N. T.
HSU,
KAZUHIKO SATO, and
B. H. SAGE
California Institute of Technology, Pasadena, Calif.
Temperature Gradients in Turbulent Gas Streams Effect of Flow Conditions upon Eddy Conductivity
T H E use of macroscopic concepts in describing the statistical characteristics of turbulence is subject to uncertainties. The basic microscopic nature of turbulence is not completely understood at present and may well require a new mathematical approach (7). Until the microscopic character (2, 20) can be predicted with certainty, simple macroscopic concepts such as eddy properties may be useful in estimating the transport characteristics of turbulent flow processes. The concept of such eddy quantities was originated by Reynolds (28). Kfirmhn (79) set forth the defining relationships for eddy viscosity and eddy conductivity. Boelter and others (41 extended these expressions on the assumption of the Reynolds analogy, which provides that the eddy viscosity and the eddy conductivity are numerically equal a t the same point in the flow channel. Only limited direct measurements of eddy conductivity are available for nonuniform flow as a resul1 of the difficulty of establishing these quantities from thermal transfer data in such flows. Eddy properties under uniform conditions may be evaluated straightforwardly (26, 29). Measurements were made to establish over a limited range of velocities the eddy conductivity in a uniform, steady, turbulent air stream flowing between parallel plates (5, 7, 26, 27). Additional experimental information is needed concerning the eddy conductivity of gases and of liquids with Prandtl numbers that differ markedly from unity. Because of the scarcity of information in this field, the behavior of gases other than air is now predicted on the basis of theoretical considerations concerning the effect of the molecular Prandtl number upon the ratio of eddy viscosity to eddy conductivity (77), called the eddy Prandtl number. Seban and Shimazaki (32), who explored in a theoretical manner the effects of asymmetric wall temperatures upon thermal transfer to a turbulent stream, also considered the influence of molecular Prandtl number on such transfer processes. Forrstall and Shapiro (72) summarized in a n extensive bibliography the information available concerning the thermal material and momentum transfer in jets.
22 1 8
Information concerning the rate of decay of turbulence is limited (3, 8, 33). None regarding the rate of increase or decrease of eddy conductivity, other than what might be surmised from Corrsin’s recent discussion (8) indicating that temperature fluctuations deteriorate more slowly than velocity fluctuations, has come to the authors’ attention. For this reason, the rates of change of eddy conductivity with time must be estimated. Further direct experimentation upon the rate of decrease or increase of eddy conductivity in nonuniform flow is required before it will be possible to predict with certainty thermal transfer under conditions of nonuniform or unsteady momentum transfer.
However, there is strong evidence that finite values of eddy viscosity exist at the axis of a flowing stream (27). Near the wall the analytical description of the velocity distribution may be evaluated from Rannie’s proposal, which was described by Dunn (71). This proposal leads to the following expressions for eddy viscosity (5):
[0.0695
(be‘)
4-71
(3,
The logarithmic velocity distribution results in the following expression (29) for the eddy viscosity in the central portion of the stream:
Eddy Properties The eddy viscosity and eddy conductivity are defined for steady, uniform transfer by the following expressions (79) :
These definitions, which were established by Kfirmhn (79) and Reynolds (28), serve to determine the quantities that must be measured to fix the values of these eddy properties. The evaluation of eddy conductivity from Equation 2 requires a knowledge of the temperature distribution and the thermal flux under steady, uniform conditions. At a distance from the walls of the channel the logarithmic velocity distribution suggested by Khrm6n (78) is often employed to evaluate the eddy viscosity. Such a velocity distribution yields zero values of the eddy viscosity at the center of the channel. The proposals of Gebelein (73) also indicate this behavior but to a less pronounced extent. At present there exists some uncertainty as to the proper value to assign to the eddy viscosity for values of Z/Zo less than 0.1 either for steady, uniform flow between parallel plates or in circular conduits.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Equation 3 applies to values of the distance parameter y + smaller than 26.7 and Equation 4 should be used for larger values. Keither expression describes the behavior near the center of the channel satisfactorily. Tabulations of the eddy viscosity as a function of Reynolds number and position based upon Equations 3 and 4 are available ( 6 ) for flow between parallel plates and in a circular conduit. The generalized values (6) are only in fair agreement with experimental measurements (22, 27) for Reynolds numbers below 20,000. With the definition of Reynolds number based upon the hydraulic radius, the relative viscosity EJV, for flow in a circular conduit and between parallel plates differs markedly at the same Reynolds number. Thermal transfer under the conditions encountered in industry is usually nonuniform, and from a mathematical standpoint the direct evaluation under such conditions of the eddy conductivity from measurements of temperature as a function of position in the stream is difficult. Even after the analysis is limited to two-dimensional steady, uniform momentum transport between parallel plates with a nonuniform temperature field, it involves the solution of a partial
differential equation of the following form (3, 29, 30): E
This expression is rather tedious to solve by common methods for values of the eddy properties (3). For this reason it is more convenient to determine the eddy conductivities under such conditions that both the thermal and momentum transfers are uniform. Under these circumstances the integration of Equation 5 results in the form of Equation 2 and it is possible to evaluate the eddy conductivity directly from the temperature distribution and thermal flux. Such measurements have been made for the flow of air between parallel plates over a range of flow conditions ( 7 , 2 5 , 2 7 ) . The behavior near the wall was studied in some detail ( 5 )and the measurements of shear and thermal flux were correlated (37).
Experimental Measurements
As a result of continuing uncertainty of the influence of position and conditions of momentum transport upon the eddy conductivity, a series of supplemental measurements of the eddy conductivity for air flowing between parallel plates under steady, uniform conditions was made. This investigation used the same equipment (7) as in an earlier study. I n principle, it consists of two parallel copper plates 13 feet in length and 12 inches in width, which were separated by a distance of approximately 0.75 inch. Air was permitted to flow between them. Circulating oil was provided behind the top and bottom plates in order to keep the temperature of the upper boundary of the air stream at a different value from that at the lower. The arrangement is shown in Figure 1 with the two copper plates a t A and A' and the agitated oil baths B and B'. The air stream flowed between plates as shown a t C. After a few feet of travel, a uniform temperature distribution was established in the stream. A 1-mil resistance thermometer shown at D was used to measure the temperature as a function of position, and a calorimeter :(37) shown a t E was used to measure the thermal flux. As the thermal and momentum transports in the air stream were nearly uniform, the measurements of the thermal flux and of the temperature gradients sufficed to establish the eddy conductivity by direct application of Equation 2. The total losses from the calorimeter were not more than 2% of the total thermal transport at a bulk velocity of 10 feet per second and a thermal gradient of 110" F. per foot. Appropriate firstorder corrections were made for these losses (37). The temperature of the air stream was measured with a precision
within 0.005" F. However, as a result of the fluctuation in temperature with time in a turbulent air stream a standard deviation of 0.05" F. was experienced. The position of the thermocouples was known relative to the wall of the channel within 0.002 inch. I n the present measurements the primary uncertainty rested in the determination of the thermal flux, which in part was based on an earlier correlation (37) directed specifically to the evaluation of the thermal flux in the flow channel under conditions of uniform thermal transport. Good agreement between the measurements of thermal flux carried out in the course of this investigation and the earlier detailed study (37) was obtained. For the most part the details of temperature, energy, and pressure measurements follow closely the methods described (7, 37) and it does not appear necessary to consider again the details of the experimental techniques and the apparatus employed or the probable error associated with
I 0
the measurement of the primary variables.
Results
A series of experimental measurements of the temperature distribution in steady, uniform flow was made with the equipment that has been described. The results were for the most part qualitatively similar to those obtained in earlier studies (26, 27). The experimental conditions are recorded in Table I ; most of the quantities necessary for the calculation of the eddy conductivity were included in this tabulation. Nineteen different sets of measurements were made. Some slight variation in the humidity of the air was experienced in the course of the investigation. The temperature difference imposed across the channel varied from 30" to 60' F., but in every case the average temperature of the incoming air was held at
100" F. A sample of the measured tempera-
II
e
I
INTERPOLATED
15.0
2 0
u
ln W
a 0 W
IW
f
10.0
W
a
s
2 0-
2 X
J
5.0
P O S I T I O N I N CHANNEL YO
Figure 2.
Experimental total conductivities VOL. 48, NO. 12
DECEMBER 1956
22 19
Table I.
Experimental Conditions
-___ 140
61
6-00
Distance between plates, foot 0.05725 0.05708 0.05758 Traverse locationh, feet 12.5 8.1 12.5 100.0 100.0 Incoming air temperature, F. 100.0 Upper plate temperature, F. 114.98 85.29 114.96 85.31 Lower plate temperature, F. 114.49 85.03 Gross velocity, feet/second 13.1 12.6 30.0 14.9 34.9 Maximum velocity, feet/second 15.2 Reynolds No. 8122 7754 18578 Thermal flux, B.t.u./(sq. ft.) (sec.) 0.0113 0.0107 0.0242 Wt. fraction water 0.0115 0.0145 0.0110 Pressure, at traverse location, Ib./sq. inch 14.374 14.308 14.260 O
O
O
T e s t Numbers .____141 143 144 14.5 146 147 148 0.05825 0.05825 0.05750 0.05750 0.05815 0.05833 0.05833 8.1 12.5 12.5 12.5 8.1 12.5 8.1 100.0 100.0 100.0 100.0 100.0 100.0 100.0 114.87 115.04 115.04 114.95 115.04 115.07 114.88 85.10 84.82 85.15 85.53 85.04 84.85 85.03 10.4 9.8 30.0 91.3 86.8 57.6 60.2 11.8 12.6 34.8 104.0 98.8 66.8 68.6 6611 6069 36115 37528 18821 56689 54680 0.0421 0.0243 0.0411 0.0558 0.0537 0.00934 0.00901 0.0129 0.0095 0.0109 0.0110 0.0089 0.0087 0.0096 14.332 14.378 14.319 14.337 14.389 14.285 14.286
Test AVumbers I I _
I96
196
197 0.06000 10.6 100.0 130.24 70.01 86.1 98.2 55657 0.1073 0.0070 14.252
198
199
coo
Distance between plates, foot 0.05983 0.06042 0.06017 0.06042 0.06050 10.6 10.6 10.6 10.6 10.6 Traverse locationb,feet 100.0 100.0 100.0 100.0 Incoming air temperature, F. 100.0 129.98 115.00 115.00 130.02 115.00 Upper plate temperature, O F. 85.00 70.02 85.00 70.02 85.00 Lower plate temperature, ' F. 28.2 57.3 27.6 85.9 57.1 Gross velocity, feet/second 32.7 32.0 98.0 65.6 65.7 Maximum velocity, feet/second 18342 37232 17766 55568 37476 Reynolds No. 0.0220 0.0438 0.0525 0.0767 0.0387 Thermal flux, B.t.u./(sq. ft.) (sec.) 0.0116 0.0103 0.0094 0.0080 0.0093 Wt. fraction water 14.246 14.270 14.243 14.319 Pressure at traverse location, lb./sq. inch 14.250 Tests 60 and 61 made to investigate in preliminary fashion effect of inverting temperature gradient. Traverse location measured from end of converging section. O
ture distribution across the central portion of the channel is recorded in Table 11. Because the qualitative aspects of these measurements are similar to those found earlier (26, 27), there does not appear to be any justification for graphical presentation of t'le temperature distribution. A record of all the measurements of temperature as a function of position is available for
Table II. Sample of Experimental Temperature Measurements (Test 199, velocity 60 feet per second) Y/YO
2220
Temp.,
'F.
0.0480 0.0619 0.0758 0.0897 0.118 0.145 0.173 0.201 0.256 0.312
91.90 92.65 93.17 93.51 94.03 94.46 94.78 95.16 95.85 96.48
0.367 0.423 0.478 0.478 0.506 0.534 0.589 0.645 0.700 0.756
97.25 98.08 98.90 98.81 99.27 99.81 100.64 101.46 102.21 102.88
0.811 0.839 0.867 0.895 0.922 0.936 0.950 0.964 0.978
103.57 103.87 104.28 104.65 105.16 105.45 105.83 106.35 107.28
each of the experimental conditions (76). The information given in Table I1 is typical of the experimental data encountered in this investigation. Measurements were made in a regular sequence in order to establish the general trend of temperature with position. Then several measurements were made in which the position was chosen at random. These random measurements did not deviate from a smooth curve of temperature with respect to position to any greater extent than in the case of the measurements carried out in regular sequence. The standard deviarion of the measurements assuming all the error in position averaged 0.000154
B8
___-
___-
601
206
203
0.05950 10.6 100.0 115.00 85.00 9.5
0.06100 10.6 100.0 130.0 70.00 9.8 11.9 6424 0.0172 0.0121 14.274
0.05996 10.0 100.0 115.01 85.03 28.8 32.0 18322 0.0219 0.0126 14.299
11.5 6060 0.0085 0.0120 14.264
foot. A more detailed study of the distribution of error is available ( 7 G ) . From the information presented in Table I and the available temperature distribution (76), values of the eddy conductivity and the total conductivity were calculated. Information concerning the moIecular properties of air was based upon a recent review of such properties (26)at atmospheric pressure which utilized Hirschfelder's values ( 7 5 ) for the effect of moisture on the viscosity of air. The detailed values of the eddy and total conductiviiies were included with the experimental temperature measurements (76). Such information establishes the eddy conductivity as a function of position and Reynolds
O3
B L 0.2 Y
4
I ."%
0.1
Figure 3. number
INDUSTRIAL AND ENGINEERING CHEMISTRY
Experimental values of e d d y conductivity as a function of Reynolds
number and when correlated with the earlier data (27) permits a reasonable evaluation of point values of the eddy conductivity for air flowing between smooth parallel plates in the range of Reynolds numbers from 6000 to 56,000. Experimental values of the total conductivity for several different Reynolds numbers are recorded in Figure 2. The experimental points were not extended to the region near the wall, because these data are already available (5), and the full curves shown correspond to the earlier evaluation of the eddy conductivities in the boundary flow near the upper wall. The general trend of the data shown in Figure 2 was similar in form to that of the data obtained earlier (26, 27). The points shown represent the calculated values of the total conductivity established from the experimental temperature dis-, tribution and thermal flux. I t is apparent that the flow is not entirely symmetrical and slightly higher maxima were obtained near the upper wall. Likewise near the boundary there was a difference between the behavior near the upper and lower walls, as was found in a n earlier study which involved some of the present conditions of measurement ( 5 ) . The reason for the lack of symmetry is not altogether clear, although the difference is somewhat greater than the probable uncertainty of measurement. Some of the possible factors influencing this lack of symmetry, which increases a t large values of l/&, were discussed briefly in consideration of boundary flows ( 5 ) . The minima at the center of the channel increase in sharpness with a n increase in Reynolds number, as was found in earlier investigations (26, 27). Two independent series of measurements were made about a year apart in order to avoid uncertainties associated with a single calibration of the instrument. The data obtained in this study, which are similar to those shown in Figure 2, were smoothed with respect to Reynolds number, which was evaluated in the following fashion : tidy = 2yaU -I
(6)
v*
The quotient of the eddy conductivity
Table 111. 1 -
n Z 0 a u n.
k
0.2
u Ly
d
v) e-
?
0.1
X 0.0.2
0.0.2
$12
6-0.6
b oe T.0.8
9.0.0
I 20,000
10,m
Figure 4.
I I
/!4
I
I
30,000 REYNOLDS
I.
sop00
40,000
NUMBER
Effect of Reynolds number on eddy conductivity
I
I
I
I
I
I
l0,OOO
20,000
--
Figure 5.
I 30,000 REYNOLDS
I
I
I
I 40,000
I
I
so,ooo
1 - - - GENERALIZED
I
I
NUMBCR
Comparison of eddy viscosities
corresponding to the transition region. At prekent no experimental data concerning the behavior a t Reynolds numbers below 5000 appear to be available. The standard deviation of all the experimental data for the conditions recorded in Table 1 from smoothed curves such as is shown in Figure 3 was 0.01345 X 10-6 square foot per second, assuming that all the error lies in the ratio of
and the Reynolds number is shown in Figure 3 as a function of the latter variable, with position in the channel as a parameter. The experimental data associated with the parametric variable are presented. Again data near the wall have not been included, as they are already available (5). A consideration of this diagram indicates that the eddy conductivity increases in nearly a linear fashion with a n increase in the Reynolds number above that
dRe. Table I11 records values of the eddy
Eddy Conductivities for Air Flowing between Parallel Plates Reynolds N u m b e r
5000
7500
1.58 1.62 1.64 1.60 1.49 1.32 1.09 0.78 0.41
2.28 2.30 2.32 2.36 2.32 2.18 1.94 1.54 0.98
10,000
12,500
15,000
17,500
20,000
25,000
30,000
40,000
50,000
60,000
11.10 11.35 12.02 12.95 13.90 14.40 14.20 12.70 9.90
12.36 12.72 13.68 15.00 16.20 16.99 16.90 15.18 11.70
E d d y Conductivity X 108,Square Feet per Second
l0
0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80
O3
$
2.94 2.98 3.06 3.16 3.19 3.08 2.82 2.37 1.55
3.54 3.60 3.70 3.92 4.06 4.00 3.73 3.18 2.15
4.10 4.17 4.36 4.66 4.90 4.92 4.68 4.04 2.78
4.63 4.73 4.98 5.38 5.71 5.84 5.62 4.92 3.37
5.16 5.26 5.58 6.09 6.54 6.74 6.54 5.78 3.95
6.20 6.32 6.72 7.39 8.02 8.35 8.18 7.22 5.01
,
7.26 7.38 7.83 8.58 9.30 9.72 9.57 8.49 5.99
9.32 9.50 10.04 10.80 11.56 11.98 11.79 10.58 7.96
VOL. 48, NO. 12
DECEMBER 1956
2221
REYNOLDS
Figure 6.
Effect of Reynolds number on eddy Prandtl number for air
conductivity as a function of Reynolds number for several different positions in the channel. These data are in agreement with earlier information concerning boundary flow (5) near the upper wall. In preparing these tabulations it was assumed that the behavior was symmetrical and for this reason the standard deviation indicated in the preceding paragraph was somewhat larger than would have been found if asymmetry had been recognized. The smoothed values of the eddy conductivity recorded in Table I11 should not involve a probable error greater than 4% of the value recorded. Figure 4 is a graphical representation of these data. The smooth curves were dotted for Reynolds numbers below 5000 to indicate the uncertainty in this region. Figure 4 also includes a comparison of the present measurements
with an earlier study made with the same equipment just after it was assembled (26, 27). The agreement is satisfactory at the lower Reynolds number but just within the combined estimated uncertainty of measurement at the higher rates of flow. A part of the difference lies in the values of thermal flux obtained in the earlier study (26, 27) as compared to the more recent correlation (37). Points showing the behavior with the original and present data for thermal flux have been included in Figure 4. A slightly greater deviation of the copper plates from a plane surface ( 7 ) which existed in the first investigation may have contributed to a portion of this discrepancy. In any event, the difference between the two sets of investigations seems slightly larger than the probable error of the present data. Such poor agreement is
P O $ I T , O N 1N C H A I I \ E L ,
Figure 7.
2922
NUMBER
2
Fractional resistance to transport for several Reynolds numbers
INDUSTRIAL ANR ENGINEERING CHEMISTRY
an indication of the sensitivity of local conditions of turbulent flow to very minor variations in experimental conditions. This is particularly apparent a t a value of Z/lo of about 0.6. By utilizing the generalizations set forth in Equations 3 and 4, the ratio of the eddy viscosity to the Reynolds number was evaluated for the conditions or interest in this study. These generalized values (6) are shown as the dotted curves on Figure 5. The discontinuity in the slope of the ratio of the eddy viscosity to the Reynolds number with respect to Reynolds number arises from the fact that Equations 3 and 4 yield identical values of the eddy viscosity at the edge of the transition region in boundary flow. At this point the rate of change of the eddy viscosity with Reynolds number is not the same, however, for Equations 3 and 4. Some earlier experimental data (27) for flow between parallel plates have been included and are in fair agreement with the generalized values, except for a continuous gradation in the eddy viscosity with Reynolds number in the transition region. The experimental data are not symmetrical about a plane midway between the center of the channel and the wall, as is the case for the generalized values based on Equation 4. Experimental information is scarce concerning the effect of conditions of flow and the molecular properties of fluids upon the eddy Prandtl number, which may be defined in the following way:
(7) In an earlier discussion (29) preliminary values of the average eddy Prandtl number were presented, based upon experimental values of the eddy conductivity (26, 27) and eddy viscosity (27) and were averaged throughout the flow channel. These results gave a small increase in the eddy Prandtl number with an increase in Reynolds number. I n Figure 6 are shown the more complete results for the eddy Prandtl number obtained in the current investigation. The experimental eddy viscosities shown in Figure 5 were employed. These data indicate an increase in the eddy Prandtl number followed by a decrease in this combination of quantities with increase in Reynolds number. Such behavior is not in accordance with elementary concepts of the mechanism of turbulent transport (77). I t appears that a basic understanding of the interrelation of momentum and thermal transport in turbulent flow must await furthcr microscopic investigations of the velocity fluctuations (7, 27) and the temperature fluctuations (8-70, 34). Consideration might well be given to extensions of the concepts
of irreversible thermodynamics (74, 23, 24) to the unsteady local situations obtaining in turbulence. The present theories apply only to steady transport. The primary cause of the discrepancy between a n earlier study (29) and the present investigation of the influence of Reynolds number upon the eddy Prandtl number is the difference in the values of the eddy conductivity found a t the higher Reynolds numbers and the use of the same values of the eddy viscosity with the two sets of eddy conductivities. The reasons for these discrepancies in eddy conductivity, which were within the estimated probable error of measurement of the earlier studies, have been described. I t will remain for further study of the velocity distribution ’and temperature distribution in steady, nearly uniform, turbulent flow to ascertain the detailed behavior of the eddy Prandtl number as a function of the microscopic conditions of flow. Resistance to thermal transport at any point in the stream is proportional to the reciprocal of the total conductivity. These resistances may be summed over the cross section of the stream to yield a comparison of the resistance in different parts of the stream. The fractional resistance to thermal transport at any point as compared to the total resistance from the wall to the axis of the stream may be evaluated from
Nomenclature
A
CP
(lb.)(’ F.) cosh = hyperbolic cosine = differential operator d = Fanning friction factor f .i? = acceleration due to gravity, feet/sec.2 = distance from c h t e r line, feet 1 = distance from center line to lo wall, feet = eddy Prandtl number p7, = thermal flux, B.t.u./(sq. feet) (sec.) Re = Reynolds number = temperature, O R. 7= temperature, O F. t = gross velocity, feet per second U = velocity, feet per second U = velocity in x direction, feet per 262 second = distance downstream, feet Y = distance normal to axis of Y stream, feet = distance from wall, feet Yd = separation between plates,feet YO Y+ = distance parameter,
d
V
b
=
EC
=
50
=
Em
=
!rn
=
K
=
Y
=
P
= =
U
7
TO
The results of the evaluation of the fractional resistance obtained from Equation 8 are shown in Figure 7 for a number of different velocities of flow. The behavior near the center of the stream has been shown on a n enlarged scale on the right hand side. The Reynolds numbers corresponding to the flow in the experimental channel were included in this figure along with the corresponding gross velocities. There exists a significant although not a marked difference in the distribution of the resistance to thermal transport as a result of increase in Reynolds number. The total resistance to thermal transport decreases rapidly with a n increase in Reynolds num bel .
Acknowledgment The assistance of Betty Kendall in connection with the rather extensive calculations associated with reducing the experimental data to a form suitable for presentation is acknowledged. W. N. Lacey reviewed the manuscript and Patricia Moen assisted in its final preparation.
= constant = isobaric heat capacity, B.t.u./
= =
P
partial differential operator eddy conductivity, sq. feet per second total conductivity, ea K, sq. feet per second eddy viscosity, sq. feet per second total viscosity, em Y , sq. feet per second thermometric conductivity, sq. feet per second kinematic viscosity, sq. feet per second density, (lb.) ( ~ e c . ~ ) / ( f t . ~ ) specific weight, lb./cu. foot shear, lb./sq. foot shear a t wall, lb./sq. foot
+
+
SUPERSCRIPT
*
= averagevalueof
.
literature Cited (1) Batchelor, G. K., “Theory of Homogeneous Turbulence,” Cambridge Univ. Press, New York, 1953. (2) Batchelor, G. K., Townsend, A. A., Proc. Roy. Soc. (London) A193, 539 (1948). ( 3 ) Berry, V. J., Mason, D. M., Sage, B. H., IND.END. CHEM.45, 1596 (1953). (4) Boelter, L. M. K., Martinelli, R. C., Jonassen, F., Trans. Am. Sac. Mech. Engrs. 63, 447 (1941), (5) Cavers, S. D., Hsu, N. T., Schlinger, W. G., Sage, B. H., IND. ENG. CHEM.45, 2139 (1953). (6) Connell, W. R., Schlinger, W. G., Sage, B. H., Am. Documentation Inst., Washington, D. C., Doc. 3657 (1953). (7) Corcoran, W. H., Page, F., Jr., Schlinger, W. G., Sage, B. H., IND.END.CHEM.44,410 (1952). (8) Corrsin, S., J . Aeronaut. Sci. 18, 417 (1951). (9) Corrsin, S., Uberoi, M. S., Natl. Advisory Comm. Aeronaut.,Rept. 1040 (1951).
(10) Corrsin, S., Uberoi, M. S., Natl. Advisorv Comm. Aeronaut.. Tech. Note 1865 (April 1949). ’ (11) Dunn, L. G., Powell, W. B., Seifert, H. S., “Heat Transfer Studies Relating to Rocket Power Plant Develogment,” Third Anglo-American Aeronautical Conference, 1951, Royal Aeronaut. SOC.,England. (12) Forrstall, W., Jr,, Shapiro, A. H., J . Appl. Mechanics 17, 399 (1950). (13) Gebelein, p., “Turbulenz,” Julius Springer, Berlin, 1935. (14) Groot, S. R. de, “Thermodynamics of Irreversible Processes,” Interscience, New York, 1952. (15) Hirschfelder, J. O., Bird, R. B., Spotz, E. L., Trans. Am. Sac. Mech. Engrs. 7 1 , 921 (1949). (16) Hsu, N. T., Sato, K., Sage, B. H., Am. Documentation Inst., Washington, D. C., Doc. 4986 (1956). (17) Jenkins, R., in (‘1951 Heat Transfer and Fluid Mechanics Institute,” 147, Stanford University Press, Stanford, June 1951. (18) K L r m h , Th. von, J . Aeronaut. Sci. 1, No. 1, 1 (1934). (19) K L r m h , Th. von, Trans. Am. Soc. Mech. Engrs. 61, 705 (1939). (20) KArmhn, Th. von, Howarth, L., Proc. Roy. Sac. (London) A164, 192 (1938). (21) Laufer, J., Natl. Advisory Comm. Aeronaut. Tech. Note 2954 (June 1953). (22) Nikuradse, J., Forsch. Gebiete Ingenieurw. Forschungsheft 356 (1932). (23) Onsager, L., Phys. Rev. 37,405 (1931). (24) Ibid.,-38, 2265. (25) Page, F., Jr., Corcoran, W. H., Schlinger, W. G., Sage, B. H., Am. Documentation Inst., Washington, D. C., Doc. 3293 (1952). (26) Page, F., Jr., Corcaran, W. H., Schlinger, W. G., Sage, B. H., IND.ENG.CHEM.44, 419 (1952). (27) Page, F., Jr., Schlinger, W. G., Breaux, D. K., Sage, B. H., Zbid., 44, 424 (1952). (28) Reynolds, 0. S., Mem. Proc. Manchester Lit. and Phil. Soc. 14, 7 (1874). (29) Schlinger, W. G., Berry, V. J., Mason, J . L., Sage, B. H., IND. ENG.CHEM.45, 662 (1953). (30) Schlinger, W. G., Berry, V. J., Mason, J. L., Sage, B. H., “Prediction of Temperature Gradients in Turbulent Streams,” Proceedings of General Conference on Heat Transfer, Inst. Mech. Engrs., September 11-1 3, 1951, London, p. 150 (1953). (31) Schlinger, W. G., Hsu, N. T., Cavers, S. D., Sage, B. H., I N D . ENG. CHEM.45, 864 (1953). (32) Seban, R. A., Shimazaki, T. T., Trans. A m . Sac. Mech. Engrs. 7 3 , 803 (1951). (33) Taylor, G. I., Proc. Roy. Soc. (London) A135, 685 (1932). (34) Townsend, A. A., Australian J . Sci. Research, Ser. A, 2, 451 (1949).
RECEIVED for review November 14, 1955 ACCEPTED June 14, 1956 Material supplementary to this article has been deposited as Document No. 4986 with the AD1 Auxiliary Publications Project, Photoduplication Service, Library of Congress, Washington 25, D. C A copy may be secured by citing the document number and by remitting $2.50 for photoprints or $1.75 for 35-mm . microfilm. Advance payment is required. Make checks or money orders payable to Chief Photoduplication Service, Library of Congress.
.
VOL. 48,
NO. 12
DECEMBER 1956
2223
