A Study of Heat Transfer by Turbulence Citation Roudebush, Bert Victor (1941) A Study of Heat Transfer by Turbulence. Engineer's thesis, California Institute of Technology. doi:10.7907/7g1q-mx90. https://resolver.caltech.edu/CaltechTHESIS:02272025-035722790 Abstract This paper consists of four main divisions. The first of these is a discussion of turbulence in a flowing fluid and of how heat and momentum are transferred thereby. The second is a presentation of the problems met in constructing the apparatus with which to measure the pertinent quantities connected with turbulent transfer. The third is a description of the method of calibration of the measuring instruments. The fourth is a presentation of the experimental results obtained; these results, however, are so limited in quantity that they serve merely as an indication of those to be expected in the future. Item Type: Thesis (Engineer's thesis) Subject Keywords: (Chemical Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Chemical Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Lacey, William Noble (advisor) Sage, Bruce H. (advisor) Thesis Committee: Unknown, Unknown Defense Date: 1941 Record Number: CaltechTHESIS:02272025-035722790 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:02272025-035722790 DOI: 10.7907/7g1q-mx90 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17030 Collection: CaltechTHESIS Deposited By: Ben Maggio Deposited On: 27 Feb 2025 17:22 Last Modified: 28 Feb 2025 00:03 Thesis Files PDF - Final Version See Usage Policy. 30MB Repository Staff Only: item control page

A STUDY OF HEAT TRANSFER BY TURBULENCE Thesis by B. V. Roudebush In Parti al Fu lfillment of the Requirements for the Degr ee of Chemical Engine e r Californi a Institute of Technolog y Pasadena, California 1 9 41 (1 ) Abstract This paper consists of four main divisions. The first of these is a discussion of turbulence in a flowing fluid and of how heat and momentum are transferred thereby. The second is a presentation of th~ problems met in constructing the apparatus with which to measure the pertinent quantities connected with turbulent transfer. The third is a description of the method of calibration of the measuring instruments. The fourth is a presentation of the experimental results obtained; these results, however, are so limited in quantity that they serve merely as an indication of those to be expected in the future. ( 2) The Theoretical Aspect W. F. Durand (1)~:- states, 11 The present position of the theory of heat transfer in eddying flow is very unsatisfactory. It appears quite certain that the exchange of quantiti8s having scalar properties, such as heat energy, does not alw a ys obey the same law as the exchang e of a quantity having vector properties, such as momentum . . . . . . . . . . . . . 11 This indicates a lack of knowledg e as to how heat and momentum are transferred by a turbulently flowing fluid . To gain this knowledge one must first have an understanding of vvhat turbulence is. The hypotheses as to what constitutes turbulence, and the functional relationships that have been used in connection therewith, have undergone two st ag es of development and are now beginning upon a third. The first of these stag es was one of strictly limited know- ledge . Recognition was given to the differences between laminar and turbulent flow, but functional relationships were restricted to the use of emp irical constants, valid onl v for limited rang es. The second stag e, however, marked a decided step forward. Osborne Reynolds' work u p on the subject, leading to the formulation of the ratio that bears his name, showed that turbulence in a flowing fluid depends upon the balance between inertial and viscous forces. When the former predo minate, the flo w is turbulent; when the latter predominate, the flo w is larninar. Thus, by t he formul a tion of the dimensionless ratio known as the Reynol d s number, corr el a tion be- tween different systems under dif f erent flo w c o nditions ~:- See pag e 55 for bib]jography was possible. Still, however, the exact mechanism of t urbulence remained unknown; within a limited ranr e of values of the Reynolds number, the flow c ould be either lamina r or turbulent, depending upon the absence or presence of di s turbances. Now the third s tag e of development of knowledg e h a s been entered u p on with the hope that the laws which g overn turbulent motion can be ascertained. ConcomMitant therewith it is hoped tha t relation- ships can be found that will p e rmit the calcul a tion of vel o city and pressure distributions in turbulent fl ow and of the ener gy transfers across a turbulent flow without resort to empirical elements. The essence of turbulence is now known to reside in the f a ct that the velocity and the pres sure at a g iven point in a turbulent flow are continuously fluctuating in an irregular fashion about a mean value. Both the variation and the irregularity are of prime importance. It is essential that the variations hav e no d e finite frequency. Howeve r, there is a definite correlation between the components of the vel.o ci ty fluctu a tions at a g iven point; for, with out that c ondition, there c ould be no transmission of she aring stress ( or transfer of the momentum which g enerates shearing stress ) across a plane parallel to the direction of mean flow, except for that contributed by laminar friction. Consequently, the fundamental equation of she aring stre s s al ong a plane parallel to the me an flow in a re g ion where the viscous forc~s are inconsequential is :-i~ ( 1) ➔:- F o r a tabulation of the nomenclature see Appendix 1, page 45 l4J or in terms of heat transferred across this plane: Q= C.f 9',v-' ( 2) The evaluation of the s e fluctuations in terms of othe r proper t ies of the flowing system requires further consideration. The mechanism of the transfer of a quantity may b e thought of in the following manner. Existing in the turbul ent flow are re g ions of equal potential, where potenti .'1 1 . r. ny ref e r ei t her to v e locity or to temperture. Also in the turbulent flow are eddies or small gusts that move across the turbulent stream with random motion and exist only momentarily. When one of these e ddies forms and leaves a given region, it leaves with the potential of that re g ion associated with it. When it has completed its motion, it finds itself, in general, in a region of different potential from that of the re g ion from which it came. As a consequence, its potential must be equalized with that of its surroundings, either by its giving up momentum (or heat) if it has an excess, or by absorbing momentum (or heat) if it is deficient in this quantity. This continuous formation, movement, and dissolut- ion of eddies is the mechanism by which turbulence effects a transfer of a quantity. The component, perpendicular to the direction of mean flow, of the aver ag e distance an e cdy travels before it loses its identity by transferring its excess potential to its surroundings has been given the name "mixing length", here denoted byL. This gen- eral idea is subscribed to by most theorists upon the mechanism of transfer by turbulent flow. Howe ver, the evaluation of this mixing leng th in terms of other properties of the fl ow has received varied treatment. L. Prandtl (2) has proposed that the essence of the transfer is that of momentum in such a manner that momentum is conserved in the direction of mean flow. Using this concept it seems justifiable ( 5) to consider u 1 as being proportional to the product of the mixing leng th and the rate of chang e of vel ~city with respect to distance perpend ic u lar to the mean flow. Also, as there must be a correla- tion between the velocity fluctuations at a given point, v 1 may be set equal to a const ant times u 1 • Thus, absorbing all constants of proportionality into L, Prandtl obtained f r om equation (1): 1,, ( J.M/ ~) ( 3) and from equation (2): ~-:; -cp L~ ( ole/~)(~/°1-) ( 4) G. I. Taylor (3), however, started upon the premise that the essential feature of transfer was that of vorticity under conservation of vorticity. Correspondingly, his equation derived from ( 1) is: ( 5) where Lhasa dif f erent numerical value from that g iven by the Prandtl equations. Th. von Karman (4) has st arted from a different base than either of his predecessors. He desired not to be limited by the assumptions of conservation of momentum or of vorticity and, instead, postul a ted the following two conditions: (A) The mechanism of turbulent interchang e is independent of viscosity except near walls. (B) The local flow pattern is statistically similar in the neighborhood of every point, and only the time and leng th scale vary. This second condition is equiva.lent to saying that the diff e rent components of the velocity fluctuations are governed by a constant correlation. Using these assumptions, von Karman was led to the same equatio n s as was Prandtl and, in addition, to the following e~uation~ all provi d ed that Lis small in comparison with the ( 6) linear dimensions of the flow: L = k \ ( Jw./ ~) / (&.,,A/ L ";:: '1 '-) \ ( 6) K \ (J.e / ~) / (J.,,'o / ½'-) \ where K is a universal c onstant equal to 0.4. ( 6) Whether these two values of Lare e 4 uivalent is not explainable by theory alone, and experimental data must be obtained t o clarify this point. The available experimental data ewe. too meag er to verify any of the foregoing formulations as to the essence of turbulent transfer. However, as a starting point for the present investiga tion, let the assumptions of von Karman be taken as the criteria for turbulent transfer. When consideration is given to transfer from a solid wall to a moving fl ,-1.id, the simplifications made with re gard to derivations in the above theories are no long er applicable; for now the transfer through the boundary layer must be considere d , and viscous forces are no longer inconsequential. In this region, the transfer of momentum is equal to the gradient of the vel o city in the direction normal to the surface upon which the boundary layer exists multiplied by a constant of proportionality, the coefficient of viscosity. Likewise, the transfer of heat is equal to the corresponding temperature gradient multiplied b y a constant of proportionality, the thermal conductivity. Thus, for the comp lete transfer from wall t o flowin g fluid, equations ( 1) and ( 2) become: ~-;: r J.v../~ - P ~· q -:. _ R ch,/1 + c.p J\r, v-' e' ( 7) ( 8) Reynolds has sug ge ste d that turbulent transfer may be expressed in a form analogous to that of laminar transf e r by the introduction of the concepts of e ddy viscosity and eddy conductivity. that these t wo quantities are numeric a lly equal, the above Assuming equations may be written: ( 9) q::, (10 ) or rewritten as ~ (✓ *e') olv-/°1 (11) ~ -('R./tf -t-e.) Je/~ (12) Thus, it may be seen that the transfer of heat is directly proportional to the transf e r of momentum if either of the foll owing conditions is satisfied: ( A) J and ~/cf ( B) J are negligible compared to e and 1\/, f are numerically equal; that is, the Prandtl number = a- -::. ,JA'-/P. ':. ' The first condition is satisfied quite generally in the main body of any turbulent flow, but not near the walls. The second c ondition is satisfied completely only in the case of an ideal ga s, but it is closely approximated b y the common g ases. For li'1.u ids, ho vve ver, it is not even approximately true. The ratio 'T / p has the dim e nsions of velocity squared; thus, the quantity ').'io/ru"', where U is s ome reference vel o city, is dimensionless. It is usually designated as the coefficient of friction, Cr• Similarly, Q/cp has the dimensions of velocity times temperature. Using a reference temperature 8, the quantity 0/cfU6is dimension- less and may be called the heat transfer number, ch. Thus: I'\ O ~ (!_ ~ p,j''- I ')._ ( 13) q = c\.. f c. u e (14) For the case whe re V equals ~/c.f , as stated in condition ( B) above, von Karman ( 5) has shown that: c."' :. C~ I 'l. Howev e r, for the cases where V is not e y_ual to ( 15) R. /c. rand where V and R/Lf are not negli g ible comp ared toe throughout the rang e of transfer, the approximations allowable in d e ri vinp; equation ( 15) no longer are applicable. In this case, von Karman has arrived at the relation: .1/c.,_ = (16) If Cf and Ch can be determined from equations (13) and (14), the validity of equations (15) and (16) can be substantiated or refuted. To do this, ho wever, 1'0 and Q must be determined experimentally for a series o f values of U and 9 . In addition, by the determination of point values of the temperature and velocity at suitably located points in a turbulently flowing stream, the mixing lengths of e 4 uations ( 6) and (6 1 ) can be computed and compared. Thus, to proceed further with the study of heat transfer by a turbul ~ntly flowing fluid, experimental data Qv~ necess ary. uonsequently, 8 discussion of the factors which were pertinent to the construction of an ap p aratus with which the quantities needed for this furtherance could be measured is appropriate. (9) Construction of the Ap paratus The prime consideration before construction wa s the decision as to the physical set-up in which the turbulent motion was to be produc ed and studied. It was decide d that the production of a turbul e nt flow between t wo parallel plates, being hel d at different t emp er atures, in such a manner that t he flo w would be t wo-dimensional would yield a maximum amount of information with a minimum number of complicating factors. main problems: This goal created the following three (1) the obtaining of two-dimensional turbul e nt flow betwe e n the p ar all e l plates, (2) the maintenanc e of th e p a rallel plates a t specified temperatur es , (3) thv ar r ang ement of devices to measure t he ess ent ial quantities connected with the flow namely, the temperature and the velocit y of the turbulentl y flowing stream at many known points between the plates, the total quantity of heat being transf e rred between the plates, and the shearing stres s existing in the fl u id at the side-wall. Dimensions of experime~tal channel. itself into two simpler ones. The first problem resolved The first wa s to nrovide the correct width and the correct leng th of plates to corr e spond to a g iven distance between them; that is, the determination of the correct dimensions of a channel in which to make experimental measurements. After study of the data bearing u pon this qu esti o n, it was fo u nd tha t the ratio of leng th to width to d epth sho u l d be abo ;1t 60 : 1 8 : 1. This wo u l d g ive a sufficient leng th s o that the turbul enc e would be full y adapted to the cha nne l by the ti me the flo w had reached its end; and the hi gh ratio of width to de p th wou ld create twodime ns j_ onal flo w in at le a s t the middle t h ir d of a cros s -section. The smallest depth with which it would be conv e nient to work was thought to be 3/4 of an inch. A s u it able width of one foot and a ( 10 leng th of five feet were then chosen, thus g ivi ng t he ratios 8 0:16:1. - - - - - - - - - - ----- --- The second part of t he first Produc t ion of the Turbulent Flow. probl em was the cre at ion of a proper stream of fluid throug h the channel. Proper in thi s case reduced to (lJ kno wn quantity, (2) correct bulk temperatur e , and (3) sufficient propulsion. r, Sinc e a i r a t bulk tempera t ure of 1 30 -·F and a pressure of approxim a te ly atmospheric was arbitrarily selected as the initial fluid to be used in these measurements, the 4 uestion as to quantity reduced t o a consideration of the values of the Reynolds with which it was desire d to work. number It was felt that if it could be conve niently done, it would be extremely inf ormative if the Reynolds number could be varied from 2000 to 100,000. Under the specified conditions, this meant that the quantity of air should vary from 0.205 cubic feet per second to 1 0 .25 cubic feet per second. This latter fi gure then g ave the maximum cap a city of the blower needvd t o circulate the air - namely, 615 cubic feet per minute. This was the first quantity nee r ed in the complete speci- fication of a blower; in addition, the deliverable static head had to be specified. For a proper determination of this latter quantity, the detailed elements of the path to be follo wed b y the flowi ng air h ad to be known, because the frictional l o sses entailed by this pa t h represented the minimum value of the st a tic head that the blower had to deliver. Any lowvr value wo u l d have provided insufficient propulsion to drive the air throu~hout the comp lete circuit, which, of necessity, had to con sist of more than the blower and the expe rimental channel. Consequently, atten- tion h ad to be turned to those details of path. Measurement of Rate of Air Flow. Somewhere in the flow path (11) there had to be a metering device to give a measurem8nt of the total quantity of air that would p8.ss through the experimental channel. It was first thoue;ht that an orifice meter v,ould be the proper choice, but, when it was realized how g re a t energy losses due to other factors were go ing to be, the Venturi meter was selected as havin~ smaller fl ,w resistances. Calculations (details of which appear in Appendix I~ showe d tha t three Venturi meters of the sizes t a bulated below were necessary in order to measure the desired range of flows. Nominal pipe-size of Venturi housing (inches) Actual Pipe Diameter (inches) Throat Diameter (inches) 2 3 2.05 3.09 3.88 0. 66 1. 21 2 .39 4 Construction of the Venturi Meters . Air Flow Rate Minimum Maximum (cubic feet/sec) 0.205 0.658 2.22 0.658 2.22 9. 2 5 In the a ctual construction of the Venturi meters, economy and efficiency dicte.tecl the use of a cement form in an iron pipe . This was a ccomplished by constructing wooo. forms, consisting of two interlocking pie c es, in the shape of the air-cavity in the finished meter. These forms were then greased to prevent the adherence of concrete and inserted in the appropriate p i p es. The concrete was then poured in, allowed to set, and the wood forms withdrawn . Flow Path of the Air. With the knowledg e of the number and the kind of metering devices to be employed, the total flow path could be more fully described. (A schematic diagram of this flo w p a th is shown in Figure 1, page 12). The blower woul d dischar g e directly into a manifold fr om which would extend the three pipes containing the Venturi meters. The dischar g e ends of these pipes, each of which would be equipped with a mechanism to cut it out of t h e flow circ u it, (12) Fiu Pitot tube e I Probe '=1:iermoc ou le Oil ath. ~ T Block lT ' r ,. ~ ~ Oil Bath I :Blowe>r S½.ut-off Plate A B C D (13) would run into another man ifol d . From this a curve d condu it, with a radius of 2. 5 feet and a 6 x 1 2 n rectangular cross-section, would 11 lead to a secti on which would conver g e directly t o the experimental channel. After the channel, there would be a diverging section ending in a circular pipe of five-inch diameter. This p ipe, by me ans of two right-angle elbows, would lead back into the blower. This information was s ufficient to permit calculation of the friction al resistances to the air flow. As information re garding these resistances was pertinent only in the specific a tions of a blower, the calcu lations were restricted to the case wherein the maximum friction would occur - namely, at the maxium rate of flow. The results of these determinations a re liste d in the fol lowing table: Portion of path Los s ,o f Head in feet of f l owing fluid I 214 1 23 .5 138.5 214 0.64 16 .7 327 30.8 161 Total 1226.14 A B C D E F G H This total drop in head corresponde d to 15.9 inches of water ( the static head at which the blowe r had to discha:rgeL and this, in turn, requ ired a blower horsepower of 1.54. Thus, the complete blower specifications were: a maximum capacity of 615 cubic feet per minute, delivered at a static head of 16 inches of water, and req u iring at least one and one-half horsepower to drive. Bulk Air Temper a ture .■_ In connection with the first main problem, there remained at this point only the ~1estion of mRintaining the 0 air at the desired bulk temperatur e of 130 F while passing throug h (14) the expe rime nt a l channel. It seemed likely tha S owing to the friction developed and to the recirculation of the air between the heated plate~ an equilibrium would be e stablished so that the air would enter the experime ntal channel at 8 temperature sufficie n tl y hi gh to obviate the neces s ity of a pre-heater. But only exp e rimental testing could show the validty of this assumption, since the factors which would affect the he a t losses from such a system wo u ld develop into equations whose complexity would out-weigh their utility. Control of the Temperature of the Par a llel Plates. The second main problem was the maintenance of t he parallel plates, which form the top and bottom of the channel, at specified temperatures. This coul d best be done by using a good heat conduc tor for the pl a tes, and a n insul s.ting ma terial for the sides. selected we re, res p ectivel y , copper and wood. The materials It was d esired t o maintain the temperature of each plate constant to within o.o5° F uniformly along the length ; and, to do this, circulati ng oil b a t h s in d ir e ct contact with the p l a tes we re neces sary. Circulation of the Oil. These conside r at ions a gain brought u n their more specific problems, which were: (1) proper circulation of the oil, (2) proper heating of the oil, (3) measurement of the temperature of the plates. The first of these was solved by the installation of two axi a l-flow pumps, which were desi ~ned and constructed (in accordance with the calculations g iven in Appendix III) to give a volumetric rate of flow of 0.175 cubic feet o f o il per second when running at 1000 RPM. Heating of the Oil. In order to provide for the proper heating of the oil, each bath was equipped with three heaters of 500 watt capacity (as specified in Appendix IV), t wo being externa lly mounted (15) and one immersed. One of the external heaters was designed solely to help bring the bath to temperature, whereas the other external unit and the immersion heater wer e to keep the bath at the desired temperature under the t::y_uilibrium condition. For this reason, one external heater on each bath was connected across a Variac so that the potential applied could be varied from Oto 130 volts. Thus, by ex- perimental test, the proper voltage could be determined so that this heater could be left on continuously to supply the heat that was withdrawn during the flow. The immersion heater was connected through a relay system to a mercury regulator which would throw this heater off when the oil became too hot, and on when it became too cold. Thus, at the equilibrium condition, one external heater would be off, one external heater would be on at some voltage lower than 130, and the immersion heater would be going on and off at 40 second intervals (the delay period of the relays). Measurement of Point Velocities. There remained then only the problem of the provision of the measuring devices. The first of these to be considered was the one which was to measure the velocity of the turbulent flow at various points between the parallel plates. It was decided to accomplish this measurement by the location of a Pitot, or impact, tube of extremely small diameter (0.0176 inches) in the flowing stream. If this tube wer e connected to one arm of a manometer, the pressure exerted upon the surface of the liquid therein would equal the sum of the impact pressure and of the static pressure at the point where the Pitot tube was located. If now it were assumed that the static pressure was constant across the width of the channel, the connection of a static pressure tap in the side-wall of the channel to the other a1·m of the manometer (16) would produce upon the surface of the liquid therein only the static pressure. The difference in the level of the liquid in the two arms of the manometer multiplied by the density of the liquid would, therefore, be a direct measure of the impact pressure. Divid- ing the impact pres sure by the density of the flowi r,g air would give the velocity head, from which the point velocity of the air stream could be determined by the use of the fundamental equation of a Pitot tube: (17) where c' could be assumed as unity for the flow under consideration. Measurement of the Point Temperature. It was decid e d to determine ti-.~ the point temperature fro m e lectromotive force developed between the " two junctions of a copper-constantan thermo_couple (hereafter referred to as the probe thermocouple), the one junction being placed at the point in question and the other being immersed in an ice bath in order to obtain a reproducible refer e nce temperature. This electro- motive force could then be directly measured by a potentionmeter; and, from a knowledge of the relationship between the electromotive force developed and temperature difference existing b~tween the junctions, the latter could be calculated. The smallest wires with t he re~uired tensile strength were desired for the constru ction of the probe thermocouple. This necessitated the us e of copper wire h a ving a diameter of 0.0037 inch and of constant a n wire of 0.0026 inch. erature of each cop p er plate, thermocouple To meas ure the temp- one junction of a copper-cons tantan was imbedded therein, the o t her junction being placed in an ice b a th. Determination of Loc a tion of the Pitot tube and of the Probe Thermocouple. For the adjustment of the position of the Pitot tube a nd that of the probe thermocouple, e a ch of which was desi g ned to move only vertically in the center of the channel, each was eyuipped with (17) a micrometer screw. Thus, the exact position of either with respect to the plates could be determined from the micrometer reading. In this connection, the sag of the thermocouple, which is 0.003 of an inch at the junction, as well as the diameter of the wires and of the Pitot tub~would have to be taken into account. Measurement of Heat Transferred. The next part of this problem was how to measure the total quantity of heat that would be transferred from the top plate to the bottom plate. To determine this value, a heater in the form of a cylindrical block was provided. This fitted into a hole in the top copper plate in such a way that the surface exposed to the flowing air was perfectly smooth. This block was in- sulated from the rest of the plate by a bakelite ring and fro m the oil by a metal cylinder and a Dewar flask filled with wax. Then, by a measurement of the amount of heat needed to ke e p this block at the same temper a ture as the rest of the top plate (temperature e 4 uality being indic Bted by a differential thermocouple) and from a knowledge of the area of the surface of the block, the heat transferred per unit area of the plate coul d be c a lculated. Measurement of Shearing Stress. How to measure the she a ring stress in the flowing fluid at the side-wall presented itself as the final problem in the construction of the appar a tus. This problem became greatly simplified when cognizance was taken of the fact that: --r O == - W\ rJ..P / a~ <1s) Thus, to determine 'T0 only the rate of chan ge of the static pres8ure in the direction of flow would be needed. This could be measured by placing a pressure tap at each end of the channel and measuring the pressure drop from end to end with a manometer. (18) Calibration of Apparatus Following the comDletion of the construction of the ap n aratus, the calibration of all me a suring devices was undertaken. In this regard, only the thermocouples and the Venturi meters needed consideration. Calibration of the Thermocouples. Because of the inaccessibility of the lower plate in the finished apparatus, onl y the probe thermocouple and the thermocouple in the top plate could be calibrated directly against a standard. This was accomplished by immersing the standard thermometer in the oil of the top bath until it was in direct contact with the copper plate (actually in a depression in the plate). Meanwhile the probe thermocouple was raised until its junction was in practically direct contact with the upper plate. Simultaneous readings of the standard and the two thermo- couples were then made. The probe thermocouple was then lowered until it touched the bottom plate, and simultaneous readings we:ce taken of it and the lower-plate thvrmocouple. This procedure was repeated a t several different temperatures, the upper and lower plates being maintaine d at constant and substantially equal temperatures durinf each set of re a dings. l1hus, there was a chance for only a ne g l i §! ibl y 1 small err o r in the calibration of the probe and lo wer-plate thermocouples, since t h e air between the copper plates could not vary appreciably from their temperature as no air wa s allowed tn flow during this calibration. Calibration Curves for the Thermocoupl~s. Because of the small range of temperature involved, the plot of electromotive force in microvolts versus temperature difference between the junctions in ° F. is a strai g ht line for e a ch of t he the r moc ouples. b:r- 8.t ir:--D ~, ,rves for the three thermocouples a re The c ali - tl9) shown as Figures II to IV. Calibration of the Venturi Meters. If the throats of the Venturi meters could be accurately measured, it would be unnecessary to calibrate them; for, as is seen from the fundamental equation of a Venturi meter: 'b-:. (19) t.'l"\V,)H M -:: A, / V-(-0,-/-D'\.--:)'t-::-_-1 ( 20) every qu a ntity, with the pos s ible exceotion of c', would be kno wn, and sufficient dat a a re available that v a lues of c 1 could be predicte d without serious error. : However, it was impossible to measure accurately the throats of t he completed Venturi meters, and it was inadvisable to employ the diameter o.f the wood forms used in their construction as correct. Conse¼uently, the re4uired c a libr a tion was accomplished while the me ters wc:;re in p osition in the apparatus, and before the diver g ent section was connected to the end of t he experimental channel, by taking Pitot tube re a ding s at various p oints across the cross-section of the channel. For several different distances from the side-wall, velocities were taken at a sufficient number of points in a vertical plane so t hat a plot of velocit y versus distance from the top plate would g ive a smooth curve. The areas under these curves were then integrated, and the integrated values plotted ag ainst distance from the sidewall. The area under this curve, which was volumetric rate of flow, was then determined. Then,by means of the Venturi e~uation (19), and by assuming a disch a rge coefficient of o. ~s , the value of M was cal- culated, and then the diameter of the throat of the meter. The cal- ibrations, sample curves of which will be found as Figures V to XII, yielded these results: ( 20) Venturi Meter Actual Throat Diameter (inches) Throat Diameter of wood form (inches) 1.18 2.41 1.21 2.39 The two-inch Venturi meter was not calibrated because the multiplying manometer needed for the reading of Pitot tube differentials at these low volumetric rates of fl c,w was not yet available. (21 ,.... ''· J :[.tl ·• .. i ..... _ •••• - ,, ' .,. • ;..!.. : (22 ) Fi__,_u re III -:T I. -l- ! :~ ; ~~ + 4 ., ' I • -~ T~k~ I I -,----t; · ,. : I ' ·1.·· .:_ 4+i"t'THrl:-ff.rrl~T4+.--f.il~f;-+..;+~+s-~& ~~--.J,....,...J_;_J,..~ .......................:,J ; 1· i ure 0 rv (23) Fi...;ure V ~ I.,t. .. _,.,...,~-r-r-r:-r.=-----r--rr---:-r-r"-r-.,....,.....-.-:--r-~~~'-rr-i'~-:c,..,..cr:-;-,-,-.-.=-,-rr,-,-,-.,....,,-,....,.....,.-,-.,--c-;-r--,-,...........,..-r-~----.---~...:;:;G5) ~ 1: 1 i I . r' l .. '•· .... : ,, .' . , iL:>ur e IX j - I I • I . i -· 1-·- -·-f- .. I ( 2o) I Fi., ure XII .. : ; ..!..i......_ : -- ·; . :;;:: ,- ' ~ . ! L! l ~ .. . ;~ • . . - ----·! .; ',t '1 ;:-: -11.: :~ :~:' '+ '!:II~ I• ~ 1 - - + - - t - - + ' - - r - - - - , - - - - ~ 1 - -..... .J [ I ~ ' .L i .::! ~:·. t- ~i l f _,.L; • ~ - --·+---+.,__-1--+--4-~1-·-·-·-·--1-;_...;f_:+·--.+...L;-'+i--:-'1'-'~....-+ '---+~-~ r 1 ... 1 ""I ~. ,-r • :: ,t:"... tt!,, .. . t .•.-l-4, . - !n;, "L ,! . - .... ....., ..... r:t ~!4: ,::r~ !Jt t- _, • ..1-r- ·-:R _.. r-: - ! 1.t:I ? ff "lr:,, , ... J ' .. ' I•. I ,,- ·--' i. (32) The Experimental Results Because of the limited time available, the only experimental data obtained consisted of several velocity and temperature traverses between the parallel plates. No measurements of Q and were possible; conse'-1uvntly, the onl y theor e tical concepts tha t co u ld be evaluated were the ~a-called mixing lengths defined in equations (6) and (6 1 ) . These mixing lengths were calc u l a ted for the several traverses and plotted as functions of the distance froM the top plate of the channel. Fi gure~ XVI to XXVII show these mixing lengths together ,,1th their correspondin,e: traverses. Unfortunately, the data are too meager to give any indication as to the corresrondence between the mixing lengths for momentum transfer and those for heat transfer, except to sho w that they are of the same order of magnitude. Thus, the results obtained must serve merely as a sample of those of the future. Fi.;ur, XVI Fi__,ure XVII (34) iuur TII~I (36) ( 37 ) Fi__,ur e XXI (3o) Fi 0 ure XXII i 0 urc TIIII 4u . ' -·~rI I . I::_ I ' :;, i....,ur ( .1) !..... -· •l 42 ) I I I I i- .,......L..J. I -1 - 1· ;••••• _ I ... II ·- .L ..~I-. i' I -· -· ·1 .• 1~ ·-~ - . -f-·7 . I --! • I.• ( 45) Appendix I Nomenclature Symbol Name Units for Calculation Al area square feet C specific heat (at constant pressure for a compressible fluid) Btu/lb •°F CI discharge coefficient friction coefficient heat transfer coefficient pipe diameter inches throat diameter of Venturi meter inches e eddy viscosity; eddy conductivity g acceleration due to gravity feet/sec.sec G mass rate of air flow lb/hour•ft h heat transfer coefficient Btu/hour•ft 2 .°F H difference in static head feet of flowing fluid H' velocity head feet of flowing fluid manometer differential inches of kerosene thermal conductivity Btu/hour•ft -°F 2 K universal constant, 0.4 L mixing length inches L' thickness of copper plate feet m hydraulic radius =-crosssectional area/wetted perimeter feet M constant in Venturi equation p pressure q rate of air flow cu. ft ./sec Q heat transferred Btu/hour•ft2 Q' heat transferred Btu/hour ( 46 )1 Symbol Name At temperature difference u,u velnc j_ ty Units for Cqlcu l~tions 0 F ft/sec velocity fluctuation in direction of nean flo w v' velocity fluctuatio n in direction perpendicular to QJ.ean fl ow mean v alue of this product, with respect to time V overall heat transfer coefficie n t 2 Btu/hour·ft -°F w mass rate of oil flo w lb/hour X distance parall e l to direction of mean flow y distance p t)rpend icular to direction of mean flow e' temp e ratur e fluctuation e temperature coefficient of viscosity lb/ho u r · ft coeff icient of vi scos ity for the air film lb/hour-ft kinemat ic viscosity Prandtl number density shearing stress shearing stresA at the wall abso lute value of A, where is any y_uantity A lb/cu ft (47): Appendix II Venturi Meter Desi s n Because of its relatively low density and its clear meniscus, kerosene was selected as the fluid for the manometer with which the Vent uri differential wa s to be read . It wa s desir e d to keep this differential between the limits of 1 inch and 20 inches of kerosene; the lower limit being dictated by the desired pl'ecis ion of reading and the up per by the deliverable static head of the small blower. Elementary c a lculations showed that for the de- sired range of flows one Venturi meter was insufficient. Further t\\~ calculations based upon I\ fundamental equation: i -: : (. M V~, '1 I ( 19) M -:. A, / \[( (), /r>'I.)~- I (20) and using the mean densit y of kerosene to be 49.5 pounds per cubic foot and that of air to be 0.0673, yielded the following table of values: ,,,_ D1 D2 2.05 3.09 3.88 0.66 1.21 2.39 c M(2g) min.Hk max.}ik min.q max.q 0.0188 0.0635 o.264 1.94 1.75 1.15 20.0 20 .0 20.0 0.205 0.658 2.22 0.658 2.22 9.25 1 M 0.00239 0.00806 0.0335 The odd values of Di were, in the first two cases, due to the fact that the most easily obtainable pipe-on-hand was selected; and, in the third case, this was the actual diameter of ordered \VOi\ four-inch galvanized tubing. " It was seen from the values of mini- mum Hk that lower values could be read with each of thesv meters; thus, the overlap ping here insure d that there would be no gaps in ability to measure rates of flow if a differential of 20 inches of kerosene could not be obtainedby the blower selected. ( 48) Appendix III Determination of Circulation Rate for the Oil As an initial basis of calculation, the temperature of the oil 0 on the top plate was tak e n as 135 F and that of the oil in contact 0 with the bottom plate as 125 F. Thus, there would exist 8. five degree temperature difference between the oil and the air stream. In order to determine the rate of heat transfer between the oil and the air, the heat transfer coefficients for the three resistances to flow-namely, the oil film, the copper plate, and the air film-had to be calculated. For the air film, an equation which has been derived from dimensional analysis (6) was used: -0,?.. (~ I c. c.) Ccr ~ I 'rt.) ,,12J = o. o l 3 ( '1 ~ ~/)A~) ( 21) However, for the common gases, this simplified to: ~ :. 0, 0\ 44 t.. C.,. o,f / (4w-.) 0 ,').. (22) It was found by calculation that values of "h 11 obtained from these equations ag reed to within 4 percent. As these particular calc ulations were subject to errors of greater magnitude th an this, it was decided that the accuracy of equation (2 2 ) was sufficient. Thus, by its use heat transfer coefficients were calculated and plotted against the Reynolds number for the flowing air as shown in Figure XIII. The heat transf e r coefficient of each oil film was obtained as ,,,t\oc.•,+1 a function of oil re ative to the plate and this relationship is "- shown in Figure XIV. At a velocity of o.7 foot per second, the values o f the oil film coefficient s were se e n to be subst antially equal to that of the air film at a Reynolds for the air. number of 1 00 ,000 Therefore, this veloci t y was s e lected for both the hot and cold oil. The corresponding valu e s of the ( 49) oil film coefficients were: ,_ At 135~F At 125 F . h:.26 .8 :Stu/hr •ft· F 'I. • h-=26.2 ·'.~Btu I hr•ft, F In calculating the overall heat transfer coefficient, there were the above-mc::ntioned resistances to be considered. I /\I and l '/\.\.)on + ( LI / ;. ) : q -:. M. ~ -\-o. \ t- (I/ I,.., \ ~;..,.. Thus: (23) V bt (24) By the use of e~uation (23) a nd Figures XIII and XIV, V was obtained as a function of the Reynolds in Figure XIII. number of the air and was so plotted In this calculation it wa s found that (Lj~) for the metal was a negli g ible quantity and that the overall coefficient from the hot oil to the flowing air wa s substantially equal to that from the flowing air to the cold oil. Then, for a given temperature differ- ential between the oil and the air, from equation (24) and from Figure XII~~ was obtained as a function of the Reynolds number of the air, and this relationship is ::ihown in Figure XV. From Fi gure XV it was seen that the amount of heat transferred from the oil in the top bath to the air ( and from the Rir to the oil in the bottom b a th) when the Reynolds number of the alr was 100,0GO would be 6 4 .5 Btu per square foot of plate per hour, thus making a total transfer of 622 .5 Btu per hour for the entire surface of fiv e square feet. The minimum advisable dc::pth of oil flowing over the plate was thought to be thre e inches; therefore, a s its velocity was to be 0.7 foot per second and its width of flow was to be one foot, 0 .175 cubic or 10 pounds of oil would be nee d ed per second past each plate. Then from the follo wing equation: (50) it was d ecuced th a t A t, the temperature drop of t h e oil in p a s s ing the leng th of the plate, would be 0.018°F, which wou ld be less than the o.o5°F drop set as the maximum allowable variation in temperature along the leng th o f t he plate. was s u it a ble. Thus, this rate of flow for the oil X lI 5 __ _ ' ---·i-- .. - -·---~- __ - ., _l '.:.I... ... i L- 1-----t-:---:---;:---''f-"""'.:--:-irc-----t,...,..-:---:--+-:-7""--rrrm·--lr---:--+----+·-.,----!f----,----+-- _ r_ _;_+-+--~~~---~~---~ =-~,,.--'-+-'-'--.;......;.. t ---'--- - 1 9--t---:-~~-...,..-,1----:---t-'----+-r---,...+~'--;--if'-pi.~-.!+L-:.....,,..:;.+:.,.-,....:..,--'--.+-:---,--t----~.\-=-~_;_L\ ..,,,...........~>-'- -l - - - - _ _ . ~.~ - 8--t--,--,-":rlr--'--+--,.--r=---+--,c=-"'·--,;1-;--=-..:t,1UJiµ+.:-+--=-==--jf---'·..:..:..__1-___.:..._.J---:....+-.::..__:__L...\....\.--.!-'----1 !- -◄-----'---'­ I j- 3-·t---+--:---+--~+---+---+---+----+---+---+---+----+---+-----U i- i .....f-·' 1 ~-~ i :7: .1 I I L 8'P ; ~ ! • .I xv 4 ___ -, --' .j.. -' -+ - ➔-- ---r 3 ___ .J-- - -·,-·, ,-~ -+ ,- -l-- ,__J T -j ..j 2 __ -T ' ; ~ ~ ; 4--""'--------~-+-----+-----+----------------+-----------+-----+--,---~- ------+--+-- -- -- - ---w, _j_ r 1______.....,......_______________.....___..._____......,_ _,..__ _..,______________________ --r (54) Determination of Heaters Needed for the 011 Baths The capacity of each oil bath was to be about 3.5 cubic feet. Taking the heat capacity of the oil at 0.5 Btu/lb?F and its density at 57 pounds p e r cubic foot, the heat needed to raise either bath one deg ree Fahrenheit would be 100 Btu (i.e.J29.3 watt-hours.) Assuminr the oil would have to be heated about 60°F initially, the total wa tt-hours needed would be 1750. '11hus it seemed advisable to use three heaters each of 500 watt capacity on each oil b a th; so they could be brought to temper atur e in a reason ably short time. (55) Bibliography (l) .•. Durand, W.F., "Aerodynamic Theory",.§, 238 (2) ••• Prandtl, L., "Z.f. ane;. Math. u. Mech.", P• 136 (1925) "Proc. of 2nd Int. Cong. for App. Mech. 11 (Zurich, 192~ (3) ..• Taylor, G.I., "Phil. Trans. A", fil:2, 1 (1915) "Proc. Roy. Soc. A , ill, 685 (1932) ( 4) .•• von Karman, Th., "Some Aspects of the Turbulence Problem", (Calif. Inst. of Tech.) (Pasadena) (1935) (5) .• •\'."on Karman, Th. : "The Analogy between Fluid Friction and Heat Transfer", (Calif. Inst. of Tech.) (Pasadena) (1939) ( 6) ..• Walker, Lewis, McAdams, and Gilliland, "Principles of Chemical Engineering ", (McGraw-Hill) (New York and London) (1937) (7) .•• Bakhmeteff, B.A., "'rhe Mechanics of Turbulent Flow", (Princeton University Press) (1936) 21., 416 (1939) (9) ..• Sherwood and Woertz, Ind. and Eng. Chem.", 21, 1034 (1939) (10) .• Towle, Sherwood, Seder, "Ind. and Eng. Chem.", 2.1, 462 (1939) (8) ..• Dryden; H.L., "Ind. and Eng. Chem~", 11 (11) .• Murphree, E.V., "Ind. and Eng . Chem.", 24, 726 (1932) (12) •• Wattendorf, F.L., "Jour. Aero. Soc.", 2, 200 (1936) (13) •• von Karman, Th., "Jour. Aero. Soc. 11 1, 1 (1934) (14). oRouse, H., "Fluid :M echanics for Hydraulic Engineers", (McGraw-Hill) (New York and London) (1938) (15) •• Rouse, H., "Trans. Amer. Soc . . Civ. Eng . 11 , 102, 463 (1937) (16) •• Schubauer, S-.B., "Natl. Adv. Com. Aero.", 524, 337 (1935) (17) •• Dryden, Schubauer, Mock, Skramstad, "Natl. Adv. Com. Aero.", 581, 109 (1937) Acknowled,gment Grateful acknowledgment is made to Doctors W. N. Lacey and B. H. Sage for their help and suggestions which made this 1 undertaking possible. Thanks also are given to Robert RBy for his assistance in connection with the experimental results.