 9, 0.65 M ethanol and M NADH is -1500 (Bauer, 1980). For a reuse number (wA) less than about 100
-
(29) The second term in the square root is small so that expansion of the square root yields
+ NADH is independent of position
(30) The time dependence of the bulk concentrations is obtained by substituting eq 30 into eq 27 and integrating.
where D I DAIDN. YAYN/YP = K
wN
+ wP = ex.[
-ak4t/(
+ I)]
(31)
K’CE/CHfbNo
(19) where K’is the reaction equilibrium constant for reaction I11 available from the work of Dickinson and co-workers (1973,1978) and K is a constant for a given pH, ethanol concentration and initial NADH concentration. B.C.: I
This result used in eq 26 yields
wA, wN, and wp can be completely determined by combining eq 31 and 32 with the equilibrium relation (eq 24), but the time dependence of the reuse number is readily obtained from eq 32 since wN 5 1.
At small values of k4t the reuse number increases linearly with time while at long times it approaches a limiting value depending upon the selectivity of the electrode reaction (selectivity = k3/(k3+ k$). It can be seen from eq 33 that the influence of the mass transfer rate on turnover number is quite small because the NADH concentration near the electrode is maintained by the homogeneous reaction. Poorer mass transfer conditions (k4/DN/Llarge) actually improve the transient reuse number somewhat by reducing
Ind. Eng. Chem. Fundam., Vol. 20,
Table I. NADH Oxidation Rate Constants and Mass Transfer Coefficient at 900 rpm ~~~
~
electrode graphite (thick) graphite (thin) platinum
k , + k,,
50
DNIL,
cm/min
cmlmin
0.11 0.11 0.19
0.11 0.11 0.29
the concentration of NADH near the electrode where it can be deactivated. The dominant factor determining turnover number is the selectivity of NADH oxidation on the electrode surface. A reuse number of 1000 requires a selectivity of better than 0.999. Results and Discussion In the course of the experiments it was quickly established that, in agreement with other investigators (Blaedel and Jenkins, 1975; Braun et al., 1975), variations in both oxidation rate and selectivity could be obtained depending upon the pretreatment and history of the electrode with the effects being particularly severe for the graphite electrodes. Coughlin and Wallace (1978) have reported on variations of the natural catalytic activity of graphite for NADH oxidation in the absence of an applied potential and suggest that at least two types of surface sites exist having different selectivities. Platinum catalytic activity is well known to be dependent upon electrochemical pretreatment which affects the particular oxides on its surface (Adams, 1969). Unless otherwise indicated, the experimental results described below are for freshly cut graphite electrodes that have been soaked in distilled water and oven dried in air. The platinum electrodes were pretreated by successively applying for 5-min periods: +1600 mV (SCE), -1600 mV (SCE), and +1600 mV (SCE) prior to use. This treatment seemed successful in restoring oxidation rates and selectivities during oxidation experimetns but not during the longer duration turnover number experiments. Similar electrochemical treatment of graphite resulted in some increase in activity but significantly poorer selectivities. NADH Oxidation and Selectivity. All of the observed rates of electrochemical oxidation of NADH whether on platinum or graphite electrodes indicated a first-order process in agreement with eq 10, which can be written in the form In W N = -ak,b,t (34) where (35)
and the film thickness dependence on stirring speed is indicated. For each of the electrodes used, experiments at different stirring speeds were analyzed by plotting experimental values of against Nf for different values of x until a linear relation was found. As illustrated in Figure 4, x = -2/3 adequately fit the results for all electrodes. Both thick and thin graphite electrodes, despite different shapes and interfacial areas, seem to be adequately represented by a single line while the platinum results indicate different surface rate constants from the graphite and a somewhat different mass trader coefficient. In Table I are tabulated values of (k3 + k4) and the mass transfer coefficient at N = 900 which will be used in the analysis of the reuse number experiments. The observed selectivity of NADH oxidation to active NAD+ on fresh graphite electrodes varied from about 75 to 95%. The selectivity never exceeded about 88% on the thin electrodes but did attain 95% on thick electrodes.
t
0
0
No. 2, 1981 145
Thick Graphite Thin Graphite
~
0
.0 1
.03
.02
04
-24
ISTIRRING SP&:/J , RPM Figure 4. Observed first-order rate constants as a function of agitation rate.
Nevertheless, there was considerable unexplained variation between runs which could not be removed by various pretreatments. The ratio k 3 / ( k 3+ k4) is apparently responding to uncontrolled variations in surface oxides present. It cannot, in fact, be established unambiguously that the formation of inactive cofactor during NADH oxidation is a first-order process because the observed first-order loss of NADH is masked by the reaction to active NAD+. It was also observed that there was a potential-dependent deactivation (oxidation?) of NAD+ at the anode near the end of an experimental run when the NADH concentration was low and NAD' high. This process also obeyed fmborder kinetics with a rate constant ( 1.5 X cm/min) much smaller than that for NADH oxidation (Bauer, 1980). The observed selectivities on electrochemically cleaned platinum at pH 9 and 700 mV (SCE) ranged from 93 to 100% with most values above 98%. It is even more difficult to establish a fmborder process for NADH oxidation to inactive cofactor at these high selectivities. If a firstorder reaction is assumed then k 4 / ( k 3 + k4) 0.02 on platinum. Platinum was also capable of deactivating NAD+ by a first-order process with a rate constant of -7 x cm/min. Reuse Number Experiments. All reuse number experiments were conducted as batch reactions at pH 9,700 mV (SCE) and a stirring speed of 900 rpm. The acetaldehyde concentration was followed as a function of time. Typical experimental reuse number results for thick graphite anodes of different surface areas per solution volume and for platinum anodes are compared to the model predictions for WA in Figures 5, 6, and 7. Reuse number values did vary significantly from run to run apparently due to changes in surface conditions with time during the experiment. Since the experimental reuse numbers are less than 100, eq 33 can be employed to predict reuse number values. The mass transfer coefficient at 900 rpm and the oxidation rate constant for NADH (k3 + k4) are given for each electrode in Table I. In the figures assumed values of (k3 + k4)/k4 were used for illustrative purposes since the experimental values varied. It should be noted that, although the rate constant for NAD+ deactivation is comparable to k4, this deactivation process is unimportant in the reuse number experiments because the concentration of NADH >> NAD+ at all times. Examination of Figures 5 and 6 indicates reasonable agreement between the model and the experimental observations provided selectivities for the NADH oxidation are somewhat better than those observed in the oxidation
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Ind. Eng. Chem. Fundam., Vol. 20, No. 2, 1981
148
k sk
periments were generally completed in less than 1 h. Based upon the very high observed selectivity for platinum-catalyzed oxidation of NADH, higher reuse numbers were expected. Yet, as shown in Figure 7, values of only about 40 could be achieved. In one case a reuse of 70 was achieved with a fresh platinum surface but could not be repeated regardless of the pretreatment of the used surface. It seems clear that during the course of the experiment the selectivity of the surface decreased. Further, the initial observed activity (k3+ k4) appears to be higher than that measured in the oxidation experiments as evidenced by the greater initial slope of the experimental curve as compared to the model predictions. It is clear from the above results that not yet understood dynamic behavior of the anodic surface during the course of a batch experiment is influencing the results. Nevertheless, the model provides a reasonable prediction of experimental behavior particularly demonstrating the importance of k 4 / ( k 3+ k4). Further study of the selectivity of NADH oxidation on better-characterized anodic surfaces is in progress. The general theme of utilizing electrocatalysis to drive and control a desired homogeneous reaction is amply demonstrated by the above results. It is hoped that other applications of this concept will be developed. Nomenclature a = electrode external area per volume of solution, cm-’ A = absorbance b = molar concentration in bulk solution, mol/cm3 c = molar concentration in film, mol/cm3 D = diffusivity, cm2/min D = ratio of acetaldehyde to NAD+ diffusivities k = surface reaction rate constants, cm/min K = dimensionless equilibrium constant L = film thickness, cm N = stirring speed, rpm r = reaction rate per unit volume, mol/cm3 min t = time, min T = dimensionless time w = dimensionless bulk concentration Y = dimensionless film concentration z = distance, cm 2 = dimensionless distance
!40 VI
20
a
L
P
2
6
36
32
24
C
42
4p
T I M E , HOURIS
Figure 5. Comparison of experimental reuse numbers (0) with model predictions (-) for four thick graphite electrodes; D N / L = 0.11 cm/min, k3 k4 = 0.11 cm/min, and a = 0.48 cm-’.
+
160
140
120
i I
U
I
6
18
‘2
36
30
24
TIKE, HOURS
Figure 6. Comparison of experimental reuse numbers (0) with model predictions (-) for eight thick graphite electrodes; D N / L = 0.11 cm/min, k3 + k4 = 0.11 cm/min, and a = 0.96 cm-’.
%
1
2c
Subscripts A = acetaldehyde Et = ethanol H+ = proton N = NADH P = NAD+ 1 , 2 = refer to forward and reverse enzyme-catalyzed reaction rates, respectively 3 , 4 = refer to electrochemical oxidation reactions of NADH producing active and inactive NAD+, respectively m = final value of absorbance Superscript 0 = initial value
-
Literature Cited “4
0
6
‘2
le
?4
1
1
I
3C
JF
42
t
45
Figure 7. Comparison of experimental reuse numbers (0) with model predictions (-) for a platinum electrode; D N / L = 0.29 cm/ min, k 3 + k4 = 0.19 cm/min, and a = 0.20 cm-’.
experiments. Indeed, for the experiments with 8 thick electrodes a selectivity of greater than 99% ((k3+ k 4 ) / k 4 = 100) is indicated while no selectivity greater than about 95% was ever observed during oxidation of NADH on graphite. It should be noted that reuse number experiments lasted of the order of 50 h while the oxidation ex-
Adams, R. N. “Electrochemlstry at Solid Electrodes”, Marcel Dekker: New York, 1969, p 139. Aizawa, M.; Coughlin, R. W.; Charles, M. Biochlm. Blophys. Acta 1976, 440, 233. Alexander, B. F.; Coughlln, R. W. Blotech. Bioeng. 1975, 77, 1379. Bailey. J. E.; Ollis, D. F. “Biochemical Engineering Fundamentals”, MceawHill: New York, 1977, Chapter 5. Bauer, R. M.S. Thesis, University of Virginia, Charlottesvllle, VA. 1980. Blaedel, W. J.; Jenkins, R. A. Anal. Chem. 1975, 4 7 , 1337. Braun, R. D.;Santhanam, K. S. V., Elvlng, P. J. J. Am. Chem. Soc. 1975, 97, 2591. Chapman, T. W.“Ionic Transport and Electrochemical Systems”, in Bird, R. B., et ai.,“ Lectures in Transport Phenomena, AIChE Contlnulng Education Series”, No. 4, 1969, 43. Coughlln, R. W.;Wallace. T. C. Bktech. Bioeng. 1878, 20. 403. Dickenson. F. M.; Dlckinson, F. M. Blochem. J. 1975, 147. 303.
Ind. Eng. Chem. Fundam. 1981, 20, 147-149 Dickinson, F. M.; Monger, 0. P. Blochem. J . 1973, 131, 261. Eberson, L. E.; Weinberg, N. L. Chem. Eng. News Jan 25, 1971, 40, 40. Friday, D. K. M.S. Thesis, University of Vkglnla, Charlottesville, VA, 1980. Janik, B.; Eking, P. J. Chem. Rev. lSS8, 68, 295. Kelly, R. M.; Kirwan, D. J. Biotech. Bbeng. 1977, 19, 1215. Mahler, H. R.; Cordes, F. 0. “Biological Chemistry”, 2nd ed., Herper and Row: New York, 1971.
147
Olander, D. R., AIChE J., 19S0, 6,233. Prescott, J. H. Chem. Eng. Nov 8, 1965. 238. Searby, P. E. M. S. Thesis, University of Virginia, Charlottesville, VA, 1977.
Received for review May 27, 1980 Accepted February 6,1981
Effects of Inertia, Surface Tension, and Gravity on the Stability of Isothermal Drawing of Newtonian Fluids J. C. Chang and
M. M. Denn*
Department of Chemical Engineering, Universe of Delaware, Newark, Delaware 1971 1
F. 1.Geyllng Bell Laboratories, Murray Hill, New Jersey 07974
The stabili of continuous isothermal drawing of Newtonian fluids has been analyzed, taking inertial, surface tension, and gravitational effects into account. The results agree qualitatively with an earlier analysis, but there are important quantitative differences. Agreement with experiment is good.
Introduction An instability known as draw resonance is sometimes observed in processes involving the continuous drawing of liquid filaments. The instability is characterized by oscillations in tension and drawn filament diameter with a well defined period and amplitude. Some illustrative data and general reviews of theory and experiment are contained in the works of Petrie and Denn (1976), Kase and Denn (1978), and Denn (1980). Most experimental studies have focused on polymeric liquids, because of applications in processes such as fiber spinning and extrusion coating. The first theoretical analyses, however, were carried out for Newtonian liquids, and the Newtonian fluid limiting case continues to be of interest because of possible applications in glass fiber drawing, as well as because it provides an analytical framework in which to explore certain mechanisms without the addition of rheological complexity. One set of experiments on draw resonance has been reported on a truly Newtonian fluid (Chang and Denn, 1979), and Donnelly and Weinberger (1975) and D’Andrea and Weinberger (1976) have reported draw resonance experiments on a silicone fluid that is so slightly viscoelastic over the deformation rate range employed that a Newtonian theory might be adequate. These experiments were carried out isothermally, so they provide some insight into underlying mechanisms, but they are not directly applicable to practical processing. Glass and polymer fiber drawing always occur under nonisothermal conditions, where the effect of heat transfer on temperature-dependent physical properties is very important. The isothermal, Newtonian theory of draw resonance predicts instability when the area reduction (drawdown) ratio exceeds 20.2 for conditions in which gravity, surface tension, inertia, and air drag are unimportant, and the experiments of Donnelly and Weinberger seem generally to codirm this value. The experiments of Chang and Denn 0 196-431 3/81/1020-0 147$0 1.2510
and DAndrea and Weinberger were done under conditions where the first three of these effects could not be ignored, however. Here, agreement with thoretical values obtained by Shah and Pearson (1972) is not good. We report here new calculations for the onset of draw resonance in the isothermal drawing of a Newtonian liquid, using two computational procedures that differ from the method of Shah and Pearson. The general qualitative effects of gravity, surface tension, and inertia reported by Shah and Pearson are confirmed, but there are important quantitative differences that bring the theory and experiments closer. The work reported here combines the results of independent studies done at the University of Delaware and at Bell Laboratories. Spinning Equations The asymptotic dimensionless stress, momentum, and continuity equations for isothermal Newtonian spinning are (Kase and Matsuo, 1965; Matovich and Pearson, 1969)
(3)
Here T, u, and a are dimensionless total axial stress, axial velocity, and area, respectively (note that some authors use a to denote radius), and 0 and 5 are dimensionless time and axial position. The dimensionless groups, Reynolds, Froude, and Weber numbers are, respectively Re = pLV~/3q (44
Fr = vQ2/gL We = 2a01/zvQ2p/a1/2a 0 1981 American Chemical Society
(4b) (44
