ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Correlation of Turbulent Velocities for Tubes and Parallel Plates R. R. ROTHFUS
AND
C. C. MONRAD
Carnegie Institute of Technology, Piftsburgh, Pa.
V
ELOCITY distributions in fluids flowing in turbulent isothermal motion through smooth round tubes have often been presented in terms of two dimensionless parameters,
and
Very near the wall of the tube, where a thin laminar film is assumed to exist, it follows immediately from the definitions of the parameters that, if the change in shearing stress across the narrow film can be neglected, (3)
u+ = Y +
Furthermore, the theory of Prandtl ( 6 ) , based on assumptions which are difficult to defend, predicts that in the main stream of the turbulent fluid,
u+ = A
+ Bln y+
(4)
I n view of Equations 3 and 4, it would be reasonable to hope that the relationship between u and y + might prove to be unique and independent of the bulk Reynolds number, D V p / p . Unfortunately. however, such is not the case. This article presents a modification of the u+, y + relationship which does result in a unique correlation of the main stream velocity distribution in smooth tubes and extende the correlation to the case of flow between parallel flat plates. Such a correlation effectively eliminates the necessity for interpolating tube data obtained a t various Reynolds numbers and establishes a reasonable basis for extrapolating the very limited number of flat plate data available a t the present time. +
An empirical method for modifyipg the conventional diagram in order to make the relationship between the coordinates unique is suggested by inspection of Figure 1. It is apparent that the necessary correction can be made through multiplying one or both of the coordinates by a factor which is essentially unity a t high Reynolds numbers and departs from this value with progressive rapidity as the Reynolds number is decreased. The ratio of the bulk average velocity to the maximum local velocityi.e., center line velocity-is such a function. The result of multiplying u+ by V / u , and y + by u,/V is shown in Figure 2. The main-stream relationship between the modified coordinates is essentially a single line over the entire turbulent range of 3000 < ATR, < 3,240,000 covered by the experimental data. There are too few data at values of the abscissa less than 30 to permit an accurate evaluation of the buffer layer profile. It is obvious, however, that while such a correlation is unique in the main portion of the stream, it cannot be unique in the immediate vicinity of the wall if a laminar layer is actually present in the latter region. More experimental information is necessary before behavior near the wall can be established firmly. The values of V / u , used in modifying Figure 1 are relat,ed to the bulk Reynolds numbers in the manner shown in Figure 3. Also included in Figure 3 is the relationship between V/u, and the Reynolds number, Du,p/p, based on the maximum rather than bulk average velocity. The following example illustrates how Figures 2 and 3 may be used. Illustrative Problem 1. Water a t 70" F. flows steadily and isothermally in a smooth 0.50-inch-inside-diameter tube a t a bulk average velocity of 2.50 feet per second. Calculate the mean local fluid velocity at a point in the stream 0.15 inch from the tube wall. For water a t 70" F., p =
Modifications of pararnethrs remove Reynolds number effect on velocity distributions for flow i n tubes
p =
It has been customary to represent the isothermal turbulent velocity distribution in a smooth tube by three lines on u+,y 4 coordinates:
< y + < 5.5 (laminar film) 5.5 < y + < 30 (buffer layer)
0
y+
> 30 (turbulent
core)
u+ = Y + u+ = -3.05 u+ = 5.5
+ 2.4 In y +
+ 2.5 In y +
62.3 pounds per cubic foot
(3) (5)
From curve a of Figure 3, V / u , = 0.740 The Fanning friction factor obtained from Seneca1 and Rothfus (11)a t a bulk Reynolds number of 4160 is
(6)
f = 0.0098
I n representing experimental data in this manner, however, the effect of Reynolds number on the correlation is overlooked. Figure 1 indicates that the relationship between u+ and y + in the main stream of the fluid is appreciably affected by the Reynolds number, especially in the range 3000 < NRe< 25,000. There are not enough accurate velocity data available a t the present time to establish the situation existing within the buffer layer and laminar film. Recent dye experiments (7, 9) have indicated, however, that the value of y + a t the edge of the laminar region is strongly dependent on Reynolds number and that the buffer layer is similarly influenced a t least as much as the turbulent core. 1144
0.00156 pound per (second)(foot)
:.
u* = V d r 2
and
'+
(?)p
From Figure 2
:.
u
=
u+u*
(2.50) d
=
=
9 = 0.175 foot per second
(0.15)(0.175)(62.3) = 18 (12)(0.00156)(0.740)
U+
y+>
m
(g)
=
13.6
=
(13.6)(0.1875)/(0.740)= 3.22 feet per second
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 6
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
30
20
f a w
LI U
a
2
-
IO
0
sw >
0 I
IO
Figure 1.
= =
+
u* ( 5 . 5 2.5 In y+) (0.175)(5.5 2.5 In 87) = 2.92 feet per second
= u + u* =
Interpolation of Figure 1
Procedure allows extrapolation of data
There is very little experimental information about fluid velocity distributions in conduits of noncircular cross section. This fact makes i t a practical necessity to formulate a workable means of extrapolating and interpolating the few data that are available. As shown in Figures 1 and 2, a considerable number of consistent points have been obtained in the main stream of fluids flowing in tubes. It seems reasonable, therefore, to seek a relationship between flow in tubes and flow in other types of conduits. The flow of a fluid between two parallel flat plates offers a satisfactory situation for comparing circular and noncircular cross sections. I n viscous flow, the shearing stress distribution is linear and the velocity distribution is parabolic with distance from the center line, just as in a tube. The position of maximum fluid velocity is fixed at the center of the stream by symmetry. Furthermore, the velocity and friction data of Sage and coworkers ( I , d , 6, IO) provide a firm experimental picture of flow in the lower turbulent range. Consider, first, the steady viscous isothermal flow of an incompressible fldid through the space between two parallel flat
I_-_I
Tp
!rO)F
= __
b
(7)
and the velocity distribution
(18.6)(0.175) = 3.25 feet per Recond
on flow between parallel flat plates
June 1955
plates separated by the distance, 2b. Integration of the NavierStokes equation yields the shearing stress distribution
+
which means a difference of 9.3%. yields a U + of 18.6, from which
u
I o5
yt
Effect of Reynolds number on u+, y f correlation of velocity distribution in smooth tubes
The same velocity calculated by means of Equation 6 is
u
I o4
I 03
IO2
FRICTION DISTANCE PARAMETER
where r is the distance from the center of the stream to the point a t which r and u are measured. For similar flow in a tube of radius rot the equation of motion yields Tp
=
(7o)P
ro
(9)
and
Therefore, if the tube is chosen to be of such size that rg = b and if, in addition, the same fluid flows in each conduit, the velocity distributions in the tube and between the flat plates must neces, the same sarily be coincident providing the skin frictions, T ~ are in both cmes. As pointed out b y MacLeod (W),it would be reasonable to postulate that the same behavior might occur in fully turbulent flow -that is, a coincidence of velocity profiles might be expected when ro = b, pp = p p , p p = p p and TO)^ = TO)^ If such an assumption is actually valid, it follows immediately that (U* ) P
=
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
(U* ) F
(11)
11-45
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
30
a
c
E
20
IO
0 I
IO
I os
I 04
I 03
IO2
MODIFIED FRICTION DISTANCE PARAMETER, y+(um/V).
Figure 2.
Modified uc, y + correlation of velocity distribution in smooth tubes
lent tubes were taken from Rothfus and coworkers (8, 11). Values of ( V/U,)Pwere obtained from Seneca1 and Rothfus ( 1 1 ) and Stanton and Pannell(12). Friction factors predicted by the hydraulic radius concept are also included in Table I for purposes of comparison. Equations 19 and 16 yield friction factors that agree with those obtained from the data of Sage and coworkers within ordinary experimental precision. The values calculated from hydraulic radius deviate by greater amounts.
Furthermore, because
in each conduit and because the maximum local velocities must be the same if the profiles are actually coincident, it can be concluded that
Table 1. ( N R ~ F
The bulk Reynolds numbers, defined in the usual manner, are
6,980 9,110 17,500 36,400 53,200
Av. absolute deviation
and
At the condition of coincident velocity distributions, the two Reynolds numbers must be related by the equation
Sage and coworkers have measured pressure drops between flat plates in the range 6960 < (NR,)a < 53,200. Table I shows a comparison of Fanning friction factors calculated directly from these data with those calculated by means of Equations 15 and 16. I n the latter computations, friction factors for the equiva-
1146
Comparison of Friction Factors Sage and coworkers 0.0077 0.0078 0.0072 0.0063 0.0056
....
Fanning Friction Factor (fp) Equations Hydraulic 15 and 16 radius 0.0078 0.0088 0,0075 0.0081 0.0069 0.0068 0.0059 0.0056 0.0053 0.0050 0.0003 0.0006
The hydraulic radius concept predicts equal values of f for tubes and flat plates a t equal values of the bulk Reynolds numbers, ( N R e ) p and ( N R e ) p . At sufficiently high Reynolds numbers, where there is little difference between (V/u,)p and (V/ u&, Equations 15 and 16 indicate equal values o f f at the condition (NRd)p= 2(NRe)P. Since the friction factor does not vary greatly with Reynolds number in the highly turbulent range, it is not surprising that hydraulic radius is reasonably accurate for such a case. In the lower turbulent range, however, the variation in V / u , values with Reynolds number causes the amount of deviation shown in Table I.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 4'1, No. 6
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 0.94
0.90
0.86
IHii
0.8 2
*
REFERENCE (3) NRE
0
€0.78
5
0
5
a 0.74
*
i
!z
u
3 w
DATA OFSAGE ETAL. NRE
6960
> 0.70 I
I
I
I
l
I
l
I
I
I
1
I
I
0
0.6 6
I03
17,500 34,600 53,200
I 05
I04
0.3002 -0
4085
Q
3062
REFERENCE (8)
x
,+
NRE 6210 24,500
IO6
23,300 43,400 (> 105,000 0 205,000 A 394,000 A 723,000 A 1,102,000 A 1,536,000 1,955,000 2,351,000
v
2.786.000
3;240,000
I 07
I oe
REYNOLDS NUMBER
Figure 3.
Effect of Reynolds number on ratio of average to maximum fluid velocity Curves a , b refer to flow in smooth tubes Curve c refers to flow between smooth parallel plates
Schlinger and Sage (IO) found that velocity data (1, 4, 6) do not differ very greatly from tube data on unmodified u+, y + coordinates. This resemblance is to be expected in view of Equations 12 and 13. It should be remembered, however, that the latter equations are strictly applicable only when the bulk Reynolds numbers are related through Equation 16. Thus the u+, y + profile obtained at a particular Reynolds number between flat plates might be expected to correspond to a smooth tube profile obtained a t some lower Reynolds number. Since the deviation from Equation 6 increases more and more sharply with decreased Reynolds number in the case of tubes, it can be predicted that the u+, y + correlation for flow between flat plates must show a stronger effect of Reynolds number than the corresponding correlation for tubes. This conclusion i s clearly verified by the data of Sage and coworkers. For example, a t a Reynolds number, (ATR& of 6960 and a ( y + ) F of 100, these authors obtain a ( u + ) F of 19.1 in contrast with the value of 17.2 obtained from Figure 1 for tubes a t the same Reynolds number, (NRe)p, and the same friction distance parameter, (y+)p. The difficultyof checking Equations 12 and 13 by interpolation on the u+,y + diagram can be circumvented through use of the relationship expressed in Figure 2. If ( u + ) p = ( u + ) g and (y+)p = ( y " ) ~ , any modification of the u+, y + diagram which results in a unique relationship for tubes must also yield a unique curve in the caae of flat plates. Figure 4 shows the data of Sage and coworkers on u+, y * coordinates modified in the manner of Figure 2. It should be noted particuIarly that the factor, (V/U,)Pis evaluated at the tube Reynolds number, (N&, obJune 1955
tained from Equation 16, rather than at the flat plate Reynolds number, (NRe)p. Friction velocity, u*, values used in preparing the graph were calculated on the basis of friction factors obtained from Equations 15 and 16 in order to maintain consistency in the method of computation. Figure 4 verifies the assumption that the velocity profiles for turbulent flows in tubes and between flat plates are coincident when the radius of the tube equals the half-clearance between the plates, the same fluid flows in both conduits, and the center line velocities (or, just as well, the skin frictions) are equal. There is ample reason t o believe that such an assumption can be used aa a basis for extrapolating the presently available velocity data on flat plates to high Reynolds numbers. I n order to proceed from a knowledge of the flat plate Reynolds to the calculation of mean local velocities, it is number, (NRB)g, necessary to know the relationship between ( V/U,)F and the Reynolds number. This is shown in Figure 3 for the range covered by the work of Sage and coworkers. The velocity calculation is illustrated in the following example. Illustrative Problem 2. Water flows steadily and isothermally a t a temperature of 70" F. and a bulk average linear velocity of 2.50 feet per second in the space between two smooth parallel flat plates separated by a distance oE 0.50 inch. Calculate the mean local velocity a t a point in the fluid 0.15 inch from one of the plates.
For water a t 70" F., f i = 0.00156 pound per (second)(foot) p = 62.3 pounds per cubic foot
INDUSTRIAL AND ENGINEERING CHEMISTRY
1147
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
I
10
I 03
IO2
I
04
M O D I F I E D F R I C T I O N DISTANCE PARAMETER y+(um/v),,
Figure 4.
Modified u+, y+ correlation of velocity distribution for flow between flat plates
From Figure 2 From curve c of Figure 3 ( V/U,).P = 0.851 Thus, ( u ~ ) F= (urn)= = (2.50)/0.851 = 2.94 feet per second But since ro = b, p~ = p p and p~ = p p when the velocities are coincident, the equivalent tube must operate a t a Reynolds number, based on the maximum local velocity, of
U
= (U+)F
( N R ~ )= P (D*p) P =
(1) Urn P
( u , ) ~= V r d f 7 2 = (2.50)(0.0613) = 0.153 foot per second
(Y+)F
1148
(F)p
(0.15)(0.153 )( 62.3)
= ( 12)(0.00156)(0.731)
-
loj
=
(u*)F
[5.5
which means a difference of 10.0%. Interpolation of the Sage data on unmodified u+, y * coordinates yields a ( u + ) F of 18.0 at a Reynolds number, ( N R ~ ) F of , 8330. This results in a local velocity of = (u+)F
( u * ) F = (18.0)(0.153) = 2.76 feet per second
Nomenclature
A B b
and
13.3
+ 2.5 In ( y L ) p ] = (0.153) [5.5 + 2.5 In 771 = 2.51 feet per second
u
Therefore
($!)p
=
If Equation 6 for tubes were used unaltered t o calculate the velocity in the flat plate case, the result would be
(4890)(0.731) = 3580
From Rothfus and coworkers ( 8 ) f p = 0.0102 at ( N R e ) p= 3580, and by virtue of Equation 15,
(U*)F
(z)p
(13.3)(0.153)/(0.731) = 2.79 feet per second
u = From curve b of Figure 3 (V/u,)p = 0.731 Thus, the bulk Reynolds number in the equivalent tube is
(u+)F
constant, dimensionless constant, dimensionless = half-clearance between plates, ft. D = diameter of tube or pipe, f t . f = Fanning friction factor, 2 r 0 / p V 2 , dimensionless N R ~= Reynolds number, D V p / p for tubes, 4 b V p / p for flat plates, dimensionless r = distance from center line of conduit to point of measurement in fluid, f t . TO = radius of tube or pipe, ft. u = mean local fluid velocity, ft./sec. urn = maximum value of mean local flugvelocity, ft./sec. u* = friction velocity 4 \ / 7 0 / p or v d / f / 2 , . ft./sec. u = friction velocity parameter, u/u*, dimensionless =
=
+
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 41, No. 6
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT V
=
y
=
g+ fi p 7 7 0
bulk average linear fluid velocity, ft./sec. distance from tube wall or plate surface to point of measurement in fluid, f t . = friction distance parameter y u*p / p , dimensionless = viscosity of fluid, lb./(sec.)(ft.) = density of fluid, lb./cu. ft. = local shearing stress in the fluid, poundals/sq. ft. = skin friction at the tube wall or plate surface, poundals/ sq. f t .
Subscripts P denotes flow in smooth tubes or pipes F denotes flow between smooth, parallel plates literature cited (1) Corcoran,
UT.H., Page, F., Jr., Schlinger, W. G . , and Sage, B. H., IXD.ENG.CHEM.,44,410 (1952).
(2) MacLeod, A. A , , Ph.D. dissertation in chemical engineering, Carnegie Institute of Technology, 1951. (3) hTikuradse, J., V D I - F o r s c h u n g s h e f t , 356, 1 (1932). (4) Page, F.,Jr., Corcoran, W. H., Schlinger, W. G., and Sage, B. H., IND.ENG.CHEM.,44, 419 (1952). ( 5 ) Page, F., Jr., Schlinger, W. G., Breaux, D. K., and Sage, B. H., I b i d . , p. 424. (6) Prandtl. L.. 2. Ver. dezct. Ina., 77. 107 (1933).
(7) Prengle, R. S., Ph.D. disiertation in chemical engineering, Carnegie Institute of Technology, 1953. (8) Rothfus, R. R., Monrad, C. C., and Senecal, V. E., IND.ENG. CHEM.,42, 2511 (1950). (9) Rothfus, R. R., and Prengle, R. S., I b i d . , 44, 1683 (1952). (10) Schlinger, W.G.,and Sage, B. H., I b i d . , 45,2636 (1953). (11) Senecal, V. E.,and Rothfus, R. R., C h e m . Eng. Progr., 49, 533 (1953). (12) Stanton, T. E., and Pannell, J. R., Trans. Roy. SOC.(London), A214, 199 (1914). RECEIVED for review September 8, 1954.
ACCEPTED October 5, 1954.
Temperature Tolerance in Frozen Food Processing Effective Temperatures in Thermally Fluctuating Systems SIGMUND SCHWIMMER
AND
LLOYD L. INGRAHAM
W e r f e r n Utilization Research Branch, Agricultural Research Service,
U. S. Department of Agriculture, Albany 6, Calif.
H. M. HUGHES University o f California, Berkeley, Calif.
C
ALCULATIONS of reaction rates in nonisothermal systems have been applied to such diverse problems as the pyrolysis of hydrocarbons in the petroleum industry ( 7 ) , thermal death rates in the food canning industry ( 6 ) , and effective scalding temperatures for enzyme inactivation in the frozen food industry (6). I n connection with temperature tolerance studies on frozen foods at Albany, Calif., laboratories of the Western Utilization Research Branch of the U. S. Department of Agriculture (,?), investigators have been concerned with the prediction of effective temperatures as related to chemical and organoleptic evaluations of the quality of the frozen foods. A theory is presented in this paper that permits calculation of reaction rates and effective temperatures in some simple periodically fluctuating temperature systems, used in the temperature tolerance studies, in which the reactions have known constant temperature coefficients in the range of interest. This theory may be of value in other industrial processes where temperature fluctuations affect the final product. Calculations from this theory permit the presentation of some simple charts that afford a rapid comparison of the mean temperature with the effective temperature. The effective temperature is that constant temperature at which the rate of the reaction of interest is equal to the mean reaction rate when the system is subjected to the given fluctuating temperature cycle. The theory presented in this article is similar t o that proposed by Hicks ( I ) , who developed an equation for the estimation of sinusoidal diurnal temperature fluctuations on reaction rates as applied to foods stored a t ambient temperatures. The present discussion may be considered as an extension of the Hicks theory, 1 Present address, Department of Biometry, School of Aviation Medicine, Randolph Field, Randolph, Tex.
June 1955
in that it develops equations for two additional modes of temperature fluctuation and provides a more convenient means for calculations of effective temperature for the sinusoidal variations by restricting the calculation to the one parameter, &A. The Hicks considerations involve &A and A separately, so that results of calculation from his equation can be presented only in tabular form. Relationships allow prediction of effective temperatures and reaction rates
Systems in which the temperature undergoes three modes of fluctuation-saw-toothed, square, and sine waves-have been analyzed. These cycles are illustrated in Figure 1, which also the mean temperature; T,, the effective temperature; shows TO, A , the amplitude (half the range of maximum and minimum temperatures); and B, one quarter of the period of fluctuation. Saw-Toothed Wave. During one half-cycle of amplitude A and duration time 2B, A T=To+-i! -B
