En:nig
Temperature Gradients in Turbulent Gas Streams
ri ng
pocess development
METHODS AND APPARATUS FOR FLOW BETWEEN PARALLEL PLATES I
W. H. CORCORAN’, F. PAGE, JR.*, W. 6. SCHLINGER, CALIFORNIA INSTITUTE
P
REDICTION of the thermal transfer and temperature distribution in turbulent fluid streams from analogies with corresponding transfer of momentum requires detailed information concerning the velocity gradients and the variations of shearing stremes with position. Such information supplemented by experimental values of the thermal flux and temperature gradients for the same conditions of flow permits a direct comparison of the transfer of energy and of momentum. Measurements of these quantities have been obtained for the steady, nearly uniform flow of air in a rectangular channel 12 inches in width, 0.70 inch in height, and 13 feet in length. The temperatures of the upper and lower surfaces of this channel were controlled within small limits a t predetermined but different values, thus imposing a temperature gradient upon the essentially two-dimensional flowing air stream. The thermal flux was measured a t the boundaries of the stream a t two points in the working section and the temperature, as a function of distance between the plates, w m established by the use of a small resistance thermometer. I n addition, the pressure gradient and the velocity profile were determined at several points along the axis of flow. These measurements were made with equipment refined from that originally utilized (6) and it is believed that the data permit the establishment of the relationship between eddy viscosity and eddy conductivity with an uncertainty of not more than 5% except near the wall and the center of the channel. Flow conditions have been obtained in the turbulent region up to a Reynolds number of 65,000. The present paper describes the apparatus and presents typical isothermal data. The apparent analogy between the transfer of momentum and of thermal energy hae received attention from the principal authorities in the field of fluid mechanics. The contributions of Reynolds (B),Prandtl (% andI) von , K A r d n (1.0 are perhaps the most interesting to the present investigators. The more recent work of Boelter (4) extends the concepts proposed by von K 6 r m h (14) and indicates a method of integrating the thermal flux m a function of position in a circular conduit. This work has been supplemented by more recent considerations by Martinelli (17) which relate to the thermal flux in molten metals. Preliminary memurements of the temperature gradients in a substantially two-dimensional air stream have been reported (6). These measurements indicate a simple relationship between the Nusselt heat transfer number determined experimentally and that predicted from the von KCirmCin analogy (14). On the bask of such assumptions, the temperature distribution in a turbulent stream of water within a heated circular conduit has been predicted (ii). The present experimental program represents an attempt to add to the background of experimental information in order to permit an extension of the application of the analogy of the ther1
AND
B. H. SAGE
OF TECHNOLOGY, PASADENA, CALIF.
mal and momentum transfers in turbulent streams to the practical prediction of thermal flux in situations of industrial interest. Recent advances in the use of the analog computer (16) have permitted the direct calculation of the temperature and thermal flux as a function of position in a turbulently flowing stream. These calculations may be accomplished without the need of the simplifying assumptions which usually are made in the interest of directly integrating the applicable differential equations. Such methods offer promise as a practical means of predicting temperature and thermal flux in turbulent streams under conditions of industrial importance. Only a knowledge of the flow condition and a definitive relationship between the eddy viscosity and eddy conductivity are required. One of the detailed objectives of the present investigation is t o obtain additional detailed information as to the relative values of eddy viscosity and eddy conductivity under several conditions of flow, The background of fluid mechanics upon which any such correlations must be based was discussed by Gebelein (9), Goldstein (IO), Bakhmeteff (8, and others. This theoretical treatment has been supplemented by the classic experimental work of Nikuradse (18)and by the thermal transfer measurements of Colburn and Coghlan ( 6 ) ,Eagle and Ferguson (8),and Reichardt (91). From the experimental information presently available it appears that it will be possible to predict the edd? conductivity from the flow conditions obtaining. Therefore, additional information of the type already acquired (6) concerning temperature gradients and thermal flux densities is useful in establishing the eddy conductivity. EXPERIMENTAL EQUIPMENT
The primary objective of this apparatus was to establish normal turbulent flow in the space between the two plates and to provide equipment for the determination of the temperature and velocity as a function of position in the working section. In order to accomplish this, it was necessary to maintain steady conditions with respect to momentum transfers and temperature distribution. In principle, the equipment utilized in this investigation was similar to that described in connection with an earlier study ( 6 ) . The general arrangement of the apparatus is shown in Figure 1. It consisted of two parallel plates, A and B, which were maintained at predetermined surface temperatures. Air was passed between these two parallel plates which were 13 feet in length, 15 inches in width, and separated by a distance of 0.70 inch. The air was circulated through the space between the plates by means of the blower, C, and its entering temperature was controlled within small limits by means of the continuous heater, D. An appropriate entrance section, E , t o the space between the plates was provided. The arrangement of the entrance section is shown in Figure 2. The principal dimensions have been included so that the influence of the entrance conditions upon the flow characteristics might be evaluated more easily. Throughout all of the meamrements the distance “downstream” is measured from
Present address, Cutter Laboratories, Berkeley. Calif.
* Present address, sierra Engineering Co.. Sierra Madre. Calif. 410
February 1952
9
INDUSTRIAL AND ENGINEERING CHEMISTRY
point A of F i p r e 2 which is located at the end of the converging section. A distance of 13.56 inches has been provided from point A to the temperatureconditioned cppper lates. The converging section has been covered with 0.15 inch ofmagnesia insulation in order to decrease the influence of room temperature upon the entrance characteristics of the flow.
41 1
strip, C in Figure 3, was aligned with the main copper plates by the series of bolts at D, which were spaced at about 2-inch ink;vals. Little difficulty was experienced with this method of attaching the copper plates to the steel sections. ment was operated at tem eratures ran in from 60" T 2 6 E q u 2 After final assembly $e flatness of d e copper plates was checked and final adjustments were made by inducing stresses in the steel oil bath by means of the struts, H , to bring the plate into the desired , The lower oil bath, A of Figure 3, was ocated within a heavier steel section shown at C, which contained the ways used to support the traversing mechanism. Shims were employed to bring the surface shown at F parallel with the ways J. I n turn the supporting structural steel members shown a t K and the adjustable struts, H , were adjusted to stress the vessel, B, in such a fashion to brin the surface I , parallel with the surface, F. After final adj'ustment the average deviation of the oopper plates from parallel planes was believed t o be 0.02 inch. This latter fact was checked frequently by mounting a dial indicator on the traversing meohanism and determining the position of each of the surfaces relative t o the ways, J. Throughout all the measurements, reference dimensions were taken relative to J,which, under the loading imposed b their own weight and that of the traversing mechanism, yieldeia deflection of a proximately 0.01 inch at the mid-point of the span. This s l i g g deviation from a straight line which was followed by the axis of the flow channel and all associated equipment is not believed t o have influenced eignificantly the nature of the flow of the air stream.
position
Measurements of the level of turbulence have been made at the center of the channel at the point 8 feet downstream. These indicate that the instantaneous velocity fluctuation parallel t o the axis of the flow is approximately 3% of the point velocity. Detailed measurements concerning the magnitude of this ratio may be made when such measurements are of particular importance to the objective of the investigation. I n order to maintain the copper plates A and B of Figure 1 at the desired temperature, oil was circulated a t a rapid rate above and below the upper and lower plates, res ectively. The oil flowed past each copper plate at a velocity SUCK that When a ternperature difference of BO" F. existed between plates A and B and the air streamwas mavin at a maximum velocity of approximately 100 feet per seconf, the temperature difference between the two ends of each plate was less than 0.75' F, At lower air velocities and temperature differences the variation was correspondingly smaller.
Figure 5 is a photograph of the working section with a portion of the movable side blocks removed upstream from t h e traveraing mechanism. The struts used to hold the upper plate in position are shown as well as the Venturi meter used t o measure the gross flow rate of the air. The details of the sliding blocks employed to close the sides of the working section are shown in Figure 6. oneof these blocks was attached to the traversing mecha&m through a linkage of one degree of freedom. Thus, when the traversing mechanism waa moved along the ways, J of Figure 3, the blocks in front of the traversing equipment slid along the wall of the channel. The remaining blocks following the travera ing mechanism were moved t o close the channel. For short longitudinal movements of the traversing mechanism it was not necessary t o interfere with the flow of air through the working
A Figure 2. Arrangement of Entrance Seotion The details of the working section are depicted in Figure 3. The copper plates were constructed from 0.375-inch specially rolled sheets, flattened as nearly as possible t o planes by a manually controlled ressing operation. This pressin operation was followed by a {mited amount of special macfinmg and polishing to ensure that the plates were planes within 0.003inch in any given square foot of surface. Measurements of the surface roughness at several positions along the channel indicate a root mean square deviation of less than 16 microinches from the plane surface of the plate. Reproductions of the actual surface contours are presented in Figure 4. The effects of the polishing on the plate surfaces may be seen in the rounded contours of t h e high spots of the plate surfaces. The lates were attached to the machine-welded steel sections, A and of Figure 3, used to confine the oil. The method of attaching the steel oil containers t o the copper plates is shown at E, R here a thin rubber seal was rovided to prevent leakage of oil between the copper plate and tge steel section. Adequate clearance for the special bolts a t D was made in the fiteel sections A and B to permit relative movement of the copper and steel because of the difference in their thermal expansions. The upper copper
%
Figure 3. Sectional View of Flow Channel
.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Figure 4. Reproduction of Surface Contours of Flow
Channel
section since the use of a few extra blocks permitted the side walls of the channel to be dosed during the movement of the traversing equipment. The side blocks could be attached t o one another by means of small circular springs mounted on the knobs shown in Figure 6. To decrease the leakage of air from the sides of the working section, felt strips were glued into the grooves provided in the plastic blocks. The traversing equipment was arranged to permit controlled motion in vertical and horizontal directions in addition t o the longitudinal movement on the ways. This equipment was designed along the general lines of machine tools. Adjustable gibs operating upon V-type ways were provided for both the vertical and horizontal secondary motions. A lead screw was used for the horizontal motion and two lifting screws were provided for the vertical traverse. Counters were attached to the lead screws to indicate the approximate position of the measuring equipment. The actual vertical position of the traveling pitot tube and hot wire anemometer was determined by means of a small cathetometer mounted upon the traversing equipment which is shown in Figure 5. The traversing mechanism was constructed from rolled mild steel structural shapes khich were welded together and annealed. Careful measurements with the traversing equipment indicated that the hot wire could be located in its position relative to the copper plates within 0.002inch. The longitudinal positions of the anemometer and pitot tube were determined by means of a steel scale calibrated with an uncertainty of less than 0.01 inch per foot. Figure 7 presents the general arrangement of the traversing equipment. Particular emphasis was placed upon the design of yoke A, used t o support the measuring equipment. This yoke was sufficiently rigid to yield a deflection of less than 0.01 inch when stressed by a tension of 100 pounds in the wires, B . The actual measuring
Vol. 44, No. 2
equipment was mounted upon two parallel piano wires which are shown a t B. These wires were stressed sufficiently, SO that when properly damped their location relative to J of Figure 3 remained fixed within 0.001 inch. I n an insert to Figure 7 is shown a perspective view of the mounting of the traversing pitot tube and the small diameter wire which was used in measuring both the speed and temperature of the air stream and which has been called a thermanemometer ( 3 ) . The elements of the circuit employed with this latter instrument have been described (%), and in the present measurement it was utilized under substantially the same conditions. In brief, the arrangement involved its use as an anemometer with a current of approximately 0.1 ampere to maintain the temperature of the wire a t a fixed value of about 150" F. above the temperature of the air stream. The readings of the anemometer were checked by the indications of the traversing pitot, impact tube. As is shown in the upper part of Figure 7, this was mounted on the same supports as the thermanemometer located to the right of the pitot tube. In order to avoid parasitic electromotive force in the measuring circuits, 36 B & S gage platinum leads were brought from the points of the needle supports depicted in Figure 7 to the exterior of the working section. The 0.5-mil pure platinum wire used for the thermanemometer was welded t o these platinum leads with sufficient tension to avoid a center sag of more than 0.001 inch in a span of 0.38 inch. Some difficultywith breakage of the fine wire was experienced until the small glass cross arm, shown between the needles in the insert to Figure 7, was installed t o prevent vibration of the needles. This addition practically eliminated the problem of the breaking of the fine wire. Provision was made for the adjustment of the relative position of the needles by the holder shown in the insert t o the figure. Some measurements of the velocity were made by utilizing a constant current technique (25) and for some purposes such an approach has a distinct advantage. However, the additional calculations required with the latter approach offset the more direct experimental measurement. The circuit permitted the relatively simple maintenanee of a uniform resistance corresponding to a constant average temperature of the wire. This technique was employed for all of the measurements reported herein. A discussion of the principles of anemometry has been presented ($5). It is believed that, with the direct calibration of the hot wire
.
February 1952
INDUSTRIAL AND ENGINEZRING CHEMISTRY
under the temperature conditions encountered in actual measurements, the velocities were known within 0.5% except in the immediate vicinity of the wall where the uncertainty may have increased. At each point where measurements were taken, the location of the wire relative to the upper plate was established with the cathetometer, J of F i g u r e 7. T h i s location was checked by reading the counter a t t a c h e d t o the vertical lead screws. When the wire was brought to within less than 0.02 inch of either wall, the distance was determined by a reflection technique that involved the measurement of the apparent distance b e tween the wire and its image in the adjacent p ol i s h ed copper surface. Plate glass windows were provided in the sliding block Figure 6. Details of Sliding Blocks forming t h e side wall of the working section adjacent t o the traversing equipment, Illumination was provided from a collimator mounted on the traversing equipment yoke on the side of the working section opposite to that from which the cathetometer observations were made. The differential pressures determined by the use of the traversing pitot tube were measured by means of a micromanometer which is shown in Figure 8. This instrument was so arranged that continuous readings of the pressure differential between 0 and 3 inches might be obtained with an uncertainty of not more than 0.001 inch. Every effort was made to keep the volume of the tubing connecting the traversing pitot tube with the micromanometer to a minimum in order that relatively rapid equilibrium readings might be obtained. The traversing pitot tube was constructed from hypodermic tubing 0.031 inch in diameter. The end was carefully honed t o the shape recommended for such devices (10)and careful attention was given t o aligning the pitot tube t o ensure that its axis was parallel to that of the flow channel. As a check upon this instrument, a second pitot tube waa installed
Figure 7.
413
near the exit of the working section from which impact pressurea were determined at single longitudinal and lateral positions, but at any location between the two plates. The micromanometer shown in Figure 8 was employed for all the pitot tube measurements at the lower velocities. I n the case of situations involving differential pressures larger than 3 inches of kerosene, steps were taken t o use a keroseneand-glass manometer of conventional design. The manometer tubes with an inside diameter of approximately 0.6 inch were carefully cleaned before use and were within 0.01 inch of t h e same diameter throughout their length. The difference in! elevation of the air-kerosene interfaces was established by means of a specially designed cathetometer which permitted direct readings of the relative elevation with a precision of at least 0.001 inch. It is believed that the difference in elevation of the kerosene-air interfaces was determined within 0.002 inch. However, the flowing system was not sufficiently stable t o permit the impact pressure to be established by other than statistical methods t o an uncertainty smaller than 0.5%. I n Figure 9 is shown a photograph of the bank of manometers and the cathetometer. The manometers were connected to t h e working section by means of appropriate copper tubes and each line was carefully checked for leaks. Small leaks in the manometer connections make disproportionately large errora in t h e measured differential pressure. This situation is especially true in the case of the small-sized pitot tube used for the impact pressure measurements. The change in pressure from one end of the working section t o the other was established by a series of three piezometer bars mounted on the copper plates. The actual connections at each bar consisted of 11 carefully machined holes, 0.032 inch in diameter, in the copper plate, connected by means of a manifold t o the tubing leading t o the manometer bank. These manifolds or piezometer bars were connected t o the entrance and exit of the working section in both the upper and lower plates, and one connection was provided in the upper plate near the center of the working section. Two additional pressure
Arrangement of Measuring Equipment
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
414
Figure B.
Vol. 44. No. 2
Photograph of Miommanometer
eormectiuiia were iruide at tbe nmvable Ride walls near the position of the traversing gear. The presaure differentials were measured with tho mioroninnorneter shown in Figure 8 or the manomdei benk of Figure 9. T h e data obtained were wed t,o ertablish the shear in the flowing fluid. The grms rate of air flow was determined by mean8 of one of a seriea of four Ventw+ inek.1~. The metom were oonatruoted of cast aluminum arid were nclchined to fairly clme tolerances with tm entrance COW angle of 10' $1'. The change in static pressure between the upstream arid throat aections wIiq determined by continuow piommeter rinings locnted at the entrance and throat of the Veduri rnotem. The characteristics of the Venturi meters were predicted from tlleir dimensions (I, Id, 19). The actual velocily of flow was established from tbe pitot meaaurements in the working section. The Venturi meter provided a convenient mcnns of maintaining stwidy flow conditions. For the mmt part the flow did not Ghmge over a perid of 15 minutes by more than 0.2% except when unususl voltage venations in the power supply were expaicnced. In such Instances the hasic flow rate w88 re-establkhed at the p.determioed value by adjustment of the variable speed motor wed to drive the blower, C, shown in Figure 1. Make-up air for the 8y8teni w~18provided %a required from outside the operating area through ~t set of steam coils that were by-pmed in part by menns of a Bmall motordriven damper that was opemted fmin the control dcnk. The makcup air %mounted to less than 1% of that circulating in the flow channel and it. temperature w m controlled within about 1" F. by means of the damper. This equipment required only adjustment w R reault of change8 in outside air temperature. For mast of the work herein deBcri1mJ tho make-up air supply was not important. IIowever, the sm&ll leakages of a k into OF from the channel through the side wslls probsbly contributed t o some of the small but systematic delintiona from uniform flow which were noted for mwurements cam& out at other thnn atmospheric pressure. The primary temperature meaauroments were accomplished by m a n s of the equipment mounted on the structure portrayed in Figure 10. This temperature bench was so arranged a8 to permit the necessary interconnection of meamring and control eireuits by means of COD DO^ d u m and iaeks. It ha8 I w n fnmd
that the parasitic eleotroinotive foiae introduccd by tiieac plu&x and jacks was somewhat less thnii 1 microvolt, under the most adverse conditions and usually lers than 0.2 microvolt. Two Mueiler-typc bridges and one \Vtlite double potentiometer of 10,oW microvolts range were provided for the me8surements. Them instruments were supplemented by three Mueller-type bridges that were constructed for tho purpose of controlling the temperature of the upper and lower oil baths and of the incoming air stream. These control bridges were wed in conjunction with strain-free resistance thermometem for this purpose. The off-balance of a elvanometer in the bridge circuit was used in eonnect,ion with a droop-corrected, modulating photoelectric circuit to control the energy input t o the oil and air streams. The finnl adjustment of the air stream temperature was accompiinhod by an electric heater of low thermal capacity located
February 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
immediately upstream of the converging section shown is Figure 2, This system proved very satisfactory in the control of the temperature of circulating fluids. The general arrangement of the components of the system is shown in Figure 11. It was found that the temperature of the upper oil bath could be maintained within 0.01' F, of the desired value, as measured by --_--.p ________ an independent platinum res i s t a n c e thermometer, even though changes in room temperature of as much as 1 0 O F . occurred in a period of 1 hour. The eauiDment just described - affords a satisfactory means of Figure 11. Droopmeaauring temperature and of Corrected, Modulated controlling the temperature of Photoelectric Control Circuit the several flowing streams inA. Resiatenca thermometer volved. All temperatures re8. bridge corded have been referred to C. Mueller Galvanometer D . Light source the international platinum scale E. Photo cell F. Thyraton circuit as determined with a strain-free G. platinum resistance thermometer A. Heater Thyraton circuit that was calibrated by the National Bureau of Standards. It is believed that the temperature values recorded were established in terms of the international platinum scale within the precision of measurement. The precision attained was limited for the most part by the stability of the flowing stream. Differential temperatures were established by the use of multilead, multijunction, copper-constantan thermocouples. The thermocouples were calibrated in place by comparison with a calibrated standard platinum resistance thermometer. The thermanemometer was calibrated by bringing both the upper and lower oil baths to a prescribed temperature with no Ltir flow in the main channel. The resistance of the instrument then was determined for a series of temperatures of the baths. In the case of measurements made under conditions of isothermal flow, the temperature calibration of the thermanemometer was obtained by extrapolation of the measured stream temperature to the known wall temperature. The corrections to the indicated temperatures were checked by direct measurement of the effect of the current flowing in the wire and the recovery factor, which W&E found t o be 0.75 a t a velocity of 30 feet per second. Efforts were made t o establish radia t i o n c o r r e c t i o n s for t.he thermanemometer, but within the range of temperatures employed the corrections were negligible. Figure 12. Schematic Drawing of Calorimeter The calorimeters used in this work were mounted in separate openingsin the upper copper plate in accordance with the general arrangement presented in Fi ure 12. This portion of the equipment consists of the block, A , wkch wag exposed to the flowing stream through an opening in the copper plate, B. The oil Bowing along the upper surface of this plate was segregated from the calorimeter by means of a jacket, C, which was attached to the upper plate, B, by means of a series of machine screws, D. The copper block, A , was located within the calorimeter vacuum jacket, E, and was provided with a "pancake" type ofheater, F, and with the cooling coils, G. The latter coils were not employed in the current investigation. The vacuum jacket, E , was located in the space between the jacket, C, and the copper block, A , and was connected to this block near the plate, B. Three small radiation shields were provided a t il within the calorimeter vacuum jacket, E. The leads from the calorimeter heater and from the thermocouples installed in the internal portion of the
415
block A were brought outside the calorimeter through a conventional wax seal outside the oil bath above oint J . In order to permit the lateral motion of txe calorimeter relative to the upper closure of the oil bath shown a t K of Figure 12, a special ring seal was provided. This seal was arranged to permit relatively free lateral motion of the jacket, C. The seal between the ring, L, and the upper closure, K ,was achieved by means of an O-ring. Vertical movement was permitted between the jacket, C, and the ring, L, by means of a second O-ring shown at M . An oil-diffusion pum was employed in connection with a conventional mechanicafpump to bring the pressure within the space between the calorimeter block, A , and the vacuum jacket, E, to a value below one micron. The calorimeter heater shown at F of Figure 12 was of the four-lead type. The energy added to the calorimeter was supplied by acid-type storage batteries of such a capacity that the rate of energy input did not change by more than 0.1% in a period of 1 hour. The electromotive force imposed upon the heater was measured by a potentiometer used in conjunction with a suitable volt box. The current was determined by measurement of the electromotive force across a standard resistance placed in series with the heater. It is estimated that the energy in ut was established with an uncertainty not greater than 0 . 0 5 g The multilead, %junction thermocouples shown at N of Figure 12 were used to bring the temperature of the calorimeter block, A , and surrounding oil bath and copper plate, B, t o the same value. By requisite control of the energy supplies to the calorimeter the temperatures of the block, A , and the plate B of Figure 12 were maintained within 0.002' F. of each other during the course of a set of measurements. PROCEDURES
In the operation of the equipment the oil baths first were brought to the desired temperatures, which were chosen arbitrarily so as to yield nominal temperatures a t the working surface of the copper plates of 70', 8 5 O , 95O, loo', 105O, 115O, or 130' F. The air stream then was adjusted to the desired gross temperature and the weight rate of flow was adjusted to predetermined values. By choosing fixed values of the plate temperatures and Bow rates the correlation of the data was simplified markedly. Equilibrium was attained in a period of approximately 30 minutes after the plates were brought to the desired temperature. Determinations of the velocity profiles were made with the anemometer and supplemented with the pitot tube. Measurements with the pitot tube were not obtained a t points closer to the wall than approximately 0.03 inch. However, the anemometer was used to within 0.003 inch of the wall. In each case, the vertical position of the anemometer and the pitot tube was determined by means of the small cathetometer mounted upon the traversing equip ment. The comparison of the indications of the hot-wire anemometer and of the pitot tube permitted a direct calibration of the former instrument under the conditions encountered in the working section. Temperature profiles were measured by the use of the anemometer as a resistance thermometer. The latter measurements usually were made at the same coordinates of the working section as were used in the velocity measurements. A set of isothermal velocity measurements based upon tests 37 and 50 is shown in Figure 13. These are among the early traverses made with the equipment but are typical as to precision of the data. Figure 14 shows upon a residual basis a comparison of the experimental velocity deficiencies for tests 37, 37A, 39, and 50 with the data of Skinner (25) a t Reynolds numbers from 21,000 to 47,000 and the average of the measurements of Nikuradee (18) a t Reynolds numbers between 4000 and 3,240,000. The similarity hypothesis proposed by von K&rmln (13) is related to the velocity deficiencyin the following way: Urn
- 74
-i.--
U*
Yo
The values shown by the full curves on Figure 14 are based upon this equation. It is seen that rather small changes in the constant k from 0.3 to 0.4 cover the range of data involved. This
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Vol. 44, No. 2
INDUSTRIPL AND ENGINEERING CHEMISTRY
POINT
VELOCITY
cant and is occasioned by a slight leakage of air into the channel through the sliding side wall sections during test 39. The data of tests 37A and 37 made at 3.1 and 8.1 feet downstream agree much more closely with the measurements of Nikuradse and Skinner than do the measurements of test 39. This agreement serves as additional evidence that the velocity distribution of test 39 resulted from the existence of nonuniform flow. The small difference between tests 37A and 37 may well result from downstream distance insufficient to establish uniform flow for test 37A. Tests have shown that the lateral velocity gradients are zero within the precision of measurement in the central 3-inch portion of the channel at a distance of 12.5 feet downstream. The basic assumption of uniform, twodimensional flow appears not to be invalidated by the minor redistribution of shearing stresses associated with the changes in velocity profile presented in Figure 15.
FEET PER SECOND
Figure 13. Typical Velocity Profiles a t 100' F.
range corresponds to the values proposed by Bakhmeteff ( 2 ) and Goldstein (IO). Tests 37 and 50 are in reasonable agreement with the recent values of Skinner for a two-dimensional air stream and the data of Nikuradse for the flowof water in a circular conduit at Reynolds numbers above 43,400. The present memurements show better agreement with the values of Nikuradse corresponding t o lower Reynolds numbers. The abscissa represents relative distances from the wall, and the data have been omitted a t values of the quantity
YO
below 0.03 which are outside the
turbulent core. Test 39 yields a somewhat smaller velocity deficiency than would be predicted from the measurements of Nikuradse, Skinner, and other tests carried out with this equipment. These values have been included to show the significant e p t of a small influx of air into the flow channel through the movable walls. This deviation from uniform flow appears to account for the difference in the slope of the data of test 39 in Figure 14 and the slope that was obtained from the other measurements. Figure 15 indicates by the use of residual techniques the variations in the velocity profiles which were experienced between three sections of the channel at the time test 39 was made. It appears that there is a gradual decrease followed by a n increase in the velocity near the upper and lower walls of the channel with downstream position. This variation accounts for the lower values of velocity deficiency shown in Figure 14 for test 39. The residual velocity depicted in Figure 15 was computed from
2
= 33.70
38.15
-
(k - 0.5)* - u
Quantity
(2)
The variation in the velocity with position was found t o be 1.7 feet per second at values of
of 0.1 and 0.9 as a result of a It0
change in position from 8.1 t o 12.5 feet downstream. This change in the velocity distribution is believed t o be signifi-
RELATIVE
Figure 14.
WSlTlON
IN
CHANNEL
Velocity Deficiencies for Several Reynolds Numbers
TABLE I. EXPERIMENTAL CONDITIONS
Units Foot Feet F. F. O F. Feet per second
Test Number 37 37A 0.0567 0.0578 8.1 3.1 100.0 100.0 100.0 100.0 100.0 100.0 28.94 29.55
50 Distance between plates 0.0580 Trsverse locationb 12.5 Incoming sir temperature 100.0 Upper plate temperature 100.0 Lower plate temperature 100.0 Bulk velocity 58.7 17700 17600 inrnn m4 Revnolds number -___ .n n Pressure gradient C -0.243 -0,230 (Pound/squsre foot) per foot -0.230 -0.774 Thermal flux 0 0 0 (B.c.u./second) per square foot 0 Weight fraction water 0.014 0.0058 0.0084 0.0156 Weight rate of flow Pound per second* 0.1313 0.2708 0.1324 0.1328 Pressure a t traverse position Pounds per square inch 14.252 14.265 14.335 14.426 Barometric pressure Pounds per squsre inch 14.263 14.340 14.427 14.261 a Nnniiniform traverse . b Traverse location measured from end of converging section. Pressure gradient is aversge of change in static pressure over 4-fOOt length of workin jection approximately 10 feet d o m t r e s m from entrsnce to channel snd messured with traversing mechanism %ownstream from ststic taps. No significant change with time observed.
_____
394 0.0554 12.5 100.1 100.0 100.0 30.02
_____
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417
the shear and the velocity gradient du - are zero at this point. dY For this reason the uncertainty in the direct evaluation of the eddy viscosity increases as the center of the channel is approached. However, this situation is not found in Equation 5 since both the thermal flux and the temperature gradient dt d!J are finite throughout the flow channel. I n Figure 16 the data of tests 37,37A, 39, and 50 are compared with a logarithmic velocity distribution defined by the equations Uf
U+
= Y+ =A
(7)
Y+ 30
(8)
Equation 8 is considered to apply to values of yf greater than 30. No formulation of the transition region for values of y + between 5 and 30 has been included. The data of several investigators have been depicted by solid curves in Figure 16. The corresponding equations describing these straight lines are as follows:
Figure 15. Residual Velocity Proiiles a t 100' 37, 37A, and 39
F. for Tests
This figure serves to illustrate that measurable variations in velocity distribution still exist even after reasonable precautions have been taken to obtain uniform flow. The details of the conditions associated with the tests reported in Figures 13, 14, and 15 are set forth in Table I. The average values of the velocity, Reynolds number, and pressure gradients have been indicated, together with the location of the traverses, the distance between the nearly parallel plates, and other pertinent information. The average velocities were obtained from the corresponding point values by means of the following expression : (3)
The Reynolds number for the section waa determined from
Re =
'yoU V U h
a8
(2) -
yf
Deissler (7)
U+ =
1 +0.360 In
*
Skinner (93)
u + = 6.0
1 + __ 0.369 In '+
Laufer (16)
u+ = 5.5
1 +0.344 In y t
3.8
Wattendorf (94) u + = 4.0
1 + 0.373 In '+
It is apparent that rather wide variations in velocity profiles have been found. The present experimental data fall well above the measurements of Wattendorf and somewhat below those of Laufer and Skinner. Again the data for test 39 yielded a slightly different slope from that for tests 37 and 50. The data for testa 37, 37A, and 50 are in good agreement with the measurements of Deissler (7) but somewhat below the values reported by Skinner (93)and Laufer (16). It appears that there still exists a significant uncertainty as t o
20.0
3 (5)
P
1 + 0.400 In
(4)
0.5
Following the convention of von K&rmln (14) the eddy conductivity and eddy visbosity a t a particular point in the flowing stream may be defmed by the following relationships:
I 7 -
Nikuradse (18) u + = 5.5
v
Except for the thermometric conductivity (14) and the kinematic viscosity all the quantities in Equations 5 and 6 are subject to direct experimental evaluation with the equipment which has been described. These expressions permit the estimation of the eddy conductivity and eddy viscosity as a function of position in the flow channel. The values of the eddy viscosity become indeterminate at the center of the stream when evaluated directly from Equation 0 since both
B
la0
120
E
3
9 ao
4.0
I
2
0
20
50
KK)
203
503
K )O
DISTANCE PARAMETER Y c
Figure 16. Logarithmic Velocity Distribution for Turbulent Flow
INDUSTRIAL AND ENGINEERING CHEMISTRY
418
TABLE11.
Vol. 44, No. 2
EXPERIMENTAL VALUESOF VELOCITY AND TEMPERATURE
Temp., YlYO
-2m M L o t r r Y GRADEM
Figure 17.
I
1
I
I
lax,
0
lax,
2 m
dY
I
FEET PER YCOND PER FCOT
F.
1,000 0.991 0.988 0.985 0.979 0.871 0.964 0.956 0.949 0.934 0,919 0.904 0.889 0.874 0.859 0.844 0.829 0.814 0.784 0.723 0.663 0.603 0.543
100.0
1,000 0.990 0.983 0.975 0.967 0.952 0.937 0,922 0.908 0.893 0.878 0.864 0.849 0.. 834 0.820 0.790 0.761 0.731 0.702 0.672 0.647
100.0 99 95 99.93 99.87 99.88 99.90 99.91 99.91 99.91 99,87 99.89 99.88 99.90 99 I90 99.92 99.92 99.92 99.92 99.93 99.89
1,000 0.992 0.987 0.982 0.974 0.967 0.953 0.938 0.909 0.881 0.852 0.823 0.794 0.765 0.707 0.649
100.0 99.96 99.95 99.93 99.89 99.90 99.92 99.85 99.81 99.78 99.76 99.77 99.75 99.77 99.76 99,76
99.97 99.89 99.95 99.97 100.01 100.07 100.05 100.09 100.12 100.07 100.06 99.99 99 I95 100.09 100.07 100. 06 100.06 100.04 100 IO1 100.05 99.96 100.01
Velocity Gradient at 100' F. for Test 50
the details of the velocity distribution in turbulent flow. The equipment was not well adapted for the study of velocity profiles since the walls were relatively close together. However, the velocity distribution found in these early measurements are all within the range of behavior which has been found by other investigators. For this reason it has been assumed that the flow obtained was steady and uniform. The small height of the channel prevents significant studies of the boundary layer except a t low velocities. The deviations from laminar flow in the boundary layer indicated in Figure 16 are smaller than the experimental uncertainty in determining the location of the wire. The actual magnitude of the distances from the wall for tests 37 and 50 is shown on supplemental scales in Figure 16. The velocity gradient as a function of distance from the lower wall is depicted in Figure 17 for test 50. In this figure the gradient determined from the experimental measurements is compared with that predicted from the differentiation of Equation 1 which yields the expression (9)
Curves have been included in Figure 17 for values of k of 0.3 and 0.4. The apparent complex variation in the velocitg gradient with respect to vertical position near the center of the flowing stream shown by the experimental curves of Figure 17 is typical. The flow is not quite symmetrical about the axis of the channel. For many of the flow conditions so far investigated with the present equipment a small asymmetry has been found. However, the distortion is within the precision of the measurement and the asymmetry under isothermal conditions varies from side to side. The values of the velocity as a function of position for the measurements presented in Figures 13 to 17 are recorded in Table 11. This detailed information has been included in case the reader wishes t o carry out computations based upon these measurements beyond those indicated in the present discussion. The temperature measurements recorded in Table I1 have been corrected for the influence of impact, making use of the experimentally determined recovery factor that has been described. It is apparent that the temperatures near the entrance section of the channel were slightly lower than those measured at the location 12.5 feet downstream. This difference in temperature resulted in part from the fact that the incoming air was maintained at 100' F. a t substantially zero velocity, which corresponds to a stream temperature of 99.92' F. after adiabatic
21.
Feet/Sec. go TEST39 0 0,483 0,423 10.52 12.31 0.368 0.362 15.40 17.88 0.308 20.21 0.302 22.41 0.248 23.63 0.242 24.90 0.242 26.36 0.218 26.97 0.212 27.67 0.188 28.14 0.182 28.64 0.152 29.09 0.122 29.52 0.092 29.86 0.062 30.15 0.047 30.84 0.032 31.83 0.024 32.79 0.017 33.30 0 33,63 TEST37 0 0 643 8.58 0.618 11.02 0.614 17.88 0.588 20.96 0.584 22.23 0.559 24.23 0.555 24.82 0.500 25.74 0.496 0.412 26.11 26.81 0.408 27.28 0.324 27.97 0.320 0.235 28.39 28.80 0.231 29.51 0.176 30.32 0.172 30.86 0.072 31.36 0.018 31.72 0 .32.4W TEST37A 0 0.592 9.43 0.534 9.73 0.481 16 08 0.477 17.91 0.423 20.40 0.g19 22.92 0.361 24.45 0,307 26.32 0.250 27.70 0.246 28.56 0.188 29.46 0.130 30.09 0.072 30.74 0.044 31.85 0.021 32.71 0
Temp.. F.
100.08 100.07 100.07 100.07 100.06 100.07 100.05 100.04 100.06 100.08 100.03 100.04 100.00 100.02 100.03 100.02 100 IO
99.89 99.87 99.94 99.91 99.90 99.86 99.90 99.81 99.83 99.84 99.99 100.0
99.77 99.79 99.79 99.77 99.78 99.89 99.98 99.93 99.99 99.96 99.98
100.0
Fee&ec 33.62 33.24 32.875 32.74 32.150 32.04 31.34" 31.23 31.37 30.670 30.67 30.220 30.20 29.39 28.39 27.20 25.12 23.88 21.84 19.92 18.20 0
32.23 32.780 32.36 33.08" 33.01 33.31a 32.95 33.360 33.39 32. 94a 33.01 32.53" 32.00 30.495 30.49 29. l g n 29.21 24.13 11.90 0
33.22 33.36 33.735 33.57 33,440 33.52 32.96 32.02a 31.15a 31.23 29.77 27.98 25.25 21.93 15.09 0
T E ~ 50 T 19.14 0.991 100.09 100.12 22.75 0.988 29.27 100.06 0.985 37.04 99.99 0.980 42,.94 99.94 0.968 43.78 99.96 0.965 45.54 99.97 0.961 47.32 99.98 0.950 49.48 100.02 0.935 51.28 100.05 0.922 52.52 0.908 99.81 54.92 99.75 0.879 56.79 99.74 0.850 58.20 99.72 0.822 60.28 99.72 0.793 62.20 99.79 0.735 99.80 63.74 0.678 0 Pitot tube measurement, all others hot wire.
99.79 99.77 99.78 99.81 99.82 99.83 99.80 99.80 99.83 99.88 99.89 99.90 99.94 99.90 99.87 99.81
65.38 66.150 66.21 66.83 66.52 65.69 63.84 61.78 59.48 56.44 52.52 50.00 47.87 45,23 42.60 40.03 38.42
acceleration to a velocity of 30 feet per second. This acceleration caused a slightly lower temperature for the air near the entrance of the channel and some delay in the establishment of a uniform temperature. It is not believed that variations in temperature indicated from point to point in t,he flowing stream at the position 12.5 feet downstream are significant. However, at the positions nearer to the entrance section the minor variations in temperature appear to result from turbulent mixing of the stream subsequent t o the decrease in stream temperature in the converging section shown in Figure 2.
February 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
TABLE 111. ESTIMATED UNCERTAINTY OF MEASUREMENTS Range of
Meaeurement Velocity feet per second Weight &f flow pgund per second Thermal flux, ’( .t.u. per second) D e r square foot Priasure gradient, (pound/aquare foot) D e r foot Tern &&re OiP 0 F. Ai; O F. Point’temperature,0 F.
Low High 10.0 100.0 0.04 0.4
0.004
-0.06
Estimated Uncertainty Low High 0.06 0.20 0.0002 0.002 0.0002 0.005
0.1
-1.6
70.0 130.0 70.0 130.0 70.0 130.0
0.004
0.004
0.01 0.07 0.07
0.01 0.07
0.07
419
y + = distance ammeter, U,vd/U YQ = vertical &stance between plates, feet = eddy conductivity, square feet per second ec = eddy viscosity, square feet per second e,, K = thermometric conductivity, square feet per second 3 viscosity, pounds seconds per s uare foot 7 u = kinematic viscosity, q / p squareqeet per second = density, pounds (square seconds) per (f00t)d p v = s ecific weight, pounds per cubic foot T = skear, pounds per s uare foot = shear at wall, poun& per square foot TO In = natural logarithm Re = Reynoldsnumber LITERATURE CITED
The pressure gradients recorded in Table I were taken from pressure measurements a t the piezometer bars located in the upper wall of the working section. The data recorded in Table I represent average values taken over a 4-foot length of the working section excluding the traversing equipment. These data correlate effectively with other measurements of pressure gradients (7, 18, 24) as indicated in Figure 16. After approximately one year of use of this equipment it is believed that the several measurements pertinent t o the establishment of the eddy viscosity and the eddy conductivity were determined within the probable errors listed in Table 111. I n this tabulation the range of each particular variable has been listed aldng with the probable error estimated to be associated with the maximum and minimum ranges of each of the variables. ACKNOWLEDGMENT
The assistance of G. W. Billman and H. H. Reamer in the design of the equipment and of W. M. DeWitt and L. T. Carmichael in its construction and assembly is gratefully acknowledged. S. D. Cavers, D. M. Mason, and D. K. Breaux contributed to the measurements reported herein. NOMENCLATURE
= intercept C P = isobaric heat capacity, B.t.u. per (pound)(O F.) a = differential k = von KArm&nuniversal constant $ = heat flux, B.t.u. per (square foot)(zecond) t = temperature, point, time average, F. 21 = velocity, point, timexerage, feet per second u* = friction velocity, 1 / ~ 0 / pfeet per second 3 - residual velocity, feet per second %+ = velocity parameter, u/u* Urn = maximum velocity, feet per second u = average velocity, feet per second l l - vertical distance from lower plate, feet ya = vertical distance from nearest wall, feet A
(1) Allen, C. M., and Hooper, L. J., Trans. Am. SOC.Mech. Engrs., 54, 1-16 (1932). (2) Bakhmeteff, B. A., “The Mechanics of Turbulent Flow,” Prinoetdn, N. J., Princeton University Press, 1936. (3) Billman, G. W., Mason, D. M., and Sage, B. H., Chem. Eng. Progress, 46, 625-35 (1950). (4) Boelter, L. M. K., Martinelli, R. C., and Jonassen, F., Trans. A m . SOC.Mech. Engrs., 63, 447-55 (1941). (6) Colburn, A. P., and Coghlan, G. A., Am. SOC. Mech. Engrs., Annual Meeting, New York (Dec. 2-6, 1940). Corcoran, W. H., Roudebush, B., and Sage, B. H., Chem. Eng. Progress, 43, 135-42 (1947). Deissler. R. G., Natl. Advisory Comm. Aeronaut.. Tech. Note 2138 (1950). Eagle, A,, and Ferguson, R. M., Proc. Roy. SOC. (London), A127, 540-66 (1930). Gebelein, H., “Turbulenz,” Berlin, Julius Springer, 1935. Goldstein, S., “Modern Developments in Fluid Dynamics,” Vol. I and 11,London, Oxford University Press, 1938. Jenkins, R., Brough, H. W., and Sage, €3. H., IND.ENC.CHEM., 43, 2483-6 (1951). von Kdrm&n,Th., J . Aeronaut. Sci., 1 , No. 1, 1-20 (1934). von K&rm&n,Th.,Nachr. Ges. Wiss. Gattingen, Math.-physik. Klasse, 1930, S . 58. von K&rm&n,Th., Trans. A m . SOC.Mech. Engrs., 61, 705-10 (1939). Laufer, J., Natl. Advisory Comm. Aeronaut., Tech. Note 2123 (1950). McCann, G. D., Proc. Natl. Elec. Conf.,2,372-92 (1946). Martinelli, R. C., Trans. Am. SOC.Mech. Engrs., 69, 947-59 (1947). Nilcuradse, J., Forsch. Gebeite Ingenieurn., 3, supplement, Fomchugsheft, No. 350, 1-36 (1932). Pardoe, W. S., Mech. Eng., 58, 60-2 (1936). Prandtl, L., Physik. Z.,29, 487-9 (1928). Reichardt, R., 2.angezo. Math. u.Mech., 20, (6) 297-328 (1940); Natl. Advisory Comm. Aeronaut.. Tech. Mem. 1047 (1943). Reynolds, O.,Mem. Proc. dfanchester Lit.& Phil. SOC.,14, 7-12 (1874). Skinner, G., thesis, Calif. Inst. of Technology, 1950. Wattandorf, F. L., Proc. Rog. SOC. (London),A148, 565-98 (1935). Willis, J. B., Australian Council Aeronaut. Rept., ACA-I9 (Oct. 19, 1945). RECXIVED December 19,1949.
(Temperature Gradients in Turbulent Gas Streams)
TEMPERATURE AND VELOCITY DISTRIBUTIONS IN UNIFORM FLOW BETWEEN PARALLEL PLATES F. PAGE,
T
JR., W.
H. C O R C O R A N , W.
HE prediction of thermal transfer between solid boundariea and fluid systems is a n important industrial problem. The background of experimental information in this field is large, and the general nature of the phenomena encountered has been summarized in a quantitative fashion by McAdams (16)and Jakob (8). Von KArm6n ( 1 1 ) , Reynolds @I), Prandtl (go), and Taylor (93) have all contributed to an analogy between thermal and momentum transfer. The concepts of eddy viscosity and eddy conductivity as set forth by von KArrn6.n (11) and Prandtl
G. SCHLINGER,
AND
6. H. S A G E
(20) have been utilized by Boelter (I) and Martinelli (fy), in conjunction with generalized velocity distributions, to estimate thermal transfer to fluids flowing in conduits. Limited thermal transfer studies have been made (4) involving the flow of air in a fairly thin rectangular channel. The results confirm approximately, under a limited range of conditions, von KArm&n’s hypothesis of a relationship between the eddy viscosity and eddy conductivity. These measurements were of a preliminary nature and left much to be desired in the way of precision,
