February 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
TABLE 111. ESTIMATED UNCERTAINTY OF MEASUREMENTS Range of
Meaeurement Velocity feet per second Weight &f flow pgund per second Thermal flux, ’( .t.u. per second) D e r square foot Priasure gradient, (pound/aquare foot) D e r foot Tern &&re OiP 0 F. Ai; O F. Point’temperature, 0 F.
Low High 10.0 100.0 0.04 0.4 0.004
-0.06
Estimated Uncertainty Low High 0.06 0.20 0.0002 0.002 0.0002 0.005
0.1
-1.6
70.0 130.0 70.0 130.0 70.0 130.0
0.004
0.004
0.01 0.07 0.07
0.01 0.07
0.07
419
y + = distance ammeter, U,vd/U YQ = vertical &stance between plates, feet = eddy conductivity, square feet per second ec = eddy viscosity, square feet per second e,, K = thermometric conductivity, square feet per second 3 viscosity, pounds seconds per s uare foot 7 u = kinematic viscosity, q / p squareqeet per second = density, pounds (square seconds) per (f00t)d p v = s ecific weight, pounds per cubic foot T = skear, pounds per s uare foot = shear at wall, poun& per square foot TO In = natural logarithm Re = Reynoldsnumber LITERATURE CITED
The pressure gradients recorded in Table I were taken from pressure measurements a t the piezometer bars located in the upper wall of the working section. The data recorded in Table I represent average values taken over a 4-foot length of the working section excluding the traversing equipment. These data correlate effectively with other measurements of pressure gradients (7, 18, 24) as indicated in Figure 16. After approximately one year of use of this equipment it is believed that the several measurements pertinent t o the establishment of the eddy viscosity and the eddy conductivity were determined within the probable errors listed in Table 111. I n this tabulation the range of each particular variable has been listed aldng with the probable error estimated to be associated with the maximum and minimum ranges of each of the variables. ACKNOWLEDGMENT
The assistance of G. W. Billman and H. H. Reamer in the design of the equipment and of W. M. DeWitt and L. T. Carmichael in its construction and assembly is gratefully acknowledged. S. D. Cavers, D. M. Mason, and D. K. Breaux contributed to the measurements reported herein. NOMENCLATURE
= intercept C P = isobaric heat capacity, B.t.u. per (pound)( O F.) a = differential k = von KArm&nuniversal constant $ = heat flux, B.t.u. per (square foot)(zecond) t = temperature, point, time average, F. 21 = velocity, point, timexerage, feet per second u* = friction velocity, 1 / ~ 0 / pfeet per second 3 - residual velocity, feet per second %+ = velocity parameter, u/u* Urn = maximum velocity, feet per second u = average velocity, feet per second l l - vertical distance from lower plate, feet ya = vertical distance from nearest wall, feet A
(1) Allen, C. M., and Hooper, L. J., Trans. Am. SOC.Mech. Engrs., 54, 1-16 (1932). (2) Bakhmeteff, B. A., “The Mechanics of Turbulent Flow,” Prinoetdn, N. J., Princeton University Press, 1936. (3) Billman, G. W., Mason, D. M., and Sage, B. H., Chem. Eng. Progress, 46, 625-35 (1950). (4) Boelter, L. M. K., Martinelli, R. C., and Jonassen, F., Trans. A m . SOC.Mech. Engrs., 63, 447-55 (1941). (6) Colburn, A. P., and Coghlan, G. A., Am. SOC. Mech. Engrs., Annual Meeting, New York (Dec. 2-6, 1940). Corcoran, W. H., Roudebush, B., and Sage, B. H., Chem. Eng. Progress, 43, 135-42 (1947). Deissler. R. G., Natl. Advisory Comm. Aeronaut.. Tech. Note 2138 (1950). Eagle, A,, and Ferguson, R. M., Proc. Roy. SOC. (London), A127, 540-66 (1930). Gebelein, H., “Turbulenz,” Berlin, Julius Springer, 1935. Goldstein, S., “Modern Developments in Fluid Dynamics,” Vol. I and 11,London, Oxford University Press, 1938. Jenkins, R., Brough, H. W., and Sage, €3. H., IND.ENC.CHEM., 43, 2483-6 (1951). von Kdrm&n,Th., J . Aeronaut. Sci., 1 , No. 1, 1-20 (1934). von K&rm&n,Th.,Nachr. Ges. Wiss. Gattingen, Math.-physik. Klasse, 1930, S . 58. von K&rm&n,Th., Trans. A m . SOC.Mech. Engrs., 61, 705-10 (1939). Laufer, J., Natl. Advisory Comm. Aeronaut., Tech. Note 2123 (1950). McCann, G. D., Proc. Natl. Elec. Conf.,2,372-92 (1946). Martinelli, R. C., Trans. Am. SOC.Mech. Engrs., 69, 947-59 (1947). Nilcuradse, J., Forsch. Gebeite Ingenieurn., 3, supplement, Fomchugsheft, No. 350, 1-36 (1932). Pardoe, W. S., Mech. Eng., 58, 60-2 (1936). Prandtl, L., Physik. Z.,29, 487-9 (1928). Reichardt, R., 2.angezo. Math. u.Mech., 20, (6) 297-328 (1940); Natl. Advisory Comm. Aeronaut.. Tech. Mem. 1047 (1943). Reynolds, O.,Mem. Proc. dfanchester Lit.& Phil. SOC.,14, 7-12 (1874). Skinner, G., thesis, Calif. Inst. of Technology, 1950. Wattandorf, F. L., Proc. Rog. SOC. (London),A148, 565-98 (1935). Willis, J. B., Australian Council Aeronaut. Rept., ACA-I9 (Oct. 19, 1945). RECXIVED December 19, 1949.
(Temperature Gradients in Turbulent Gas Streams)
TEMPERATURE AND VELOCITY DISTRIBUTIONS IN UNIFORM FLOW BETWEEN PARALLEL PLATES F. PAGE,
T
JR., W.
H. C O R C O R A N , W.
HE prediction of thermal transfer between solid boundariea and fluid systems is a n important industrial problem. T h e background of experimental information in this field is large, and the general nature of the phenomena encountered has been summarized in a quantitative fashion by McAdams (16)and Jakob (8). Von KArm6n ( 1 1 ) , Reynolds @I), Prandtl (go), and Taylor (93) have all contributed to an analogy between thermal and momentum transfer. The concepts of eddy viscosity and eddy conductivity as set forth by von KArrn6.n (11) and Prandtl
G. SCHLINGER,
AND
6. H. S A G E
(20) have been utilized by Boelter (I) and Martinelli (fy), in conjunction with generalized velocity distributions, to estimate thermal transfer to fluids flowing in conduits. Limited thermal transfer studies have been made (4) involving the flow of air in a fairly thin rectangular channel. The results confirm approximately, under a limited range of conditions, von KArm&n’s hypothesis of a relationship between the eddy viscosity and eddy conductivity. These measurements were of a preliminary nature and left much to be desired in the way of precision,
INDUSTRIAL AND ENGINEERING CHEMISTRY
420
Vol. 44, No. 2
eddy quantities in steady, uniform flow for the range of Temperature, F. Reynolds numbers here inProperty Units Reference 70 100 130 vestigated and for Prandtl 3.82 3.98 4.14 (18, 14, 18, 19) (Lh.)(sec.)/sq. ft. Viscosity X lo7 numbers near unity. S ecific heat 0.2403 0.2406 0.2409 (6, 18, 19) B.t.u./ lh ) ( " F T?l ermal conductivity X 10s The equipment e m p l o y e d B.t.u./)se&(o d.)(ft.) 4.14 4.34 4.54 (19) S ecific volumes 13.348 14.105 14.862 7 18, 19) Cu. ft./lb. was described in some detail Sq. ft./sec. 2.30 2.54 2.80 Tgermometric conductivity X 1W {1.b) Kinematic viscosity X 104 (18,14, 18. 19) Sq. ft./sec. 1.64 1.81 1.98 (3) and the nature of the velocity distribution encountered under isothermal conditions of flow was indicated. I n principle, the apparatus involved a flow channel 0.70 inch in height, 12 inches in width, and 13 feet in length. Air was circulated through this rectangular duct under carefully controlled entrance conditions and the temperature a t each point was kept invariant with respect to time. The temperatures of the upper and lower surfaces of the duct were maintained a t constant but different values, thus imposing a uniform temperature gradient normal t o the flowing stream. Pressure gradients were determined by means of piezometer bars and a traversing piezometer tube, used in conjunction with kerosene-in-glass manometers observed with a cathetometer. The velocity was established from a traversing pitot tube and a hot wire anemometer, while the gross flow rate was measured by the use of a Venturi meter in the supply duct. The point temperature of the air stream was ascertained from the resistance of a short length of platinum wire so mounted that VELOCITY U FEET PER SECOND Figure 1. Velocity Distribution at Temperatures between 95" and it traverse the conduit. The flux was determined by calorimetric techniques ap105" F., 12.5 Feet Downstream plied t o thermally insulated sections of the upper surface of the rectangular duct, As has been indicated ( 3 ) , the velocity and shear were each Recently, equipment has been developed (3) which permits a measured with a probable error of less than 2.0'%,while the temmore precise measurement qf thermal flux, temperature, velocity, perature was determined with a precision of 0.02' F. and a n and shear as related t o position in a substantially uniform, accuracy of 0.05' F. relative to the international platinum scale. turbulently flowing air stream. The present discussion deThe thermal flux was ascertained with a probable error of 2%. scribes measurements of these quantities and the derived values Some d s c u l t i e s were experienced in the measurement of the of eddy conductivity. There is available (10)anapplicationofan details of the velocity distribution of tests 31 t o 34 inclusive as a analog computer, using estimated values of eddy conductivity, result of small changes with time in the calibration (3) of the hot t o the prediction of thermal transfer between solid boundaries wire anemometer. For this reason, the uncertainties in the and a turbulently flowing liquid. It is hoped that by extension computed values of the eddy viscosity for these tests were larger of such methods it will prove feasible ultimately to predict than those for tests 37 and 40. thermal flux and temperature as functions of position in a flowing The pertinent intensive properties of dry air, which were chosen stream in many situations for which only the velocity and shear for this work, are recorded in Table I for 70', loo', and 130" F. distributions are known, The relationship of the eddy viscosity The sources of these data are indicated by appropriate references and the eddy conductivity may be a function of the Prandtl in this table. The primary uncertainty associated with the number (9, 17). For this reason, the present measurements application of these values resulted from the presence of moisture should be considered only as indicative of the relationship of these
TABLE I. PROPERTIES OF DRYAIR AT 14.696 POUNDS PER SQUARE INCH
TABLE 11. EXPERIMENTAL CONDITIONS Quantity
Units
30
..
32 0.0626
Test Numbers 33 0.0618 10.3 100.0
34 0.0625 0.0616 Distance between plates Foot 10.3 10.3 10.3 Traverse locationb Feet 100.0 100.0 100.0 ' F. Incoming air temperature .. Upper plate temperature ' F. Lower plate temperature F. 8: 89 54:2 7912 29:3 16:s Average velocity Ft./sec. 5980 52900 36400 19900 10850 Reynolds number -0.0166C -1.180 -0.588' -0.138C Lb./cu. f t . -0.0708c Pressure gradient 0,0865 0.116 0.0173 0.0555 B.t.u./(sa. ft.)(sec.) 0.0353 Thermal flux 0,0088 0.0062 0.0088 0.0088 0.0072 Wf. fraction water 14.446 14.336 14.381 14.368 Pressure a t traverse location LK/sq. in. 14.352 14.447 14:398 14.373 14.375 Baromettlc pressure Lb./sq. in. 14.356 a This material reproduced from reference (3)for completeness. b Traverse location measured from end of converging section. 0 Pressure gradient estimated from friction factor correlations. d Pressure radient is average change in static pressure over &foot length of working section approximately 10 feet downstr eam and measurecfwith traversing mechanism downstream from the statio taps. No significant change with time observed.
..
31 0.0616 10.8 100.0
.
I
37a 0.0567 8.1 100.0 100.0 100.0 28.94 17500 -0.230d 0
0.0058 14.335 14.340
450.0571 12.5 100.0 104.7 95.3 28.13 17100 -0.216d 0.00730 0.014 14.316 14.312
from enti'ance to channel
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1952
in the air used. The available information concerning the effect of small p a n t i t i e s of water upon the viscosity of air-water mixtures is contradictory (18,,%). For this reason, the viscosity of the fluids in the equipment was established from available (2) theoretical considerations. The influence of composition upon specific volume of the air-water system in the gas phase a t atmospheric pressure was determined on the basis that the phase was an ideal solution (16). The specific volume of these mixtures was computed from available data concerning the 1.0
$
o.8
z
9 0.6 Y
Y
z
5
z
5
421
this reason, corrections were made on the basis of elapsed time of flow after calibration. A few corresponding. values of velocity were obtained a t a given point in the flow channel with both the hot wire anemometer and the pitot tube. These comparisons were independent of and in addition t o the pitot tube information used t o establish the ealibration of the hot wire anemometer. Figure 1 sho,ws the velocity distribution obtained for the conditions set forth in Table I1 and identified as test 40. These data are for approximately the same weight rate of flow as previous measurements made under isothermal conditions (9), which are identified as test 37. Tests 37 and 40 give smaller velocity deficiencies at a specified distance from the center than would be expected for uniform flow. This difference appears to.result from the small influx of air into the flow channel through the movable side walls. These deviations from uniform flow were sufficiently small as not t o impair the analysis of the thermal transfer measurements. At most the
0.4
0.2
e 0.0
97
98
99
100
TEMPERATURE
101
02
99
I03
O F
I
Figure 2. Temperature Profile between 95" and 105' F., 12.5 Feet Downstream
specific weight of air (7, 11) and gaseous water (19). For the latter substance, the formulation of Keenan and Keyes ( l a ) was extrapolated t o the pressure prevailing within the flow section. The values of the specific volume computed from these data on the basis of ideal solutions compared well with available measured values for gaseous mixtures of air and water. The variations in pressure experienced within the working section as a result of changes in barometer and of variations in flow conditions made it necessary t o apply small corrections t o the values of specific volume of dry air recorded in Table I. This was accomplished on the basis of the perfect gas laws without significant loss in accuracy. The primary uncertainty in thermal data was associated with the thermal conductivity. The data of Keenan
0.6
f 0.4 6 -1
6 0.2 I-
2: 0.0
Figure 4. Temperature Gradient between 95' and 105' F., 12.5 Feet Downstream
TABLE 111. SAMPLE
Y/UQ
Figure 3. Residual Temperature Profile a t Temperatures between 95" and 105' F., 12.5 Feet Downstream
and Kaye (I&'), representing a weighted mean of several investigatom, were used in t h e present calculations and are recorded in Table I. The heat capacity of the air-water system was computed on the basis of ideal solutions (16) using the values employed in a previous publication (6)for air and those of Gordon (6)fpr water. The details of the procedures followed for calibrating the e q u i p ment used for the measurement of the several quantities involved have already been described (3). The only instrument which presented a problem in this series of measurements was the hot wire anemometer, as it was found that the calibration changed materially with the time of use in t h e moving air stream. For
0.8
O F EXPERIMENTAL VALUES O F AND TEMPERATURE
TEST40 Temperature, F. 103.66 103.55 103.35 103.06 102.82 102.72 102.55 102.44 102.23 102.17 101.96 101.92 101.83 101.77 101.67 101.62 101.54 101.43 101.15 100.95 100.69 100.40 100.09 99.75 99.47 99.16 98.90 98.79 98.62 98.53 98.15 98.21 98.01, 97.85 97.74 97.49 97.33 97.16 96.57 96.30
VELOCITY
Velooity, Ft./Seo.
8.96 9.60 11.96 15.43 18.47 19.55 21.21 22.50 23.82 24.85 26.05 26.45 27.20 27.48 27.83 28.23 28.61 29.26 30.81 31.21 31.70 32.17 32.30 32.18 31.56 30.97 29.86 29.51 28.65 27.99 27.02 26.21 24.93 23.81 22.81 20.68 19.41 17.80 11.65 8.87
INDUSTRIAL AND ENGINEERING CHEMISTRY
422
difference involved 1.5 feet per second in the maximum velocity. The velocity gradieht was very similar t o that found for isothermal conditions already described (3). The data upon which Figure 1 is based are all recorded as test 40 in Table 111 and the experimental conditions are stated in Table 11. The detailed
TABLEIV. POINTVALUESOF EDDY CONDUCTIVITY AND
VISCOSITY X 10s
Test
30
Eddy Conductivity 31 32 33 34
Eddy Viscosity
40
40
37
Y/YO
0.98 0.96 0.94 0.92 0.90
9
4 0.8 z
2 0.6 %
4
La4 5
2
$ 0.2
I-
Y 5 QO'-a&
o.dos
0602
0.604
aios
EDDY CONDUCTIVITY A N D SQUARE FEET PER SECOND
vlscoslr~
Figure 5. Comparison of Eddy Viscosity and Eddy Conductivity between 95' and 105' F., 12.5 Feet Downstream
The experimental points in Figure 2 do not deviate from the smooth curve by more than 0.02' F. The rather complex temperature profile in this figure is typical of most of the measurements obtained in this channel. The temperature gradient obtained from the data of test 40 is shown in Figure 4. F r t m the thermal conductivity recorded in Table I and the thermal flux indicated in Table I1 the temperature gradient at the upper plate was estimated to be 1680" P. per foot. This value is in good agreemcit with the limiting thermal gradient estimated from the temperature traverse measurements. The data from test 40 which are recorded in Table I11 permit, b y application of the following definitive expressions, calculation of values of the eddy viscosity and eddy conductivity. fm
= PI ( % )-
om1 E m WXOSlTY
,
w 2
Y
a003
I pD00
0.690 1.58 2.05 0.05 '0.21 0.18 o:iz 1.63 3.90 5.90 0.20 1.22 0.88 i:iia 2.26 1.38 2.61 5.92 8.21 0.38 2.08 1.58 2.12 3.78 7.65 11.7 0.58 2.92 2.23 3.00 3.64 2.81 4.84 9.14 13.5 0.78 3.57 2.79 3.25 6.75 11.4 15.5 1.22 4.89 3.79 4.02 4.16 7.52 12.4 16.1 1.60 5.66 4.40 4.21 4.35 7.61 12.4 16.0 1.83 5.92 4.66 4.13 4.21 7.60 11.8 15.4 1.98 5.90 4.69 4.00 3.86 3.98 6.37 10.8 14.3 2.00 5.49 4.45 3.88 6.00 9.95 13.2 1.90 4.98 4.01 3.75 0.69 3.84 5.88 9.27 12.5 1.82 4.67 (3.59) (3.90)n 3.91 5.87 9.04 12.3 1.83 4.51 0.50 0.45 4.01 6.01 9.09 12.7 1.90 4.50 CS.'56) (2:98) 0.40 3.42 4.17 6.34 9.79 13.7 2.20 4.68 3.83 2.22 5.12 4.35 3.71 4.45 6.98 11.0 15.1 0.35 4.00 0.30 4.84 7.70 11.9 16.2 2.05 5.58 4.55 4.52 5.06 8.06 12.2 16.6 1.68 5.75 0.25 4.21 4.71 7.66 11.6 16.3 1.17 5.52 4.22 0.20 4.27 0.15 4.03 3.81 6.47 10.4 14.8 0.79 4.64 3.65 0.10 3.19 2.54 4.23 8.77 12.4 0.48 3.42 2.71 0.94 1.95 3.22 7.58 11.0 0.37 2.94 2.26 0.08 ..._ 1.31 2.10 5.70 9.33 0.25 2.31 1.78 0.06 0.04 . . 1.16 3.99 6.95 0.17 1.50 1.22 0.02 . . 0.48 1.85 . . . 0 07 0.41 0.39 a Eddv conduotivitv exoressed in souare feet Der second. b Eddy viscosity eipres'sed in squar'e feet per second.--viscosity in parentheses are subject to much larger unC Values of edd certainties than otKer values as a result of proximity to the axis of flow.
.. ..
experimental results for all of the experimental conditions listed in Table I1 are available (19)in the same form as Table 111. The temperature profile for test 40 is shown in Figure 2. The residual temperature distribution presented in Figure 3 was computed from the data recorded in Table I11 by application of the expression
r
Vol. 44, No. 2
The results of such calculations are indicated in Figure 5. Corresponding values of the eddy conductivity and the eddy viscosity are presented for test 40 in Table IV along with eddy viscosity data for test 37 from an earlier study ( 3 ) . Values of the eddy viscosity are not recorded in Table IV for the other tests because of uncertainty as to the absolute values of the velocity. However, if desired, the values may be computed from the available velocity data (19). The relationship between "the point values of the eddy conductivity and eddy viscosity shown in Figure 5 for test 40 indicates a nearly constant value of 0.79 for the ratio between them throughout the central portion of the channel. The accuracy of the data was such that the ratio could not be established with certainty near the wall, where both the eddy conductivity and eddy viscosity approached aero or a t the center where the value of eddy viscosity becomes indeterminate. Figure 6 compares the point values of eddy viscosity for tests 37 and 40: The average value of the Reynolds number as defined by Equation 4 was not greatly different for the two cases.
I
SQUWE FEET PER SECMlD
Figure 6. Comparison of Eddy Viscosity under Isothermal and Nonisothermal Conditions
Figure 7. Typical Temperature Distribution for Several Reynolds Numbers
INDUSTRIAL A N D ENGINEERING CHEMISTRY
February 1952
423
.
(4)
It is not believed that the small differences In symmetry between the isothermal measurements of test 37 and the conditions of test 40 are significant. The temperature distributions for tests 30 t o 34 inclusive are shown in Figure 7. I n order t o aid in the graphical represent* tion of the data, they are portrayed in terms of an arbitrary temperature at the center of the channel. The average temperature at this point for tests involved was 100.3’ F. The precision of t h e measurement of temperature was approximately 0.1’ F.
OD 05 EDW CONDUCTIVITY
aoio SQUARE
MI5 FEET PER SECOND
Figure 9. Influence of Reynolds Number upon Eddy Conductivity ACKNOWLEDGMENT
The assistance of H. H. Reamer in connection y i t h the supervision of the laboratory work associated with this program is acknowledged, S. D. Cavers, D. M. Mason, and D. K. Brertux aided in obtaining data recorded herein. The general guidance, inspiration, and leadership of W. N. Lacey have made this work possible. I 500 TEMPERATURE
I 1000 GRADIENT
& $ *f
I I500 PER FOOT
NOMENCLATURE
1
Figure 8. Typical Temperature Gradients for Several Reynolds Numbers
Cp
=
d
= =
k
&
=
Re = but somewhat large uncertainties in the absolute values may exist as a result of changes in the calibration with time of the small resistance thermometer. The corresponding values of the temperature gradients for tests 30 t o 34 inclusive are presented in Figure 8. The markedly higher gradients encountered at values of V / ~ of O 0.1 and 0.9 a t the lower Reynolds numbers are evident. Near the wall a nearly inverse relationship of the gradients to the gross velocity is encountered. The gradients in the central portion of the channel were nearly inaependent of the Reynolds number. The point values of eddy conductivity for tests 30 to 34 are recorded in Table IV and presented in Figure 9. The increase in the eddy conductivity with an increase in Reynolds number is evident and the marked decrease m this eddy quantity near the wall follows the relationship suggwted by von Kgrrngn (11) for eddy viscosity. However, in contradistinction t o von K&rm&n’spredictions the values of eddy conductivity were finite near the center of the channel. At Reynolds numbers above 10,000 the behavior is similar t o that proposed by von K&rm&n (11) and Prandtl (go) as t o the influence of Reynolds number upon the eddy properties. It is desired t o emphasize that this discussion relates primarily t o an evaluation of the eddy conductivity and that the data comparing eddy values of viscosity and conductivity are illustrative. Material additions t o the measurements reported in test 40 must be made before it will be possible to ascertain the relationship of eddy conductivity and eddy viscosity for uniform flow with a fluid similar t o air. The data concerning eddy conductivity are not believed t o include uncertainties larger than 8%.
j t u
= = =
U
=
y yo
= = = = =
e,, K Y
=
p
=
u 7
=
isobaric heat capacity, B.t.u. per (pound)(’ F.) differential o erator thermal confuctivity, B.t.u. per (second)(feet)(’ F.) heat flux, B.t.u. per-(square foot)(second) Reynoldsnumber tem erature, point, time average, O F. resiBua1 temperature, 0 F. velocity, point, time average, feet per second average velocity = 1 s ” u dy, feet per second Yo 0 vertical distance from lower plate, feet vertical distance between plates, feet eddy conductivity, square feet per second eddy viscosity, square feet per second thermometric conductivity = k/(Cpu), square feet per second kinematic viscosity, square feet per second density, pounds (square seconds) per (foot)4 s ecific weight, pounds per cubic foot skear, pounds per square foot LITERATURE CITED
(1) Boelter, L. M. K., Martinelli, R. C., and Jonassen, F., Tram. Am. 800. Mech. Eng~s.,63, 447-55 (1941). (2) Chapman, S., and Cowling, T. G., “The Mathematical Theory of Nonuniform Gases,” pp. 230-1, Cambridge, Cambridge University Press, 1939. (3) Corcoran, W.H.,Page, F., Jr., Schlinger, W. G . , and Sage, B. H., IND.ENG.CHEM.,44, 410 (1952). (4) Corcoran, W.H.,Roudebush, B., and Sage, B. H., Chem. Eng. Progress, 43, 135-42 (1947). (6) Gerhart, R.V., Brunner, F. C., Miokley, H. S., Sage, B. H., and Lacey, W. N., Mech. Eng., 64, 270-3 (1942). (6) Gordon, A. R., J . Chem. Phys., 2 , 65-72 (1934). (7)“International Critical Tables,” VoI. 3,p. 3, National Research Council, New York, McGraw-Hill Book Co., 1928. (8) Jakob, M., “Heat Transfer,” Vol. I, New York, John Wiley & Sons, 1949. (9)Jenkins, R., Ph.D. thesis in ohemical engineering, Calif. Inst. of Tech., 1949. (10) Jenkins, R., Brough, H. W., and Sage, B. H., IND. ENG.CEEM.. 43. 2483-6 (1951).
INDUSTRIAL AND ENGINEERING CHEMISTRY
424
(11)von KkmBn, Th., Trans. Am. SOC.Mech. Engrs., 61, 706-10 (1939). (12) Keenan, J. H.,and Kaye, J., “Thermodynamic Properties of Air,” New York, John Wiley & Sons, 1936. (13) Keenan, J. H.,and Keyes, F. G., “Thermodynamic Properties of Steam,” New York, John Wiley & Sons, 1936. (14) Kellstrom, G.,Phil. Mag., 23, 7th Series, 313-38 (1937). (15) Lewis, G.N.,J. Am. C h m . Soc., 30, 668-83 (1908). “Heat Transmission,” New York, McGraw(16) McAdams, W. H., Hill Book Co., 1942. (17) Martinelli, R. C., Trans. Am. SOC.Mech. Engrs., 69, 947-59 (1947). (18) Milliltan, R.A., Phil. Mag., 19,6th Series, 209-28 (1910). (19) Page, F.,Jr., Corcoran, W. H., Schlinger, W. G., and Sage,
Vol. 44, No. 2
B. H., Washington, D. C., Am. Doc. Inst., Doc. No. 3293 (1950). (20) Prandtl, L., Phgsilc. Z.,29, 487-9 (1928). (21) Reynolds, 0.S.,Mem. Proc. Manchester Lit. & Phil. Soc., 14, 7-12 (1874). (22) Stearns, S.C.,Phys. Rev., 27, 116 (1926). (23) Taylor. G.I., Advisory Comm. Aeronaut., London, Tech. R D ~ . . 2, 423-9 (1916-17). RECEIVED January 30, 1950. For material supplementary to this article order Dacument 3293 from American Documentation Institute, 1719 N Bt., N.W., Washington 6, D. C.,remitting 51.00 for microfilm (images 1 inch high on standard 35-mm. motion picture film) or $1.80 for photooopiea (6 X 8 inches) readable without optical aid.
(Temperature Gradients in Turbulent Gas Streams)
POINT VALUES OF EDDY CONDUCTIVITY AND VISCOSITY IN UNIFORM FLOW BETWEEN PARALLEL PLATES F. PAGE,
JR.,
W. G. SCHLINGER, D. K. BREAUX’,
T
HE prediction of the distribution of temperature and thermal flux in a flowing stream under specified boundary conditions is a matter of engineering interest, The ability to carry out such predictions permits an approach to the treatment of thermal transfer similar to that followed in the field of fluid mechanics from a consideration of point values of velocity and shear. Macroscopic concepts and dimensional analysis (9) in the correlation of thermal transfer data have proved t o be of great industrial utility. The work of McAdams (12)and Jakob (7) is outstanding in this regard. The present discussion deals with experimental measurements of the nearly uniform (18) flow of air in a rectangular conduit under conditions where a fixed transverse temperature gradient could be imposed. Sufficient data were obtained t o establish experimentally the point values of thermal flux, temperature, velocity, and shear for each set of conditions. Measurements of this type supplement earlier investigations by Sherwood (19) and others (3, 4, 16) upon material and thermal transfer. The quant.ities eddy conductivity and eddy viscosity are discussed b y von K & r m h (9). Their values vary with the nature of the turbulent flow and the position in the stream. Such properties should be considered a t a point and must be determined as a function of position in the vertical section of a twodimensional uniform stream (18). For present purposes the eddy conductivity and eddy viscosity may be defined in accordance with the conventions established by von K6rmAn (IO), who did not differentiate between the numerical values of the eddy conductivity and the eddy viscosity.
Under conditions of uniform flow (3,18) with respect t o both shear and thermal flux, the terms in Equations 1 and 2 may be considered constant with respect t o the direct,ion of flow. T h e shear may be determined as a function of the position in the flow channel by-the general expression for uniform flow ( 1 ) 7=1-
1
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Present address, AiResearoh Manufacturing Co., Loa Angeles, Calif.
(3)
AND
B.
H. SAGE
EXPERIMENTAL METHODS AND PROCEDURE
The evaluation of the eddy viscosity involves only the measurement of the pressure gradient and the velocity distribution. On the other hand, the determination of the eddy conductivity in the case of uniform flow requires information concerning the temperature distribution and the thermal flux a t the boundary, The equipment employed for the measurement of these quantities under substantially uniform conditions of flow for a turbulent air stream has been described (S), and the accuracy with which each of the pertinent variables was measured has been reported. This apparatus was employed in the present instance to establish corresponding temperature and velocity distributions for average temperature gradients from 0” to 510’ F. per foot and for average velocities from 10 to 90 feet per second. The flow channel (3) was 13 feet in length, 12 inches in width, and approximately 0.70 inqh in height. The temperatures of the upper and lower plates were controlled by the use of circulating oil baths, and suitable calorimeters were provided in the upper plate to establish the thermal flux a t two points a t the boundary. The pressure gradients were determined by means of piezometer bars used in connection with kerosene-in-glass manometers. The difference in elevation of the arms of the manometers was determined with a cathetometer. I n the case of small pressure differences an inclined tube, null reading, kerosene-in-glass manometer ($) was employed. Traversing ear was provided to permit the measurement of the velocity a n 8 temperature as a function of position in the flowing stream. For the present work, all measurements were taken a t the vertical axis of flow and at distances of 8.1 and 12.5 feet downstream from the end of the converging section. The data used for the calculation of eddy quantities in most cases were taken a t the latter position. Tlie measurements reported were carried out in a fashion similar t o that which has been described previously (3, d ) , and the conditions under which they were made are recorded in Table I. Each set of conditions has been identified by a test number which is used later in reference t o a particular traverse. An effort has been made to choose the operating conditions at fixed nominal values of the primary variables. These quantities are considered to be the average velocity of flow and the imposed temperature difference between the upper and lower boundaries of the flow channel. In Table I1 is recorded the status of the experimental measurements upon uniform flow which have been carried out with this equipment (3). I n each instance the associated literature reference from which the primary data may be obtained has been indicated. A sample of the. experimental data obtained in the present
