AGITATION Heat and Mass Transfer Coefficients in Liquid-Solid Systems' ARTHUR W. HIXSON AND SIDNEY J. BAUM Columbia University, New York, N. Y.
A n equation is derived, expressing the rate of melting of the solid in a two-phase, single-component, liquid-solid system, based on the rate of heat transfer at the liquid-solid interface. This equation resembles the approximate, integrated form of the cube root law, and the surface coefficient of heat transfer, h, is comparable to the dissolution constant, K . The equation correlating heat transfer data for several liquids in a series of geometrically similar agitators is similar in form to the expression correlating data on mass transfer in the same equipment, and an analogy between heat and mass transfer in liquid-solid agitation systems is indicated. Applications of this analogy to studies of reaction kinetics are suggested. A method for correlating heat transfer data in jacketed and coil-heated agitation vessels is proposed and confirmed by the data of Icing and Howard (7) on heat transfer from a platinum wire in an agitated vessel.
P
REVIOUS investigators of transfer processes, interested in the estimation of coefficients t o aid in the design of engineering equipment, have confined themselves to studies of gas-liquid and gas-solid systems. Few have attempted similar studies of the equally important liquidsolid system. A knowledge of transfer rates for this phase combination is of extreme importance in such widely used operatims as leaching, dissolution, crystallization, and adsorption. The first paper of this series (6) showed that known concepts of chemical engineering could be applied t o a study of mass transfer in liquid-solid systems where the solid phase is suspended in the liquid by the action of an agitator. This work is extended in the present paper to a study of heat transfer in liquid-solid agitation systems t o show that an analogy exists between heat and mass transfer for this type of phase combination. Previously proposed analogies between heat and mass transfer have been confirmed mainly by data on liquid-gas systems (1, 2, 5, IO). The value of such analogies lies in the ability t o predict coefficients for transfer of material by diffusion on the basis of the usually more abundant heat transfer data, and to estimate coefficients when the processes of heat and mass transfer are occurring simultaneously. 1
T h e firat article in this serics appeared in April, 1941 (6).
Theoretical Considerations I n 1897 Osborne Reynolds, as quoted by Stanton ( I S ) , proposed an analogy between heat transfer and fluid friction in turbulent flow, based on the ratio of the fluid friction to the momentum of the moving stream. This led to an expression which fitted experimental data on the flow of heat t o a gas in a pipe within 10 per cent. Sherwood (IO) applied a similar line of reasoning t o derive an equation for mass transfer to a fluid flowing in a pipe. Prandtl (9) modified Reynolds' equation for heat transfer by separating the flow in a conduit into two sections-a laminar film in which heat flow was treated as a process of pure conduction, and an eddy zone t o which the Reynolds analogy was applied separately. Colburn (8) used Prandtl's modification of the Reynolds analogy t o derive an equation for mass transfer in gas-liquid systems. Prandtl's equation for heat transfer was found to correlate data rather poorly, and Colburn (4) later modified this equation empirically by introducing the dimensionless group, cp,/lc, known as the Prandtl group, which resulted in the semitheoretical equation:
The original Colburn equation (8) for mass transfer was later modified by Chilton and Colburn (2) by introducing the dimensionless group, pL/pD, which is equivalent to the Prandtl group in heat transfer. For use in a liquid, the recommended equation can be written (7) :
Application of Equations 1 or 2, or that proposed by Sherwood based on the Reynolds analogy, t o a study of liquidsolid agitation systems is impossible a t present since a knowledge of pressure drops, friction factors, and mass velocities is involved, and these variables cannot be determined or even defined for a liquid undergoing agitation. However, the well-known Nusselt equation, derived from dimensional analysis, is available for correlating data on heat transfer. I n this type of equation a velocity term appears in only one dimensionless group:
The various dimensionless groups are usually expressed as power functions. For heat transfer in conduits, McAdams (8)recommends:
For correlation of their data on the rate of vaporization of water and several organic liquids from a wetted-wall column into a turbulent air stream, Gilliland and Sherwood (6) use an equation which is similar to 4 but in which the Prandtl group is replaced by the equivalent group, p l p D :
1433
INDUSTRIAL AND ENGINEERING CHEMISTRY
1434
Vol. 33, No. 11
Solvingfor T , The first paper (6) showed that data on mass transfer in a series of dimensionally similar agitators could be correlated by a similar expression when the Reynolds number for a circular conduit shown in Equations 3, 4, and 5 , is replaced by the equivalent group for an agitator, ndZp/p. For mass transfer in the turbulent zone for the turbine type of agitator, the data were correlated by:
T = T o - L(W0 - W ) Mc
Since A = a W 2 / 3Equation , 7 can be written
or
(9)
urhere m = TO- T p - ~ W O
P = L/Mc For heat transfer in the same equipment, it is proposed that the analogous equation be used:
Equation 9 is exactly similar in form to the rate equation for mass transfer previously derived ( 6 ):
I n comparing processes of heat and mass transfer in liquidsolid agitation systems, it is necessary t o have conditions for either process as nearly identical as possible. The mass transfer process involves the free suspension of soluble solid material in an agitated fluid. To study the heat transfer process, it is necessary to use freely suspended material whose dissolution is accompanied by an exchange of heat. This can most easily be achieved by using one-component, two-phase systems. The requirements for any component for test work are: cheapness, since comparatively large amounts are necessary; purity, since a sharp melting point is desirable; and a melting point fairly close t o room temperature to prevent excessive heat gain from external sources. The liquids which were found to comply with these requirements are water, acetic acid (melting at 16.6" C.), benzene (melting a t 5.8" C.), and nitrobenzene (melting a t 5.7" C.).
wheref
Derivation of Rate Expression I n the derivation of the rate expression for mass transfer in liquid-solid systems, it is assumed that equilibrium conditions are attained at the liquid-solid interface, and that the driving force is equal to the difference between the saturation concentration and the concentration in the bulk fluid. If a similar assumption is made for a single-component, two-phase system-namely, that equilibrium is attained a t the liquidsolid interface-it follows that the interfacial temperature must be the melting point of the solid, T,. Since the rate a t which the solid dissolves depends on the rate with which heat is transferred to the interface, and this in turn is dependent on the temperature gradient across the "effective film" between the liquid and solid, Newton's law for heat conduction can be written for this case as
2
=
RA(T
-
Tp)
(7)
This is similar to the equation used by Wilderman t o express the rate of solution of an ice cube in water (14). If experimental conditions are maintained so that the weight of solid that dissolves is small compared to the weight of liquid, and the initial temperature of the liquid is fairly close to the melting point of the solid, so that the heat gained by the liquid formed on melting is small compared t o the total heat exchanged, then essentially all the heat transferred is available for supplying the heat of fusion of the solid, L,and the differential heat balance is The total heat balance is approximately2 L(Wo - W ) = Mc(To
- T)
W,
- Wo
Integration of Equation 10 leads to the algebraically complicated form of the cube root law, and integration of Equation 9 leads t o a similar expression. The first paper ( 6 ) showed that starting with the differential form of the cube root law, dW - KAA,dO an approximate integrated form can be derived:
Wo
-W
=
KA,A,,B
(11)
(12)
where and
Acm
Aco
=
-
4 c
2.303 log
A,
which correlates the experimental data as well as the exact integrated form. By a similar line of reasoning, starting with the differential form of the equation for heat transfer, dQ = hA Ad@ (13) an approximate integrated form was developed in the same manner :
Wo - TV
h
= - AmAm@
L
where
(14) (15)
and
Am =
Ao
- A
2.303 log
2 A
(16)
The following limitations should be observed in using Equation 14: 1. The ratio of the weight of liquid t o the weight of solid that dissolves is large. 2. The heat gain from the surroundings - is small and taken into account. 3. The solid has a temperature approximately equal to its melting point at the start of the process. * I f the weight of t h e solid t h a t melts in a one-component syatem is included in the heat balance, then
(WO- W ) C ( r - TP) f L ( W 0
- w)= hfC(T0 - T )
B u t t h e term ( W o - W ) c ( T - T,) is kept small in comparison to t h e term L(Wo W ) in this investigation and can be omitted in t h e integration of Equation 9. An example of t h e error introduced in t h e total heat balance by omitting this t e r m appears in Table 11, where t h e heat gained by t h e liquid formed on melting in being raised t o t h e temperature of the bulk fluid is only (194/3860 X 100) = 5.0 per cent of the total heat transferred.
-
d Q = -LdW
=
November, 1941
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1435
4. The area of the solid can be expressed by A = C Y W ~ ' ~ mercury thermometer for recording tem eratures in a rate process, during the dissolution owing to the large heat capacity and resufting time lag of the instru5. NO splitting of soG:Zicles occurs. ment, a series of runs was made in which a multijunction fine-wire thermocouple was used as the temperature measuring device for comparison with the thermometer. The large surface and low Experimental heat ca acity of the thermocouples tend to minimize the time 26 B. and S. gage copper and constantan wires were lag. The series of dimensionally similar turbine agitators and the soldered together t o form ten hot and ten cold junctions. Both apparatus assembly described in the first paper (6) were used the hot and cold junctions were spread out so that the individual in the heat transfer measurements. The smaller vessels were couples would not interfere with one another. The cold juncinsulated somewhat by resting them on corks inside the next tions were immersed in a mixture of ice and water in a Dewar larger vessel and sealing the concentric o ening with cotton flask, and the hot junctions were suspended directly in the a itawaste. The 61.0-cm. vessel was uninsulatecf and the single run tion vessel. The electromotive force was measured wi& a in the plant-size (119-om.) vessel was made under a similar condiLeeds & Northrup type K potentiometer, using a type HS galtion. To minimize external heat effects further, the room vanometer which has a low eriod of oscillation (5 seconds). temperature was maintained as close t o the temperature of the The multijunction thermocou ye was calibrated against a Bureau run as possible. For runs in which acetic acid was used, a 15.2of Standards thermometer. b i t h this apparatus temperatures cm., flat-bottom, glass battery jar and a glass stirrer, built to could be estimated to better than 0.01' C. The series of runs conform to the dimensions of the corresponding metal stirrer, with the multijunction thermocouple was carried out in the were employed. 20.6-om. vessel with the system ice-water, and the results were The acetic acid, benzene, and nitrobenzene were frozen in compared with those obtained using the thermometer (Table I). ordinarv aluminum ice-cube travs in a refrigerator. These trays m;ere kept tightly covered to revent cond;nsation of any moisture on the chemical compoun$. Since the solid hase for these compounds is denser than the liquid phase, no otfer treatTABLEI. COMPARISON OF THE THERMOCOUPLE AND THERment was necessary. In some cases appreciable shrinkage took MOMETER AS TEMPERATURE MEASURINQ DBVICBS place when the liquid froze, but this was always confined to the Speed Initial upper surface where a holIow would form. When this ha pened, R. P. hi. Temp., C. Measuring Device h* the hollow was filled with the particuIar liquid and rezozen a 200 8.28 Thermometer 0.062 sufficient number of times until the resulting upper surface was 8.29 Thermocouple 0.062 250 7.95 Thermometer 0.068 flat and the final desired dimensions of the shape'were attained. 7.98 Thermocouple 0.066 For each of these compounds the equilibrium tem erature be300 8.18 Thermometer 0.074 tween liquid and solid was determined in an insuEted Dewar 8.11 Thermocouple 0.072 flask fitted with a thermometer and stirrer, For acetic acid, a * For calculation of h, see Table 11. temperature of 15.34' C. was found; for benzene, 5.00" C.; and for nitrobenzene, 5.70" C. Commercially available liquids were used, and the equilibrium temperatures of these differ It is evident that the thermometer and thermocouple give somewhat from the true melting point of the pure compounds. comparable results. Because of the ease of manipulation, the Since these equilibrium tem eratures re resent the interfacial thermometer was used throughout the remainder of the work. tem erature, they were u s e f i n the evafuation of the surface It was further found that the position of the thermometer in coe8cient of heat transfer, h. the agitation vessel had no effect on the recorded results. In The water was frozen around cylinders of brass or lead. The most runs it was suspended midway between the stirrer shaft metal cylinders were drilled with a '/az-inch hole along their and the vessel wall and the depth of immersion was approxiVertical axes, and suspended by a thread passed through these mately 3 inches. borrection was made for stem emergence in holes and knotted at one end. The thread was attached to a all runs. horizontal rod which rested on the ed es of the ice cube tray. The length of thread and the position of the su porting rod were Calculation of Heat Transfer Coefficient so adjusted that, when the water was frozen, t i e metal cylinder would be in the center of the solid shape. The weight of the From the time-temperature relation, the calculated surface metal cylinder was so chosen that the resulting solid shape had area, and available physical data, heat transfer coefficient an effective density greater than 1. F, was calculated for each run. A typical run and the method Runs were made with cubes measuring 2.70 and 3.18 cm. on of calculation are shown in Table 11. one- edge, with rectangular shapes measuring 3.18 X 3.18 X 5.56 om., and with cylinders having diameters and heights of 1.70 X 1.70,2.86 X 2.86, and 2.86 X 3.81 om., respectively. In the one run made with ice and waterin the 119-cm. vessel, 7.78 X 7.78 cm. cylinders COEFFICIENT TABLE 11. CALCUL.4TION O F THE HEATTRANSFER were used. With the exce tion of their effect on the (Results of experiment 68, rate of melting of solid bensene in liquid benzene: d 20.6 surface area, the sizes anzshapes had no apparent 6900 cc., weight of li uid at start = 6060 cm., speed = 300 r p m. volume of li uid influence on the results obtained. $,ma, sp. heat of bk;end = 0.396 cal.~gram[" C. melting point ofBensene = 5.00" C eat of fusion of benzene 30.3 cal./gram, weight Gf can = 1443 grams, sp. heat of can 2 In any particular run the can was filIed with a 0.11 cal./gram/o C room temp. = 13.0' C. Four 3.18 X 3.18 X 3.18 om. oubes used; depth of liquid equal to its diameter. Initial liquid initial wkght of oubes = 121.1 grams. do 243 eq. cm.a, 01 9.95b) temperatures below that of the room were usually Temp.d Corrected Am, Am h desirable, and the li uids were frequently precooled Time, Teomz:C, Cor:ection, Teomz;, Equa- EquaEqhaSeo. C. (Wo-W)* W A* tion 15 tion 16 tion 14 in the refrigerator. %he stirrer was started and allowed to come to speed. The trays were removed 121.1 243 0 ...... 11 00 .. 14 28 2... .,. ... 0.0244 .... 10 48 from the refrigerator and warmed slightly by stand9 2 . 6 204 30 10.13 - 0.0087 8.5 222 5.30 6 5 . 5 161 5 . 1 4 0.0272 60 5 5 . 6 201 0 . 0 1 8 9 . 7 8 9 . 8 0 ing at room temperature until the shapes slid easily 45.5 127 9.53 75.6 182 5.05 0.0277 -0.026 90 9.56 from their compartments. This allowed the solid 31.6 166 4.93 0.0276 -0.035 99 9.36 9.40 89.5 120 12.1 0.0280 to reach a temperature close to its melting point. 52 9.12 9.17 136 4.81 -0.052 109.0 180 108 4.73 -0.070 17 118.8 2.3 9.00 9.07 a40 The re uired number of particles was then rapidly 121.1 . . . ... ... ... 0.0293 8.97 9.05 -0.078 .... 270 weighej in a sieve resting on a large beaker so that any 300 9.05 . . .. .. ... ... . . . . . ... ... .... liquid formed during the weighing operation would 5 No material was lost in the weighing operation. drop into the beaker. The particles were then rapidly b Q &a =3 s 9.95. thrown into the agitated liquid and at the same time iimiz/a a stop watch was started. At various time intervals C CorreLttkd for & ; emergence and calibration. 10.48 .- 9.05 = 1.43' C. Heat lost by can = 1.43 X 0.11 X drop d Total tem the temperature of the liquid was recorded. Heat lost b y liquid benzene = BO60 x 1.43 X 0.396 3430 cal. Total 1443 230 For most of the temperature measurements a heat lost by liquid + can = 230 + 3430 = 3660 cal. Heat gained by solid on melting = thermometer graduated in tenths of a degree was 121.1 X 30.3 = 3670 cal. Heat gained by the liquid formed,on,melting = 121.1 X 0.396 X (9.05 - 5.00) = 194 cal. Total heat gained by solid and 1wmd formed = 190 + 3670 = used. I t was Calibrated against a Bureau of Stand3860 - 3660 = 200 cal. Benzene equiva3860 cal. Heat absorbed from surroundings ards thermometer and, with a calibration curve, lent of aan + liquid = 6060 + 0.11/0.396 X 1443 7 6460 grams. Assuming that the could be used to estimate temperatures to the nearest heat flow from external sources is a t a uniform rate during the entire run, the effect of this external heat flow in e-ach 30-sec. interval on the thermometer reading = -200/(9 X 6460 X 0.01 C. Because of the rapidity with which tem0.3j6) -0.0087" U. peratures were recorded, particularly at the start of (Wo- W )is based on the total heat balanae; (Wa- W )L30.3 + 0.396(T 5.00)l = a run, any temperature reading had a probable ac6460 X. 0.396(!Z'0 T)or WO- W 2560 10.48 T)/30.3 + 0.396(T - 5.00). * A IS based on A uWe/a 9.95W253. curacy of t0.02O C. Since an objection may be raised to the use of a
&.
O
-
-
-
E
-
I
-
-
E
-
E
O
-
-
-
-
-
I N D U S T R I A L A::-. - N G I N E E R I N G C H E M I S T R Y 3
SYSTEM Water
2
V E S S E L SYMBOL 15,2 20.6 X 26.0
+
359
457 Benzene
Id
Acetic A c i d qitrobenzene
6 1.0 119 2 0.6
2 6.0 15.2 260
Vol. 33, No. 11
I
0
v v
6 0V
9 8
7 6
I
I
l
l
I
I
I
E q u a t i o n of solid line:
5 4
Eauation o f b r o k e n line: nJp 0.62 LO'S =D Q.I60(T)
3
(
2 5 6 78QlOs
2
3
4
5
Re-&
FIGURE 1.
CORRELATION O F
6
7 8glO6
Correlation of Results The dimensionless groups involved in Equation 6A were calculated from the experimental results and available physical data. Specific heat and viscosity data for all liquids were obtained from the International Critical Tables. The thermal conductivity of water was taken from the same source since this agrees with the more recently determined values. The thermal conductivity of benzene is based on an average line drawn through the data of Smith (12) and Shiba (11). The thermal conductivity of acetic acid is based on the single value a t 20' C. given in International Critical Tables. The thermal conductivity of nitrobenzene is based on the average of the values reported in the same source and by Shiba.
3
P
HEATTRANSFER DATAI N
Omitting the first and last calculated values for h, the mean value is 0.0276 and the average deviation is 0.8 per cent. This constancy is some indication of the validity of the rate equation and the experimental method. The heat transfer coefficient, calculated from the same data but based on intermittent values of A , and A, (i. e., the average from the 30and 60-second readings, the average from the 60- and 90second readings, etc.), has an average value of 0.0291 with an average deviation of 3.2 per cent. This value is shown in Table 111. I n most runs h remained substantially constant (if the value for the first time interval is not considered) until 80 per cent or more of the solid was melted, and then usually increased. A similar phenomenon was noted in the calculation of the dissolution constant from data on mass transfer. This may have been due to rounding off of corners and edges during the runs, in which case the area is not expressed accurately . values of h shown in Table I11 are based upon as C X W * / ~All the average value for the melting of the first 80 per cent of the solid material.
2
TURBINE-'rYPQ ACITATOKS
For correlation of hest transfer data, the function
(3)' is usually plotted against the Reynolds number.
(?)I The
value for y recommended by McAdams (8) is 0.4. I n this investigation a value of 0.5 is used to give a correlation comparable to the mass transfer data correlation (6) where an exponent of 0.5 was used on the group, ,uL/pD. Choice of 0.4 in place of 0.5 makes little difference in the grouping of the points for heat transfer. The experimental values are shown in Figure 1 where a single straight line correlates all points with an average deviation of less than 10 per cent. The equation of the solid line shown is
This correlation includes data for fluids having a threefold range in viscosity, a fourfold range in thermal conductivity, and a threefold range in specific heat. The size factor d varied eightfold, and the corresponding range in volumes is five hundred fold. The speed range is 160 to 450 r. p. m.
Discussion The similarity between Equation 17 for heat transfer and Equation 6 for mass transfer is evident. Since the Reynolds number appears t o almost the same power in both equations, it can be eliminated and the resulting equation simplified to K = 0.774h
(E)"'
This equation can be used t o predict mass transfer coefficients for agitators on the basis of heat transfer coefficients measured in the same apparatus. It is problematical whether the
I N D U S T R I A L A N D E N G I N E E R I N G C H E - I”,TI;Y
November, 1941
1437
L
DATAIN GEOMETRICALLY SIMILAR TURBINE AGTATORS(No BAFFLES) TABLE 111. HEATTRANSFER System Water
45.7
61.0
0
r
E
0.095 0.064 0.074 0.076 0.085 0.093 0.069 0.074 0.084 0.085 0.088
2.7 6.9
0.0163 0.0143 0.0148 0 0138 0.0142 0.0147 0.0141 0.0143 0.0146 0.0148 0.0150
1.00 1.00 1.00
1.00
4.17 6.00 6.83 6.67 7.50 4.17 5.00 6.83 6.67 7.50 5.00 3.33 6.67 4.17 3.33 5.00 6.67 3.33 5.00 4.17 4.17 5.00 6.83 6.67 7.50 4.58 4.17 5.00 5.83 6.67 4.17 6.00 6.83 6.67
0.078 0,062 0.086 0.068 0.062 0.076 0.087 0.062 0.072 0.066 0.070 0.080 0.082 0.096 0.096 0.074 0.069 0.076 0.078 0.083 0.068 0.074 0.081 0.086
7.7 7.9 7.6
1.00 1.00 1.00 1.00
6.0
0.0139 0.0138 0.0140 0.0140 0.0157 0.0155 0.0154 0.0142 0.0143 0.0143 0.0146 0.0147 0.0150 0.0149 0.0151 0.0153 0.0146 0.0150 0.0146 0.0150 0.0140 0.0142 0.0144 0.0148
1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00
3.33 5.00 5.83 2.67 6.67 3.33 4.17 5.00 3.33 4.17 5.41 6.67 4.17 5.83
0.066
6.4 5.6 6.0 7.0 6.7 14.4 14.0 14.2 6.6 6.5 6.2 5.8 8.4 8.3
0.0150 0.0149 0.0147 0.0143 0.0144 0.0116 0.0117 0.0116 0.0146 0.0146 0.0146 0.0147 0.0137 0.0138
1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00
3.33 4.17 5.00 6.67 4.17 5.41 6.67 4.17 5.83
0.081 0.098 0,100 0.112 0.082 0.091 0.100 0.086 0.102
6.0
1.00
1.00
5.9 7.7 7.9 7.3 6.9 6.6 6.6 6.7
0.0148 0.0148 0.0140 0.0139 0.0142 0.0143 0.0145 0.0145 0.0145
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00
3.33 5.00 4.17
0.087 0.100 0.090
5.8 9.1
6.6
0.0148 0.0134 0.0145
1.00 1.00 1.00
3.33 5.00 4.17
0.096 0.104 0.098
6.0 5.0 8.5
0.0152 0.0152 0.0137
0.084 0.086 0.061 0.089 0.067 0.069 0.078 0.071 0.084 0.091 0.103 0.079 0.084
6.0
8.3 7.3
6.1 ’
7.4 6.9 6.4 6.0 5.5
7.6
3.9 4.2 4.3 7.24 6.9a 6.9a 6.3 6.8 5.4 5.6 6.2 4.8 6.4 5.5 6.4 8.4 7.8 7.2 6.7
I
1.00
1.00 1.00 1.00 1.00 1.00
1.00 1.00
1.00 1.00
1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00
1.00
1.00 1.00 1.00 1.00 1.00 -1.00 1.00
1.00
I k
hd k
(,)”~
*P P
0.00135 0.00135 0.00135 0.00135 n.00135 0 00135 0 00135 0.on135 0 00135 0.00135 0.00135
12.1 10.6 11.0 10.2 10.5 10.9 10.4 10.6 10.8 11.0 11.1
1070 720 832 845 956 1050 777 833 946 956 991
308 221 251 266 295 318 241 256 288 289 297
9.62 X 6.76 7.80 9.75 1.09 x 1.18 6.84 x 8.09 9.21 1.04 X 1.16
0.00135 0.00135 0.00135 0.00135 0.00185 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00135 0.00136
10.3 10.2 10.4 10.4 11.6 11.5 11.4 10.5 10.6 10.6 10.8 10.9 11.1 11.0 11.2 11.3 10.8 11.1 10.8 10.7 11.0
1190 946 1310 1040 946 1160 1330 946 1100 1010 1070 1220 1250 1470 1470 1130 1050 1160 I190 1270 1040 1130 1240 1310
370 296 406 323 278 339 393 293 338 310 325 370 375 443 440 336 319 348 362 381 323 349 379 395
1.63 1.02 2.02 1.27 9.01 x 1.37 X 1.84 9.95 x 1.48 X 1.24 1.22 1.44 1.65 1.90 2.11 1.27 1.21 1.41 1.69 1.89 1.27 1.49 1.72 1.92
1270 1620 1660 1180 1710 1260 1300 1470 1370 1620 1750 1980 1510 I610
380 487 502 363 623 435 445 506 419 495 532 600 475 504
1.50 2.27 2.68 1.26 3.13 1.94 2.41 2.91 1.55 X 106 1.94
0.00135 0.00135 0.00135 0.00135 0.00136 0.00135 0.00135 0.00135 0.00135
11.1 10.4
10.5
104
IO’ 104 106
104 106 104 10’
1 .oo 1.00 1.00
0.00135 0.00135 0.00135 0.00135 0.00135 0.00138 0.00138 0.00138 0.00136 0.00136 0.00135 0.00135 0.00136
1.00 1.00
0.00136
11.1 11.0 10.9 10.6 10.7 8.4 8.5 8.4 10.7 10.7 10.8 10.9 10.1 10.2
1.00
0.00135 0.00135 0.00135 0.00135 0.00136 0.00135 0.00135 0.00135 0.00135
11.0 11.0 10.4 10.3 10.5 10.6 10.7 10.7 10.7
2160 2610 2660 2980 2180 2420 2660 2290 2710
651 787 825 930 672 744 812 700 829
2.90 3.62 4.60 6.18 3.78 4.39 5.94 3.71 5.19
1 .oo 1.00 1.00
0.00135 0.00135 0.00135
11.0 9.9 f0.7
2950 3390 3050
890 1080 932
4.70 7.82 6.01
1.00
1.00
1.00 1.00
1.00
1.00‘
0.00135 0.00135 0.00135
11.3 11.3 10.2
4330 4700 4420
1290 1400 1290
8.16 1.22 x 108 1.13 3.55
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00
2.50 3.07 2.06 2.85
3.38
0.087
8.6
0.0136
1.00
1.00
0.00136
10.0
7620
2410
20.6
5.83 3.33 4.17 5.00 6.83 6.67 4.17 5.83
0.036 0.021 0.026 0.029 0.031 0.034 0.027 0.030
11.4 11.6 10.6 9.6 10.9 9.7 10.8 10.1
0.0074 0.0074 0.0076 0.0076 0.0074 0.0076 0.0074 0.0076
0.878 0.878 0.878 0.878 0.878 0.878 0.878 0.878
0.307 0.397 0.396 0.395 0.396 0.395 0,396 0.395
0.00041 0.00041 0.00041 0.00041 0.00041 0.00041 0.00041 0.00041
7.2 7.2 7.2 7.3 7.2 7.3 7.2 7.3
1810 1050 1310 1460 1560 1710 1360 1510
676 391 489 540 581 632 506 559
2.94 x 10’ 1.68 2.08 2.46 x 105 2.92 3.28 2.09 2.88
26.0
4.25 6.00 3.33 5.83
0.027 0.033 0.026 0.034
14.0 13.5 13.0 12.5
0.0071 0.0072 0.0072 0.0073
0.878 0.878 0.878 0.878
0.400 0.399 0.399 0.398
0.00040 0.00040 0.00040 0.00040
7.1 7.2 7.2 7.3
1750 2140 1690 2210
656 799 630 818
3.54 4.13 2.74 4:73
16.2
8.33 4.17 5.00 5.83 6.67 7.50 4.17 5.00 5.83 6.67
0.024 0.027 0.032 0.035 0.037 0.042 0.032 0.034 0.037 0.043
23.0 21.3 22.8 21.7 20.5 19.6 22.8 21.8 20.9 20.2
0.0120 0.0123 0.0120 0.0123 0.0124 0.0126 0.0120 0.0122 0.0124 0.0125
1.05 1.05 1.06 1.05 1.05 1.05 1.05 1.05 1.05 1.05
0.490 0.488 0,490 0.489 0.487 0.487 0.490 0.489 0.488 0.487
0.00044 0.00044 0.00044 0.00044 0.00044 0.00044 0.00044 0.00044 0.00044 0.00044
13.4 13.6 13.4 13.7 13.7 14.0 13.4 13.6 13.8 13.8
829 932 1110 1210 1280 1450 1110 1170 1280 1480
226 253 303 327 346 387 303 317 344 400
8.20 6.73 x 104
4.17 5.00 6.83
0.025 0.024 0.026
13.5 13.4 12.9
0.0232 0.0232 0.0234
1.20 1.20 1.20
0.354 0.354 0.354
0.00035 0.00035 0.00035
23.5 23.5 23.8
1860 1780 1930
383 367 396
1.46 1.75 2.03
119
Nitrobenzene
P
6.67
35.9
Acetic acid
II
16.2
26.0
Benaene
TW.
n
20.6
t
h
d
26.0
Temperature measured with multijunction thermocouple.
1.01 x 105 1.14 1.30 1.44 8.45 x 104 9.90 1.14 X 105 1.30
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
1438
Vol. 33, No. 11
ance, the temperature coefficient of resistance having previously been measured. The dimensionless groups involved in Equation 6A were calculated from the data of King and Howard, and plotted as against
(?),/($)0'4
($)
in Figure 2. The points for water and a series of sucrose solutions, ranging in concentrations up to 1.5 moles per liter, are correlated by a line having the equation, 2
3
4
5
6
7
0
9
I o4
2
3
4
OF HEATTRANSFER DATAOF KINQAND HOWARD (7) FIGURE 2. CORRELATION
same constant (0.774) holds for other designs, but it is likely that the exponent on the group in parenthesis will be the same since this group involves physical properties only. For prediction of mass transfer coefficients from heat transfer data, Chilton and Colburn (2) recommend that j h us. Re for heat transfer be used to predict j , for diffusion through gas films for similar equipment. In effect this means that at ' equal Reynolds numbers, j h = j d . Using this equality, Equations 1and 2 reduce to
The value 0.4 on the Prandtl group gave the best correlation of the experimental data. Jacketed and coil-heated vessels with stirrers find widespread use in the chemical and food industries. It is likely that the heat transfer coefficients obtained in these vessels can be correlated by equations similar to 21. A few experimental points over a range of impeller speeds with water would serve to establish an experimental curve for any apparatus which could be used to indicate the expected coefficients for other fluids. Or here again, laboratory models could be used to predict the characteristics of large-scale equipment.
Summary in which the exponents agree fairly well with Equation 18. By rounding off the exponent on the Reynolds number in Equation 5 to a value of 0.8 and eliminating this group from Equations 4 and 5, we obtain
in which the exponents are in better agreement with Equation 18than those in 19. Snalogies of this type would be of help in interpreting the kinetics of chemical reactions carried out in the presence of a heterogeneous catalyst. For example, in the hydrogenation of vegetable oils to form edible fats, a solid catalyst is suspended in the oil by the action of an agitator. Knowledge of the heat of reaction and the heat transfer coefficient between a solid and liquid under similar conditions of turbulence would enable calculation of the true surface temperature of the catalyst at the liquid-solid interface. Or for the case where a chemical reaction is known to depend on the rate of diffusion of components to and from a n interface, data on heat transfer from solids to liquids under similar conditions of turbulence could be used to predict the coefficients of mass transfer and the resulting reaction rate. Equation 6A can also be used to correlate heat transfer data in agitated vessels when the heat source is fixed in place. This is illustrated b y the results of King and Howard ( 7 ) who measured the rate of heat flow from a n electrically heated platinum wire to water and sucrose solutions. The wire was centered in a hard-rubber stirring cylinder so that every part of the wire's surface was exactly 1 cm. from the inner surface of the cylinder. Four grooves (2 X 2 mm.) were cut lengthwise on the stirrer surface and the length of the stirrer was 8.3 em. Both the wire and the cylinder were submerged in the liquid contained in a one-liter beaker. The mire temperature was determined by measuring its resist-
1. The various theoretical and semitheoretical equations for correlating mass transfer data have been reviewed, and a lack of experimental data to confirm these for liquid-solid systems has been indicated. 2. An equation, expressing the rate of melting of the solid in a two-phase, single-component, liquid-solid system, has been derived and experimentally verified. 3. The results on heat transfer between several liquids and the corresponding solids in a series of geometrically similar turbine agitators have been correlated by a modified form of the Nusselt-type equation. 4. The equation correlating heat transfer data has been compared to the equation correlating data on mass transfer in the same equipment, and a n analogy between heat and mass transfer in liquid-solid agitation systems has been indicated. 5 . B method for correlating heat transfer data in jacketed and coil-heated agitation veksels has been proposed and applied to available experimental data on heat transfer from a platinum wire to agitated liquids.
Nomenclature a = A = A, = AO = b
c
C C,
d dp d,
D e,j G h
= = = = = = = = = = =
constant surface area, sq. em. av. surface area, sq. om. initial surface area, sq. em. constant specific heat, cal./gram/' C. concentration, grams solute/cc. solvent saturation concentration, grams solute/cc. solvent vessel diameter, cm. pipe diameter, cm. stirrer diameter, cm. diffusivity, sq. cm./sec. constants mass rate of flow, grams/sec./sq. cm. surface coefficient of heat transfer = k / X h , cal./(sec.) (sq. crn.)(' C.)
November, 1941 jh
INDUSTRIAL AND ENGINEERING CHEMISTRY
= Chilton-Colburn heat transfer factor defined by Equation l
j, = Chilton-Colburn mass transfer factor defined by Equation 2 k = thermal conductivit cal./(sec.) (cm.)(O C.) K = mass transfer coe&ient, grams/(sec.) (sq. cm.) (unit concn. change in grams/cc.) L = heat of fusion, cal./gram - constant weight of liquid, grams n = rotational speed of stirrer, r. p. s. Q = total heat transferred, cal. Re = Reynolds number = (dpup/p) for pipes or (n#p/p) for agitation systems T = temperature, O C. To = initial temperature, O C. T, = melting point of solid, C . u = fluid velocity, cm./sec. volume of liquid, cc. weight of undissolved solid at time 8, grams = initial weight of solid, grams = weight of solid required for saturation, grams x h = effective film thickness for heat transfer, cm. = effective film thickness for mass transfer, cm. cy, 8, y = constants A = temperature difference at time e, C. Am = log mean temperature difference, C. A0 = initial temperature difference, ’ C . Ac = concentration driving force = (C8- C), grams/cc. Aem = log mean concentration driving force, grams/cc.
1439
ACo = initial concentration driving force, grams/cc. a, a’ = mathematical symbols representing- “function of” = density of liquid, gramslcc. = time, sec. viscosity of bulk fluid, grams/(sec.)(cm.) = av. viscosity of film, grams/(sec.)(cm.)
% =
O
v = w wo = w, x,
O
Literature Cited Arnold, J. H., Physics, 4, 255 (1933). Chilton, T. H., and Colburn, A. P.,IND.ENG.CREM..26. 1183 (1934).
Colburn, A. P., Ibid., 22, 967 (1930). Colburn, A. P., Trans. A m . Inst. Chem. Engrs., 29, 174 (1933). Gilliland, E. R., and Sherwood, T. K., IND.ENQ.CHEM.,26, 616 (1934).
Hixson, A. W., and Baum, S. J., Ibid., 33,478 (1941). King, C. V., and Howard, P. L., Ibid., 29, 75 (1937). MoAdams, W. H., “Heat Transmission”, p. 169, New York, McGraw-Hill Book Co., 1934. Prandtl, L., 2. Physik, 11, 1072 (1910). Sherwood, T. K., “Absorption and Extraction”, pp, 32, 37, New York, McGraw-Hill Book Co., 1937. Shiba, Sci. Papers Inst. Phys. Chem. Research (Tokyo), 16, 205 (1931).
Smith, J. F. D., IND. ENQ.CHDM.,22, 1246 (1930). Stanton, T. E., Trans. Roy. SOC.(London), A190, 67 (1897). Wilderman, M., Brit. Assoc. Advancement Sci. Rept., 1896, 751; Phil. Mae., [6] 2, 50 (1901); [6] 4, 271, 468 (1902).
DISTILLATION By Giovanni Stradano (15361605)
No.
131 in the Berolzheimer series of Alchemical and Historical Reproductions comes to us through the courtesy of Prof. E. C. Watson of the California
Institute of Technology, being the third plate of chemical import in his very rare “Nova Reperta”, which he placed at our disposal. The original painting was one of a series made by Stradano especially for this publication. Its present location is not known. Thereupon the three Brothers Galle of Antwerp engraved the plates, this particular one being by Joannes Galle. As in the other two plates (Nos. 98 and 102 in the series), Stradano places his emphasis on the center of the picture. The use of the bent sapling to act as a spring to raise the pestle is ingenious for such an early date. Note particularly the differences between the several stills shown and those which appear in almost all of Teniers’ alchemical paintings, representing a difference of about fifty years. D. D. BEROLZHEINER 50 East 41st Street New York, N. Y. The lists of reproduotiqna and directions for obtaining copies appear 88 follows: 1 to 96, January, 1939, issue, page 124; 97 to 120, January, 1941, page 114. An additional reproduction appears each month.
