November 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
0.5% under best conditions. Thus bottom hole pressure of 3100 pounds per square inch may be 3085 or 3115. Also, the
process of taking a bottom hole pressure requires that the well be shut in for a minimum of 3 days; this entails a loss of production for that period, and there is also extra hazard involved in the actual measurement of the bottom hole pressure; many times the pressure bomb has been lost in the hole and this involves expensive fishing jobs. In consequence of these items, only a few bottom hole pressure measurements can be expected, and these will not be of an extremely high order of accuracy. With regard to the anzlyzer itself, there are several sources of error. The first of these is leakage of electricity from capacitors. This is of the order of 1% of the voltage in 10 minutea. Since few problems require more than 2 or 3 minutes to run, during which time the voltage is being reduced, this factor is small. One basic source of error is the representation of the distributed parameter system of the reservoir by the lumped parameter system of the analyzer. The magnitude of this source of error depends on the particular problem being computed, but it is instructive to consider the ideal problem of a closed circular reservoir producing at a constant rate from a single well a t the center. Although data from any naturally occurring fields of this description are not available, it is possible to compute the behavior of such a field, as Muskat (6)has done, and to compare the results obtained with the results given by the analyzer on the same problem. Here the important criterion is the comparison of pressure distribution in the theoretical reservoir and in the analyzer pool unit at any stage of depletion. At the beginning
2483
of production from the reservoir, the pressure will be uniform throughout the reservoir so that the two curves will be identical. At a later stage, the situation will be that shown in Figure 8. Here both theoretical curve and analyzer curve are plotted on dimensionless coordinates so that the results are applicable to any stage of depletion. The average difference between the two curves amounts to about 1%. As to the reliability of the instrument, no consistent source of trouble has been found; the electrical leakage of capacitors hes been noted. I n order to keep this at a minimum it has been necessary to install desiccating devices. Another intermittent source of trouble has been corrosion of slide contact wires by chemical fumes. The room in which the instrument is installed is also used for other experimental work which occasionally releaaes sulfur and other corrosive fumes. This merely occasions the periodic cleaning of the slide wires and contacts. LITERATURE CITED (1) Bruce, W. A,, Trans. Am. Imt. Mining Met. E m s . . Petroleum DQ., 151, 78-85 (1943). (2) Ibid., pp. 112-24. ( 3 ) Bubb, F. W., Nisle. R. G., and Carpenter, P. G., Ibid., 189, 143-8 (1950); Muskat, M., and MoDowell, J. M., IW.. 186, 291-98 (1948). (4) Morgan, T. D., and Crawford, F. W.. Oil Gas J . , 43, 100-6 (Aug. 28. - -,-1944). - - -,.
(6) Muskat, M., “Flow of Homogeneous Fluids through Porous Media,” pp. 657-61, New York, McGraw-Hill gook Co., 1937
RECEIVED April 4,
1951.
Prediction of Temperature Distribution in Turbulent Flow Application o f the Analog Computer RODMAN JENKINS’, H.W.BROUGH, AND B. H. SAGE California Institute of Technology, Pasadena 4, Calif.
1
A
knowledge of the temperature distribution within a turbulently flowing stream is of industrial interest. Such data are particularly valuable under conditions where the time-temperature history of the fluid is desired. A preliminary calculation of the temperature distribution of a turbulently flowing stream of water during its passage through a heated circular conduit has been made with the aid of an analog computer. The point velocities and the eddy viscosities were established from generalized equations of fluid mechanics and the corresponding values of eddy conductivity were obtained by analogy. The temperature in the flowing stream was determined as a function of radial position and downstream distance. A
comparison was made between the calculated average temperature rise and the temperature rise determined experimentally. The analog computer appears to afford a direct means of solving equations describing the energy balance in a turbulently flowing stream. I t permits information concerning the thermal flux and temperature to be estimated on the basis of a knowledge of the eddy conductivity and velocity as a function of position. The calculations may be made in a relatively short period of time and with a rather simple analogous electrical circuit. The primary uncertainty rests in the estimation of the eddy conductivity from the conditions of flow existing.
T
The electrical circuit considered to be analogous to the physical situation consists of the resistance-capacitance grid shown in Figure 1. In this circuit, the successive junction points between resistors Corresponded to successive points along a radius of the tube, and the resistances connecting them represented the thermal resistance of the intervening liquid. The time dimension of the analogous electrical network corresponded to distance in the direction of flow, and the current flow to tho condensers waa analogous to the rate a t which energy was transferred thermally
EMPERATURE distributions within circular conduits used in heating liquids under turbulent flow conditions have been calculated by means of an analogous electrical circuit. The generalized velocity distributions as predicted by von KBrmh (6)have been employed to evaluate the momentum exchange in the turbulent region. These equations have been used to obtain the eddy viscosity, which in turn haa permitted the estimation of the eddy conductivity, Preeent address, Megnolia Petroleum Co., Ddss, Tex.
’
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
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Vol. 43, No. 11
by the flowing stream. The temperature of the fluid .at a given radius was represented by the voltage a t the corresponding point in the electrical network. The variation in temperature with distance downstream a t a fixed radius was indicated by variation of the voltage with time after the closing of the circuit.
Figure 1.
Schematic Circuit of Electrical Analogy
Throughout this discussion, which is limited to flow in a circular conduit, the eddy viscosity, ea, and the eddy conductivity eo, are defined as follows:
The estimated velocity distribution was calculated by the method of von K&rm&n(6). The flow stream was divided into three regions, the laminar layer, the buffer layer, and the central portion, or turbulent core. The laminar layer was considered Td
to lie in the region between Td
u
=
=
45 p= 5. In this region
0 and 70
(3)
Td-
PV
DISTANCE FROM
Figure 2.
The buffer layer was considered to extend from the laminar rd
45 Y
velocity was described by =
ds[ P
-3.05
+ 5.00 I n (
F)]
(4)
In the turbulent core of the stream it was assumed that
+
u = e F . 5 2.5 In
( J")] Td
(5)
The foregoing equations describing the velocity distribution were taken directly from von K&rm&n(6) and are open to some uncertainty. A number of other distributions might have been used, notably that proposed by Cox (3). However, the distribution suggested by von K k m b ( 5 )has been widely accepted and was employed here as a simple means of estimating the velocity as a function of radius. Variations in the predicted velocity distribution will make corresponding differences in the computed temperature distribution. Combining Equations 3, 4, and 5 with Equation 1yielded values for e,, in the central core and in the buffer layer. The eddy viscosity in the laminar layer was taken Table I.
INCHES
For conditions corresponding to Rcynolds number of 50,800
layer to a value of -2- = 30. In this portion of the flow, the
u
WALL
Estimated Eddy Conductivity
Comparison of Experimental and Calculated Temperature Rise Conditions A B 50,800 82,300 1.891 1.826 181.8 184.5 203.7 203.1 192.0' 192.3 193.2 194.6 10.2 7.8 11.4 10.1
as zero. The eddy conductivity was obtained on the assumption that it was equal to the eddy viscosity a t the same point in the stream, The velocity distribution indicated by Equation 5 yielded values of the eddy viscosity, reaching a maximum a t r d / R = 1/2 and becoming zero a t r d = R. Data recently obtained in this laboratory indicated clearly that this was not the true situation. Furthermore, the assumption of an expression for the velocity distribution in the central core only gave markedly dissimilar values of eddy viscosity in this region. The eddy viscosity
Table 11.
Computed Values of Temperature Rise in Flowing Streams
Distance Downstream, Feet 0.01
Distance from Wall, Inches" 0.05 0.10 0.15 0.20 C A S E A, REYNOLDS NUMBER 50.800
0.02
0.0
0.00
0.00
0.1 0.2 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0
1.40 2.60 4.10 5.85 6.70 7.75 8.80 9.85 10.55 11.20 12.05 12.55
0.80 1.65 3.05 4.85 5.75 6.95 8.05 9.15 9.90 10.55 11.45 11.95
0.00 4.05 5.05 6.60 7.70 8.95 9.60 10.65 11.35 " Deficiency radius. b Center line of channel. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.00
0.00
0.00
0.00
0.10 0.00 0.00 0.00 0.80 0.25 0.05 0.00 1.80 0.95 0.55 0.30 3.50 2.40 1.80 1.50 4.55 3.55 3.00 2.75 5.75 4.70 4.20 4.00 6.90 6.00 5.60 5.40 8.00 7.10 6.70 6.40 8.95 8.10 7.70 7.45 9.75 9.10 8.80 8.60 10.40 9.80 9.50 9.40 11.15 10.60 10.30 10.20 CASEB, REYNOLDS NUMBER 82,300 0.00 0.00 0.00 0.00 0.00 3.20 0.90 1.75 0.60 0.50 4.40 3.35 2.10 2.50 1 .go 5.95 4.75 3.85 3.55 3.40 7.05 5.25 5.95 4.90 4.70 8.40 7.20 6.50 6.20 6.05 8.25 7.30 9.10 7.60 7.25 10.10 9.30 8.80 8.50 8.45 9.60 10.20 9.70 10.95 9.50
0.2466 0.00
0.00 0.00 0.30 1.40 2.70 3.90 5.30 6.30 7.35 8.50 9.40 10.10
0.00 0.50 1.80 3.25 4.60 6.00 7.10 8.40 9.50
t
November 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
apparently remains substantially a t the maximum value throughout the central portion of the channel, as would be obtained from the velocity distribution as suggested by Cox (a), On the basis of this experimental information, differentiation of Equation 5, and substitution of this value for the rate of change of velocity with radius in Equation 1, the expression for eddy conductivity has been modified to the following form:
2485
The voltage measurements were made with a cathode-ray oscilloscope which registered the electromotive force as a function of time. A typical photographic record from this instrument is shown in Figure 4. The data of this type were smoothed graphically with respect to the analogs of temperature and radial distance. Figures 6 and 6 present the relation of temperature to position for a Reynolds number of 50,800; similar information was obtained for the conditions corresponding to case B of Table I. The smoothed values of the temperature rise as a function of position are recorded in Table 11.
The variation of eddy conductivity with radial position is shown in Figure 2. In this instance the difference in the function assumed for the buffer layer and for the turbulent core ia evident from the indicated discontinuity. The analogy of the electrical circuit shown in Figure 1 to the flowing fluid may be established in the following way. From the definition of eddy conductivity as given by Equation 2, there may be obtained by differentiation
This differential equation may be approximated by the finite difference expression (11) indicated below:
40 80 RATIO OF DISTANCE DOWNSTREAM TO DIAMETER OF FLOW TUBE
Figure 3. Change in Heat Transfer Coefficient with Longitudinal Position in Flow Channel For Reynolds number of 82,300
In Equation 8 the subscripts refer to a series of sequential points equally spaced along the radius at a given longitudinal position in the stream. It has been assumed that the flow is symmetrical about the axis a t the conduit. In the analogous electrical situation, the behavior of the circuit shown in Figure 1 was governed by the following relationship: v 3 - v z ---
v2
88.2
- VI
Shl
-
cz (%)*
(9)
Inspection of Equations 8 and 9 yields the following analogies: Temperature ru Ar Ar/r(u f E , ) Downstream distance
--
N
N
Results of the calculation of the changes in average temperature aa presented in Table I are in only fair agreement with the experimental data. It appears that these discrepancies may be ascribed to the following uncertainties in the calculations: Variations in the redicted values of eo, since marked dissimilarities in value in e&y viscosity result from minor changes in the assumed velocity profile. Differences between numerical values of eddy conductivity and eddy viscosity, assumed equal in the present instance. Distortion of symmetry by natural convection resulting from temperature gradients in the stream. and ignored in the computations.
voltage capacitance electrical resistance time
Such analogies have been utilized in an analog computer (6) to estimate the temperature distribution in a turbulent stream on the basis of the assumed distribution of eddy viscosity obtained from Equations 4 and 5 with the modification near the center described by Equation 6. In order to compare results with available experimental data, conditions were chosen which corresponded to the over-all situation reported by Sherwood and Petrie (9). These conditions and the rise in average temperature obtained in each case are presented in Table I. The thermal flux a t the wall was equal to the product of the thermal conductivity and the radial temperature gradient. The latter quantity was determined from the change in voltage across the first resistor of the electrical network shown in Figure 1. This flux a t the boundary was combined with the corresponding average temperature difference between the fluid and the wall to yield an effective thermal transfer coefficient. The average temperature of the stream was determined by the integration of the product of the temperature and the point velocity, divided by the average velocity. This procedure established a temperature Corresponding to that of the fluid if equilibrium were reached a t a particular cross section. The effective thermal transfer coefficient obtained in this fashion for a Reynolde number of 82,300is shown in Figure 3, which shows a significant variation in the coefficient with longitudinal position near the entrance.
Figure 4. Typical Oscilloscope Record
Use of finite difference approximations in the solution of the differential equation, which may have resulted in uncertainties in temperature changes of approximately 2%. Deviations in the resistances and capacitances used from values stipulated by analogy, and the relatively small size of the oscilloscopeemployed. In the present case, the variation in ec is believed t o be the predominant source of uncertainty. As it has been assumed, as suggested by von K4rm&n (b), that cc and cu are numerically equal, minor discrepancies in the velocity distribution, as described by Equations 3,4,and 5, markedly influence the values of eddy conductivity employed. From velocity measurements, Nikuradse ('7, 8 ) indicated that e,, approached zero a t the axis of
INDUSTRIAL A N D ENGINEERING CHEMISTRY
2406
I
T E M F E R A T U R L RISE AT W A L L -
21.9'F
1
Vol. 43, No. 11
1
li LL
m
t: IO a
3.2
0
t;W
8
8
LL
w
9 a 24
w 6
a 2 e
-3
$ 4
a
W
\P-L-.lJ i
W
a
s
W
I
+ 2
-i
v)
$ 8
1.6
w
0
z
DISTANCE FROM W A L L
Figure 5.
INCHES
Temperature Rise in Flow Channel
FB
a
0.8
A t Reynolds number of 50,800
flow. In his theoretical work, Gebelein ( 4 ) predicted a maximum value of e. a t the axis. The work of Sherwood and Woertz ( 1 0 ) showed that the eddy diffusivity, which is analogous in many ways to eddy conductivity, reached a maximum near the axis of symmetry. This was substantiated by recent measurements of thermal transfer (I, 8) under conditions of uniform flow. For this reason it is not surprising that some discrepancy appeared between the predicted and experimentally determined changes in average temperature. By the choice of some other velocity distribution (9),closer agreement with the experimental temperature rise might have been obtained. The equipment used for the electrical analogy measurements only permitted the circuit paramekrs to be established with an accuracy of about 5%. This large discrepancy in certain regions was a result of the wide variation in relative values of resistance and capacitance needed in the solution of the problem. By the use of larger circuit elements and a correspondingly longer time for each measurement, the maximum uncertainty could be decreased. ACKNOWLEDGMENT The friendly cooperation of G. D. McCann and members of his staff, who made the analog computer available for this work, is much appreciated. The assistance of W. G. Schlinger in the preparation of the manuscript and financial support to the senior author from the National Institutes of Health are g r a t e fully acknowledged. NOMENCLATURE heat capacity a t constant pressure, B.t.u./(lb.)(O F.) = capacitance, farads d = total differential k = thermal conductivity, B.t.u. (foot)/(sec.) (sq. foot) In = natural logarithm 4 = thermal flux, B.t.u./(sec.)(sq. foot) r = radial distance, feet rd = dietance from wall, deficiency radius, feet (Td = R - r ) R radius of conduit, feet S = electrical resistance, ohms T = temperature, F. TI= inlet water temperature, O F. T, average tube wall temperature, a F. Ta = measured outlet temperature, F.
C, C
=
-
0.08
0.16
DISTANCE
FROM W A L L
0.24
INCHES
Distribution of Temperature Rise in Flow Channel
Figure 6.
A t Reynolds number of 50,800 2'4 = calculated outlet temperature, O F. u = velocity, feet per second V = electrical potential, volts z = downstream distance, feet ev = eddy viscosity, sq. feet er second eo = eddy conductivity, sq. g e t per second 8 = t i e , seconds Y = kinematic viscosity, sq. feet per second u = specific weight, pounds per cu. foot p = density, lb.(sec.Z)/foot K = thermometric conductivity, k/C9u, (sq. fbot)/(sec.) 7 = shear,pounds e r s T O = shear a t wall ofcon?;fi::iounds per sq, foot
LITERATURE CITED (1) Corcoran, W. H., "Temperature Gradients in Turbulent Air
Streams," Ph.D. thesis, California Institute of Technology,
1948. (2) Corcoran, W. H., Roudebush, B., and Sage, B. H., Chem. Eng. PTOUreS8.43, 135-42 (1947). (3) Cox, A. B., Mech. Eng., 56, 7,135 (1934). (4) Gebelein, H., "Turbulena, Ann Arbor, Edwards Brothera, 1944. (5) KBrmBn, Th. von, Tram. Am. SOC.Mech. Engrs., 61, 705-10 (1939) I
(6) McCann, G. D., Wilts, C. H., and Locanthi, B., Proc. Inat. Radio Enure., 37, 954-61 (1949). (7) Nikuradse, J., "GesetsmBssigkeiten der turbulenten StrGmung in glatten Rohren," Forschungsheft 356, Berlin, VDI, 1932. ( 8 ) Nikuradse, J., "Strljmungsgesetze in rauhen Rohren." Forsohungsheft 361, Berlin, VDI, 1933. (9) Sherwood, T. K., and Petrie, J. M., IND.ENQ. CHEM.,24, 736-45 (1932). (10) Sherwood, T. K., and Woerts, B. B., Trans. Am. Inst. Chem. EnW8.. 35, 517-40 (1939). (11) Southwell, R. V., "Relaxation Methods in Theoretical Physics," London, Oxford University Press, 1946.
R ~ C E I V EJuly D 12, 1949.
(End o f Symposium)
