I
A.
B.
METZNER and P. S. FRIEND'
University of Delaware, Newark, Del.
Heat Transfer to Turbulent Non-Newtonian Fluids The analogy between turbulent heat and momentum transfer is extended to include non-Newtonian fluid systems. Theory is checked by the first extensive data on turbulent slurries and polymer solutions
TEE
heat transfer characteristics of purely viscous non-Newtonian fluids ( 7 7 ) flowing turbulently inside smooth, round tubes are discussed here. "Purely viscous" behavior includes all fluids for which the shear rate is dependent on only the imposed stress. Materials classified as Bingham plastic, pseudoplastic, dilatant, and generalized nonNewtonians all fall into this group; Newtonian behavior is merely one special case. Thixotropic and rheopectic materials may, under limiting conditions, also approach this behavior. The prior art (70, 74) has dealt only with slightly non-Newtonian fluids or limited ranges of conditions. Recent fluid mechanics studies (2) now permit a semitheoretical analysis of the heat transfer problem by analogy between heat and momentum exchange ( 7 , 5, 8, 9 ) ) in contrast to purely empirical approaches used heretofore. Theory By using earlier derivations for Newtonian fluids (72, 76) as models or special cases to which the present analysis must reduce for the special case of Newtonian behavior, a theoretical equation was derived for the coefficient of heat transfer ( 4 ) . It was assumed:
tance, if the analysis is restricted to fluids of high Prandtl number. As nonNewtonian fluids are usually highly viscous systems, the assumptions are even more justifiable in the present analysis, if one excludes slurries in which the continuous phase is a liquid metal, as sometimes in nuclear reactor work. T h e final equation was:
The c factor is defined by the integral :
As the region of low Prandtl numbers (Nprw< 1.0) is excluded, & and 8, are of secondary importance. Reichardt (76) has shown that 8, varies between 0.84 and 0.95 over the Prandtl number range of 1.0 to 10, all at a Reynolds n u a b e r of lo4. At higher Reynolds numbers and for non-Newtonian fluids having flow behavior indexes of less than unity the velocity profile is flatter and 8, is even closer to unity. As most Prandtl numbers of interest in non-
Newtonian systems are above 10, 0, was chosen equal to 1.0. Values of $%, calculated from the fluid mechanics analysis for non-Newtonian systems (2) are given in Table I. As one common characteristic of nonNewtonian fluids is that the fluid consistency generally increases greatly as the flow behavior index n' decreases, it becomes progressively more difficult to obtain high Reynolds numbers as n' moves to lower values. Therefore an average value of & = 0.83 will be valid for most studies. These considerations give : Table I. Values of Dimensionless Mean Velocity at Various Reynolds Numbers and Flow Behavior Indexes An average of 0.83 is valid for most studies
dh n' =
nt
=
71'
=
N'Re
1.0
0.7
0.5
5 X 108 1 X lo4 3 X 104 1 X lo5
0.79 0.81 0.83 0.85
0.82 0.83 0.85 0.87
0.84 0.85 0.87
0.89
Heat flux is constant along the length of the exchanger. Heat flux is a linear function of tube radius-Le., q/q, = 1 - y / R . Eddy diffusivities of heat and momentum transfer are equal a t all radii. T h e molecular viscosity of the fluid, as defined by P
= 7gc/(du/du)
(1)
may be expressed as a function of the shearing stress, 7 , only. The last assumption is merely a restatement of purely viscous behavior; the others have been used in the analogous development for Newtonian fluids and already subjected to critical test. Prior work has shown that they are either nearly correct or of little imporPresent address, Engineering Department, E. I. du Pont de Nemours & Co., Wilmington, Del.
REYNOLDS NUMBER, Nk,
m
D"'v*-~
--Tf Figure 1. Pressure-drop (friction factor) design chart for purely viscous non-Newtonian fluids VOL. 51, NO. 7
JULY 1959
879
400
If this equation is to be used well beyond the range of the present experimental data, substitution of the actual value of l/+m (Table I) for the constant 1.20 is recommended. Use of Equation 4 presupposes a knowledge of the friction factors for nonNewtonian fluids (Figure 1) (2). T h e c factor may now be evaluated empirically from experimental data. Equation 3 shows that it would a t most be a function of the Prandtl number of the fluid as evaluated at the wall shear stress, the flow behavior index (which will determine the variation of p7 with radial position or u + ) , and possibly the Reynolds number. The dependence of c upon Reynolds number would be a result of the upper limit of the integral, which may vary slightly with the Reynolds number. The integrand of c is equal to the ratio of the molecular to the total heat flux,qmo/q, multiplied by the factor Np,,
(Ew)
-
1. The
ratio qnLo/qdecreases rapidly from unity a t the wall to low values (7, 8, 72, 76). If at the edge of the wall region of the tube (approximately at j / R = 0.1) qmo/q = 0.01 or less, the integrand can be neglected beyond this point. In this case c will be independent of the generalized Reynolds number, if it is assumed that the turbulence within this wall region is independent of the Reynolds number, as in the prior art for Newtonian fluids (7, 8, 72, 76). Therefore, u + at
Y I R = 0.1
Figure 2. The curve represents the best known equation for c for the special case of Newtonian behavior
or
Table I1 summarizes the lower limits of the Prandtl number for which c will be independent of the generalized Reynolds number. As the generalized Prandtl number is related to the Prandtl number a t the wall,
880
C.
URRY POINTS:
0.39-0.64 IO
100
400
NP,.
both Prandtl numbers are given in the table. These conditions under which c is clearly independent of the generalized Reynolds number are much more stringent than in either Metzner and Friend's (72) or Reichardt's (76) analysis. This increase in stringency does not appreciably limit the applicability of c, and provides a desirable safety factor, as the fluid mechanics of non-Newtonian materials are not as well defined as for simpler fluids. Development of Equations 4 to 7 required choice of a relationship between fluid viscosity (or shear stress) and shear rate. The empirical "power law" 7
The point a t which j / R = 0.1 is within the region of the main stream or fully turbulent velocity profile for all fluids with an n' of unity of less. Using the calculated velocity profiles for nonNewtonians ( 2 ) one may show (4)that c will be independent of Reynolds number, if
100
=
K(du/dy)"
(9)
was chosen in view of its proved ability to portray experimental data correctly over adequate ranges ofshear rate (2. 7 7). This choice of a particular rheological equation is not critical: Equations 6 to 8 only define the inequalities of Table 11. While the numerical constant of 1.20 in Equation 4 depends directly on the assumed fluid behavior, it is usually overshadowed by the second term of the denominator. The non-Newtonian friction factors of Figure 1 are also very insensitive to this choice of a kind of rheological behavior (2). Another way of concluding that the rheological equation chosen is unimportant is to note that for fluids of high Prandtl number (to which the present analysis is restricted) the temperature gradients tend to be confined to narrow regions next to the tube wall (9). Within this region almost any two-constant equation will correctly portray the laminar behavior of real fluids. This, together with the extreme insensitivity of friction factors to deviations from the
INDUSTRIAL AND ENGINEERING CHEMISTRY
power law, again suggests the accurate, if not rigorous, conformation of nonpower-law fluids to the present analysis. As parameters K' and n', which are similar but not identical to K and n of Equation 9, are rigorously correct under laminar flow conditions (70, 73), all the equations have been cast into this form. Equipment a n d Procedure
The steam-heated heat exchanger, which consisted of three concentric pipes, has been described (5). The temperature differences between the wall and the fluid were always kept low15" F. on the average. When operating in this way, the rise in temperature of the fluid in passing through the heat exchanger was very small and good heat balances were difficult to obtain. However, the fluxes checked within 9.47" on the average. To reduce this and other experimental errors, all runs having mean temperature differences of less than 8" F., heat transfer coefficients above 4000 B.t.u./(hr.) (sq. ft.) (" F.), or heat balances that did not check within 157, were discarded. The viscometric properties of the fluids used were measured on a capillary tube viscometer. The detailed flow curves (4)show only a negligibly small change in the flow properties of the fluids over the range of temperatures within the heat exchanger. As a result, the properties were always evaluated at the average bulk temperature of the fluid and no corrections for nonisothermal effects were applied. Experimental measurements of specific heat were smoothed by plotting the results as a function of fluid density. T h e maximum correction was 6%; in only one case did the correction exceed 3%. The measured thermal
NON-NEWTONIAN FLUIDS conductivities were corrected upward 7’ because calibraby approximately 5 tion with water indicated a constant heat loss of this magnitude which could not be readily eliminated. The dependencies of density, heat capacity, and thermal conductivity on temperature, like those of the flow properties, were also shown to be negligible over the small range of temperatures encountered. Results
The physical properties of the fluids studied are listed in Table 111. Carbopol was chosen as typical of polymeric solutions which are not appreciably elastic and Attagel as a typical slurry. Three dilutions of corn sirup were used to explore the region of high flow behavior indexes and high Prandtl numbers. All properties were evaluated at the average bulk temperature in the heat exchanger. The important variables in problems of heat transfer to non-Newtonian fluids are: either the Nusselt or the Stanton number, the generalized Reynolds number, the flow behavior index, and either the generalized Prandtl number or the Prandtl number at the wall. Table IV lists the ranges of variables studied for each fluid used. Additional runs were made with water to check technique and equipment at periodic intervals. The average deviation of the water data from the Colburn (9) equation was 8.8%. Correlation of Non-Newtonian Data
Figure 2 depicts the values of c as determined by Equation 4 from the experimental measurements of the heat transfer coefficients (3, 4, 6, 75) and the friction factors given in Figure 1. The curve in Figure 2 is not a “best-fit” line through the data, but instead represents the known equation for c for the special case of Newtonian behavior, as given by Friend and Metzner (5). The possible dependency of c on the flow behavior index of the fluid is less than the scatter observed in the data. As a result, the equation previously developed for Newtonian fluids is also applicable to non-Newtonian systems, if the difference in friction factors is taken into account. The value of the integral which defines c is determined primarily by conditions very near the tube wall for fluids of high Prandtl numbers. I n this region the shear stress, hence the fluid viscosity, is very nearly constant even for highly non-Newtonian materials and the observed behavior of c is hardly surprising. For Prandtl numbers well below those studied here, c may become a function of n‘, but such systems would usually be excluded from the present analysis by Equation G or 7.
T h e final correlation proposed, therefore, becomes : NBt =
1.20
+ 11.8 z/zf/ 2
(NprW -
l)(A’prw)-’J*33
(10) The friction factors in Equation 10 are functions of the flow behavior index of the fluid; hence any design chart would contain this factor as a parameter. As the Prandtl number at the wall, Npr,, is directly related to the “generalized” Prandtl number by means of Equation 8, Equation 10 could as well
have been developed in terms of this group. Figure 3 shows the observed agreement between the experimental coefficients and those predicted by Equation 10. T h e average deviation is 17.6y0 and the standard deviation is 23.6%, compared with a standard deviation of 9.4Yo for “smoothed” data in the previous correlation for Newtonian fluids ( 5 ) . Such smoothing, while conventional in heat and mass transfer correlations for Newtonian fluids (77), was not used in the present case, because the original data at various Reynolds numbers are desirably
i
Table 11.
The c Factor Is Independent of Reynolds Numbers in These Regions w’
1.o
0.8 WPr
5 1 3 1
x x x x
103
5 1 3 1
x x x x
10’
14.2 8.05 3.06 1.07
16.5 9.14 3.54 1.25
103 104 104 106
14.2 8.05 3.06 1.07
12.3 6.01 1.91 0.540
104 104
0.6
0.4
19.4 10.9 4.25 1.52
24.3 13.7 5.42 1.96
>
N ~ r> a 9.98 4.08 0.956 0.194
8.16 2.35 0.296 0.031
Properties of Fluids Studied
Table 111. Concn., Fluid Carbopol A
B C Attagel D
%
n’
K‘
P
CP
k
0.1 0.3 0.6
0.87 0.73-0.81 0.60-0.65
0.000085 0,00122-0.00052 0.018-0.011
60.0 60.0 60.0
1.00 1.00 1.00
0.373 0.373 0.373
0.92 0.44-0.72 0.59-0.64
0.00009 1 0.0232-0.0014 0.00960.0058
65.2 65.9 67.5
0.92 0.89 0.87
0.355 0.355 0.355
72.5 83.6 81.9
0.79 0.60 0.63
0.325 0.270 0.280
71.4 60.0 60.0 64.2-65.6
0.79
0.325 0.373 0.373 0.330
10 13 16
E F Corn sirup CSA CSB
csc
e . .
...
+
Corn sirup Carbopol (CSA C) Data of Farmer (3) Data of Raniere (16) Data of Haines (6)
+
...
... .. . ... ...
1.00 1.00 1.00
0.89 0.42-0.81 0.48-0.63 0.39-0.70
Table IV. Fluid
Carbopol A
B C Attagel D
E F Corn sirup CSA CSB
csc
+
0.00028 0.00055-0.155 0.0057-0.058 0.0011-0.090
1.00
1.00 0.84-0.96
Ranges of Variables Studied
n’
N ’ R ~X 10-8
0.87 0.73-0.81 0.60-0.65
6.10-37.3 10.4-40.3 7.74-11.9
74-398 193-632 295-394
12.6-10.4 52.2-32.4 186-158
10.40-6.88 27.1-17.4 78.4-64.4
0.92 0.44-0.72 0.59-0.64
71.4 6.30-30.9 11.9-28.5
634 243-664 314-690
14.2 89.0-40.3 92.0-68.6
10.7 28.1-15.8 30.6-19.4
1-00 1.00 1.00
7.29-18.2 5.82 6.54-13.9
76.5-167 171 146-274
11.3 264 122
5.65-13.0 42.7-129 2.92-58.4 2.98-24.1 7.40-36.5
102-218 190-380 78.4-766 94-549 494-676
39.2-36.2 1.88-2.09 307-30.6 271-70.4 151-29.8
Corn sirup Carbopol (CSA 0.89 C) Water 1.00 Data of Farmer (3) 0.42-0.81 Data of Raniere (16) 0.48-0.63 Data of Haines (6) 0.39-0.70
+
0.000040 0.001025 0.000476
N N ~
Nprf
VOL. 51, NO. 7
11.3 264 122 32.5-27.7 1.88-2.09 186-15.6 14&20.1 33.8-6.26
JULY 1959
881
shown to determine the absence of any systematic deviations a t the extremes of high or low Reynolds numbers. In this Light the differences between the present 23.6% standard deviation and the 9.4y0 deviation for Newtonian fluids is seen to be largely due to the methods chosen to present the results. More importantly, no distinction exists between the slurry data points and those for polymer solutions-the correlation applies equally well to both types of systems. T h e results would not be expected to apply to polymer solutions that exhibit appreciable viscoelasticity. The discussion so far has been solely concerned with the "isothermal coefficient of heat transfer and the experimental technique was carefully chosen to approach such conditions as closely as possible in order to confirm the theory. In practice, however, finite and frequently large temperature driving forces are employed. I t is suggested that the "isothermal" coefficients calculated from Equation 10 be correcred to the actual flow conditions by multiplication by either (y/y,,)O.'4 or ( y ~ ' y ~ , . )co:-rection o.~~~ factors analogous to those suggested by Sieder and Tate (78) and Kreith and Summerfield (7) for Kewtonian fluids. Because most non-Yewtonians are much less temperature-sensitive than similarly viscous Neivtonian fluids, these correction factors. as yet not subjected to rigorous test under turbulent conditions, are likely 10 be small. Correlation of the present experimental data by an empirical approach (14) showed an average deviation of the experimental coefficients from the predicted ones of 26%> compared to 17.6% for Equation 10. Thus, the only priorart empirical approach not already discredited (70) is of some utility, but Equation 10 represents an appreciable improvement.
IC
Figure 3. Agreement between experimental and predicted coefficients has a standard deviation of
23,670
,
882
SYMBOL
0
070-089
A
092 070-072
A
t
z
W -
0
039-064
LL LL W
0
u
8 0 data points are included. All points in the crowded region of good agreement do not show up. Coefficients expressed as B.t.u./(hr.) ( s q . ft.) ( " F.)
n W
G1
SOLID SYMBOLS POLYMER SOLUTIONS: OPEN SYMBOLS
3
Ya
V
I
1000
)
EXPERIMENTAL COEFFICIENT
consistency- index defined by Equation 9: (lb.fl) (set.")/ sq. ft. = consistency index, (lb.p) (sec.. '),/ sq. ft. defined, under laminar flow conditions, by T,v =
=
K'(8V/D)"' flow behavior index defined by Equation 9, dimensionless = floiv behavior index, dimensionless (for definition, see K ' ) = total heat flux, a t any radial position, B.t.u./(hr.) (sq. ft.) = flux due to molecular conduction = flux a t wall = tube radius, feet = fluid point velocity, ft./sec. = friction velocity (Z),ft./sec. = fluid velocity parameter (2): dimensionless = centerline velocity, ft.,/sec. = volumetric or bulk mean velocity, ft./sec. = distance from tube wall, feet denominator of generalized Reynolds number, AT'Re.y = g,K'8" '-l(lb..w)/(ft.j ( ~ e c . z ')- ~ ~ as evaluated at fluid bulk temperature = y evaluated a t wall temperature = dimensionless mean temperature difference = (Tbulk- T,vsll)/ ( Toenterline - L J = fluid viscosity evaluated a t the shear stress T , lb.,/(sec.) =
Nomenclature
Typical units are given. but any consistent set may be employed. c = dimensionless factor in heat transfer correlation for Newtonian and non-Newtonian fluids, defined bv Equations 2, 3. and 10 C, = heat capacity. B.t.u./(lb..lr) (" F.) D = diameter. feet E = eddy diffusivitv (assumed equal for heat and momentum transfer). sq. ft./hr. f = Fanning friction factor, dimensionless g, = conversion factor, 32 2 (ft.) (lb.lI) /(lb.F) (sec2) G = mass velocity, lb.,f/(hr.) (sq. ft.) h = heat transfer coefficient. B.t.u./ (hr ) (sq. ft.) (" F:) k = thermal conductivity, B.t.u./ (hr.) (ft.) (" F.) or B.t.u./ (sec.) (ft.) (" F.)
n' IO0
I
fft.) \-
viscodity evaluated a t wall shearing stress = fluid density, lb.,w/cu. ft. = shear stress, lb.p/ft. = shear stress a t tube wall = dimensionless mean velocity,
=
v/u
DIMEXSIOULESS NUMBERS .I=,. = Nusselt number. hD,'k AV'Fr = generalized Prandtl number, (C,rlk) ( v / D ) " ' - ' ,TI,,-. ., .. = Prandtl number as evaluated using viscosity of fluid a t wall shearing stress, C,pT,v/k
INDUSTRIAL AND ENGINEERING CHEMISTRY
-\'IRe
= generalized Reynolds number
o'"v~-'l' P
= Stanton number. h
C,G
Literature Cited (1) Deissler, R. G., Natl. Advisory Comm. Aeronaut., Rept. 1210 (19553. (2) Dodge, D. i V . , Metzner, .A. B., A.I.CIi.E. .Journal 5, (1959). (3) Farmer, R . C., Iv1.Ch.E. thesis, Uni-
versity of Delaware, Newark, Del.
(4) Friend, P. S., M.Ch.E. thesis, Univ. of Delaware, Newark, Del., 1958. 15) Friend. W. L.. Mrtzner. '1. B.. A.Z.Ch.E: Journal 4; 393 ;,19j8). \
,
(6) Haines, R . C., B.Ch.E. thesis, Univ. of Delaware, Newark, Del., 1937. (7) Kreith, F., Summerfield, M., Trans. Am. SOC.Meek. Encrs. 72, 869 (1950). (8) Lin, C. S., Modton, R. W., Putnam,
G. L., IND.ENG.CHEW.45, 636 (1953).
(9) Mc-ddarns, W. H., "Heat Transmission," 3rd ed., hlcGraw-Hill, New
York, 1954.
110) Metzner, A. B., "Advances in Chem-
ical Engineering, vol. I, T. B. Drew and J. W. Hoopes, Jr., eds., Academic Press, New York, 1956. (11) Metzner, A. B., Rheoi. A c h 1, 205 (1958). (12) Metzner, A. B., Friend, W . L., Can. J. Chem. Eng. 3 6 , 235 (1938). (13) Metzner, A . B., Reed, J. C., A.Z.Ch.E. Journal 1 , 434 (1955); IKD. ENG.CIIEM. 49, 1429 (1957). (14) Metzner, A . B.: Vaughn, R. D., Houghton, G. L., A.I.Ch.E. Journal 3, 92 (1957). (15) Raniere, F. D., B.Ch.E. thesis, Univ. of Delaware, Newark, Del., 1957. (16) Reichardt, H., "Fundamentals of Turbulent Heat Transfer," from Arch. Ges. Warmetechnik, No. 6 / 7 (1951) ; Natl. Advisory Comm. Aeronaut. Rept. TM1408 11957) N-41947 (1956). (17) Sherwood, T. K., A.1.Ch.E. Golden Jubilee (Philadelphia) meeting, 1958. (18) Sieder, E. N., Tate, G. E., IND.ENG. CHEM.28, 1429 (1936).
. Y
REcEivm for review January 14, 1959 ACCEPTED April 14, 1959 Financial support obtained from the Office of Scientific Research, U . S. Air Force.
