HEAT TRANSFER AND DIFFUSION RATES AT
Solid-Liquid Boundaries CECIL v. KING AND PAUL L. HOWARD Washington Square College, New York University, New York. N. Y.
assumed a thin film next to the interface in laminar flow with the temperature gradient extending farther into the turbulent region. Prandtl’s relation and a similar one for mass transfer derived by Colburn (6)are developments of the Reynolds analogy between momentum and heat transfer. Both involve the friction factor and a ratio uB/21mof the mean rate of flow a t the outer edge of the supposedly laminar layer, to the mean rate of flow of all fluid, both along the main axis of flow. This method of attack avoids the direct introduction of film thickness, but the ratio u,/u, is actually calculated empirically or semi-empirically. I n the case of water in turbulent flow in a tube, Fage and Townend (7) have shown that turbulence persists up t o the interface itself. Although flow very near the wall is essentially laminar (but not rectilinear), there is no reason to believe that the v component of the turbulent motion (normal to the wall) shows a sharp break a t any point a short distance from the wall. Rather it shows a regular, though not linear, rise with distance. Only this v component of turbulence can be effective in the transport of material by convection to and from the surface, and the introduction of flow rates parallel to the surface is necessarily empirical except in so far as the latter may be related to the former. A rather successful semi-empirical method of predicting heat and mass transfer coefficients has been developed by Colburn (6) and by Chilton and Colburn (4). The experimental rate coefficients, h and k , in gas-liquid or gas-solid systems are expressed in terms of dimensionless groups which contain the necessary variables and are simple functions of the Reynolds number. The heat and mass transfer factors are presumably equal to each other and equal or closely related to the friction factor. This treatment avoids the question of film thickness and of any flow rate except the mean rate of all fluid. The factors, in form suitable for use i n aqueous solutions, are.
Measurements of the rate of heat transfer from a thin platinum wire to rapidly stirred water and sucrose solutions have been made and compared with the dissolution rates of thin copper, zinc, and cadmium wires in aqueous reagents in which the rate is believed to be independent of the chemical reaction velocity. I t is shown that these and other measurements agree reasonably well with some of the requirements of the semi-empirical relations of Chilton and Colburn, which were derived from data for heat and mass transfer processes in gaseous systems. The calculated “effective film thickness’ * i s shown to increase regularly with increasing values of the diffusivity.
M
ASS transfer a t an interface is closely related to heat transfer at a similarinterface, and both problems have received considerable study in connection with gas-solid and gas-liquid systems, especially those of industrial importance. I n spite of numerous attempts, it has been impossible to disentangle completely the roles played by convection and by diffusion or heat conduction. In the case of solid-liquid systems in which a chemical reaction takes place a t the interface, its rate may also enter as another complicating factor; on the other hand, in some dissolution processes, including those used in the present research, the chemical rate is believed to be so rapid that it need not be considered in comparison with convection and diffusion rates. The film concept of heat and mass transfer has been of great value in interpreting these processes, but has proved difficult of theoretical development because of the lack of knowledge of the laws of fluid motion near an interface and the consequent lack of a clear picture of the layer in which a temperature or concentration gradient exists. Lack of information concerning diffusivities of reagents in solution, especially in the presence of other solutes, has also hindered treatment of the dissolution problem. The best that can be done a t present is to calculate an “effective film thickness”-the thickness necessary to give the observed rate if the film were completely stagnant and the fluid outside the film were kept completely homogeneous by the stirring. The picture of a stagnant film to which the gradient is confined was developed for liquid systems by Brunner and Nernst (2, 12) but is inconsistent with present knowledge of hydrodynamics. Prandtl (IS) and others, in the heat transfer problem, have
The heat transfer and dissolution rate measurements deb scribed in this paper were made with the hope that they could be correlated satisfactorily and that they might throw mme light on the general theory of the transfer processes invdved. The rate of heat transfer from a thin platinum wire to rapidly stirred water and cane sugar solutions was measured, and for comparison the rate of solution of copper, zinc, and cadmium wires in appropriate aqueous reagents was measured. Thin wires were chosen because of the ease of making the heat transfer measurements; the dissolution rates are considerably less precise than with other experimental arrangements..
Heat Transfer A platinum wire 0.0055 em. in diameter and 50.0 cm. long wa5 mounted on a glass holder in such a way that every art of it was approximately 1 cm. from the surface of a hard ruxber stir75
INDUSTRIAL AND ENGINEERING CHEMISTRY
76
VOL. 29, NO. 1
ring cylinder which was 2 cm. in diameter and 8.3 cm. long, and had four grooves (2 X 2 mm.) cut lengthwise in its surface. The wire and stirrer were surrounded by liquid in a liter beaker. The wire was heated with an accurately measured current (about 0.5 ampere) from a set of storage cells; its temperature could be determined at any moment with an accuracy of .t0.02"C . by measuring the potential drop across it. This was done with a Leeds & Northrup type K potentiometer; the resistance of the wire and its temperature coefficient had been measured previously. The temperature of the liquid was read simultaneously on a thermometer, to the same accuracy. From the temperature difference ( A T ) , the rate of energy dissipation (d&/ dt), and the surface area of the wire ( A ) , the thermal transfer coefficient h was calculated: dt
= hAAT
In order to measure the current accurately, the platinum wire was connected in series with a calibrated resistance (17.580Q) of negligible temperature coefficient. The current through both was determined by measuring the potential drop across this resistance just before throwing the potentiometer across the platinum wire. Since the current fluctuated somewhat, three or more sets of readings were taken for each solution. The temperature difference was usually 2" t o 4" C.; the value of h does not vary rapidly with temperature or temperature difference. The temperature of the liquid was kept near 25". Good distilled water and rock-candy sugar solutions were necessary to minimize conduction; they were deaerated to prevent deposition of gas bubbles on the wire. Table I shows details of the experiments with water. Table I1 gives average values of h for water and the sugar solutions and also values of K , c, p, and p . The values of K , the thermal conductivity, were obtained by combining the value for water given by Bates (1) with those from the International Critical Tables for sugar solutions. TABLE mI. RATEOF HEATTRANSFER TO WATER ,Stirring Speed
R. p .
m.
2000 3000 4000
E
E
across 17.5800
P t Wire
&or088
T of Water
Volts 8.945 8.948 8.953 8.948 8.927 8.918 8.955 8.953 8.953
Volts 15.354 15.351 15.364 15.335 15.303 15.298 15.339 15.341 15.340
C. 24.88 24.77 24.80 24.64 24.76 24.94 24.91 25.01 24.93 a
AT C. 3.09 2.98 3.04 2.76 2.72 2.77 2.30 2.31 2.37
FROM
dQ/dt
WIRE5
h
Cal./s% Cal./sec. cm. sec. C. 1.868 0.899 1.868 0.724 1.871 0.711 1.866 0.780 1.858 0.792 1.866 0,776 1.868 0.941 1.868 0.936 1.868 0.914
FIQURE 1. DISSOLUTION RATECOEFFICIENTS vs. DIFFUSIVITIES Lowerline (right-hand soale), data at 2000 r. p. m (left-hand soale), data at 3000 r. p. m:'
-
upperline
weighed. Dissolution rate coefficients k were calculated from the equation: dn - = kACreagent dt
where dn/dt = gram equivalents of metal dissolved per min. A = mean area of wire surface at beginning and end, sq. cm. C = concn. of reagent, gram moles/cc. Little error is introduced by using the differential form of the equation since only a, small fraction of the reagent was used in each experiment. Theoretically the value of k should vary with the diameter of wires as thin as these; wires of the same diameter were not available, and the diameter decreased appreciably during the run. No consistent trend of IC with wire diameter was noticed, however: although the rates are decidedly different from those obtained with flat metal surfaces. I n comparing the rate coefficients with diffusivities, the greatest error prob-
a Wire 0.0055 am. in diameter, 60.0 cm. in length, 0.864 sq. cm. in area. R at 20° C. = 29.484% temp. ooeffiicient = 0.08700/O C.
TABLE11. AVERAQE VALUESOF h FOR WATERAND SUCROSE~ SOLUTIONS M 105
P
0.3 143 135 0.93 1.04 0.0089) 0.0127
1':
0.6 128 0.88 1.08 0.0162
0.9 121 0.83 1.12 0.0234
1.2 114 0.78 1.15 0.0329
1.5 108 0.74 1.19 0.0523
Value of h at: 2000 r. p. m. 3000 r. p. m. 4000 r. p. m.
0.711 0.783 0.931
0.615 0.681 0.839
0.639 0.731 0.846
0.544 0.618 0.708
0.570 0.635 0.723
0
Csucrose,
K x c
P
0.692 0.796 0.904
0
-1
Solution Rate Measurements The zinc, cadmium, and copper wires were mounted on a glass holder similar to that used for the platinum wire and suspended in the same manner in one liter of the appropriate reagent. Dilute acetic and hydrochloric acid solutions containing potassium nitrate as a depolarizer were used with the zinc and cadmium and dilute ferric nitrate with the copper, since the rate is believed t o be free from any chemical factor in these reagents (8, 10). After cleaning and weighing (on the microbalance) the wires were immersed in the solution for a suitable length of time, and then washed, dried, and re-
Q.
z0
5
Y
w 0
c
-2 AND HEATTRANSFER COEFFICIENTB FIGURE 2. DISSOLUTION ws. ,THERMAL AND REAGENT DIFFUSIVITIES Lower line (left-hand soale), data at 2000 r. p. m.; upper line (right-hand soale), data at 3000 r. p. m.
JANUARY, 1937
INDUSTRIAL AND ENGINEERING CHEMISTRY
ably lies in the difficulty of obtaining measurements under comparable conditions. As shown elsewhere (3, 9),the diffusivities of hydrochloric acid and ferric nitrate are greatly influenced by salts; the best comparison would be obtained by measuring all dissolution rates in salt solutions of such concentration that the reaction products would have little effect on the rates. Table I11 summarizes the results of these experiments. OF EXPERIMENTS AT 25" C. TABLE111. RESULTS
Mg. Av; Lost -lo x 102Length Wire Diem., in 5 3000 2000 4000 r.p.m. r.p.m. r.p.m. M Cm.' Cm. Min. Zinc Wire in AcOH; 0.05 M KNOa Present; DAOOH = 1.27 X 10-6 Sq.Cm./Sec. 9.06 1.74 .. .... 52.0 0.02494 0.013 9.00 1.72 .. . . . 0.02498 0,01275 52.0 7.87 1.85 . .. . 47.4 0.02264 0.013 43.8 0.02282 7.83 1.99 . .. 0.013
Creagent,
.. .
. .. .. ..
. - .. .
Av. 1.83 Zinc Wire in HC1; 0.05 M KNOs Present; DHCl 0.02178 24.26 0.01155a 52.0 0.02214 23.66 0.01081a 52.0 0.02366 25.61 52.0 0.01155 52.0 0.02384 22.02 0.01086 0.02456 28.23 52.0 0.01155 0.02406 23.01 53.0 0.01067
= 5.56 X 6.04 6.15 5.85 5.31 6.21 5.49
sq. Cm./sec.
.... .... .... .... ....
.... .... .... ....
Cadmium Wire in AcOH; 0.05 M KNOa Present 29.6 0.00848 3.64b 2.10 1.80 33.0 0.00856 3.696 1.88 1.70
2.01
- ....
....
....
Av. 5.84 0.013 0.013
1.85 - - -
Av. 1.99
1.75
1.93
Cadmium Wire in HCl; 0.05 M KNOa Present 0.01155 33.0 0.00826 5.32C 7.55 6.68 0.01155 33.0 0.00844 6.34c 7.41 6.49
8.43
- - 7.89 -
Av. 7.48
Copper Wire in Fe(N0a)a; DFe(N0a)s = 0.64 X 10-5 5.06d 2.12 53.6 0.007072 0.0105 6.38d 2.31 52.0 0.007124 0,0105 1.92b 1.78 53.6 0.008398 0,0105 1.88b 1.80 0.006404 0.0105 52.0
6.59
8.16
8s. Cm./Sec. 1.77 1.93 1.47 1.62
2.37 2.71 2.22 2.33
- 1.70 2.41
Q
b
With 0.3 M KC1 added. I n 5 minutes.
Av. 2.00 C In 2.5 minutes. d In 10 minutes.
The diffusivities, D, given in Table I11 were selected from measurements by Cathcart (3) as the most probable values for the reagent concentrations used here. The value for hydrochloric acid is the maximum reached in magnesium chloride solutions; that for ferric nitrate, the maximum in potassium chloride solutions. The diffusivity of acetic acid is little changed by salts.
Discussion In Figure 1 the dissolution rate coefficients of Table I11 are plotted against the diffusivities, D. So far as these and other published (8) and unpublished data obtained in this laboratory are concerned, the values can be represented, within experimental error, by straight lines. These lines do not, however, pass through the origin, as would be required by a constant film thickness, independent of D; nor do they pass through the points for the heat transfer coefficients. Two explanations are possible: (1) the extrapolated values of k a t D = 0 are rate coefficients which would be obtained with reagents of zero diffusivity whose transfer to the surface would be by convection alone (this is highly improbable); (2) the points really lie on an exponential curve which is nearly linear over the range of values plotted in Figure 1. That the rate coefficients, including the heat transfer points, can be represented by a simple exponential relation is shown in Figure 2, where k and h / c p are plotted on a loglog scale vs. D and K/cp. The data are reasonably well represented by straight Lines with a slope near 0.7. If the
17
values were plotted, then, against the 0.7 power of D and K / c p , they would lie on straight lines passing through the 0rigin.l This is in accord with the concept of a diffusion layer which increases in thickness with increase in the value of the diffusivity, suggested by King (8) from a consideration of the variation in the v component of turbulent motion with distance from the interface. It is also in reasonable accord with the Chilton-Colburn relations mentioned earlier, which also require an effective film thickness, increasing with the value of the diffusivity. The Chilton-Colburn relations may be put in the form, h/cp = c r ( K / c p ) 2 / 3 and k = crDZ/s where CY = jG '/' and is constant at a given flow rate, denP
(;>
sity, and viscosity.2
The exponent "3 has no theoretical derivation and seems somewhat too low for these and other similar systems. For example, when the rate coefficients of King (8) (obtained from experiments on the dissolution rates of metal and benzoic acid cylinders rotated with a peripheral speed of about 18,400 cm. per minute) are plotted against diffusivities of the reagents on a log-log scale, they lie on a straight line with a slope near 0.8. As shown in Figure 3, the rate coefficients plotted against the 0.83 power of the diffusivities lie on a straight line passing through the origin. This likewise agrees with a film thickness increasing as D increases. The decrease in heat transfer coefficients closely parallels the decrease in thermal conductivity of the sucrose solutions. In 1.5 M sucrose, 12 is 20 to 23 per cent less than in water, K is 24 per cent less. The accuracy of the data is insufficient to w a r r a n t a n y further conclusion. This is exactly a n a l o g o u s to the direct proportionality between k and D in dissolution rate e x p e r i m e n t s w h e n the v i s c o s i t y is increased by adding sucrose (8), and it indicates that the effective film thickness is p r a c tically independFIGURE 3. DISSOLUTION RATECOEFe n t of t h e v i s FICIENTS OF KING(8) us. 0.83 POWER c o s i t y . This is OF DIFFUSIVITIES probably true, however, only with the type of stirring used here, because of the manner in which the turbulent motion varies with the viscosity a t constant stirring speed in revolutions per minute. The effective film thickness for thin wires can be calculated from the equations: h = K / r In k = D / r In
- 21, r
r
r
- xd
~
The first was deduced by Langmuir for heat transfer from thin wires in gases ( I I ) , and the second was similarly derived 1 h/cp and K / c p have the same dimensions as, and are otherwise comparable with lo and D , respectively. Sinoe the specific heat and density of all the solutions used, except those oontaining sucrose, were practically unity, values of h and K are actually plotted in Figure 2. * Since for a flat surface xh = K / h and X d D / k . i t follows that x h
-
2 ( K / c p ) l / S and x d
=
1 (D)'/a;
-
i. e., the effective film thickness should
vary with the $ / I power of the diffuaivity. The d a t a presented here support this relation if the power is altered slightly.
INDUSTRIAL AND ENGINEERING CHEMISTRY
7 8.
from Fick's law of diffusion applied to this Case where zd is comparable in magnitude to r . The zinc wires used dissolve 2 to 2.5 times as fast, per Unit area, as flat disks 1.2 cm. in diameter, having only one face exposed to the solution a t the same distance from the stirrer. They also dissolve 20 to 25 per cent faster than metal cylinders of the same diameter as the stirrer and rotating at the same speed.
Nomenclature A = surface area of wire, sq. om. (in Table 111, average of initial and final values) Crcapent, M = concn. of reagent, gram moles/liter (unlessotherwise noted) c = specific heat, cal./gram, C. D = diffusivity of reagent, sq. cm./sec. C = rate of mass flow, grams/sq. cm. sec. h = heat transfer coefficient, cal./sq. cm. sec. ' C. j = Chilton-Colburn heat and mass transfer factors, dimensionless k = dissolution rate coefficient, cm./sec. K = thermal conductivity, cal./cm. sec. C. O
VOL. 29, NO. 1
r = rad_iugof yire, cm. (in Table 111, average of initial and
final values)
v = turbulent motion normal t o interface effectivefilm thickness in dissolution process x3 x h = effective film thickness in heat transfer a = constant with dimensions of I/sec.
1$~~&f'c. Literature Cited (1) Bates, IND. ENQ.CHEM.,28, 494 (1936). (2) Brunner, 2.physik. Chem., 47, 56 (1904). (3) Cathcart, W. H., Thesis, New York University, 1936. (4) Chilton and Colburn, IND.ENQ.CHEM., 26, 1183 (1934). (6) Colburn, Ibid., 22, 967 (1930). (6) Colburn, Trans. Am. I n s t . Chem. Engrs., 29, 174 (19331. (7) Fage and Townend, Proc. Bop. ASOC.(London), A135, 656 ( 1,932). (8) King, J. A m . Chem. S O C . . 828 ' ~ ~(1935). , (9) King and Cathoart, Ibid., 58, 1639 (1936). (10) King and Weidenhammer, Ibid., 58, 602 (1936). (11) Langmuir, Phys. Reo., 34, 401 (1912). (12) Nernst, 2. physik. Chem., 47, 52 (1904). (13) Prandtl, Physik. Z.,11, 1072 (1910). RH~CH~IVED November 12, 1935.
Controlling Gin Flavor
F
ROM a technical point of view, the beverage alcohol or distilling industry in the United States has been stagnant for a t least thirty years. During the years of prohibition the industry in the United States was legally dead and, for fifteen years prior to that time, large inventories, tradition, and fear of governmental interference and prohibition were not conducive to research and development. During this thirty-year interval other industries experienced marked advancement along all technical lines. Aca-
MECHANICAL CONTROLLINQ AND RECORDING EQUIPMENT ON
A
GIN STILL
H. F. WILLKIE, C. S. BORUFF,
AND
DARRELL ALTHAUSEN Hiram Walker and Sons, Inc., Peoria, Ill.
Gin production in the past has been characterized by lack of control over many of the important variables such as quality of spirits, quantity and quality of flavor in the various botanicals used, variable types and methods of operating the stills, etc. Critical study of these variables disclosed valuable information which led to standardization of spirits and operations which, with proper selection of botanicals and regulation of the quantity of each ingredient used in the formula in accordance with its flavor value, now permits the production of gin under technical control which guarantees uniformity and quality of final product.
'
demic and pure research findings had been adopted by the majority of industries, and a technic of industrial research had been developed that had been accepted by practically all plants. This acceptance led to new products, modernization, improvement in quality and uniformity, and greater profits to the management. The liquor industry, which possesses a heritage that dates back beyond recorded history, has always shunned research and development, accepting only those advances that were imperative, such as the classical work of Pasteur, the development of the Coffey still, and a few others. The industry and the consuming public were again robbed of the benefits of industrial, chemical, and biological advances of the age by re-establishing in most plants the old generation of distillers in power following repeal. The industry as a
