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
dP
ax
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
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1952
425
Figure 1 presents the temperature distribution in the Test Numbers flowing stream for an average 4w 37a 41 43 44 Units Quantity velocity of 15 feet per second 0.0571 0.0567 0.0568 0.0570 0.0564 Foot Distance between plates and a temperature difference 12.5 12.5 12.5 12.5 8.1 Feet Traverse looationb 100.0 100.0 100.0 100.0 100,o F. Incomin air temperature between the plates of approxi104.5 104.7 104.6 100.0 100.0 F. Upper & L e temperature 95.4 95.3 100.0 100.0 95.4 F. Lower plate tsmperature mately 30" F. These condi.O 61 28.13 88.7 88.4 28.94 Ft./sec. Avera e velocity 37300 tions are identified as test 46 17100 53200 52500 17500 ReynoTds number -0.868 -1.580 -0,216 -1.628 -0.230 Lb.jcu. ft. Pressure y d i e n t c in Table I. The corresponding 0.0126 0 0.0156 0,00730 0 B.t.u./(sq. ft.) (sec.) Thermal ux 0.0081 0.014 0.0128 0.0139 0.0058 Wt. fraction water velocity distribution for this 14.297 14.315 14.073 14.108 14.335 Pressure at traverse location Lb,.isq. in. 14.310 14.312 14.103 14.148 14.340 Lb./sq. in. Barometric prewure condition is shown in Figure 2. Test Numbers The data obtained for the other 45 46 48 49 50 conditions set forth in Table I 0.0576 0.0574 0.0575 0.0580 0.0574 Foot Distance between plates were comparable in detail and 12.5 12.5 12.5 12 5 12.5 Beet Trsverse locationb 100.0 100.0 100.0 100.0 100.0 F. Incoming air temperature precision to those presented in 114.4 100.0 100.0 104.7 100.0 F. Upper plate temperature 15.7 100.0 100.0 100.0 Figures 1 and 2. 95.2 F. Lower plate temperature 15.29 14.84 11.32 14.77 58.7 Ft./sec. Average velooity From data of the type re9370 9110 6960 8980 36400 Reynolds number -0.0642 -0.0370 -0.774 -0.0693 0.0680 Lb.jCU. f t . Pressure gradiente corded in Table 111, values of 0.0139 0 0 0.0045 B.t.u./(sq. ft.)(sec.) 0 Thermal flux 0.0118 0.0114 0.0149 0.0154 0.0156 Wt. fraction water the eddy conductivity and eddy 14.199 14.282 14.277 14.156 14.265 Pressure a t traverae location Ll;.jsq. in. viscosity were established for 14.203 14.285 14 283 14.159 Lb./sq. in. 14.261 Barometric pressure the conditions described in a This material reproduced from referenoe (16) for completeness. b Traverse location measured from end of converging section. Table I by application of c Pressure gradient is average of change in static pressure over +foot length of working section approximately 10 Equations 1 and 2. Eddy confeet downstream from end of converging section. It is measured with traversing mechanism downstream from static t R p S . No significant change with time was observed. The values recorded were correlated with data obductivities were not obtained tained a t other flow conditions, for isothermal c o n d i t i o n s since the thermal flux was TABLE 11. STATUSOF EXPERIMENTAL MEASUREMENTS UPON substantially zero. The eddy UNIFORMFLOW conductivities and viscosities were smoothed with respect t o Gross Temperature Difference, F. position in the flowing stream and the results are given in Table Velocity, Ft./Seo. 0 10 30 60 IV. Figure 3 presents values of the eddy conductivity and the Teat" Ref.b Test Ref. Test, Ref. Test Ref. eddy viscosity as a function of position in the flowing stream for 33 (16) 43 44 6 90 test 46. 32 (16) 50 .41 60 Near the axis of flow the eddy viscosities are subject to larger 31 (16) uncertainties and become indeterminate at the axis when directly evaluated from Equation 2. For this reason the curve for eddy 15 48 C 45 46 0 30 (16) viscosity in Figure 3 has been dotted in this region. Some lack 10 49 e of symmetry about the horizontal axis of the flow channel was Av. temperafound in the eddy values derived directly from experimental data. ture radient, 0 %./ft. o 170 510 1030 This asymmetry apparently resulted in part from gradual minor For aescription of test conditions, see appropriate table of reference. changes in flow conditions duringa partioulrtr set of measurements. b Reference listed a t end of paper. Such small drifts in experimental conditions induced significant 0 Bee Table I of this paper.
TABLE I. EXPERIMENTAL CONDITIONS
O O
-
C
Q
study is presented in Table 111. Similar information for all of the conditions indicated in Table I is available (IS). This includes measurements made at a point 8.1 feet downstream. The major part of these latter measurements has not been employed in this discussion.
i &a
OIZ
Figure 1.
Temperature Distribution in Stream for Test 46
TABLE 111. SAMPLE OF EXPERIMENTAL DATA TEST46 Temperature, Y/UO
F.
0.990
112.98 112.72 111.45 110 71 109.69 109.16 108.66 107.75 106.55 105.87 105.30 104.30 103.56 102.79 101.93 101.22 100.35 99.47 98.58 97.79 96.97 96.23 95.29 94.58 93.79 93.28 92.22 91.63 90.40 89.67 88.57 87.20
0.986
0.977 0.971 0.961 0.954 0.949 0.935 0.906 0.877 0.848 0.789 0.733 0.675 0.614 0.559 0.501 0.443 0.385 0.327 0.267 0.212 0.164 0.123 0.096 0.081 0.058 0.048 0.035 0.029 0.022 0.015
Velocity Ft./Bec.' 2.89 3.34 4.97 6.43 8.20 9.30 10.48 11.75 13.62 14.54 15.28 16.22 16.83 17.46 17.83 18.00 18.20 18.01 17.80 17.44 16.92 16.32 15.37 14.61 13.60 12.94 11.34 10.17 8.34 6.80 4.93 2.96
INDUSTRIAL AND ENGINEERING CHEMISTRY
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Vol. 44, No, 2
D
I O S
$
Ofi
oA
i
E
O2
BO 120 VELOCITY FEET PER SECOND
40
Figure 2.
Velocity Distribution for Test 46 1.0 20 E[XN CONDUCTIVITY A M ) VISCOSITY,
variations in the computed eddy values. The velocity and temperature in the flow stream are functions of the weight rate of flow and the temperature of the plates, both of which vary with time. This fact may be expressed by the following statements of the general equation of partial differentiation:
SOUARE F E E T P E R
Figure 3.
3.0
E
I
,'OI
SECOND
Eddy Conductivity and Eddy Viscosity for Test 46
conditions. It follows that Equations 4 and 5 reduce to
The data which are presented in Table IV have been adjusted to a symmetrical basis about the horizontal axis of the flow channel. The symmetrical values obtained by averaging the experimental data corresponding to points equidistant from the horizontal axis were cross-plotted and smoothed with respect to Reynolds numbers. The symmetrical data are shown in Table V. It is possible that a part of the lack of symmetry indicated in Table IV resulted from the configuration of the equipment (3). However, for the purpose of further analysis of the data, the symmetrical patterns of Table V have been employed. No eddy viscosities have been recorded for values of 1/20 of 0.0 and the
I n the present experimental work it was assumed that measurements were obtained under two-dimensional steady uniform (18)
TABLEIV. VALUESOF EDDY VISCOSITY AND EDDYCONDUCTIVITY X lo3 Test Numbera V/YQ
37 ern6
40 em
41 CE*
tm
43 eo
ern
440 tm
45 €E
tm
46 tc
fm
0.98 0.96 0.94 0.92 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30
(0.lS)O (0.21)c (0.8O)C (1.14)C (0.90)* (2.06) (0 . O l ) O (0.10)O (1:ie) 0.88 1.22 2.29 2.75 3.18 (4.'64)c 3.88 (o:is)c 0.30 (o:i4)0 0.30 0.59 1.58 2.08 3.60 4.18 4.95 5.71 5.74 0.39 0.64 0.35 2.26 2.92 4.58 5.34 6.30 6.70 7.42 1.05 0.68 0.95 3.00 2.23 0.70 2.79 3.57 5.27 6.30 7.50 7.56 8.90 1.15 1.50 1.15 1.40 3.64 4.02 3.79 7.85 9.03 4.21 4.40 8.58 9.75 4.13 4.66 9.00 9.97 9.17 9.61 4.69 4.00 9.38 4.45 8.95 3.86 8.37 8.95 4.01 3.75 (3.90) (3.59) 4.67 (6.85) 7.79 (9.48) (8.95) 9.59 (2.04) 2.95 (1.99) 3.00 4.51 7.69 9.52 2.69 2.84 (2.'98) (3150) 4.50 (6:41) 8.09 (9:52) (8:95) 9.76 (2:07) 2.68 (2102) 2.89 3.42 3.83 4.68 6.32 8.75 9.69 10.38 10.25 2.17 2.82 2.19 3.08 3.71 4.35 5.12 6.46 9.30 10.00 10.70 11.14 2.30 3.05 2.45 3.28 4.00 4.55 5.58 6.78 9.65 10.45 10.87 12.73 2.44 3.23 2.64 3.40 9.82 4.21 10.57 9.80 10.26 4.27 9.36 9.48 4.03 8.21 7.05 3.19 5.64 7.22 (0.94) 4.12 5.18 .. 0.04 .. 1.22 1.50 1.99 2.40 3.33 (2.87) 3.88 (0.14) 0.25 (0.18) 0.35 0.02 , (0.39) (0.41) (0.45) (0.58) (0.90) , (1.25) (0.04) (0.04) Eddy viscosity for teat 44 oomputed from traverse data obtained 8.1 feet downstream from end o€ converging section. Eddy viscosity and eddy conductivity expressed in square feet per second. Values in parentheses are subject to larger uncertainties than the other data.
.
b 6
48 EC
.
..
..
ttn
49 Ern
..
50 em
(0.34)O (o:iwc .. 2.36 (0.06)c 3.75 0.46 0.76 0.19 4.73 0.38 5.46 1.12
(2.03)
(1.39)
(7.48)
(1.'81)
1.83 1.95 2.16
(l:33) 1.40 1.46 1.49
(6:?6) 6.62 6.63 6.79
(0.16)
... .
(2.42)
..
..
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1952
Teat Numbers 1/10
48
46
49 em
cm
0.96 0.92 0.88 0.84 0.80 0.70 0.60 0.60 0.40 0.30 0.20 0.10
0:26
0:i)8
0.69 1.01
0.27 0.46 0.86 1.17 1.43 1.63 1.62 1.46 1.4
..
..
0.00
number of significant iigures was reduced for values of 1/L leas than 0.2 as a result of uncertainty in the evaluation of Equation 2 in this part of the channel.
42Z
so
Figure 4 presents the data for tests 37, 43, 48, 49, and SO on a semilogarithmic 0.6 diagram with u + and y + as ordinate and 2.26 3.66 abscissa. Some variation with Reynolds 4.77 6.62 number is indicated for the lower 7.02 velocity. T h e solid curve for the laminar 7.81 8.12 flow near t h e wall has been made single7.96 valued b y assuming the shear constant, 7.69 7.27 aa was done by von Kdrmdn (8). For 7.1 .. comparison, a c u r v e r e p r es e n t i n g laminar flow between parallel plates at a Reynolds number of 2000 with values of shear, which waa established from Equation 1, has been included. The expression for this condition of flow is 4-
-A]
u+LI+[l
ISOTHERMAL FLOW
As indicated in Table 11,a series of measurements were made at on upper and lower plate temperature of 100' F. for nominal gross velocitia between 10 and 90 feet per second. Under these nearly isothermal conditions and relatively low velocities the thermal flux was negligible (13). The only temperature gradients are those necessary t o transfer from the flowing stream t o the walls the energy resulting from the friction associated with the
I
2
I
5
I
10
I
20
MSTAKE
I
x)
PAWMETER
I 1w
I 2w
I
Mo
I
1x0
Y'
Figure 4. Experimental Velocity Distributions Shown upon Generalized Coordinates
A marked difference between the two curves for laminar flow is obtained as a result of taking into account the change in shear with position. A comparison of the present data with those of other investigators is shown in Figure 5. T h e data of Figure 4 fall within the range of values that have been reported for the flow of air between
I
I
I
2
I J
I 0
20
I 50
DISTAXCE P&R@METER
I Kx)
x10
5cQ
Y'
Figure 5. Comparison of Velocity Distribution as Determined by Several Investigators
I loo0
VOl. 44, No. 2
INDUSTRIAL AND ENGINEERING CHEMISTRY
428
8.0
6.0
40
Q
b
TEST49 EST%
EE-
20
cw5 RELATNE
aio
ox,
I
POSITION IN CHANNEL
I 0.2
04
Figure 6. Comparison of Experimentally Determined Velocity Deficiencies
I
I
OB
POSITION IN STREAM
OB
!/I*
Figure 8. Influence of Temperature Gradient upon Eddy Viscosity
I
1
I
J
0.2 0.4 0.6 POSITION I N S T R E A M
p/is Figure 7. Comparison of Experimental and Predicted Eddy Viscosity
parallel plates (11, 20, 81). In addition, the measurements of Nikuradse (14) and Deissler (6) for flow of water and air in circular tubes are depicted. At Reynolds numbers above 20,000 the present data are in good agreement with the measurements of Deissler (6) and Nikuradse (14) and are bracketed by the work of Skinner (20) and Wattendorf (21). At lower velocity near the transition region the curves are no longer independent of Reynolds number but approach the laminar values calculated from Equation 10. The velocity deficiency (1) for tests 37, 43, 48, 49, and 50 is shown in Figure 6. The values obtained by Skinner (20)and Nikuradse (14) have been depicted. For comparison, the behavior of a laminar stream a t a Reynolds number of 2000 has been included. The analytical expression for this type of flow is
The present data are in good agreement with the measurements of Nikuradse and Skinner near the center of the channel. A number of investigators have suggested relationships which
POSITION IN STREAM
Figure 9. Eddy Conductivities at Gross Velocity of 15 Feet per Second
establish the variation of the eddy viscosity with the position in the flowing stream. The proposals of Prandtl (17), von K B r m h (IO), and Gebelein ( 6 ) are of particular interest. I n Figure 7 is presented a comparison of the values predicted by the two latter proposals with the present experimental data for a gross velocity of 90 feet per second corresponding t o test 43. It appears that the predictions of Gebelein agree most closely with this experiment. The formulations of Prandtl, von KBrmBn, and Gebelein indicate a sharp minimum of zero in the eddy viscosity a t the axis of flow. Each of the correlations yields a value of zero a t this point and a discontinuity in the first derivative of the velocity with transverse position in the flow channel. NONISOTHERMAL FLOW
A series of measurements a t a nominal velocity of 15 feet per second have been made for average temperature gradients of Oo, 170°,and 510' F. per foot corresponding to telts 48,45, and 46. I n Figure 8 is shown the influence of this temperature gradient upon the eddy viscosity a t a point 12.5 feet downstream from the entrance to the uniform section. The changes in eddy viscosity
INDUSTRIAL AND ENGINEERING CHEMISTRY
February 1952
1.0
429
I
WALL
I
WSITION
IN
STREAM
-
t
Figure 11. Influence of Reynolds Number on Turbulent Prandtl Number MLOUN
FEET PER SfCOND
Figure 10. Variation of Temperature with Velocity for Test 46
shown in Figure 8 between one temperature gradient and another are believed t o be insignificant. It also appears t h a t the values of the eddy conductivity are not influenced b y a transverse temperature gradient. This independence is depicted in Figure 9 where the eddy conductivity is shown as B function of position for average temperature gradients of 170' and 510' F. per foot. Figures 8 and 9 indicate that transverse temperature gradients as high as 500' F. per foot do not affect significantly the value of the eddy properties. I n the central portion of the stream, values of the ratio of e,,,/~~ and were obtained from the smoothed data tabulated in Table V. Difficulties occur in establishing the former ratio from Equations 1 and 2 near the wall where relatively rapid changes in the temperature and velocity are encountered. The eddy conductivity and eddy viscosity become zero a t the wall, and the direct establishment of their ratio leads to an indeterminate value. Since a substantial part of the thermal resistance is located in the boundary layer, it is of interest to establish this quantity with some certainty near the wall. The turbulent Prandtl number may be determined from the following expression which is obtained by a combination of Equations 1and 2 under conditions of uniform flow:
I n Figure 10 is presented the variation of the velocity and temperature for test 46 based upon the data in Table 111. The temperature gradients and shear imposed upon the stream markedly influence the slope of the curve in this figure. From Equation 12 i t is apparent that near the wall the nearly constant values of the velocity-temperature gradient yield substantially constant values of Prt. This ratio would be expected t o approach the Prandtl number a t the laminar boundary layer and t o remain constant throughout that region. I n Table V the values of :JfC are presented as a function of position for each of the several conditions which are recorded in Table I. The uncertainty associated with the establishment of this ratio is estimated t o be 8%. For values of I/& near unity the ratio was established from Equation 12 and curves of the type shown in Figure 10, while for values of Z/& less than 0.7 the ratio was derived from the smooth eddy properties presented in Table V. The effect of transverse position in the stream upon the turbulent Prandtl number is shown in Figure 11 for several Reynolds numbers. This figure indicates a consistent variation with the Reynolds number which more than compensates for the decreasing importance of the molecular properties of the fluid
9
5
WALL
---)
0.4 POSITION IN STREAM
t
Figure 12. Ratio of Eddy Viscosity to Eddy Conductivity
a t the higher velocities. The standard deviation of the simple curves of Figure 11 from the data of Table V is 6.201,,which is comparable t o the estimated total uncertainty of the measurements. The ratio of the eddy viscosity t o the eddy conductivity is recorded in Table V. Simple curves corresponding t o those presented in Figure 11 are shown in Figure 12. This figure contains the value of the ratio obtained from data involving several velocities ranging from 15 t o 90 feet per second and average temperature gradients between the walls varying from 170" t o 510' F. per foot. The variations in this ratio caused by changes in the average transverse temperature gradient are believed t o be insignificant. However, ,the ratio does appear t o approach unity a t the higher Reynolds numbers in the central portion of the stream. The values of the ratio presented in Figure 12 may involve uncertainties from 5 t o 10%. T h e larger error applies t o the region near the wall where the ratio reaches an indeterminate limiting value. I n this region the ratio em/ec was established from the values of Prr obtained from Equation 12 by use of the equation
The values of eo were taken from Table V. The appropriate values of K and Y together with other physical properties of air were obtained from Table I of an earlier study (26). ACKNOWLEDGMENT
The assistance of H. H. Reamer in carrying out the reported experimental measurements was of benefit t o the program. The
INDUSTRIAL AND ENGINEERING CHEMISTRY
430
help of F. H. Wright and C. L. Thiele of the Jet Propulsion Laboratory of the California Institute of Technology isacknowledged in connection with the measurements of the axial fluctuating velocities. The constructive criticisms and suggestidas of H. W. Liepmann and W. N. Lacey in connection with the review of this manuscript are acknowledged. NOMENCLATURE
isobaric heat capacity, B.t.u. per (pound) ( ” F.) = differential acceleration of gravity, feet per square second distance from center linemormal to flow, feet = distance from center line to boundary of channel, feet thermal flux, B.t.u. per (square foot) (second) = weight rate of flow, sounds per second pressure, pounds per square foot = temperature, ” F. = velocity, feet per second = velocity parameter, u/u* p feet per second = friction velocity, m = maximum velocity, feet er second = distance along channel, Peet = distance from lower wall normal to axis of flow, feet = distance from nearer wall = distance between channel walls, feet = distance parameter, YdUL / v = distance from side of channel, feet = eddy conductivity, square feet per second K, square feet per second 3 total conductivity, cc = eddy viscosity, square feet per second total viscosity, srn Y , square feet per second = time, seconds thermometric conductivity, square feet per second = = kinematic viscosity, square feet er second = density, pounds (square seconds7 per (foot)4 specific weight, pounds per cubic foot = shear, pounds per square foot = shear a t boundary of channel, pounds per square foot = turbulent Prandtl number = Reynolds number
CP =
d
f -:
l o
Q =
L t u u+ u* Urn
x
Y vd
yo
2/+
z 4
-e,
“= em
e K
v P 0
r 70
Pn Re
+
+
LITERATURE CITED
(1) Bakhmeteff, B. A,, “The Mechanics of Turbulent F l o ~ , ” Princeton, N. J., Princeton University Press, 1941. (2) Bridgman, P.W., “Dimensional Analysis,” New Haven, Conn., Yale University Press, 1937. (3) Corcoran, W. H., Page, F., Jr., Schlinger, W. G., and Sage, B. H.. IND.END.CHEM..44. 410 (1952). (4) Corcoran, W.H., Roudebush,‘B., and Sage, B. H., Chem. Erie. Progress, 43, 135-42 (1947). (5) Deissler, R. G.,Natl. Advisory Comm. Aeronaut., Tech. Note 2138 (1950). (6) Gebelein, H., “Turbulens,” Berlin, Julius Springer, 1935. (7) Jakob, M., “Heat Transfer,” Vol. 1, New York, John Wiley & Sons, 1949. (8) von K&rm&n,Th., J. Aeronaut. Sci., 1 No. 1, 1-20 (1934). (9) von KBrm&n,Th., Mech. Eng., 57, 407-12 (1935). (10) von K&rm&n,Th., Trans. Am. Soc. Mech. E m s . , 61, 706-10 (1939). (11) Laufer, J., Natl. Advisory Comm. Aeronaut., Tech. Note 2123 (1950). (12) McAdams, W. H., “Heat Transmission,” New York, McGrawHill Book Co., 1942. (13) McAdams, W. H.,Nicolai, L. A., and Kennan, J. H., Z ’ T ~ ~ O . Am. I w l . Chem. Engrs., 42,907-25 (1946). (14) Nikuradse, J., Forsch. Gebeite Ingenieurw., 3, supplement, FOTSChUWShej?, NO. 356, 1-36 (1932). (15) Page, F., Jr., Corcoran, W. H., Schlinger, W. G., and Sage, B. R., IND.ENG.CHEM.,44, 424 (1952). (18) Page, F., Jr., Schlinger, W. G., Breaux, D. K., and Sage, B. R., Washington, D. C., Am. Doc. Inst., Doc. No. 3294 (1951). (17) Prandtl, L.,Physik. Z . , 29, 487-9 (1928). (18) Rouse, H., “Fluid Mechanics for Hydraulic Engineers,” New York, McGraw-Hill Book Co., 1938. (19) Sherwood, T. K.,and Woertz, B. B., Trans. Am. I w t . Chem. Engrs., 35, 517-40 (1939). (20) Skinner, G., thesis, Calif. Inst. of Technology, 1950. (21) Wattendorf, F. L., and Kuethe, A. M., Physice, 5, 153-64 (1934).
.-
-
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I
RECEIVED November 7, 1950. For material supplementary to this article order Document 3294 from American Documentation Institute, 1719 N St., N.W., Washington 6, D. C., remitting $1.00 for miorofilm (images 1 inch high on standard 35-mm. motion picture film) or $3.60 for photocopiea (6 x 8 inches) readable without optical aid.
Subscripts 1 = lowerplate u = upperplate
EngFnering
Vol. 44, No, 2
Precoat Filter Filtration of Phosphate-
Process development
Defecated Affination Sirup I
L. E. WEYMOUTH
AND
R.
S. MONTGOMERY’
JOHNS-MANVILLE RESEARCH CENTER, MANVILLE, Ne J.
W
HILE the use of diatomite filter aids in the clarification of affiation sirup is widely adopted in the sugar refining industry, i t is a relatively difficult type of filtration, because of the character and amount of colloidal impurities present. In the treatment of afEnation sirup generally, filtration rates are low, filter cycles short, and filter aid consumption relatively high. The treatment of a f i a t i o n sirup has been considered as a possible useful application of the Oliver Precoat filter, since this type of filter is adapted to the handling of many liquids that are difficult to filter. I n this filter, a precoat of straight diatomite filter aid is formed on a rotating drum under vacuum until a layer Present address, The Daw Chemical Co., Midland, Mich.
of as much as 2 inches in thickness is formed. In use, the turbid liquid to be clarified is passed through this rotating precoat layer, from the surface of which a scraper, automatically advancing with each revolution of the drum, removes the accumulated solids plus a very thin layer of the precoat itself, and thus presents a fresh filtering surface to the sirup. A preliminary study of the Oliver Precoat filter filtration of affination sirup in this laboratory was described by Cummins and Morris (1). This earlier work indicated definite advantages for the use of this filter in the processing of afKnation sirup with use of phosphate treatment. The phosphate-defecated sirup showed both better clarification and faster flow rate than was obtained
