D = NH, decomposition, mole % L = N H ? leakage (unreacted), mole % R i = C H 4 / N H 3mole ratio Rr = air/ (CH, + N H ? ) mole ratio S = variance Y = ultimate yield = HCN/[(,UH3), - (NH7)"I, mole L& 2 = reactor performance, C , D , L , or Y u = standard error of estimate i, 0 = feed inlet and reactor outlet
the decreasing of N H s in the feed. NH3 decomposition will also decrease to zero as the limit. These extremities become more pronounced when XH3/CH4 becomes smaller, as shown in broken lines a t the right of Figure 2 . Since the above extrapolations are based on R1 = C H 4 / NH3 from 0.7 to 1.7, the two additional sets of data of R I = 0.6 and 2.0 a t Rz = air/(CH, + N H 3 ) = 3.09 can be regarded as confirmations of the correct trends of the above extrapolations. Acknowledgment
The author thanks Fred Applegath for many suggestions and computer work. Nomenclature an, a l
---- as C
Literature Cited
Ostle, B., "Statistics in Research," 2nd ed., pp. 190-1, Iowa State University Press, Ames, Iowa, 1964. Pan, B. Y. K., Roth, R . G., IND. ENG.CHEM.PROCESS DESIGNDEVELOP.7, 1-53 (1968).
= coefficientsof polynomials = once-through yield = HCi%/(i%H3),, mole %
RECEIVED for review June 20, 1968 ACCEPTED December 16, 1968
HEAT TRANSFER BETWEEN COILS AND NON-NEWTONIAN FLUIDS
WITH PROPELLER AGITATION A .
H .
P .
S K E L L A N D
A N D
G .
R .
D I M M I C K '
University of Notre Dame, Notre Dame, Ind. 46556
THEdirect involvement in non-Xewtonian products and processing in the chemical process industries of the United States probably exceeds 12 to 15 billion dollars annually (Brasie, 1964). I t is therefore somewhat remarkable that so few papers have appeared in established journals on heat transfer to or from non-Newtonian materials in agitated vessels. Carreau et al. (1966) studied heating and cooling of power law pseudoplastic non-Newtonian fluids in a jacketed vessel agitated by a four-bladed, 45" pitched turbine. They correlated about 109 data points with a mean deviation of 19.3% by the equation
D2"L h,Di= 1.474 K k
[
"'(=)I
n
"
for D J D T = 315; 0.34 5 n i 0.63; 100 2 "NRegen" - 5000. The differential viscosity pd = d.r,,g,/d ( d u l d y ) , 5 and subscript 03 denotes high shear rates. These authors expressed doubts about the validity of their Prandtl num. ber evaluated using p d - . Hagedorn and Salamone (1967) correlated extensive heating and cooling data, again specifically for power law non-Sewtonian liquids in a jacketed vessel using, in turn, anchor, paddle, propeller, and turbine agitation. They Present address, Atomic Energy of Canada, Ltd., Chalk River, Ontario, Canada.
achieved correlation with a mean deviation of 20% using the equation
for 1.56 5 D f D r 5 3.5; 0.36 5 n 5 1.0; 35 5 D ' N ? - " p / K 5 680,000. The constants C' and a to g were tabulated for the various impeller types used. The quantities K and n appearing in Equations 1 and 2 are those used to characterize the well-known empirical Ostwald-deWaele or power law model:
No studies on heat transfer between coils and nonKewtonian fluids have yet appeared. I n addition to being confined to jacketed vessels, Equations 1 and 2 are further restricted to non-Newtonian materials of the power law type. The present study was therefore undertaken to provide data on coil heat transfer and to obtain a generalized correlation applicable to all time-independent nonNewtonian fluids, thereby eliminating restrictions associated with particular empirical relationships such as the Bingham plastic, Ellis, or power law models. VOL. 8 NO. 2 A P R I L 1 9 6 9
267
A design equation has been developed by correlating the data from 123 runs with a mean deviation of 35.5% for heat transfer between coils and nonNewtonian fluids with propeller agitation. Heating and cooling data are included, with both upthrusting and downthrusting agitation for substantial ranges of impeller speed, equipment geometry, and fluid physical properties. Although experimental data were confined to power l a w materials, the correlation should not be restricted to any empirical relationship between shear stress and shear rate, but should apply to all time-independent non-Newtonian fluids. The equation reduces to the Newtonian form recommended by Uhl with respect to the quantitative effects of six key variables. Scale-up relationships are presented for achieving quantitative duplication of various patterns of heat transfer in two vessels of different size.
To allow for variation in apparent viscosity, g c T y x / ( d u / d y ) , with shear rate and location in an agitated system, Metzner and Otto (1957) and Magnusson (1952) identified an average shear rate ( d u / d r ) Ain the vessel. The apparent viscosity PA corresponding to ( d u / d r ) aequals the viscosity of that Newtonian fluid which would show exactly the same power consumption for agitation under identical conditions, a t least in laminar flow. I t was found empirically that
where hs has been tabulated for a wide variety of impellers and agitation systems (Skelland, p. 338, 1967) ( h s = 10 for propellers). Thus for a proposed impeller speed, N, the corresponding p~ is evaluated via use of Equation 4 and the flow curve (Skelland, pp. 5, 334, 1967) as PA = g,T/(du/dr)a= g,i/ksN. The Reynolds number D'Npl FA has been very successfully used for correlating power consumption in a diversity of agitated non-Newtonian systems, ensuring coincidence with the corresponding Kewtonian correlation in the laminar region. By analogy with the generalized Reynolds number for Newtonian and non-Newtonian flow in tubes, Metzner and Otto suggested that the following expression might constitute a generalized Reynolds number for agitation of power law fluids:
This is evidently the form used by Carreau et al. (1966) in Equation 1 and it is helpful to consider the proportionality between D2Np/paand NRe,gen as follows:
D2Np
=$
D2N2- "P
( 6nn+
(6)
PA
where $ is the proportionality constant
(7) For a power law fluid,
Substituting Equation 4 for ( d u / d r ) , 4 , TI
and combining Equations 7 and 8,
I t follows from Equations 6 and 9 that if plots of power number Np us. DLNp/pAfor power law fluids coincide with the appropriate line for Newtonian materials-at least under laminar conditions-plots of N p us. NRe,gen must deviate from this line by an amount dependent on $ and consequently on hs and n. I n contrast to D2Np/ p a , therefore, the "generalized Reynolds number" of Equation 5 will not correlate Np on a unique curve for a given type of impeller for all values of n. I t was accordingly decided not to correlate in terms of the Reynolds number of Equation 5, but instead to adopt an extension of the form proposed by Skelland (1967):
Such an expression should be generally applicable to all time-independent materials, since FA is not dependent upon any particular model. I n the special case of power law fluids, pA is given by Equation 8. The temperature change of the fluid in the vessel was of the order of 40°F. in the course of a run. A problem therefore arises with regard to selecting a representative average value of p A for a given run. All the liquids studied here turned out to be power law fluids. Furthermore, for a given fluid, n was effectively constant over the relevant temperature range and a plot of In K against temperature was linear with negative slope. Thus PA
PA
268
=
gcT
(du/dr)*
K ( d u i d r )1 (du/dr)a
K (du/dr)\-"
I & E C PROCESS D E S I G N A N D DEVELOPMENT
=
( h s N ) " - ' K = ( h s N ) n - exp l (A
-
BT)
where A and B are constants. The integrated mean over the run is then
PA
VARIABLE
SPEED
MOTOR
INSULATION
The average of the run is
FA
corresponding to the mean temperature
The ratio of the "true" mean given by Equation 12 is
pA
(Equation 11) to that
Borrowing from the experimental section of this paper, run 4 corresponds to the fluid with the highest K values (fluid 6), also TI = 72" F., T ?= 111"F. For fluid 6,
I
Figure 1. Diagram of apparatus
K at 86°F. = 1.19 lb., sec."-2 ft.-' K at 131"F. = 0.957 lb., secn- 'ft.-' n = 0.528 In K = A - B T , so that
A = 0.4339; B = 0.00485 Insertion in Equation 13 shows that p ~ , is~ within ~ . 2% of pa,, for these data. The apparent viscosity, p A , was accordingly evaluated a t the mean run temperature, ( T I+ T2)/2, throughout this work. T o detect any differences between the directions of agitation, 15% of the runs used upthrusting propellers and the rest were downthrusting.
INSULATION
BAFFLE
\
/
Equipment and Procedure The apparatus is shown in Figure 1. The vessel was an 18.25inch diameter cylindrical tank, with a flat bottom. Four lj/"inch wide vertical baffles were fixed to the vessel walls, equally spaced round the circumference. Insulation was provided by 2 inches of %-inch vermiculite. Agitation was provided by a Reeves variable-speed agitator, with the shaft along the axis of the vessel. The shaft diameter was l % f i inch. Three-bladed square pitch marine propellers of 3-, 6-, and 9-inch diameter were attached to the bottom of the shaft. The 6- and 9-inch propellers were 5.5 inches from the bottom of the tank. A special adapter was made to fit the 3-inch propeller to the shaft and this lowered the propeller to 4.5 inches from the bottom. Heating and cooling of the tank fluid were by helical coils of copper tubing, mounted concentrically with the shaft. Three coils, constructed of %-, %-, and %&-inch20 B.W.G. tubing, were used with a helix diameter to the center of the tubing of 12.25 inches. The separation between the coils was 2.25 inches and 7.75 turns were used. The lowest turn of the coils was 1.5 inches above the bottom of the tank a t its lowest point. The flow rate of the water was measured, using a rotameter calibrated for both hot and cold water. The tank temperature was measured by five 20 B.W.G. copperconstantan thermocouples (Figure 2), fixed in position by means of 16 B.W.G. galvanized iron wire, stretched between baffles. One
Figure 2. Top view of vessel showing thermocouple locations Thermocouple heights Above Tank Boiiom, Inches
3 4 5 7
a
11 10 1.5 16.5 5.5
VOL. 8 N O . 2 A P R I L 1 9 6 9
269
exit and two inlet copper-constantan thermocouples were used to measure the coil water temperatures. The eight thermocouples were connected to a Honeywell -0.05to +1.05-mv. range 8-point recorder, reading one point every 5 seconds. Readings greater than 1.05 mv. were obtained by using a potentiometer in conjunction with the recorder. The cold junctions for the thermocouples were in an ice bath. Cold water was taken from the main supply, and hot water was provided by a steam heat exchanger. Fluid rheological properties were determined using a specially constructed capillary tube viscometer (Figure 3), which consisted of a vertically mountea stainless steel tube 27 inches long and 0.060 inch in internal diameter. The tube had a drilled ,%-inch pipe plug silver-soldered to one end, and this was screwed into a fluid reservoir of approximately 250-cc. capacity. The top of the reservoir was detachable and was bolted down while under pressure. A rubber 0 ring provided the seal. A gas pressure of up to 60 p.s.i.g. could be applied to the reservoir. A mercury manometer was used to measure pressures up to 25 p.s.i.g. and a Bourdon-type gage above that. The reservoir pressure was provided by compressed nitrogen. The volumetric flow rate through the tube was measured, and temperature control of the fluid was achieved by using a water jacket connected to a constant temperature bath. The fluids used were neutralized aqueous solutions of Carbopol 934, supplied by the B. F. Goodrich Chemical Corp. Fluid properties and uses are given in Tables I and 11. The tube viscometer was operated with the water bath and jacket at the desired temperature and in accordance with usual procedures (Skelland, Chap. 2, 1967). The resulting rheological data were represented by plots of T~ = D P / 4 L us. 8V/D, where the frictional pressure drop through the viscometer tube was obtained from the expression (Skelland, p. 34, 1967)
g - 1.12 P V L (14) AP = P,- Pa+ p ( L + L') -
Table I. Physical Properties of Fluids
K gc,
Flu id
Lb.,Sec."/Sq Ft
Temp., F.
n
1
0.01432 0.01152 0.00215 0.00176 0.001496 0.000171 0.0001142 0.00725 0.0058 0.000296 0.000217 0.0370 0.0298 0.0169 0.0136 0.000574 0.000516 0.000468 0.000444 0.000474 0.000392 0.000348
86 131 86 113 131 86 131 86 131 86 131 86 131 86 131 86 104 119 131 86 113 131
0.553
2 3
4
5 6
7 8
9
0.714 0.910 0.584 0.870 0.528 0.556 0.822
0.835
Table 11. Fluids Used with Each Coil
inch Cod
k-lwh Cod
' < - I n c hCod
1
1 3 5 8
2 3 5 9
3 4 6
~
2gc
gc
This equation contains allowance for kinetic energy and end effects. For a typical run the vessel was filled to a depth of between 18 and 19 inches with a known weight of fluid-usually between
7EAL
b
"
B
E
MANOMETER
I
165 and 175 pounds. Agitation occurred a t a known impeller speed, and runs were alternately heating and cooling. For heating runs the water temperature entering the coil was typically about 150" F., and for cooling runs it was around 50" F. The change in temperature of the vessel contents was usually 40" to 50" F., and the minimum temperature difference between the agitated fluid and coil water was 15O to 20°F. These temperatures were closely known and automatically recorded throughout a given run which lasted, on the average, about 18 minutes after discarding the initial 2 minutes or so to permit uniform mixing of the vessel contents. The solutions contained between 0.1 and 0.3 weight 5 of Carbopol 934, and the densities, specific heats, and thermal conductivities were therefore taken to be the same as for water. A typical plot of the data from the tube viscometer is shown in Figure 4 for liquid 8. The plots were all linear, often over a wider range of 8 V /D than shown in Figure 4, indicating that the fluids conformed to the power law and that (Skelland, p. 172, 1967)
These values are listed in Table I for all the fluids used, some of which resulted from degradation of the four original polymer solutions due to prolonged use. Table I1 shows the fluids used with each coil. For a given fluid, a plot of In K against temperature was linear with negative slope, facilitating interpolation for K , and hence a t the coil wall temperature. Calculation of h,
Figure 3. Diagram of viscometer 270
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
The over-all coefficient U , was calculated from the equations below for the unsteady-state conditions prevailing throughout all the runs (McCabe and Smith, p. 494, 1956):
Figure 4. Typical viscometer data for liquid 8
T h e individual coefficient for the water inside the coil, h,, was calculated in all cases by the expression (Coulson and Richardson, 1956)
(The exponent on D is here corrected from its misprinted form of 1.43 in the reference.) I t is essential that any non-Newtonian correlation will reduce to the Newtonian form as a special case. The exponents in Equation 10 should accordingly be constants-or reducible functions of n when considering power law fluids. The exponents in Equation 21 therefore correspond t o the following values in Equation 10: a = 0.62 /3 = 0.32
y = 0.2
6 = 0.1 t = 0.5
The wall thickness of the clean copper tubing was known, enabling evaluation of the individual agitatedside coefficient, h,, from the relationship (McCabe and Smith, p. 436, 1956)
x, De +-U,, - DthL k m Dim 1
De
1
+ -h,
The term pA& was estimated in a manner analogous to that described by McCabe and Smith (1956, p. 444) using Equation 8, the relative constancy of n with temperature, and the linearity of the plot of In K L;S. temperature. Results
The experimental program provided data from 123 runs on power law fluids for correlation using some form of Equation 10. Uhl (1966) has critically surveyed all of the published work available in the literature on heat transfer between coils and agitated Newtonian fluids. As a result of this study he has recommended the following values for exponents on six key variables:
Seven different correlations were tried, employing variations in these CY to c values. The calculated data for each run were substituted into a given correlation and the constant C was found for that run. The average of all the C values was then used to find a calculated Nusselt number for each set of data using the experimental values of Reynolds, Prandtl, and other groups. This was done for heating and cooling data separately and for both combined, using a Univac 1107 digital computer. Mean and standard deviations were also calculated for each correlation as
% standard deviation =
VOL. 8 NO. 2 APRIL 1 9 6 9
271
Table Ill. Correlations with Heating and Cooling Data Combined
Correlation NO. 1 2 3 4 5 6 7
% Deviation B
a
0.62 0.62 0.62
0.32 0.32 0.32/n 0.32 0.62n 0.32 0.62 0.32 0.62 0.32
"0.62 i032 (1
-
Y
6
6
0.2 0.2/n 0.2 0.2 0.2 0.2 0.2
0.1 0.1 0.1 0.1 0.1 0.1 0.08
0.5 0.5 0.5 0.5 0.5 0.4 0.4
Const. Mean
Std.
0.258 0.253 0.165 0.158 1.095 0.178 0.174
51.4 51.3 128.1 67.6 114.7 51.0 50.5
35.5 35.2 85.5 45.5 83.9 34.8 34.6
n)/(2 - nj.
Table IV. Correlations with Heating and Cooling Data Separately
Corre-
c;
lation
No.
Y
8
Deviation
Const. Mean
Std.
0.290 0.288 0.191 0.195
34.5 34.8 85.2 33.3
50.5 51.2 134.6 47.3
0.225 0.227 0.138 0.152
33.0 32.8 78.0 32.7
46.0 46.1 112.2 46.9
Heating Data Only 0.62 0.62 0.62 0.62
0.32 0.32 0.32in 0.32
0.62 0.62 0.62 0.62
0.32 0.32 0.32/n 0.32
0.2 0.2/n 0.2 0.2
0.1 0.1 0.1 0.08
0.5 0.5 0.5 0.4
Cooling Data Only 0.2 0.1 0 . 2 : ~ ~0.1 0.2 0.1 0.2 0.08
0.5 0.5 0.5 0.4
The results for the heating and cooling data combined appear in Table I11 and separately in Table IV. I n correlation 4 the exponent 0.62 + 0.32 (1 - n ) / ( 2 - n) was used on the Reynolds number to obtain the same dependency on impeller speed as was found by Carreau et al. (1966). Of the seven correlations tried, the best fit was obtained using correlation 7 with a mean deviation of 34.6%. However, this does not reduce exactly to the recommendations of Uhl (1966) regarding effects of key variables for the Newtonian case. Correlations 1 and 2 give almost the same deviation, but correlation 2 is restricted to power law fluids because of the n in the exponent on the apparent viscosity ratio. The mean deviation using correlation 1 is not significantly greater than the lowest deviations obtained with either the combined or the separate heating and cooling data. The correlation is not restricted to power law materials, but applies in general to any timeindependent non-Newtonian fluid, and it reduces exactly to the recommended Newtonian form with regard to the effects of six key variables indicated in Equation 2 1 (Uhl, 1966). Correlation 1 is therefore recommended for both heating and cooling as follows:
h,De -= 0.258 k
D'Np
A plot of calculated us. experimental Nusselt numbers using Equations 22 and 23 is shown in Figure 5. Fifteen per cent of the runs were with upthrusting propellers (shown as solid points). There appears to be no stratification of these data indicating that the correlation is valid for both upthrusting and downthrusting propellers. Table V summarizes the range of variables studied in this investigation. The mean deviation of 35.5% between the data and correlation 1 is primarily a consequence of two factorsthe complexity of the flow patterns and the non-Newtonian character of the fluids. Even with the much simpler flow patterns involved in flow through straight tubes, McAdams (1954) states that his recommended heat transfer equation has a mean deviation of &20'%. Carreau et a1 (1966) correlated their data with a mean deviation of 19.3%, but their geometry was also considerably simpler, being a jacketed vessel, compared with the immersed coils of the present study. As a result of the non-Newtonian nature of the liquids used, the variation in shear rate from point to point in the vessel causes corresponding variations between the local apparent viscosity, pa, and the over-all average value, p A , used to characterize the run. The extent of this deviation between local pa and average pA depends upon coil and propeller diameters and shaft speed. Heat transfer rates, which depend on local pa, therefore show corresponding deviations from run to run. This source of deviation is not present in Newtonian systems and might be expected to increase with increasingly non-Newtonian behavior-e.g., decreasing n. This is confirmed by the data of Hagedorn and Salamone (1967) for heat transfer in the simpler geometry of a jacketed vessel. Thus their mildly non-Newtonian results (0.69 S n 5 1.0) were correlated with an average deviation of &14%, whereas for their highly non-Newtonian data (0.36 S n 5 0.69) the average deviation from their correlation increased to 12OCZ. Scale-up
One or both of the following criteria may be required to reproduce some product characteristic when scaling up from pilot plant to industrial scale: CRITERION A. Equal rates of heat transfer per unit surface of coil in the two vessels. CRITERIONB. Equal rates of heat transfer per unit volume of vessel contents.
Table V. Range of Variables cppa
Flow behavior index, n (7)""Fluid consistency index, K/g,, 41-
1
lb., sec."/sq. ft. Reynolds number, D2Np/p4 Prandtl number, c , p a / k Viscosity ratio, P A / P A = Propeller diameter/ vessel diameter Coil tube diameter/vessel diameter
I n the special case of power law fluids this becomes 272
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
0.528 to 0.91 1.14 x 1O-'to 3.7 x l o - > 332 to 2.6 x lo" 11.8 to 1110.3 0.707 to 1.36 0.164 to 0.493 0.014 to 0.041
i
/
0 0-
cu
0
0
0
0
2-
__
0
-
m-
0
OD-
-
12-
V
0 v,
-
0-
-
v)
-
-
cZ $ 0
I
z
FLUID-
I
2
3
4
POINT-
0
0
W
'
KEY
SOLID POINTS
IO
20
30
50
40 Nusselt
60
5
6
7
0
8
9
0
*
UPTHRUSTING
70 80 90 100
200
OXP
Figure 5 . Calculated vs. experimental Nusselt numbers using recommended Equation 22 '
If attention is confined to power law fluids, Equation 23 is expanded to
hc -- 4 De-o5 D134 DrO 6 NO 92 - 0 3n
To achieve Criterion B,
(24)
where
4 = 0.258 k " 6 8 ~ ' 6 2 ~ ~ 3 2 ( K / K I L ) 0K2- h "O 3 (25) S31-n
or
It is assumed that the same fluid is heated or cooled over temperature ranges with the same mid-point and that the tube wall temperatures have the same average value in the two different-sized vessels, denoted by subscripts 1 and 2. Therefore @av,l = $av,2 and for Criterion A,
h,iATi, 1 = h,zATi,
2
(26)
Combining Equations 24 and 26 and rearranging,
where L, is the length of the submerged coil and H is the height of liquid in the vessel. Equation 24 is substituted for hcl and h,, and the result solved for the ratio of shaft speeds needed to achieve Criterion B on the two scales as
which defines the ratio of impeller shaft speeds in the two vessels to obtain Criterion A. VOL. 8 NO. 2 A P R I L 1 9 6 9
273
For simultaneous duplication of Criteria A and B, Equations 27 and 30 give
ATh
[(TI - t_c) - ( T ? - t , ) ] ! l n [ ( ~ ~- < ) / (T?-t,)]
=
u = velocity in direction of shear, ft./sec. = over-all coefficient of heat transfer based on external surface of coil, B.t.u./sec. sq. ft. O F . V = average veocity, ft./sec. V, = volume of liquid in vessel, cu. ft. W , = water flow rate through coil, lb.,/sec. W , = width of impeller, ft. x, = thickness of coil tube wall, ft. y = distance in direction normal to shear, ft.
U,
so that Equation 31 and either Equation 27 or 30 must both be satisfied. Equation 31, however, cannot be satisfied when strict geometrical similarity prevails on the two scales, so that Criteria A and B cannot both be achieved in such a case.
GREEKLETTERS
Nomenclature DI PA,
constant in K - T relationship external surface of coil, sq. ft. constant in K - T relationship constant in Equation 10 specific heat, B.t.u./lb., F.
K k, k m
KO ks
K, L L’ L, M
N n
274
propeller diameter, external tube diameter, helix diameter, internal tube diameter, log mean of De and D,, tank or vessel diameter, ft. acceleration due to gravity, ft./sec.’ conversion factor, 32.174 lb., ft./lb., sec.’ height of liquid in vessel, ft. individual coefficients of heat transfer on agitated side of a coil, inside a tube, and on agitated side of a jacket, B.t.u.1 sec. sq. ft. O F . fluid consistency index, lb., sec.” - ‘ft. thermal conductivity, of metal wall, B.t.u./sec. sq. ft. F./ft. defined by Equation 18 constant defined by Equation 4; k s = 10 for propellers K evaluated a t wall or coil surface temperature length of viscometer capillary tube, ft. vertical distance between capillary tube inlet and liquid surface in viscometer reservoir, ft. length of submerged coil, ft. mass of agitated fluid, lb., impeller shaft speed, r.p.s. flow behavior index Nusselt number, h, D r / k or h,D,/k number of experimental runs frictional pressure drop, lb., /sq. ft. atmospheric pressure, gas pressure in reservoir, lb.,/sq. ft. rate of heat transfer, B.t.u./sec. agitated fluid temperature, initial, final, OF. coil inlet temperature, F. average coil fluid temperature a t any instant, F.
I & E C PROCESS D E S I G N A N D DEVELOPMENT
pA , a y ,
to
= exponents in Equation 10
p.41,,
(du/ d r J A and defined above Equation 4, a t coil surface temperature, integrated average W A between TI and T 2 , p.4 a t ( T I+ T 4 / 2 , lb.,/sec. ft. differential viscosity = dT,,g,/d(du/dy) lb.,/sec. ft. pd a t surface or wall temperature density, lb.,/cu. ft. shear stress, in x-direction on surface normal to y , a t the wall, lb.,/sq. ft. proportionality constant, defined by Equation 7
= apparent viscosity corresponding to
@.Aa,
I d
=
pdu
p
= =
T , T.,~, T W
= =
SUBSCRIPTS c = coil fluid = very high shear rates
m
Literature Cited
Brasie, W. C., Chem. Eng. Progr. 60, 37 (August 1964). Carreau, P., Charest, G., Corneille, J. L., Can. J . Chem. Ew. 44, 3-8 (1966). Coulson, J. M., Richardson, J. F., “Chemical Engineering,” Vol. 1, 1st ed., p. 201, Pergamon Press, London, 1956. Hagedorn, D., Salamone, J. J., IND.ENG.CHEM.PROCESS DESIGNDEVELOP. 6, 469-75 (1967). McAdams, W. H., “Heat Transmission,” 3rd ed., p. 202, McGraw-Hill, New York, 1954. McCabe, W. L., Smith, J. C., “Unit Operations of Chemical Engineering,” pp. 436, 444, 494, McGraw-Hill, New York, 1936. Magnusson, K., I V A 23 (2) 86-99 (1952). Metzner, A. B., Otto, R. E., A.1.Ch.E. J . 3, 3 (1957). Skelland, A. H. P., “&on-Newtonian Flow and Heat Transfer,” pp. 5, 334, 338, Chap. 2, 34, 172, Wiley, New York, 1967. Uhl, V. W., “Mixing,” V. W. Uhl and J. B. Gray, Eds., Vol. 1, pp. 294-5, Academic Press, New York, 1966.
RECEIVED for review July 3, 1968 ACCEPTED October 28, 1968
