528 Y4U Eh./Hr., hv
Av. iw
t$m2.,
V of jw, ft./sec. 24.25 17.0 11.75 8.13
59
60 61.6 63.6
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
_ _ --
1800 hid, Fig. 10 3410 2620 1990 1540
580 Lb./Hr., hv = 3140 ' AV. j w
temp., O
F.
50.3 61.6 62.9
V of jw, ft./sec. 23.9 17.08 11.6
5. The Dittus-Boelter equation can be nsed fo calculate the film coefficient on the Votator jacket, even though She path is helical. The results check expcrimentd values within TOTo
hji,
Fig. 10 3580 2680 1980
Even though the average temperature inside the Votator is considerably different, this alone does not seem to explain the marked change in h" with change in rate. Further work ir necessary to clarify this point.
ACKNOWLEDGMENT
The author wishes to thank Bruce E. Aclams for his l d r l w.ath the calculations and figures presented here.
-
SUMMARY
1, A good average over-a,ll coefficient is about 800 B.t.an./ (hour) (square foot) (" E'.). The coefficient varies as follows: Mutator Speed, R.P.M. 300 I900
Velocit of jw, Ft.&W. 5 24
Over-all Coefficient L' 520 1120
2, A minimum jacket-water velocity of 7-10 feet per second and a minimum mutator speed of 600 r.p.m. (7.8 feet per second) should be used for efficient operations on waterlike materials. Values greater than these are beneficial and should be used if other factors such as power load, jacket-water pressure drop, blade and tube wear, do not make the operation uneconomical. 3. The film coefficient on the jacket side varies about as follows :
&R
\'tlority of jw, Ft /Sei.
h,, R.T,U./(Hr.)(Sq. Ft.)(' F.)
7 5 26
1500 4100
Vol. 36, No. 6
NOMENCLATURE
cooling surface area, sq. ft. equivalent diameter of jacket hascrl 013 recttcngular section mass velocity, lb./sq. ft./sec. film coegcient of heat transfei, I-3.t.u /(hr )(sq -it. (" F.) jncket-wa-ie1 film coefficient based on inside &rea on heat transfer wall jacket-water film coeficient based on outside hrlat ti ansfer wall m r ~ a wall l expressed as film cor.Hicien1 j:Lcket water = log mean temperature differenue, * i' thickiiesa of metal hlumi,or = shaft with scraper blades AI' = pressurt: drop through water jacket, lb./sq. in. Pr =: Prnndtl number = e,u/k (;2* = heat transferred, based on heat removed f:ronn voLated water, B.t.u./hr. Re == Reynolds r?umober = DG/u T = temperature, F. Ci* = over-all coefficient. based 011 Q*, B.t.u./(hl..)lsq.-fb.! ( " F.j 1' = velocity, ft./sec. * ,'ee footnote Table I. ' 09
4. The film coefficient in the votatod water side varied abour. follows: Mutator Speed,
R.P.M. 300 2000
Peripheral Speed,
Ft./Sec 3 9 18 1
(HrJ (Sq. Ft.) 1100 2200
, -
r and Chemical Keaction in . HE concept oi an "effective film thickness" has bean of inestimable value in analyzing problems involving heat and mass transfer. Application of this concept to a study of dissolution rates of solids in liquids in a series ot dimensionally similar agitation systems was illustrated in earlier papers (3). The work is extended in the present paper to include a system in which diisolution and chemical reaction proceed simultaneously. For the case where a rapid chemical reaction takes place i n the liquid phase of a liqnid-solid agitation system, the Nernst-Brunner double-film concept (1) is generally accepted as indicating the mechanism which determinev the reaction rate. Briefly, the theory assumes the existence of a liquid layer adjacent to the solid surface in which concentration gradients of the reactants exist because of their reaction within this layer. An idealized sketch of the conditions existing in such a film when a solid acid, . A, is dissolving and reacting with an aqueous solution of an alkali, B, is illustrated in Figure 1. (X, X b ) is the total thickness of the diffusion layer; X , is the distance from the solid surface at
+
1
Present address, American Resinous Chemicals Corporation, Peabody,
Mass
a
.
. .
COLUMBIA UNIVERSITY, NEW YORK, N. Y
which neutralization takes place; C, is the saturation solubility of the solid acid in a solution of the reaction product AB of cox)oentration q; no is the initial concentration of the alkali in the main bulk of fluid and 71 is the roncentration at time e. Remtants A and B diffuse toward the neutralization plane whew product AB is formed which, in turn, diff'usea out toward t h o main bulk of fluid. The concentrations of both A and B a t the neutralization pla,rw can be considered zero although the concentration gradients are not necessarily linear with distance os shown. The sctuai dopw of the gradient cumes in the experiments dmcribed in tbis pap81 are unimportant as long as X, and X , are considered "effectiw film thicknesses" based on an assumed linear gradient. The concentration of AB in the main bulk of fluid a,t time 8 is (no -if a monobasic acid is reacting with an alkali containing one hydroxyl group. At any time the rate of dissolution of A will depend, among other factors, on the rate of diffusion of \A through a solution of AB with a concentration q, aod on the rate of d z u sion of B through a solution of AB in which a concentration g m dient between q and (no n) exists.
-
529
June, 1944
Because of the complicated nature of the film conditions, this work was limited to a study of a system for which extensive diffusivity data were available, The combination, solid benzoic acid-dilute aqueous sodium hydroxide, was chosen since it nearly meets this prime requirement (4) and possesses other additional advantages: (a) Nonfriable pellets of known surface area can be prepared from the acid; ( b ) the solubility of the acid is low, and the dissolution rate can be accurately determined; (c) comparison can be made between the data for benzoic acid dissolving in caustic solutions and in pure water (8). DERIVATION O F RATE E Q U A T I O N
Assuming, aa an approximation, that the Nernst-Brunner concept is asentially correct for the system studied, the rate of dissolution of the acid will be given by Fick’s law of diffusion,
snd the rate of neutralization of the alkali will be given by
Fhrmnging Equations 1and 2,
+ Xa - A(D4CI 4-Dan) dM4
de XO Replacing n by
(3)
(w - M 4 / V )and integrating,
+
To obtain (X, X,)in cm. when concentrations are expressed in molers per liter, tima in seconds, area in sq. cm., diffusivity in sq. cm. per second, anbndvolume in liters, the right-hand side of Equation 4 must be multiplied by 1000:
The double film concept i s applied to a study of the rate of dissolution of benzoic acid pellets in dilute aqueous sodium hydroxide. A rate equation i s derived which i s evaluated by sraphical integration. Results are expressed in terms of a total effective Film thickness, X4 X,, which attains a constant value For initial a l k a l i concentrations greater than 0.03 M. Below 0.03 M the data extrapolate at zero concentration to the film thickness For dissolution 01 benzoic acid in water. Approximated values for the individual film thicknesses indicate that the location of the neutralization plane is not stationary but moves away from the liquid-solid interfxe as dissolution proceeds. For the apparatus and the system studied here, the total effective Film thickness i s inversely proportional to the stirrer speed over a range of 900 to 450 revolutions per minute.
+
A stock solution of sodium hydroxide was diluted to give the normality desired and, unless otherwise stated, 2.76 liters of the diluted solution were used in each run. This gave a liquid depth equal to the vessel diameter. The benzoic acid pellets were the same as those used in previous experiments (3). The initial area of the pellets was calculated from the measured surface dimensions, the average area being 1.500 sq. cm. per pellet. The average weight was 0.212 gram per pellet. I n any particular run the impeller was brought up to speed, and a small amount of phenolphthalein added to the alkali solution. After allowing a t least 3 minutes to attain a condition of equilibrium, a known number of pellets was introduced. The time required for the phenolphthalein color to disappear was then noted. n some runs rate constants were checked progressively by withdrawing small samples of the solution intermittently from the same point and then titrating them for unreacted alkali. All runs were made a t a room temperature of approximately 25’ e., the variation from this temperature never exceeding 0.5” C. Duplicate experiments indicated that the time for the indicator color to disappear could be checked within 5 seconds. As an interestin side light, in several experiments the color in the main b u k of %uid disappeared a few seconds earlier than the color in a column of liquid immediately adjacent to the agitator shaft. This may indicate a “dead spot” in the s stem. The time for the color to disappear in the main bulk of Juid was recorded in the experiments. T O T A L EFFECTIVE F I L M THICKNESS
King and Brodie ( 4 , studying the same system, integrated Equation 3 for the case of constant area, and expressed their results in teof dissolution constants KOand & defined as
Kb
m
Db
b
(7)
The validity of Equation 5 and the assumed reaction m e c h s nism can be partially confirmed by the constancy of ( X , Xb) over a range of initial alkali concentrations and also during any one particular dissolution experiment. I n the first series of runs the agitation conditions were held constant, and the initial sodium hydroxide concentration was varied from 0.005 to 0.08 M . Experimental and calculated data am shown in Table I. Total effective film thickness (X, X,)WBB
+
+
I n the present investigation the industrially important factor of varying surface area during dissolution is treated, and the r a sults are expressed in terms of effective film thicknesses. It is believed that this method is superior to King and Brodie’s inasmuch as D, and Db vary considerably in any one experiment, and a better picture of the reaction mechanism can be obtained from consideration of the film thicknesses. EXPERIMENTAL PROCEDURE
The 15.2-cm. (6-inch) diameter can and four-bladed turbine type agitator, described in detail in an earlier paper (3) was duplicated in glass as closely as possible. The bottom 01 a 6inch diameter battery jar was built up with paraffin wax to provide a flat surface simulating that in the metal can. A flatbladed turbine agitator was shaped from glass and had the following dimensions: shaft diameter, l/g inch; blade length, 1 inch; blade width, , l / 2 inch; blade angle, 45’. This agitator was mounted vertically in a variable-speed drive and located centrally with respect to the 6-inch jar. A clearance of 1 inch was maintained between the bottom of the jar and the lowest part of the agitator. The direction of rotation was such that the impeller tended to force the liquid up from the bottom of the vessel.
Figure 1. ldeallzed Sketch of Concentration Gradients in Double Film for Neutralizetion of Solid A c i d A by Solution, of Alkali B
INDUSTRIAL AND ENGINEERING CHEMISTRY
830
Vol. 36, No. 6
introduce an appreciable error in l)b and thus in the initial valuefi for the integral, but this error disappears as dissolution proceeds and the term &(no &fa/v) approaches zero. Values for c, were based on an assumed linear relation between solubility and sodium benzoate concentration, the straight line being established by Scidell's value (6) for solubility in water at 25" C.namely, 0.0282 mole per liter; and by the value for the solubility of the acid in 1 d l sodium bcnzoatc solution at 25" C. reported in International Critical Tables to be 0.044 mole pcr liter. Solubility C, was also assumed to be dependent on the sodium benzoate concentration in the main bulk of fluid. Calculation of ( X , 4- xb)for a typical run is illustrated in Table 11. Table I indicates that the term ( X , Xb) is essentially constant for initial alkali concentrations of 0.03 M and greater. Extrapolation of the data to zero alkali concentration (Figure 3) yields a value for the total cffectivefilm thickness of 2.78 x 10-8 cm. This can be considered the effective film thickness for benzoic acid pellets dissolving in pure water and corresponds to the value obtained in earlier dissolution experiments (3). For n stirrer speed of 300 r.p.m. in the 6-inch apparatus, mass transfer coefficient K was found to be 0.00399 cm. per second. Using King and Brodio's value for diffusivity of benzoic acid in water:
-
a'
N
' a
-+
SODIUM BENZOATE CONCENTRATION, M O L E S / LITER
Figure 2. Diffusivity Data of King and Brodie ( 4 ) for Benzoic A c i d (lower curve) and Sodium Hydroxide (upper curve) into Sodium Benzoate Solutions
calculated by graphical integration of Equation 5. Surface area A was evaluated from the relation, A = uWZ/
(8)
a being determined for each experiment from the initial area and weight of the benzoic acid pellets. Diffusion coefficients are considerably influenced by the presence of salts, and values for D. and Db used in Equation 5 were based on the data of King and Brodie (4) for diffusion of benzoic acid and sodium hydroxide into sodium benzoate solutions, shown graphically in Figure 2. I n the absence of data on diffusion of sodium benzoate into scdium hydroxide solutions, the values for D, and Db used in determining (X, xb)were assumed to be dependent on the sodium benzoate concentration in the main bulk of fluid instead of concentration g and some average concentration between g and ( n o - n). This does not introduce too large an error in D,, which is fairly insensitive to changes in sodium benzoato concentration. It may
+
Table 1.
Rate of Dissolution of Benzoic A c i d Pellets in Sodium Hydroxide Solution (Speed 300 r.p.m., temperature 25 * 0.5O C., volume, 2.76 liters) No. of Pelleta
Man
L
~
IM
0.0060
50
0.0200 0.02RO 0.0300 0.0400 0.0500
100 120 120 120 150 150
0.0100
0.0800
50
6
rroV
1520 3097 2732 2988 2095 3810 341.5 5980
6.30 3.14 3.14 2.61 2.52 1.89 1.89 1.18
.YO C X B X.3 x 10-1 2 . 4 9 X 10-8 2.39 2.04 1.79 1.71 1.52 1.34 1.10
2.88 3.02 3.10 3.31 3.27 3.37 3.43 3.41
Xb
0.39 X 10-3 0.63 1.06
1.52 1.56 1.85 2.09 2.31
+
Xh* Table 11. Calculation of X a (m=- 0.0500; No. of peliets = 750; AG = 1.506 X 150 = 226 eq. om.; V 2 76 liters. !To = lo0 X 0.212 31.8 grams: D = 226/(31.8)*/a =
-
J
22.6: &id disaoived
N,
3
0.0500 X 2.76
0.00
A 226
0.02 0.04 0.060.08 0.10 0.12 0.138
215 0.0072 0.0283 203 0.0145 0.0284 191 0.0217 0.0285 178 0.0290 0.0287 104 0.0362 0 , 0 2 8 8 151 0.0435 0.0290 138 0.0500 0 . 0 2 9 0
c*NaB
C'r
0.0000 0.0282
* M a ia plotted
0.138 mole: 0 = 3415 aaconds)
Db
1000
Da 1 . 1 1 X l.,9,13X 1 n -5
1.i6 1.08 1.07
1.06
1.04 1.03 1.02
&a abscissa against
[
A DOC. 3. De(m
0 . 3 4 9 X 10'
2.42
0.346 0.3W 0.453 0.578 0.804 1,270 2.450
2.78 3.08 3.18 3.32 8.43 3.51
__ A[D&
- F)]
1000
C Da(m
- $)]
-
aa urdi-
uate, and the area under the curve determined. A value of 9.95 X 106 is-ob3.43 X 10-8 om. tained. Then X, Xb = 8/area = 3415/(9.95 X 10')
+
If the total efiective film thickness is a function of the degree of agitation and the physical properties or the fluid, it might logically be expected that, for the dilute solutions used in these experiments, the total thickness would remain constant over thr entire range of alkali concentrations used. As a partial explann tion for the increase of ( X , X,)for the initial alkali concentrtttion range up to 0.03 M , tho semiempirical considerations of Chilton and Colburn (8),Lhe empirical correlations of Hixson and Baum (S), and the experimental data of King and Howard (.5) have indicated that effective film thicknesses in mass transfei processes vary as a fractional power of the diffusivity. Examination of Figure 2 indicates that, up to a sodium benzoate concentration of 0.05 M ,there is almost a twofold increase in the diffusion coefficient for sodium hydroxide and above this concentration it approaches a maximum value. The diffusion coefficient for benzoic acid is fairly insensitive to sodium benzoate concentration. The initial low values for ( X , Xb) in Table 1 may be partially attributed to the comparatively low average values for sodium hydroxide diffusion coefficients in these runs. Tho constancy of (X,f Xb) over a wide range of alkali concentration and during any one experiment was checked by making a run in which the initial sodium hydroxide concontrntion wm 0.2 M , this coneentration being the upper limit for the available diffusivity data. Progressive values for ( X , x b ) were determined by withdrawing samplas of liquid intermittently and titrating them. Results are shown in Table 111. The apparatus used in this run differed from that for the results shown in Figure 2; hence the values for Xa Figure 3. Effective Film Thicknesses for Xi) are not comDissolution of Benzoic A c i d in Sodium Hydroxide Solutions parable.
+
+
+
+
531
lune, 1944
+
Xb during Dissolution of Table 111. Progressive Values for X. Benzoic A c i d Pellets in Sodium Hydroxide Solution (3-inch diameter ~tainlesssteel propeller on l/,ineh diameter shaft rotated vertically in 6-inLh diameter glass jar; clearance between propeller and bottom of jar 1/* inch; speed, 316 r.p.m * direction of rotation, forcing li uid up from b&tom of vessel; no = 0.2lO'k; No. of ellets 400; Wo 1 4 . 8 grams; v = 2.00 liters? innn
-
Table
IV. Effect of Stirrer Speed on Total Film Thickness (X.
-
+
xb)
(no-0.0300: temp.==25*0.5' C.; V a 2 . 7 6 liters; No. of pellets= 120) Speed, . 8, x o f XbD R.P.M. Sec. C m. 200 3276 3 . 5 8 X IO-* 250 3090 3 . 3 8 X 10-1 300 2995 3 . 2 7 X 10-1 350 2817 3 . 0 8 X 10-3 450 2541 2.78 X 10-8
+
Except for the initial calculated value for ( X , x b ) , the total film thickness is essentially constant throughout the dissolution process. It was found that this constancy held for an intermittent run made with an initial alkali concentration of 0.03 M . I N D I V I D U A L FILM THICKNESSES
Table I1 indicates that the product DaC, remains fairly oonstant throughout the run, varying from 3.13 X lov7at the start t o 2.96 x 10-7 a t the end. For the dilute solutions considered in the present work, little error in calculating Xa from Equation 1 will be introduced if an average value for DaCa is used. Furthermore, an earlier paper showed that, for the case where the concentration driving force is constant, Equation 1 (assuming that Xa is constant) can be integrated to
stant, i4 therefore not valid, and the values for Xa and x b shown in Table I should be considered only as approximate average values for the particular rango of alkali concentration in oach experiment. The variation of individual film thicknesses with allcali concentration indicated by the above data seems logical in the light of a simple experiment. If a pellet of potassium hydroxide is s u 5
thickness is considerably less, and alkaline streamers appear to extend several millimeters into the fluid. With 2 M hydrochloric acid, the pink layer i s hardly discernible. EFFECT OF SPEED AND A R E A
STIRRER SPEED. I n a second series of runs the effect of stirrer Xb) at constant initial thickness (Xa as studied. Results are shown in Table IV 4. The film thickness is linear with speed . Previous data on the rate of dissoldtion series of agitators, dimensionally simiesent investigation, indicated that the y proportional to a fractional power of the rotational speed. The fact that the data obtained in the present paper lie on both sides of the critical Reynolds number (6.7 x 104) for dissolution in this apparatus may explain the linear relation obtained. SURFACE AREA. The number of pellets had no apparent effect on the calculated a m thickness except as this factor entered into the definition of surface area. The ratio of initial moles of acid to initial moles of alkali (Ma,/noVin Table I) varied from 1.18 to 6.30 without apparent influence on effective film thickness. The run described in Table 111, in which 61% of the original weight of tEe pellets was dissolved, was repeated with half the number of pellets. The average values for (Xa xb) in both runs checked within 4.2% for dissolution of the same percentages of the original acid weight.
+
+
NOMENCLATURE
(9)
Film thickness X a was calculated from Equation 9, and the values are shown in Table I. Film thickness Xb was determined by difference. When plotted against the initial alkali concentration (Figure 3) the X , curve also extrapolates to a thickness of 2.78 X lo-* em. at zero concentration. However, there is no indication that Xa or Xa is constant over a range of alkali concentration. Calculation of progressive values of Xa, in the runs where this was possible, also indicated that this , thickness did not attain a constant value. Integration STIRRER SPEED- R.P.M. of Equation 1 to Figure 4. Variation of Total Effective Equation 9, a w m Film Thickness with Stirrer Speed ing that Xa is con-
A = surface area of solid, sq. em. A,, = average surface area defined by Eq. 10, sq. em. A . = initial surface area of pellets, sq. cm. CN~= B Na benzoate concentration a t time e, molesfliter C, = solubility of acid, moles/liter D, = diffusion coefficient of acid, sq. cm./sec. Dz, = diffusion coefficient of alkali, sq. cm./scc. K , = reaction velocity constant defined by Eq. 6, cm./sec. Kb = reaction velocity constant defined by Eq, 7, cm./sec. Ma = arid dissolved in time e, moles Mb = alkali dissolved in time 8, moles Moo = initial solid acid, moles n = alkali concentration a t time 0, moles/liter no = initial alkali concentration, moles/liter = sodium benzoate concentration in acid layer, moles/liter = volume of solution, liters W = weight of solid acid, grams Xa = effective film thickness of acid layer, cm. Xb = effective film thickness of alkali layer, cm. 01 = constant defined by Eq. 8 e . = time, sec.
't
LITERATURE CITED
(1) Brunner, E.,2.phyaik. Chem.,47,56(1904). (2) Chilton, T. H., and Colburn, A. P., IND.ENQ.CHEM.,26, 1183 (1934). (3) Hixson, A. W.,and Baum, 9. J., Ibid., 33, 478, 1433 (1941); 34, 120 (1942). (4) King, C.V.,and Brodie, 9. S., J. Am. C h m . Soc., 59, 1375 (1937). (5) King, C. V.,and Howard, P. L., IND.ENQ.CHEM.,29, 75 (1937). (6) Seidell, A.,"Solubilities of Inorganic and Organic Compounds", New York, D. Van Nostrand Co.,1919.
CCd 4 g
-v1
