Nonuniform Flow
0
W. G. SCHLINGER, V. J. BERRY1, J. L. MASON2,AND B. H. SAGE California lnsfitufe of Technology, Pasadena, Calif.
REDICTIOK of temperature distribution in a flowing stream is of industrial importance. Pannell (2%) and Jurges (8) considered this matter in some detail. The prediction of the temperature distribution requires a knowledge of the velocity profile associated with the flow. I n addition, a means of estimating the eddy conductivity as a function of position in the flow channel must be available. More recently, analog computers have been applied ( 7 , 25) to the solution of the equations representing the energy balance in a turbulent stream. KArmAn (10) and Prandtl (83) proposed analogies bctwccn the eddy viscosity and eddy conductivity. Boelter, Martinelli, and Jonassen (8)extended the Kftrmhn analogy and obtained useful values of the macroscopic thermal transfer coefficients. Recent studies have determined more fully the effect of Reynolds number and position in the flow channel upon the ratio of the eddy conductivity to the eddy viscosity for air flowing between parallel plates (4. $0, 21). On the basis of the latter information i t appears possible to predict the eddy conductivity for uniform, steady, turbulent air streams from a Irnowledge of the eddy viscosity. As a first order approximation the velucity distribution between parallel plates or in a circular conduit may be estimated from available generalizations ( 1 , 9 ) .
is evidence that systematic deviations from a single-valued relationship between U + and y + may exist at Reynolds numbers below 15,000 (21). Equations 1 and 2 yield identical values of du +
u+ and 7a t a value of y + of 26.7. The velocity distribution
dY
between parallel plates and that in a circular conduit are not identical. as is indicated by Equations 1 and 2 , but data available a t present ( 5 , 10, 12, 18) fail to differentiate the behavior satisfactorily. Until these matters are resolved, i t is believed that these equations may be used to describe the velocity distribution for uniform, steady flow between parallel plates and in circular conduits. The numerical constants in Equations 1 and 2 appear to apply a t Reynolds numbers above 15,000. The eddy viscosity may be coniputed from Equations 1 and 2 with satisfactory accuracy by use of the definition (10).
A combination of Equations 1, 2, and 5 yields the following relations which have been expressed in terms of the Fanning friction factor ( 1 ) :
Eddy Properties The data of Skinner (26), Laufer (I$), and Deissler ( 5 ) for flow between parallel plates and those of Nilruradse ( 1 8 ) for flow in a circular conduit were used to evaluate the coefficients in the two folloxing expressions: U + =
=
U
tanh ( z / k ; y + ) = __ tanh ( 0 . 0 6 9 5 ~ ~ ) (1) 0.0695
(7) Equation 7 yields zero values of the eddy viscosity a t the center of the conduit. At present, uncertainty exists as to the proper value of eddy viscosity in this region. Experimental evidence (80, g1) indicates that in a turbulent air stream flowing between parallel plates the eddy conductivity is finite at the center of the channel. I n addition, the direct evaluation of eddy viscosity from experimental data for parallel plates by use of Equation 5 indicates values near the center of the channel that are only slightly smaller than the maximum ( 4 , $1). For these reasons it does not appear desirable to utilize Equation 7 t o establish 1 the total viscosity sma t values of - less than 0.3. Figure 1 prelo
(4) V
sents the relative viscosity, which has been defined as the ratio of the total viscosity to the kinematic viscosity, for flow in circular pipes as a function of radial position for several Reynolds numbers. These estimates are based on experimental indications ( % I ) near the center of the channel and upon Equations 6 greater than 0.3. ro Values of the ratio of total viscosity t o kinematic viscosity as determined from Equations 6 and 7 have been computed as a function of position for Reynolds numbers between 5000 and and 7 for values of
The variations in the “constants” of Equations 1 and 2 as determined from several investigators are considerable (21). There 1
(v)
a
These equations describe the velocity distribution in the laminar, transition, and turbulent regimes for flow between parallel plates and in circular conduits. Equation 1 is applicable a t values of the distance parameter y + less than 26.7, whereas Equation 2 is applicable throughout a large part of the turbulent core. The proposal of Rannie ($4) for describing the velocity distribution in the laminar and transition regions has been employed in Equation 1. The dimensionless terms in this equation may be defined as follows: U tb- = (3)
2
_I_IC cosh2 L0.0695
Present address, Stanolind Oil and Gas Co., Tulsa, Okla. Present address, AiResemoh Manufacturing Co., Los Angeles, Calif.
662
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1953
Values of the eddy conductivity in a turbulent air stream flowing between parallel plates are depicted in Figure 2. These data are based upon direct experimental measurements ( 2 1 ) and correspond to a level of turbulence (6) of 3.8%. The experimentally determined ratio of the eddy viscosity t o the eddy conductivity (21) is presented as a reciprocal function of Reynolds number in Figure 3. I n this instance the average value of this ratio throughout the channel for a particular Reynolds number has been employed because at present it does not appear possible on the basis of existing experimental information ( 2 1 ) to determine with certainty the variation in the ratio of eddy viscosity t o eddy conductivity with position in the flow channel. There is some experimental indication (21) that this ratio decreases slightly as the wall is approached. Direct experimental evaluation of this quantity near the wall is difficult since the ratio becomes indeterminate a t the boundary of the stream.
160
E
663
80
a W
40
=
4"
0.2
0.4 RELATIVE
0.6 POSITION
;
0.8
2k
0.010
LL
n,
Figure 1. Relative Viscosity as Function of Position in Flow Channel
100,000 for flow between parallel plates and in circular conduits. A sample of the data which are available ( 5 ) constitutes Table I. The Reynolds numbers were computed on the basis of hydraulic radius and were evaluated for circular conduits and parallel plates in the following ways:
Re =
; 6.005 2 c
B
0.4
0 2
2r0U
RELATIVE
Va
Figure 2.
410 U Re = -
0.6
POSITION
IN S T R E A M
e
0.8
10
Eddy Conductivity as Function of Position in Flow Channel
YO
The eddy conductivity is defined by
Table 1. Sample of Generalized Values of Relative Viscosity for Flow of Fluids between Parallel Plates or in Circular Conduits Re
Pipes Relative position 0.3000 0.3500 0.4000 0.4500 0.5000 0.5500 0.6000 0.6500 0.7000 0.7500 OI8O0O 0.8400 0,8800 0 . 9200 0 . 9600 0.9800 0.9900 0.8920 0.9940 0.9960 0.9970 0.9980 0.9990 0.9995 1.0000
Relative
vise osity 66.67 72.38 76.19 78.73 79.37 78.73 76.19 72.38 66.67 59.69 50.80 42.54 33.65 23.49 12.06 2.74 1.32 1.20 1.11 1.05 1.02 1.01 1.00 1.00 1.00
-
30.000
Parallel Plates Relative Relative' viscosity position 0.3000 0.3500 0.4000 0.4500 0.5000
33 33 36 19 38.10 39 37 39.68
0.5500 0.6000 0.6500 0.7000 0.7500
39.37 38 10 36 19 33.33 29.84
0.8000 0.8400 0.8800 0.9200 0 9600 0.9800 0.9900 0.9920 0.9940 0,9960 0.9970 0,9980 0.9990 0.9995 1.0000
25 40 21.27 16 83 11 75 2 69 1.31 1.07
40
Figure 3.
Effect of Reynolds Number upon Ratio of Eddy Properties
Application of the data of Figure 3 to a fluid with a Prandtl number which differs greatly from that of air is open t o question. However, it appears that until additional data are available the ratio of eddy viscosity to eddy 'conductivity may be assumed for most gases to follow the relationship shown in Figure 3. From information such as t h a t recorded in Table 1 and the ratio of eddy properties depicted in Figure 3, the total conductivity may be established from
1.04
1.00 1.00 1.00 1.00 1.00
(i)(2) Y
1.02
1.01
60
GEYNOLOS NUMBER)-^ xio6
Eo
=
Y
+K
(11)
A tabulation of the ratios of eddy properties is available (3). As a result of the use of a n average value of the ratio of the eddy
INDUSTRIAL AND ENGINEERING CHEMISTRY
'664
properties some discrepancy is likely to exist between the values of total conductivity computed from Equation 11 and those determined experimentally ($1 ). The rough generalizations which have been described serve only as an approximate means of estimating the total conductivity of uniform, steady streams of gases. Near the center of the conduit Equation 7 is subject to uncertainty in the evaluation of the ratio of total viscosity to kinematic viscosity, and the value of total conductivity can at best be approximated.
Vol. 45, No. 3
reported, 25 circuit elements were employed with resistors of 46.0 ohms and a time-length conversion factor ( 2 5 ) of 1000 inches per second. I n order to determine the temperature a t the boundary of the flowing stream it was necessary to take into account t h t resistance t o heat transfer imposed by the conducting wall and the oil or other heat transfer medium circulating behind it. The overall thermal. resistance of the conducting wall shown in Figure 4 and of the circulating fluid mag be approximated b r the laqt member of the following expression:
Nonuniform Flow In nonuniform steady flow the distribution of temperature may be related to the total conductivity in the following way:
Equation 12 is limited to situations in which changes in elevation, kinetic energy, and pressure in the fluid may be neglected. In the case of two-dimensional, steady, uniform flow between parallel plates, with a nonuniform temperature field, the equation assumes the following simple form:
a
- (Ec
+ K ) aYat = -
uz
bt -
~~p
WAVE
TIMING
(13)
ax
I n this equation the diffusion in the direction of flow and the variation in the physical properties of the fluid with position have been neglected. For many situations the simplifications which were applied in obtaining Equation 13 are justified and offer a simple method of predicting the temperature distribution in a flowing stream. Such an expression is particularly applicable under conditions for which the imposed temperature gradient does not influence materially the physical properties of the fluid.
Figure 5.
Schematic Diagram of Analog Circuit
\Vhen voltage is considercd analogous to tempcraturc., im elcctrical network of the type shovvn in Figure 5 furnishes a 6 0 1 ~ tion to Equation 13 if the derivative
a
-- (ec
ax
+ K ) atax - is written in at
ADIABATIC
WALL
7
terms of finite differences. The derivative - is represented in a
A
I
TTHERMALLY
CONDUCTING
WALL
ax
continuous manner in the analog circuit by the rate of change of voltage with respect to time. The product of the conductive K ) and the velocity uz a t a given and convective transport ( E < distance from the boundary is analogous to the time constant or to the product C,,AR for a given circuit component. The relation between the physical situationand the analog maybe expressed by the following equation ( 2 5 ) :
+
Figure 4.
Schematic Diagram of Two-Dimensional Air Stream
Figure 4 presents a diagram of a physical situation to which Equation 13 is applicable. This figure represents a n air stream which flows turbulently between two thermally insulated walls and which abruptly enters a region between two thermally conducting walls where a temperature gradient is imposed upon the stream. All longitudinal positions were measured with point A of Figure 4 as a reference. Equation 9 may bc solved by a varicty of methods for the boundary conditions indicated in Figure 4. The general field solution recently described (19) in connection with the transient flow of a compressible fluid is a satisfactory but somewhat time consuming approach. Analog methods ( 1 1 , 1.4, 17) afford a somewhat less tedious means of solving such problems. Recently, a general purpose analog computer (15) was applied ( 7 , 25) to this physical situation. I n using these methods it was assumed that the temperature of the body of the fluid circulating in the bath did not change with longitudinal distance. For the work
Equation 15 requires a knowledge of the variation of the total conductivity ( E , K ) with distance from the boundary and this may be established from the information presented in Figure 2 or Table I. This particular analog solution applies to situations in which changes in the physical properties of the fluid with respect to distance downstream may be neglected. This assumption was made since no simple means was available for varying the characteristics of the circuit components with respect to voltage or time, which were analogous to temperature and downstream distance, respectively. If the effect of variation in the physical properties of the fluid need be taken into account, it would be more convenient to employ a two-dimensional network of circuit components in representing the flowing stream. By following a procedure analogous to that which has been described ( 2 5 ) , the electrical resistance, Rw, in Figure 5 corresponding to the reciprocal of the conductivity of the wall may be established from the following equation:
+
R, =
CpuAR
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1953
665
0.9
0.8
0.7
0.6 W
z z a
a
6
0.5
-z Z
Figure 6.
Typical Oscillograph Trace with Timing Line
Equation 16 permits the value of R, to be established for each set of boundary conditions. The values of the thermal transfer coefficient between the circulating fluid in the bath and the inner surface of the conducting wall next to the air stream may be estimated by conventional methods (IS). Figure 6 presents a typical voltage time record obtained with EI double-beam oscillograph. The oscillating trace was employed to record elapsed time on the photographic record. This transcendental function generated by the electrical components shown in Figure 5 has a frequency of 426.2 cycles per second and one cycle corresponds to a downstream distance of 30 inches. Measurements were made a t even-numbered nodes in a network, a part of which is shown in Figure 5, and results similar to Figure 6 were obtained throughout the network.
9 t
0.4
y1
0.3
0.2
0.1
0
Figure 8.
Velocity Distribution in Flow Channel
Experimental Results The information obtained a t node zero of Figure 5 corresponds to the temperature at the wall. Figure 7 depicts the variation in the upper and lower wall temperatures with downstream distance as determined from the analog solution and as measured experimentally. The upper oil bath and the air stream were maintained a t 100" F. and the lower oil bath a t 85 O F., with a gross air velocity of 30 feet per second between the plates. T h e agreement between the experimental and predicted wall temperatures is satisfactory. The predicted curves were based upon estimated thermal resistance ( I S ) of the plate and oil streams shown in Figure 4. 100 LL
UPPER
k 3
WALL-/
I
W
RELATIVE
95
U
ANALOG
w
EXPERIMENTAL
TEMPERATURE
OF
Figure 9. Temperature Distribution in Flow Channel with Upper Oil Bath Temperature at 11 5 " F.
E 90 2
s 85 0
25 DISTANCE
Figure 7.
50
75
DOWNSTREAM
100
125
INCHES
Variation in Wall Temperatures with Downstream Position
The measured velocity distribution a t three different downstream positions is shown in Figure 8 (16). The curve corr6sponds near the wall to the generalized velocity distribution outlines in Equation 1 which is in good agreement with these and other measurements in this flow channel (21). The temperature distribution in the channel was measured at downstream positions which corresponded t o the velocity data. These temperature measurements were made under the same
conditions of flow and a t the same positions as are presented iii Figure 8. The upper oil bath was a t 115" F., the lower oil bath was at 85" F., and the air was introduced a t 100" F. A comparison of the calculated and experimentally measured temperatures a t four downstream locations as a function of position bctween the upper and lower walls is shown in Figure 9. T h e diagram has been presented with a relative temperature scale in which the different curves are arbitrarily displaced from each other for convenience and the center line temperature corresponding t o the analog solution has been indicated on each curve for reference. The average deviation of the experimental data from the calculated values is 0.2' F. Figure 10 presents the temperature distribution at three downstream positions in which the lower oil bath was a t 85" F. and the upper oil bath and entering air stream were both at 100" F. Again the temperatures have been presented in terms of a relative, displaced temperature scale and the temperature corresponding to the analog solution is indicated a t the center line. Good agreement between the calculated and experimental values
666
INDUSTRIAL AND ENGINEERING CHEMISTRY
u 0.9
0.8
u+
= velocity, feet/second = velocity parameter, defined by Equation
z
=
y y+ z
= distance normal to axis of stream, feet = distance parameter, defined by Equation = distance across stream, feet
b
=
A /3
= difference operator
eo 66
0.7
em
n.]R -I W
ern
K Y
0.6
p
u r
2
< U 0.5
0.4
t
2
= = = = = = = = =
3
4
partial differential operator
conversion vector, second/foot eddy conductivity, square feet/second total conductivity, square feet/second eddy viscosity, square feet/second total viscosity, square feet/second thermometric conductivity, square feet/second kinematic viscosity, square feet/second density, pounds (square ~ e c o n d s ) / ( f o o t ) ~ specific weight, pound/cubic foot shear, pound/square foot
Subscripts a average B refers to upper oil bath B‘ refers to lower oil bath n component of network o refers to values a t the solid boundary of the flowing stream 2 ~ ’ refers to wall z refers t o x coordinate direction y refers t o y coordinate direction z refers to a coordinate direction
z
5
=
distance downstream, feet
Vol. 45, No. 3
0.3
0.2
Literaiure Cited 0.I
IS
10
RELATIVE
TEMPERATURE
OF
Figure 10. Temperature Distribution in Flow Channel with Upper O i l Bath Temperature at 100” F.
was obtained. The average deviation for three curves corresponding to 23.2, 50.0, and 82.1 inches downstream distance was 0.12’ F. Similar agreement (26) was found when the walls were both a t 100” F. and the air was introduced a t 85’ F. The methods outlined here indicate a means of predicting the timetemperature history of an air stream in passage through a heated conduit. Under such conditions t h e velocity distribution was assumed to be uniform. It should be realized that such methods are not applicable t o the flow in wakes or other conditions where the exchange of momentum is nonuniform.
Acknowledgment The assistance of H. H. Reamer in carrying out the experimental work and of W. iY.Lacey in reviewing the manuscript is acknowledged.
Nomenclature A C, C, cosh d
f-h h
= constant of proportionality = capacitance of analog circuit component
= isobaric heat capacity, B.t.u./(pound)( O F.) = hyperbolic cosine = differential operator =i
= thermal transfer coefficient, B.t.u./(square foot)( O
K
=
&-
= = = =
1 In
$R T
Re t
= =i
= = =
tanh =
U
Fanning friction factor
= over-all thermal transfer coefficient, B.t.u./(square foot) . ( F.)(second)
=
(second) constant of proportionality thermal conductivity, B.t.u./(second)(feet)( O F.) distance from center of channel, feet natural logarithm total number of increments in analog network thermal flux, B.t.u./(square foot)(second) electrical resistance in analog network, ohms radial distance, feet Reynolds numober temperature, F. hyperholic tangent gross velocity, feet/second
F.)
(1) Bakhmeteff, B. A , , “Mechanics of Turbulent Flow,” Princeton, Princeton University Press, 1951. (2) Boelter, L. M. K., Martinelli, R. C., and Jonassen, F., Trans. Ant. Sac. Mech. Engrs., 63, 447-55 (1947). (3) Connell, W .R., Schlinger, W. G., and Sage, B. H., American Documentation Institute, Washington, D. C.,Doc. 3657 (1952). (4) Corcoran, \T. H., Page, F., J r . , Schlinger, W. G., and Sage, B. H., I K D . ENG.C H E M . , 44, 410-19 (1952). (5) Deissler, R. G., Natl. Advisory Comm. Aeronaut., Tech. Note 2138 (1950). (6) Goldstein, S., “Modern Developments in Fluid Dynamics,” Vols. I and 11,Oxford University Press, 1938. (7) Jenkins, R., Brough, H . W., and Sage, B. H., IND.ENG.CHEM., 43, 2483-6 (1951). (8) Jurges, W., Gesundh.-Ing., 1, Suppl. 19, 1 (1924). (9) KBrm&n,T h . von, J . Aeronaut. Sei., 1, No. 1, 1-20 (1934). (10) KBrmBn, T h . von, Trans. Am. Sac. Mech. E’nais., 61, 708-10 (1939). (11) Kayan, C. F., Ibid., 71, 9-16 (1949). (12) Laufer, J., Katl. Advisory Comm. Aeronaut., Tech. Note 2123 (1 9.50). ~-...,
(13) IClcAdams, IT. H., “Heat Transmission,” New York, McGrawHill Book Co., 1942. (14) McCann, G. D . , and Wilts, C. II., J . A p p l . Mechanics, 16, 24768 (1949). (15) McCann, G. D., Wilts, C. H., and Locanthi, B. X., Proc. I . R. E. (Inst. Rccclio E n g r s . ) , 37, 954-61 (1949). (16) Mason, J. L., Ph.D. thesis, California Institute of Technology (1950). (17) Moore, A. D., J . A p p l . Mechanics, 17, 291-8 (1950). (18) Nikuradse, J., Forsch. Gebiete.Ingenieuru., 3, Suppl., Forschungsheft 356,l-36 (1932). (19) Olds, R. H., and Sage, B. H., Petroleum Technol., 192, 217-22 (1951) (Tech. Publ. 3081). (20) Page, F., Jr., Corcoran, \V. H., Schlinger, W.G., and Sage, B. H., IXD.EKG.CHEM.,44,419-24 (1952). (21) Page, F., Jr., Schlinger, 55’. G., Breaux, D. K., and Sage, B. H.. Ibid., 44, 424-30 (1952). (22) Pannell, J. R., Tech. Rept. Natl. Advisory Comm. Aeronaut., Memo. 243,22-30 (1916). (23) Prandtl, L., “Der Masseaustausch in freier Luft und verwandte Erscheinungen,” Vienna, Hamburg, 1925. (24) Rannie, W. D., J . Aeronaut. Sci., in press. (25) Schlinger, W. G., Berry, V. J., Mason, J. L., and Sage, B. H., Inst. Mech. Enols. (London), J . , in press. (26) Skinner, G. T., thesis, California Institute of Technology, 1950.
ACCEPTEDNovember 24, 1952. RECEIYED for review June 23, 1952. F o r material supplementary t.o this article order Document 3657 from American Documentation Institute, Library of Congrew, Washington 25, D. C., remitting $1.00 for microfilm (images 1 inch high on standard 35-mm. motion picture film) or $4.65 for photocopies (6 X 8 inches) readable without optical aid.
