THE EFFECT OF ULTRASONIC IRRADIATION ON THE RATE OF HEAT TRANSFER THROUGH LIQUID by Harry Bernard Pfost AN ABSTRACT Submitted to the School for Advanced Graduate Studies at Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1959 Approved aw” Q/J/ 0/1/47 ABSTRACT Previous investigators have shown that vibrations, sonics, and ultrasonics may be used to increase the rate of heat trans- fer between a solid and a fluid. All of the known investiga- tions have combined the effect of oscillating motion with nat- ural or forced convective effects. This research attempted to isolate the effects of the ultrasonic action. The determination of apparent thermal con- ductivity was made from transient temperature response to changes in boundary conditions. The governing differential equation for heat flow in one dimension and with internal energy absorption is solved and the results are used for cal- culation of apparent thermal conductivity and ultrasonic in- tensity. Parallel plate equipment was utilized and the heat trans- fer coefficient was determined from transient responses to changes in boundary conditions. The liquid under investiga- tion was confined between-two horizontal copper plates. The test cell was brought to a uniform temperature, and then the lower boundary was cooled quickly. The temperature change at the center of the liquid, as measured by a thermistor, was t then used to compute the coefficient of conductivity. Sound intensity was computed from measurements of the steady-state temperature rise at the center with both boun- O daries held at the same constant temperature. Harry B. Pfost iii Provision was made for ultrasonic irradiation either per- pendicular or parallel to the direction of heat flow. Test results are given for the effects of ultrasonics in the two directions, at frequencies of approximately one megacycle and uoo kilocycles, and at intensities up to 2.u8 watts/cm.2 in water. Tests with progressive waves and a glycerine-water solu- tion were made at one megacycle with the irradiation perpen- dicular to the direction of heat flow. The results show increased rates of heat transfer under all conditions with a maximum increase from 0.328 Btu/hr.ft. 0F. to 1.7 Btu/hr.ft. OF. The increase in the rate of heat transfer is less, at the same intensity and frequency, with progressive waves and for the glycerine-water solution indicating that the increases are probably due to convective-type currents. The effect of temperature gradients upon the refractive bending of sound beams is analyzed with reference to heat transfer applications and shown to be an important factor for application considerations. Harry B. Pfost THE EFFECT OF ULTRASONIC IRRADIATION ON THE RATE OF HEAT TRANSFER THROUGH LIQUID by Harry Bernard Pfost A THESIS ‘ Submitted to the School for Advanced Graduate Studies at Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirement for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1959 G 113er é~$~4o ACKNOWLEDGEMENTS The author expresses his great appreciation to Dr. Carl w. Hall, Professor of Agricultural Engineering, for his un- failing guidance and encouragement throughout this research. The author also wishes to acknowledge the valuable sug- gestions and aid contributed by the other members of the Guidance Committee, Dr. J. Sutherland Frame, Head of the Department of Mathematics, Dr. Egon A. Hiedemann, Professor of Physics and Astronomy, and Dr. Merle L. Esmay, Professor of Agricultural Engineering, throughout this graduate program. A debt of gratitude is owed to Dr. Arthur W. Farrall, Head of the Department of Agricultural Engineering and to Dr. John w. Hoffman, Director of the Division of Engineering Research, for their support in obtaining the funds required to complete this project. Harry B. Pfost Candidate for the Degree of Doctor of Philosophy Final Examination: September 1, 1959, 10 A.M. Dissertation: The Effect of Ultrasonic Irradiation on the Rate of Heat Transfer Through Liquids. Outline of Studies: Major subject: Agricultural Engineering Minor subjects: Mathematics, Physics Biographical Items: Born March 8, 1919, Cowgill, Missouri Undergraduate studies, University of Missouri, 1936-1940 Graduate studies: .Alabama Polytechnic Institute, M.S. 1998 Michigan State University, 1957-1959 Experience: Junior Engineer, Rural Electrification Administration, 19uo-19u1; Assistant Professor of Agricultural Engineer- ing, Alabama Polytechnic Institute, 1941-1943; Lieutenant (junior grade), United States Navy, 19h3-19h6; Associate Professor of Agricultural Engineering, Alabama Polytech- nic Institute, 19u6-19u9; self-employed farmer, Cowgill, Missouri, 1999-1953; Division Engineering Manager, Green Giant Company, 1953-1956; Product Planning Analyst, Ford Motor Company, 1956-1957; Graduate Research Assistant, Michigan State University, 1957-1959. Honorary Societies: Tau Beta Pi The Society of Sigma Xi PI‘Dfessional Societies: American Society of Agricultural Engineers TABLE OF CONTENTS INTRODUCTION.............. ........ REVIEW OF LITERATURE ........ . .......................... ANALYTIC SOLUTION OF THE PROBLEM OF TRANSIENT TEFIPEIKATURE EFFECTS0000000..0....0..00000...0000.. PROCEDURE General... ...... . ............ Test Equipment............... Temperature Calibration........................... Sound Absorption and Intensity Measurement........ RESULTS AND DISCUSSION Ultrasonics Perpendicular to Heat Flow in Water... Natural Convection Response.. Progressive Waves Perpendicular to Heat Flow in water0000000000.000000000000.00.00.0000000000.. Ultrasonics Perpendicular to Heat Flow in a Glycerine-Water Solution..... 0.0.00.0 000000000000. Effect of Temperature Gradients upon Absorption and Conductivity Measurements..................... Ultrasonics Parallel to Heat Flow..... ....... ..... General........................................... SLHAMARY..................... ...... ..................... CCHQCLUSIONS.................... .............. .......... 51K}GESTIONS FOR FURTHER RESEARCH..... APPENDIX 0.....000000. A1 - Mathematical Derivations - Transient Heat Flow in One Dimension..... ..... ................... Uniform Internal Heat Generation............. Non-uniform Internal Heat Generation, Ultra- sonics Perpendicular to Heat Flow............ A2 - Specifications of Equipment..... ........... .. BEFERENCESOOOOO. 00000 O 000000 O 00000 000.000.000.000000000 11 13 27 3O 39 41 43 Q6 52 58 61 64 66 67 68 71 73 83 Table A1. 4A2 4A3 [Ah LIST OF TABLES Steady-state temperature rise caused by ultra- sonic absorption at various levels of ultrasonic generator plate current.......................... Comparison of observed and theoretical rgsponse with no ultrasonics, k = 0.328. D = 26.1 F....... Steady-state temperature rise due to ultrasonic absorption in the glycerine-water mixture, one megacyCIe rarlge000000.00000000000000.000000000000 Comparison of observed and theoretical response, no ultrasonics, glycerine-water solution, k = 0.218 BtU/hl‘.ft.oF., D = 31.20Foeoooooooooooooooo Relation between ultrasonic generator plate cur- rent, sound intensity and rate of heat transfer, glycerine-water mixture, one megacycle frequency rame..00000000..00000000.0000000.000..00..0.0... Relation between a computed temperature response for non-uniform absorption and the observed data for one megacycle, 100 ma., in glycerine-water SOIution....O000000.0.00000...00.00.............. Relation between ultrasonic generator plate current and steady-state temperature rise, ultrasonics parallel to heat flow in water....... Relation between generator plate current and ultrasonic intensity, ultrasonics parallel to heat flow in water000000000000000000000.000000000 Tables in Appendix Physical properties of water..................... Physical properties of a glycerine-water mixture. Temperature response at center, five tests with no ultrasonics and four with low intensity in the one megacycle range.............................. Temperature reaponse at various levels of ultra- sonics intensity perpendicular to direction of heat flow in water, one megacycle range.......... Page 30 32 43 #4 Q6 50 53 56 71+ 7a 75 76 ix Table Page A5 Temperature response at various levels of ultra- sonic intensity perpendicular to heat flow in water #00 kilocycle range........................ 77 A6 Relation between generator plate Current and ultrasonic intensity, ultrasonics perpendicular to heat flow in water.000.00.000.000000000000000. 77 A7 Relation between ultrasonic intensity and appar- ent coefficient of conductivity, ultrasonics per- pendicular to heat flow in water................. 78 A8 Temperature response with natural convection (upper plate cooled) in relation to ultrasonic ~ intenSity00000000000000000000.00000000000000.0... 79 A9 Temperature response with progressive waves per- pendicular to heat flow in water, one megacycle range...00000.0.000.00.00.0000.0.0.0...00.0.0...0 80 A10 Temperature response as a function of ultrasonic intensity, one megacycle range, glycerine-water mixture, ultrasonics perpendicular to heat flow.. 80 A11 Temperature response as a function of ultrasonic intensity, ultrasonics parallel to heat flow in "ater00000000000000000000000000000000000000000 81 A12 Relation between ultrasonic intensity and appar- ent thermal conductivity, ultrasonics parallel to heat flow in water............................ 82 Figure (DflChKnt: 10 11 12 13 111 15 16 17 LIST OF FIGURES Schematic view of test cell. Ultrasonics per- pendiculartOheatf10Woooooooooooo000000000000. Schematic view of test cell. Ultrasonics para- 1181toneatfIOW............................... Test cell. Ultrasonics perpendicular to heat f10w0000000000000000.000...00000000000000.00000. Test cell. Ultrasonics parallel to heat.flow... Test cell in operating position.................. Thermistor bridge circuit....................... Amplifier and grounding circuit................. Calibration chart, lower plate thermistor, hori- zontal ultrasonics.............................. Calibration chart, center thermistor, horizontal ultrason10800000000000.000.00000000000000000000. Temperature response, two intensities........... Temperature response at various generator plate currents, 1 mo. horizontal ultrasonics in water. 0 Temperature response at various generator plate currents, 400 kc. horizontal ultrasonics in water00000.0.000.00....00...0....0..00000.000... Relationship of generator plate current to ul- trasonic intensity, horizontal cell, water...... Relationship of ultrasonic intensity to apparent thermal conductivity. Ultrasonics perpendicular ‘tOheat flow1nwater...00000000000000.000000.0. Comparison of natural convection and ultrasonics perpendicular to heat flow in water.............. Temperature response with progressive waves per- pendicular to heat flow in water, one mc........ Temperature response at various generator plate currents, 1 me. horizontal ultrasonics in glyce- rifle-water SOIUtion0000000000000000.00000000000. Page in 15 16 17 18 21 22 25 26 34 35 36 37 38 #0 #2 “5 Figure 18 19 20 21 xi Page Temperature response at various generator plate currents, 1 mo. parallel to heat flow in water.. 54 Temperature response at various generator plate currents, #00 kc. parallel to heat flow in water. 55 Relationship of ultrasonic intensity to apparent thermal conductivity. Ultrasonics parallel to heat flow in water..00000000000000.0000000000000 57 Two types of steady currents caused by sound waves.oeooeoooocoo00000000000"°°0°000'000000000 59 (fir-«Urn U) kc ma mc X 03 NOMENCLATURE Internal heat generation, Btu/hr.ft.3 Steady-state temperature rise due to sound absorption, 0F. . Temperature difference between boundaries, OF. Ultrasonic intensity, Btu/hr.ft.2 Distance between boundaries, ft. Length of sound path, ft. Temperature, 0F. Specific heat, Btu/lb.°F. Cycles/second Henry Thermal conductivity, Btu/hr.ft.°F. Apparent thermal conductivity (Combined effect of con- dzcgévity and convection due to ultrasonics), Btu/hr. Kilocycles/second Micro-farad Milli-ampere Megacycles/second Distance along heat flow path from cooler surface, ft. Time, hr. Density, lb./ft.3 INTRODUCTION This research was initiated to study the effect of ultra- sonic irradiation upon the rate of heat transfer through li- quids which are confined between two planes. The accepted theories regarding heat transfer attribute the transfer to radiation, conduction and convection. A high rate of radiant heat transfer requires that the temperature differences be- tween two surfaces be large and that any material between the two surfaces shall not readily absorb electromagnetic radia- tion. In this, as in most other research with liquids, it is assumed that the radiant heat transfer is negligible. The rate at which heat is conducted between two surfaces is a function of the material separating the surfaces and the temperature gradient. In the case of non-metals, it has been assumed that the heat is conducted from molecule to molecule 111 the material as molecular collisions occur. In the case of metallic melts it is assumed that there is an additional heat transfer through direct electron motion. It has not been shown previously that ultrasonics will affect the molecular mot ion in such a way that an increase in the coefficient of thermal conductivity will occur. Unlike conduction, in convection it is assumed that the molecules of a fluid are transported from one temperature re- gion to another and that during this flow they carry heat energy with them. This steady flow, more or less, is in addi- tion to the vibratory motion of the molecule which contributes to the conduction type of heat flow. These convection cur- rents may be either natural or forced. Sound waves in a liquid cause motions of the molecules which are large compared to the normal radius of oscillation of the molecules. If the sound waves produce motions which have certain types of velocity components parallel to the direction of heat flow, then this motion should produce a type of convective heat transfer. With convective heat transfer it has been assumed that a film of stationary or slow moving molecules exists near the boundary and that heat transfer through the film is largely conductive. Sound waves may aid in reducing the thickness or effectiveness of the film. This research covers the effect of intensity, frequency and direction of ultrasonic waves in water, and the effect of intensity in a glycerine-water solution, which has ahigher vis- cosity, on the rate of heat transfer. , If ultrasonic effects increase the rate of heat transfer 111 liquids significantly, they offer the possibility of in- creasing the capacity of heat exchangers, increasing the rate 0f heat transfer between the fluid and solids suspended in, or enclosing, the fluid. By minimizing temperature differences, 1-11<:reased rates of heat transfer offer the possibility of im- Pr‘oving quality and increasing reaction rates during process- 11Ng and chemical operations. REVIEW OF LITERATURE Reports of work regarding the effect of ultrasonics, sonics and vibrations on the rate of heat transfer indicate that little work has been done in this field and most of the work has occurred since World War II. In a survey of the metallurgical effects of ultrasonics, Hiedemann (1951+) reported that Sokoloff, in 1935, had found increased cooling rates for metallic melts due to ultrasonic treatment but that Schmid and Ehret, in 1937, had reported Iiittle effect in the cooling rate of silium and cadmium. These reports of cooling were incidental to the main purpose of the investigations. Martinelli and Boelter (1939) reported a five fold in- crease in the coefficient of heat transfer for natural con- duction in water when vibrating a 0.75 in. diameter cylinder. West and Taylor (1952) reported increases of up to 70 Percent in the rate of heat transfer, in a double heat ex- chaunger, caused by the pulsations from a reciprocating feed pmnp, Lemlich (1955) vibrated heated wires of 0.0253, 0.0396 and 0.0810 in. in diameter in air at frequencies of 39 to 122 CPS. and amplitudes of 0.07 to 0.19 in. Heat transfer coef- t'1<:i.ents increased with both frequency and amplitude up to fOur times the rate for non-vibrated wires. Lemlich explained these effects in terms of the convective film coefficient. Isakoff (1956) immersed a heated platinum wire in water and increased the heat transfer rate by about 60 percent; when a 10 kilocycle sound field was applied he converted film boil- ing to nucleate boiling. The effects of ultrasonics in increasing the rate of heat 'transfer through a liquid were reported by Deshmukh (1956). lie immersed a tubular heater in a vertical tube of water and applied #2 5 kilocycles and i megacycle ultrasonic irradiation parallel to the axis of the tube. He reported that steady- state conditions were reached in 10 percent less time with, the Iiigher frequency and no effect from the lower frequency. A 600 kilocycle tubular transducer was used to generate sound waves radially toward the heating element and this again re- duced the time to reach steady-state conditions by about 10 percent. Deshmukh reported that the temperature was more uniform throughout which would indicate increased convection currents. Scanlon (1958) reported work in which e 1 in. x 1 in. heated plate was placed to heat a liquid from below. The Plate was vibrated at 20 to 600 cps. with amplitudes of 0.001 to (3.00h in. The rate of heat transfer increased with ampli- tude and with frequency up to about 100 cps. and then de- creased. - A maximum increase of transfer of three times the I ullVibrated rate was observed. Effects of sonic vibrations in gas were reported by Tailby and Beikovitch (1958) who applied 600 to 1700 cps. vibrations to a burning gas stream in a five ft. vertical tube. The sound doubled the rate of heat transfer in the lower part of the tube. Investigations of Jackson, Harrison and Boteler (1958) were intended to determine the effects of sound upon the film «coefficient in a column of flowing air. At intensities above 1118 decibels they obtained an increase of Nusselt Modulus of river 50 percent (equivalent to an increase of heat transfer coefficient through the film of 50 percent). Early investigations to determine the thermal conductivity <3f liquids utilized long columns of liquid, cooled at the bot- tom to minimize natural convection currents. Thermometers at various vertical locations were read at time intervals and the thermal conductivity was calculated. Weber (1903) used parallel horizontal plates with liquid iJi'between. The lower plate was cooled and the heat input to the: upper plate was measured. Thermocouples located verti- cally 0.92 cm. apart were used to determine the temperature gradient. From the heat input and temperature gradient he caltnilated a coefficient of conductivity which was within seven percent of presently accepted values. To reduce the errors caused by vaporization and other heat, losses, Martin and Lang (1933) used parallel plate equip- ment, with very thin liquid layers. Smith (1936) utilized concentric cylinders with a 1/6h in. radial clearance to determine thermal conductivity in liquids. An analytic solution of the problem of vortex currents within the nodes of a standing sound wave was worked out to a first approximation by Rayleigh (1900). He showed that there should be four vortex currents between each two nodes. The circulatory vortex currents in standing waves in an air filled tube were studied by Andrade (1931). He used to- bacco smoke to trace the motion of the air currents and photo- graphed the vortex currents within nodes and around obstruc- tions in the tube. The actual vibratory motion of the smoke particles caused by the vibration of the air were photographed to give the sound wave amplitude. The vortex observed within the node verified the analysis made by Rayleigh. Axial streaming, sometimes referred to as quartz wind, inas considered by Eckart (1998). Using the same basic equa- ‘tions for viscous compressible flow as those used by Rayleigh, 116 considered certain second order effects which Rayleigh had Imaglected and derived a formula for axial streaming in a pro- gressive wave. He showed that the velocity is proportional tCJ the sound intensity and inversely proportional to the Scnaare of the wave length. Acoustic streaming near a boundary and the micro-currents ineam'a vibrating bubble were investigated by Nyborg (1958). 1118 results confirm those of previous investigators. Thermal, acoustic and other physical properties of water and glycerine-water mixtures are shown in Appendix Table A1 together with the references from which the data was taken. ANALYTIC SOLUTION OF THE PROBLEM OF TRANSIENT TEMPERATURE EFFECTS The basic differential equation governing the temperature response of a body to a transient boundary condition in one 1dimension when heat generation and thermal conductivity is liniform within the body is given by (1) PCM:[K&TAX19’]+ 09 6X2 Here p a density C 2 specific heat at constant pressure (T's temperature at a point x a: coordinate of the point at which the temperature is given 9 = time k,= thermal conductivity A I rate of internal heat generation. Churchill (1941) treats a similar problem with different boundary conditions and absorption. This equation may be solved (see Appendix A1) to give (2) T1x,e):_ 2&5:+[.E. 2_|-_]x+ M 23;. [‘"’n"]+m +_2_o *x nw'sz SID— ex _ where D a temperature difference between boundaries L_= distance between boundaries. The midpoint temperature is given by (3) T1L/26)=-‘$- -8—K+ -2- +-D-n+2={2$: —"‘?"[‘ ”n ']+%'3}5'" '21 The midpoint temperature consists of two components, one component due to internal heat generation, and the other due to heat transfer if the boundaries are at different tempera- tures. This is shown more clearly if equation 3 is rewritten in the form:, __AL2[_32 _1_’2__k9+32 91r2k9 ] T 9- —ex _exp__’o-oo z 9 +2[.+;exp_w:k9_ 4 ___._.x, M221L+] 2 LPC 377' L PC For practical use, only the first exponential term in each series is of significant magnitude for time intervals 01‘ ten minutes or longer under the conditions experienced in tnie tests reported here. This rapid convergence greatly fa- | _ _ s3: 1 L 1 L ' ' o 40 50 0 70 80 Temperature , F. Fig.8. Calibration chart, lower plate thermistor, horizontal ultrasonics. 26 4 7-Observed data 0‘ l N l Voltage, v., l l 3’ 40 $50 so ? 70 so Temperature, ° F. LI 1 Fig.9. Calibration chart, center thermistor, horizontal ultrasonics. 27 Sound Absorption and Intensity Measurement With upper and lower plates held at a temperature of 62.60 F., the signal generator was adjusted for the desired power out- put and frequency. When operating at the one megacycle range the frequency was adjusted to give a maximum sound absorption‘ which was indicated by an increase of temperature in the cen- ter thermistor. This absorption is to be eXpected since the thermistor diameter is approximately one-sixth of a wavelength at a one megacycle frequency; since it is slightly less than one-fifteenth of a wavelength at a #00 kilocycle frequency, there will be negligible absorption at the lower frequency. When operating in the #00 kilocycle range the frequency was adjusted to give maximum cavitation in the horizontal sound cell. For the #00 kilocycle frequency in the vertical cell, the top plate was replaced with a glass plate and a thin water film added; the frequency was then adjusted to give maximum' agitation of this film. The generator was operated for #i minutes and then turned off for 30 seconds to allow the thermistor to return to ambient temperature before a reading of the temperature was taken. This procedure was repeated until no further increase in tem- perature occurred. The maximum increase in temperature for each of four trials was used to obtain an average temperature increase "B" at each power level. 28 Because of a slight high frequency pick-up in the ampli- fier circuit, it was found necessary to check the calibration of the lower plate meter while the ultrasonic generator was operating. The meter "Zero" adjustment was adjusted to give correct temperature readings. This pick-up amounted to about 0.1 volt at the highest intensities. The original design of the temperature sensing circuit had made provision for using a Brush Model 520 strain gage amplifier and oscillograph. It was found that the excessive high frequency pick-up would make the use of this equipment impractical. The thermistors had been selected to match the output of this equipment. A simpler temperature sensing cir- cuit could have been designed using higher resistance, 30,000 ohm, thermistors, a #5-volt bridge signal and the vacuum tube voltmeters without the intermediate amplifier. Determination of the Rate of Heat Transfer With and Without Ultrasonics For the non-ultrasonic test the procedure was as follows: 1. The test cell was brought to a uniform temperature of 77° F. throughout. 2. The lower plate was cooled with dry ice to the de- sired boundary temperature, #1° F. (This tempera- ture was selected to avoid the density inversion point of water.) .3. 29 When the lower plate temperature reached 59° F., in about one minute, the stop watch was started and this was considered to be the zero time for the test. While holding upper and lower plate temperatures con- stant, at 77° F. and #10 F. respectively, midpoint temperature readings were taken at 5 minute inter- vals.. Dry ice was always used for the initial cooling of the lower plate. With lower ultrasonic intensities in the horizontal sound cell, the lower plate could be held more easily at a uniform temperature with ice than with dry ice. For the ultrasonic tests, the same procedure was followed except that: 1. 2. 3. The generator was adjusted for desired power and frequency as discussed in the section on sound in- tensity. The generator was turned on at the time the lower plate cooling was begun. The generator was turned off for 30 seconds before a midpoint temperature reading was taken. RESULTS AND DISCUSSION Ultrasonics Perpendicular to Heat Flow in Water The steady-state temperature rise due to sound absorption for two frequency ranges was determined using the method des- cribed in the Procedure section. The results of these tests are shown in Table l. The temperature response curves, shown in Figures 10, 11 and 12, have been plotted with the absorp- tion-temperature rise subtracted from the observed readings to give a better comparison of the response caused by the in- creased rate of heat transfer. Table 1. Steady-state temperature rise caused by ultrasonic absorption at various levels of ultrasonic generator plate current. Generator current Frequency range ma. #00 kc. 1 mo. 25 0.20 F. 0.20 F. 50 0.3 1.1 100 0.6 3.6 150 ‘ 1.1 6.3 200 1.2 Note: Each point is an average of four tests. The actual measured midpoint temperatures during five tests for the condition of no ultrasonics are shown in Figure 31 10, and the effect of an intensity corresponding to 25 ma. generator plate current, at one megacycle, upon the rate of temperature change as compared to the condition of no ultra- sonics. Figure 10 has been plotted with all measurements shown to provide an indication of the variability between tests. Later data will be given with all tests at one inten- sity averaged for each time period. The effect of ultrasonics in the one megacycle frequency range at various intensities is shown in Figure 11 and the effect of #00 kilocycle ultrasonics at various intensities is shown in Figure.12. From these figures it is seen that the effect of the ultrasonics appears to reach a limiting value. This will be shown in more detail later. Figure 10 shows the actual measured midpoint temperatures for the condition of no ultrasonics. When these data were analyzed to determine the coefficient of thermal conductivity, 1the value was lower than the value given by Jakob (19#9). Five thermocouples were mounted at various locations on the lower plate with a reference junction near the lower plate thermistor and a simulated test was made. Readings from the five test thermocouples showed temperature variations over the lower plate which varied as to location and time during the test as the ice was placed in contact with the plate and then removed. A value of K = 0.328 Btu/hr.ft.°F for water was used to calculate an effective average of lower plate temperatures. 32 Since all tests were conducted at the same observed plate tem- perature differences, it was assumed that the effective tem- perature differences were also constant for a given midpoint temperature change. For the remainder of this discussion, the value of plate temperature difference, D.°F., will refer to this effective value. Table 2 shows the relation of the observed temperature with no ultrasonics to a theoretical curve when a value of k a 0.328 Btu/hr.ft.°F. is used with an effective plate tem- perature difference of 26.1°F. Table 2. Comparison of observed and theoretical reaponse with no Ultrasonics, k = 0.328, D = 26.10Fe Time, Observed Calculated minutes temperature temperature 0 77.0°F. 77.0°F. 5 7#.# 72.8 10 70.3 A 68.7 15 67.8 66.5 20 65.8 ' 65.3 25 6#.7 6#.7 Note: Each point is average of five tests. With the value of the effective plate temperature differ- ence established for the condition of no ultrasonics, the time required for the same midpoint temperature change to occur was 33 determined for the conditions with ultrasonics. For example, referring to Figure 11, the midpoint temperature at 25 minutes for the condition of no ultrasonics was 6#.7o F. This same temperature was reached in: 17 minutes with 25 ma., 10 minutes with 50 ma., 5 minutes with 100 ma., and # minutes with 150 ma. It was assumed that during a given midpoint temperature change the average effective lower plate temperature was the same for all tests. The "apparent thermal conductivity" was calculated using equation (3) for the tests with ultrasonics. The term "apparent thermal conductivity" is used here, and will be used throughout, to indicate that both conduction and convection effects are present. Using the steady-state midpoint temperatures at various levels of plate current (Table 1) equations 5 and 7, and the apparent thermal conductivity as computed, the ultrasonic in- tensities were calculated. Figure 13 shows the relation be- tween generator plate current and ultrasonic intensity at the two frequencies. The effect of ultrasonic intensity on the value of the apparent thermal conductivity is shown in Figure 1#. The in- crease in apparent thermal conductivity is a function of in- tensity only, under the conditions studied here. 3# 78 1 74s - 3-70 — l - . i, . Om a. 2 a P. 266 - ‘ .. o 25mo * a. E ,2 62 - .. .. E '6 O. E 258Jt' 'i VI: 1 J J l 1 v55 0 5 IO IS 20 25 Tlme,min. Fig.lO. Temperature response, two intensities, compensated for ultrasonic absorption. 78 74 q 0 r 35 Each point average 4 tests O J ‘5 ‘6 :66 _ . .\ X E ISOma A 0 50m:\ ' E 62 r- A\ \o _ D g- lOOma x\x\u i \u 58 l- -l \J 3; l l l 1 1‘1" 0 5 l0 I5 20 25 Time,mln. Fig. ll. Temperature response at various generator plate currents, I ma horizontal ultrasonics in water, compensated for ultrasonic absorption. 36 78 . 1 \ Each point average 4 tests N. b NI 0 03 O) m N Mid point Temperature,’ F 0| m l 1 Ol§( ( l l l 5 l0 I5 20 25 Time, min. Fig.|2. Temperature response at various Generator plate currents, 400kc horizontal ultrasonics in water, compensated for ultrasonic ‘ absorption. Intensity, watts/cmz. 37 2 rEach point average _, 4 tests. l. .. I - ' - I ma. Range 400 l( 9 0—1—4' " fir" : so lOO use 200 Generator plate current, ma. Fig. l3. Relationship of generator plate current to ultrasonic intensity, horizontal cell, water. ' Apparent thermal conductivity, Btu Ihr.tt.°F. 2" Each point average 4 tests. ' '1 38 .. / - .. I mc. Range / o- 400kc.Range l 1 l l I 2 Intensity, watts /c m2. Fig. l4. Relationship of ultrasonic inten- sity to apparent thermal con- ductivity. UltrasoniCs perpendi- cular to heat flow in water. 39 Natural Convection Response Natural convection usually occurs in a fluid which is being heated on the lower boundary; the decrease in density of the fluid as it is being heated causes it to rise, and the natural convection currents increase the rate of heat transfer through the fluid. Similar results occur if the upper boundary of a fluid is cooled. The increase in the rate of heat transfer in liquids by ultrasonics, as compared to other methods of agitation, is of great practical importance. To obtain an estimate of the mag- nitude of the effect of natural convection as compared to ul- trasonics, a series of tests were made with the cell used for the ultrasonics perpendicular to heat flow. For these tests the top plate was removed and cooled to #1°F. and the remainder of the cell was brought to a temperature of 77°F. The top plate was then placed on the cell and temperature responses recorded. It was impossible to maintain as accurate 3 control of plate temperatures as in the usual test; the top plate tem- perature rose by about #OF. but was brought back to about #1° F. by the end of the five minute period. The results of these tests are shown in Figure 15 where they are compared with the results at one megacycle at the highest power level; the temperature responses are approxi- mately the same order of magnitude for both conditions. #0 73 Each point average 4 tests. " 74 - LL70 '\ 1 O 8" . 3 E66 No ultrasonics)\. s 3 \ (lower Plate cooled) \. S x l50ma.,lmc.(lower plate p62_ °\ \C,‘ cooled) _ E °\§\Natural convection “5 (upper plate cooled) :3 258l- ' Vi, v“ \t . . . i .w 0 5 l0 IS 20 25 Time, min. Fig.l5. Comparison of natural convect- ion and ultrasonics perpendicu- lar to heat flow in water, com- pensated for ultrasonic absorption. #1 Progressive Waves Perpendicular to Heat Flow in Water Progressive waves are produced when waves travel into an infinite media or are absorbed rather than reflected at a boundary; no stationary node points exist. Standing waves require a reflector, and stationary node points exist. In practice the terms progressive and standing are relative as it is difficult to secure a perfect reflector or non-reflector. To test for possible differences in action which might exist between standing waves and progressive waves, one test was made in which the reflector was removed and replaced by a 1 in. x 1 in. x 3 in. block of paraffin, a relatively good sound absorbing medium. The face of the block toward the sound was cut with a grid of V-grooves about 3/8 in. in depth and spacing to break up reflected waves. Three absorption tests were made at one megacycle with a generator plate current of 150 ma.; the average steady-state temperature rise was 3.7OF. Three tests to measure temperature response were made, and the results are shown in Figure 16. The apparent thermal conductivity was calculated to be 0.73 Btu/hr.ft.°F. and the intensity was calculated to be 0.5# watts/cmz. From Figure 1# it is seen that at an intensity of 0.5# watts/cm2 an apparent thermal conductivity of about 1.2 Btu/ hr.ft.°F. would be expected with standing waves. #2 73 Each point average 4 tests. ‘1 74 ~ ' ‘ 70- s 5", No ultrasonics. Q) '5 x - é ab x .\. m Progressive waves I'- x "562‘ 7 ’6 O. :9 2 58 — ._ w l l l l l 315 O 5. l0 _ IS 20 25 Time, min. Fig.l6. Temperature response with progressive waves perpendicular to heat flow in water, one rnc., compensated for ultrasonic absorphon. * “3 Ultrasonics Perpendicular to Heat Flow in a Glycerine-Water Solution If the effect of the ultrasonics on the rate of heat flow 143 primarily a result of convective-type current flow, then a higher viscosity should inhibit these currents and reduce the effect of the ultrasonics. To test this hypothesis, a solu- ‘tiori of 50 percent, by volume, glycerine in water solution was prepared and tested in the one megacycle frequency range. The viscnosity of this mixture is 6.3 centipoise as compared to 1.0]. centipoise for water. Absorption tests were made, using this mixture, at one megacycle and with various intensities. The steady-state tem- perature rise due to absorption is shown in Table 3. Table 3. Steady-state temperature rise due to ultrasonic ab- sorption in the glycerine-water solution, one mega- cycle range Generator current Temperature rise 25 ma. 1.20 F. 50 2.7 100 5.5 The temperature response without ultrasonics was deter- mined. and, using a value for thermal conductivity of 0.218 Btu/hr. ft.°F., an effective plate temperature difference of ## 31.20 F. was calculated. A theoretical curve was computed, and the results are shown in Table #. Table #. Comparison of observed and theoretical response, no ultrasonics, glycerine-water solution, k = 0.218 Btu/hr.ft.0F., D = 31.20 F. Time, Observed Calculated minutes temperature temperature 0 77.0°F. 77.0°F. 5 75.7 74.9 10 71.1 71.8 15 68.5 68.9 20 66.7 66.9 25 65.3 65.4 30 6#.3 64.3 35 63.6 63.# The results of the temperature response with and without ultrasonics are shown in Figure 17.. It is immediately appar- ent that the results for the highest intensity are not consist- ent with the other data of these or previous tests. This point will be discussed in the next section. Using the temperature response data for 25 ma. and 50 ma. as shown, but disregarding the highest intensity, the values of apparent thermal conductivity and intensities were calculated. The results, Table 5, show that there is a very small increase in apparent thermal conductivity which substan- tiates the original hypothesis. 45 78 'l Each point average 3 tests. 74 ~ ' 7 ° Oma. $70 " . 2 -' q;- lOO ma. ‘ .5. x7 O A 366" 25 ma. 3 \4\ 7 g 50 ma.>\;§. A 262 - - 6 O. '9 2 58 - 1 gk 1 l L 1 J 1 1 S} 0 IO 20 30 Time, min. F ig.l7 Temperature response at various generator plate currents, lmc horizontal ultrasonics in glycerine- water solution, compensated for _ ultrasonic absorption. #6 Table 5. ~Relation between ultrasonic generator plate current, sound intensity and rate of heat transfer, glycerine- water solution, one megacycle frequency range Generator Sound Apparent thermal current intensity conductivity ma. w/cm2 Btu/hr.ft.°F. 0 0 0.218 25 0.06 0.26 50 0.14 0.25 Effect of Temperature Gradients upon Absorption and Conductivity Measurements By referring to Figures 11 and 17 it can be seen that the temperature response curves at the highest intensities cross one or more of the other curves. One possible explanation of this behavior leads to consideration of the possible effects of temperature gradients within the liquid. Temperature gradients and their effect on the path of sound travel have been the subject of much study in sonar work, where the gradients have not been extreme but the ranges have been long enough to warrant consideration of the effect of dis- tortion of the beam. Seifer (1938) has given a value of 2.5 m/sec.°C. as the increase of sound velocity in water with increasing temperature with a velocity of 1497 m/sec at 25°C. Freyer, Hubbard and Andrews (1929) determined the velocity gradient in glycerine 47 to be -1.8 m/sec.°C. with a normal velocity of 1923 m/sec. at 20°C. and obtained a sound velocity gradient of 2.6 m/sec.°C. for water. Wood (1955), pp. 323, derived the formula for the radius of curvature of a sound beam in a refractive media as (9) 5h» ll 0 Ir» Sis where r = radius of curvature c = velocity of sound. Applying this formula where a 20°C. temperature differ- ential exists across a 2.5# cm. layer during a heat transfer *test, the radius of curvature is r r a 76 cm a 30 in. with this radius of curvature the beam will bend through one inch in a distance of 7.7 in.. or in one passage across the cell. Hence, the sound intensity near the lower plate 'will be very much greater than at the center and should be negligible at the upper plate. For sound absorption measurements, where a temperature rise of 3.5°C. was observed during the test at one megacycle with 150 me. of generator plate current, the radius of curva- ture is r a 218 cm. s 86 in. With this radius of curvature the beam will be bent through 0.5 in. in a travel distance of 9.3 in.. or again, #8 in one passage across the cell the intensity should be concen- trated at the lower and upper plates. These refraction effects tend to turn the beam toward the cooler region, downward during a heat transfer test, and away from the center and toward the upper and lower plates during a sound absorption test. Calculation of the exact intensity distribution across the height of the liquid layer would become difficult; esti— mates would have to be made of the number of times a wave is reflected as well as viscosity and stray reflection effects. To estimate the possible magnitude of the effects which might be encountered due to a shift of the intensity distri- bution, two cases will be considered. The first case to be considered is that of sound absorption test. If the assumption is made that the absorption rate, a function of intensity, varies as 2 (10) Absorption rate =lgé? [x-HE% 7 L 2 this will give the same total absorption over the volume as,a uniform absorption of value A. The distribution curve is a segment of a parabola with its axis parallel to the direction of the ultrasonic beam and its vertex at the midpoint. As shown in Appendix Al, the steady-state temperature at the mid- point is given as 2 (11) 8:..A—I:.. l6k 1+9 Comparing equations 11 and 5, it is seen that the ob- served temperature rise is only one-half as great for this case of non-uniform absorption as for uniform absorption. During a heat transfer test the beam will be refracted toward the lower, cold, plate; if a non-uniform sound absorp- tion rate of (12) Absorption rate = .3E2-(x— L)2 is assumed to exist, then the same total absorption will occur over the volume as in the case where a uniform absorption of value A exists. As shown in Appendix Al, the temperature at the midpoint may be expressed as (13) T(L/2,9): 7-°-{'+[L_:365 31/81:] 11%? m} 77 ,423.{,+_ p123- .....} for this condition of non-uniform absorption. Considering the case of 100 ma. intensity in the glycerine- water solution, as shown in Table 3, the steady-state tempera- ture rise due to sound absorption is 5.5°F. Using equation 13 and the following relations B = 5.50s. D = 31.207. k = 0.25 Btu/hr.ft.°F., a theoretical temperature response curve was computed. Table 6 shows the relation of this theoretical response curve to 50 the observed curve. The close relationship between the two curves shows that non-uniform absorption patterns, as given in equations 10 and 12, can account for the discrepancies which were observed as shown in Figures 11 and 17. Table 6. Relation between a computed temperature response for non-uniform absorption and the observed data for one megacycle, 100 ma., in glycerine-water solution - Temperature, °F. Time, min. Observed Computed 0 77.0 77.0 5 75.9 77.6 10 74.9 75-7 15 73.9 74.2 20 73.2 73.2 25 72.2 72.6 30 71.7 72.1 35 70.9 71.7 This analysis has shown that refraction effects in water will: 1. Affect the determination of sound intensity in such a way that the intensity will be underestimated. 2. Affect the temperature response during a heat trans- fer test, but the main source of error in determin- ing the coefficient of conductivity will be due to an underestimation of intensity. 51 The effect of refraction should be considered in the ap- plication of ultrasonics to increase the rate of heat transfer. Consider, for example, a tube filled with water which is being heated; the refraction would cause the beam to concentrate at the center of the tube where its effect would be largely lost. 'If the water were being cooled, then the ultrasonics would be concentrated near the tube walls where its value would be greater. It is possible that the most valuable effect of ul- trasonics will occur if the beam can be concentrated in the slow moving film of fluid which exists near the boundary when convection effects are large away from the boundary. RESULTS AND DISCUSSION Ultrasonics Parallel to Heat Flow The construction of the cell to be used with vertical ultrasonic irradiation presented difficulties not encountered with the horizontal cell. The crystal was mounted to contact the liquid from the lower boundary in order to minimize the danger of an air film preventing liquid to crystal contact if the liquid level should become low. This necessitated build- ing a reflector into the upper plate. With the crystal mounted on the lower plate the amount of plate area left exposed for cooling was reduced and hence cooling was more difficult. To cool the lower liquid boundary it was necessary to have a copper plate between the crystal and the main body of the liquid. The thickness of this plate, about 1/16 in.. could not be reduced too much without interfering with the heat trans- fer from the sides. However, it presented a definite obstacle to the passage of the ultrasonic waves and absorbed a large part of the crystal output. Some gas bubbles may have collected on the lower side of this plate as deaeration occurred, in which case, considerable blocking would have been present. The steady-state temperature rise, due to ultrasonic ab- sorption, was measured and the results are shown in Table 7. 53 Table 7. Relation between ultrasonic generator plate current and steadyustate temperature rise, ultrasonics par- allel to heat flow in water Generator current Frequengy:range ma. #00 kc. —1 mo. 25 ' m 0.0°F. 50 0.1 0.5 100 0.3 2.4 150 _ 0.5 200 1.2 The results of various tests at different intensi- ties and frequencies and their effect on the rate of tempera- ture response are shown in Figures 18 and 19. Using the data for the condition of no ultrasonics an effective plate temperature difference of 28.2°F. was calcu- lated. This effective plate temperature difference was then used to calculate the apparent thermal conductivity. Table 8 shows the relation of generator plate current to calculated sound intensity. 78 — Each point average 3 tests. 74 e V0 m0. 5‘70 4 8 3 °\ :3 ’\ 266’ 25\ mo. 7 ,5, IOO ma. A "" .YJ)\X A¥00m\a o 262 7 - '5 O. E 2 58 - 1’ v \J L J 1 1 1 - 1 ‘1" O 5 l0 IS 20 25 Time, min. F ig.l8. Temperature response at various generator plate currents, Imc. parallel to heat flow in water, compensated for ultrasonic absorption. 78 int Te moperatu re, °F 01 «1 s1 N 0) O 43 l Midpo 01 a) l 071% 55 , Each point average 3 tests. l 5 l0 IS 20 25 Time, min. Fig.l9. Temperature response at various generator plate currents, 400kc. parallel to heat flow in water, compensated for ultrasonic absorption. 56 Table 8. Relation between generator plate current and ultra- sonic intensity, ultrasonics parallel to heat flow in water Plate current __ Frquency_range ma. #00 kc. 1 mo. 0 0 0 25 - 000 50 0.001 0.007 100 0.00# 0.051 150 0.006 200 0.021 Figure 20 shows the effect of various ultrasonic inten- sity levels upon the value of the apparent thermal conducti- vity. The rate of heat transfer appears to be a function of intensity only in this case. 57 0.817 1 3: E 0 g 0:6 _ 5 . oil": --| 1110. Range '3 t 0.4 o— 400kc. Range :13 f = 3430.2 _ Each point average_ 0 3 3 tests. 0. O. < 0 L l I l 0 0.02 004 0.06 Intensity, watts/cm? F ig.ZO. Relationship 0f ultrasonic inten- sity to apparent thermal con- ductivity. Ultrasonics parallel to he at flow in water. RESULTS AND DISCUSSION General It has been shown that the rate of heat transfer through a liquid can be increased by the use of ultrasonic irradiation. There has been no demonstration that the coefficient of con- ductivity is changed; in fact, the small increase in heat trans- fer rate in the glycerine-water mixture, with its high viscos- ity, indicates that any increase in the heat transfer rate is probably directly related to convective effects. For convec- tive transfer there is a mass transfer of material along the direction of heat flow. 1 Figure 21 shows the general paths of two types of cur- rents, axial-streaming and intra-node, which are known to exist in conjunction with sound waves. Other currents, such as those around bubbles present during cavitation, may exist within a liquid. Any and all of these currents will probably contribute to an increased rate of heat transfer. For a few special cases, the paths and velocities of the steady currents have been calculated or measured. However, in a complex sound field, such as exists here with the bending of the sound beam due to temperature gradients, an analytic solution is impossible with present available information. An effective approach to a similar problem in convective heat transfer phenomena has utilized the methods of dimensional analysis. t C 1: 4; Sound source Axial streaming caused by progressive waves in a tube. Intro-node vortex pattern caused by standing waves. Fig.2l. Two types of steady currents ' caused by sound waves. 60 In the case of the effect of ultrasonic irradiation all of the factors present in convective heat transfer would occur plus additional factors suggested by the ultrasonic phenomena. The ultrasonics would introduce factors depending upon fre- quency, intensity and ultrasonic absorption. There will pro- bably be at least nine variables requiring six Pi groups to establish a relationship for the rate of heat transfer. The question of the effects of ultrasonics in addition to natural convective effects could not be answered because of equipment limitations. When an attempt was made to com- bine the two effects, the heat transfer rate was so large that it was impossible to hold plate temperatures within a range which would give results which could be duplicated. SUMMARY Previous investigators have shown that the rate of heat transfer through liquids could be influenced by sonic or ul- trasonic effects in fluids or in bodies suspended in liquids. Most of the investigations have dealt with the effect on films at boundaries; a few have considered combined effects with forced or natural convection. To investigate the possible effects within the body of a static liquid, equipment was constructed which would cause heat to flow downward through a body of liquid; thus natural convection was minimized. Most parallel plate equipment which has been used for determining the rate of heat transfer through liquids has been of a calorimetric type. Because of ultrasonic absorption this equipment was constructed to use a transient measurement and to handle the problem as a time dependent boundary value problem. Provision was made to introduce standing ultrasonic waves in a direction perpendicular to the heat flow with a horizon- tal ultrasonics cell, or parallel to the heat flow with a ver- tical ultrasonics cell. Frequencies in the range of #00 kilo- cycles and one megacycle were used. A glycerine-water solu- tion was tested in the horizontal cell; progressive waves in water were tested in the same cell. Temperature measurements on the lower plate and at the midpoint of the liquid were made with thermistors to reduce 62 errors due to time lag and location. It was found that with a thermistor having a diameter of 0.01 in. there was a very large absorption of ultrasonic energy by the thermistor at the one megacycle range. The equipment was operated at various intensity levels to determine the effect upon the rate of temperature response. Calculation of the value of the coefficient of thermal conduc- tivity with no ultrasonics showed that measured variations of the lower plate temperature over the plate were introducing an error. This error was compensated for by assuming values of the coefficient of conductivity as obtained by other in- vestigators and then calculating an effective plate tempera- ture difference. With equal temperature on the upper and lower plates the steady-state temperature increase at the center of the liquid due to ultrasonic absorption was determined at various inten- sity levels. This steady-state temperature increase was used to calculate the ultrasonic intensity. The effect of temperature gradients on the ultrasonic beam is investigated from a theoretical standpoint. The bend- ing of the beam is shown to be a probable cause of certain apparent inconsistencies in the results and an important fac- tor to be considered in application of this method of increas- ing heat transfer rates. . The results of this investigation show that ultrasonic irradiation, either perpendicular or parallel to the direction 63 of heat flow, will increase the rate of heat transfer through liquids. The increase is apparently not directly proportional to intensity and may reach an effective saturation point at intensity levels much above those used here. The observed values of the rate of heat transfer increased up to five times that of the coefficient of conductivity when no ultrasonics was present. This increase is approximately equivalent to the increase due to natural convection with a temperature gradient of 36°F./in. At equal intensity levels in the one megacycle range, standing waves caused a larger increase in apparent thermal conductivity than progressive waves. This was expected, since there is less-intra-node current in progressive waves. Ultrasonic irradiation was less effective in a glycerine- water solution. This is explained by the effect of viscosity in reducing convective type currents. CONCLUSIONS The use of ultrasonics to obtain convective type current flow within a liquid will increase the rate of heat transfer. Some of the factors which have been investigated and the re- sults obtained are as follows: 1. 2. 5. The rate of heat transfer is a direct, but not necessarily linear, function of intensity over the ranges of intensity investigated. An increase of viscosity obtained by using a glycerine-water solution greatly reduced the effect of the ultrasonics. Irradiation perpendicular and parallel to the di- rection of heat flow indicated the latter to be slightly more effective. Standing waves are more effective than progressive waves of the same intensity. The temperature gradients which exist in heat trans; fer applications may cause severe refraction of the beam pattern. This bending could cause the ultra- sonics to be less effective if it is toward the center of the fluid body. The greatest effects obtained gave increased rates of heat transfer which were of the same order of magnitude as those obtained with natural convection. 65 The use of ultrasonics to increase the rate of heat transfer would be best adapted to applications where other mechanical means of agitation could not be utilized. The greatest increase in the rate of heat transfer occurred at a frequency of one megacycle and an intensity of 2.13 watts/cm.2; the heat transfer rate between two paral- lel horizontal plates increased from 0.33 Btu/hr.ft.°F. to '1.7 Btu/hr.ft.°F. SUGGESTIONS FOR FURTHER RESEARCH Future research in this area should consider the effects which may be obtained in both static and flowing liquids. With flowing liquids, where film effects are important, the possi- bility of focusing the sound beam into the film by normal tem- perature gradients warrants consideration. The effect of ultrasonics in the presence of other con- vective currents should be studied to determine if the effects are additive. This might be done with parallel plate equip- ment by reversing the temperatures of the upper and lower plates. Based upon the research done here the following improve- ments are suggested for future design of parallel plate equip- ment: 1. Automatic plate temperature control should be pro- vided. 2. Temperature readings should be recorded. 3. An intensity probe should be used to measure the sound intensity over the cross section. #. Consideration should be given to the use of liquid solutions in which the sound velocity is independent of temperature. 5. A wider range of frequencies should be used in an attempt to set practical operating frequency limits. APPENDIX A1 MATHEMATICAL DERIVATIONS - TRANSIENT HEAT FLOW IN ONE DIMENSION Uniform Internal Heat Generation Churchill (19#1) presents the solution for transient heat flow in one dimension with internal heat generation; he has considered a case in which the heat generation and boundary conditions are different than the ones encountered in this research. The following derivations are similar to the one made by Churchill. The governing differential equation for one-dimensional heat flow with uniform internal heat generation is GT“, 8) (1) Pc5_8 akm+A. x The applicable bouand:ry conditions are: (2) T(0,9)=0 (3) TlL.91=D (#) T(x,0)=D. Making a substitution for the dependent variable: (5) Tune) 1- th,91+ W (it) then (6) aTixfi— _6Y(x,9) 6 9 a 9 68 and (7) dthx,Oi_ a‘vu,91 62W1x) z 7" z + 2 ' 6x 0x 6x Placing this substitution in the original equation and boundary conditions gives “(3) pc OYHQ): kazY1Xfii+ k Sim-FA 09 6x2 6x2 (9) YtO,G)+W(0)= 0, or Y(0,9)=0, W(O)-'=0 ‘ (10) Y(L,9l+W(L) = D, or YlL.9)=O, WlLt=D (11) le,O)+W(xi=D, or th,Ol= D—thl. Now from Equation (8) let 2 (12) |(f%_1%£§1.+lx==() X hence sz I I The constants B' and C' of Equation (13) may be evaluated by using boundary conditions (9) and (10) to give the steady- state temperature, (14) wm =—5—§+[-°— + £2] 11 2 k’ L- 21k Equation (8) is now reduced to lexBl 021m 9) 1 ) LC __..._.L._= ‘ . ( 5 k 69 6x2 69 This may be solved by a separation of variable technique ' by substituting if (16) Y(X.9)= Swim) into Equation (15) to give as 88(9)_ 1 azxm_ 2 1 _—_ — _ ( 7) ((0018 6 xnndxz ’ where a2 is simply a separation constant. Thence 2 (18) 0(9)= exp— LE9 pc and (19) Xlxt=Esinax+ F cos ax. The boundary conditions (9) and (10) will be satisfied (20) F=O and I nr 21 = —— - ( ) 0 L Hence‘Y(x,9) may be expressed by z (22) Yix,0)= 2 En sinl‘lx exp _ 2.1,...” . 03' L> LglpC: Using boundary condition (11) and Equation (22) sz 0‘ AL - nvx (2 ) ____“_.[__. ___.] :: sun 3 D+2k L +2" 1: 1:12:16" L 70 The value of En may be determined from «MM—:- -[° w m -- D L 2k sInde Ensdex which reduces to L <25) fi‘t,[(-n“_u]+%k=en-2—- Substitution of the values obtained into Equation (5) gives (26)T\x,9)=£— f+fi+_ M] ,§|{:AL-: 3[(-l)—l]+ 20} 2 L2 pc Substituting the value of x = L/2 into Equation (26) gives the midpoint temperature as =_A__L D ZAL 4&9. (27)T(L/2,9) 8-—k-+— +2 -—:"-,[\-I) '4‘] M} 2 nsl kn 3 22 9 n1 27k sin— ex "T'— 2 p L pc The steady state temperature 'B' at the midpoint for the condition of internal heat generation and with both boundaries at the same temperature, D = 0, is Al} (28) 8= n 01‘ 8kB (29) A-T . Non-Uniform Internal Heat Generation, Ultrasonics Perpendicular to Heat Flow During a sound absorption test in water the sound beam will be refracted toward the upper and lower boundary. Assum- ing that the same total energy, AL, is absorbed and that the distribution is given by I . 2 L (30) Absorption rate =L2Ee-[x— ? t the governing differential equation for steady-state conditions dzT (2A L2_ . (31) “Wan—E'P—j—O The boundary conditions are (32) T\0)=Tu.)= 0. Solving equation 31 and substituting in the boundary con- is ditions yields 4 a 2 2 (33) T‘x,=_l2A[.x_.-L_X_ +31 +6—in $126 a 2k The steady-state midpoint temperature is 2 (311) TiL/2)='B= AL 0 6k and l6k8 During a heat transfer test the sound will be refracted toward the lower boundary. Assuming that the same total energy, AL. is absorbed and that the distribution is given by (36) Atsorption rate =._L!_ (x-L)z the governing differential equation is 6mm 627;): we) x—L 69 k6 )(2 +( )2 The boundary conditions remain the same as shown in equa- (37) pc tions 2, 3 and 4. Following the same procedure as outlined above, the steady-state temperature is determined to be ( ) w __ 3A [51%; l32_x_’]*L[3H 3_A_L] 38 “I re '2 Following the same procedure for the transient term, the temperature is found to be 3A L2): 34AL] ‘ T(x.8)-- —--—+ + (39) India” 3 ]x 2 X I-(—I)" GALLE+—k n=l{-z-e;LL :[ 21:73 7} sin—"1'1";k 9 “L:3 pc For the midpoint, equation 39 reduces to , 2 2 (L10) T L/29 2+7-9—L—+n13§-'=- _. "]_5AL ‘ )= 2 +64k glknsfl) ( I) “1313+ n2 I” sin-9L exp- ":2“. 2 L pc 73 Combining equations 35 and 40 yields I928 ['_ H n] _968 D 73 a =— — —— (Lu) TiL/Z, ) 2 +4 4-3;???“ "3,: 22- sin-"l exp- "afie- nt 2 L pc A2 Specifications of Equipment Ultrasonic generator: Model BU-ZOh Brush Electronics Co. Cleveland, Ohio Vacuum tube voltmeter: Model WV-98A Radio Corporation of America Harrison, N. J. Thermistor: Type 31A1, 1200 ohm, 0.01 in. dia. Victory Engineering Corp. Springfield Road Union, N. J. Barium titanate crystals: 1 in. x 2 in. - 400 kc. and 1 mo. 2 in. dia. - #00 kc. and 1 mc. Gulton Industries, Inc. 212 Durham Ave. Metuchen, N. J. 70 Table A1. Physical properties of water Property Value Reference Density .999 g/cm.3, 15°C. Int.Crit.Tab., Vol. v. Specific heat 4.190 Joules/g.,15°C. ' ' ' Viscosity 10.09 millipoise, 20°C. ' ' - Thermal conductivity 0.347 Btu/hr.ft.°F.,3o°c, I n a 0.353 Btu/hr.ft.°F.,30°C. Jakob (19u9) 0.328 Btu/hr.ft.°F.,15°C. Jakob (1949) Acoustic 17 absorption 39 x 10' , 15°C. Hall (1948) Table A2. Physical properties of a glycerine-water mixture (50 percent by volume) Property Value Reference Density 1.130 g/om.3, 20°C. Int.Crit.Tab.,Vol.V Specific heat (glycerine) 1.235 cal/g., 20°c, - a . . . Viscosity 37.34 millipoise,p==1.1010 ' ' ' 153-6 ' ,P =1.1699 I lo a Thermal conductivity .0009“ cal/hr.cm.°C. v - - Acoustic absorption 75 x 10-17, 20°C. Willis (19h?) 75 Table A3. Temperature response at center -- with no ultra- sonics and with low intensity in the one megacy— cle range Time from beginning of test, min. Condition 0 5 10 15 20 25 No 77.0°F. 70.7°F. 70.7°F. 68.5°F. 67.00F. 65.8°F. ultrasonics 77.0 70.2 70.1 67.5 65.6 60.5 77.0 70.3 70.2 67.0 65.5 60.5 77.0 70.0 70.0 67.6 65.5 60.3 77.0 70.3 70.0 68.0 65.7 60.5 25 ma. 77.0 73.9 68.9 65.7 63.3 61.9 plate 77.0 73.0 68.5 65.1 62.9 61.8 current 77.0 73.5 68.9 65.0 63.7 62.3 77.0 73.0 68.3 65.1 62.9 61.6 76 Table A0. Temperature response at various levels of ultra- sonic intensity perpendicular to direction of heat flow in water, one megacycle range Generator plate current, ma. Time, min. 0 25 50 100 150 0 77.009. 77.0°F. 77.009. 77.002. 77.002. 5 70.0 73.6 70.0 65.1 63.7 10 70.3 68.7 60.3 60.9 62.2 15 67.8 65.3 61.1 60.0 20 ‘65.8 . 63.2 59.7 25 60.7 61.9 58.9 Note: Each point is average of four tests. Steady state temperature rise due to ultrasonics was subtracted from observed reading. ’ 77 Table A5. Temperature response at various levels of ultra- sonic intensity perpendicular to heat flow in water, 000 kilocycle range Generator plate current, ma. Time, min. 0 25, _50 100 150 200 0 77.0°F. 77.0°F. 77.002. 77.007. 77.002. 77.007. 5 70.0 73.6 73.3 71.3 68.6 67.7 10 70.3 69.5 69.0 66.9 65.2 65.6 15 67.8 66.7 66.2 60.1 60.0 63.9 20 65.8 60.9 60.3 25 60.7 63.5 63.1 Note: Each point is average of four tests. Steady state temperature rise due to ultrasonics was subtracted from observed reading. Table A6. Relation between generator plate current and ultra- sonic intensity, ultrasonics perpendicular to heat flow in water Intensity, watts/cm2 Plate Current ma. boo kc. {2239 1 mo. range 25 0.02 0.02 50 , 0.03 0.18 100 1.06 0.07 150 0.10 2013 200 0.17 Note: Each point is average of four tests. 78 Table A7. Relation between ultrasonic intensity and apparent coefficient of thermal conductivity, ultrasonics perpendicular to heat flow in water Intensit , Thermal conductivity, watts/cm Btu/hr. ft. 0F. 000 kc. range 0.0 0.328 0.02 0.00 0.03 0.02 0.07 0.56 0.10 0.65 0.17 0.70 1 me. range 0.0 0.328 0.02 0.07 0.18 0.78 1.06 1.5 2.13 1.7 Note: Each point is average of four tests. 79 Table A8. Temperature response with natural convection (upper plate cooled) in relation to ultrasonic intensity O Generator plate current, Time, ma._Li mo.) Natural min. 0 150 convection 0 77.0°F. 77.0°F. 77.0°F. 1 69.1 2 65.0 3 62.9 a 61.8 5 70.0 63.7 61.0 10 70.3 62.2 15 67.8 20 65.8 25 60.7 Note: Each point is average of four tests. 80 Table A9. Temperature response with progressive waves per- pendicular to heat flow in water, one megacycle range Time, Qgggggtor_plate current, ma. min. 0 150 0 77.0°F. 77.00F. 5 70.0 67.7 15 67.8 63.0 20 65.8 25 60.7 Note: Each point is average of three tests. Table A10. Temperature response as a function of ultrasonic intensity, one megacycle range, glycerine-water mixture, ultrasonics perpendicular to heat flow Time, Generatorgplgte current,flmg. ‘__ min. 0 25 fifi' 50 100 0 77.009. 77.002. 77.002. 77.002. 5 70.7 73.7 72.0 70.0 10 71.1 70.2 69.7 69.0 15 68.5 67.8 67.3 68.0 20 66.7 66.0 65.7 67.7 25 65.3 60.5 60.0 66.7 30 60.3 63.7 60.0 66.2 35 63.6 63.1 63.0 65.0 fiote: Each point is average of three tests. 81 Table A11. Temperature response as a function of ultrasonic intensity ultrasonics parallel to heat flow in water Time, Generator plate current, ma. min. 0 v 25 50 100 150 200 000 kc. range 0 77.0°F. 77.0°F. 77.002. 77.002. 77.002. 5 73.9 73.0 72.9 72.7 70.2 10 69.7 68.3 68.0 68.5 66.0 15 67.1 65.5 65.8 65.0 63.3 20 65.0 63.9 60.0 63.5 61.0 25 63.7 62.6 62.8 62.1 1 mc. range 0 77.0 77.0 77.0 77.0 5 73.9 73.0 72.0 69.6 10 69.7 68.5 68.1 60.6 15 67.1 65.5 60.7 62.1 20 65.0 63.6 62.8 25 63.7 82 Table A12. Relation between ultrasonic intensity and apparent coefficient of thermal conductivity, ultrasonics parallel to heat flow in water Intensitg, Heat transfer, watts/cm Btu/hr.ft.°F. 000 kc. ragge 0.00 . 0.328 0.001 0.01 0.000 0.01 0.006 0.03 0.021 0.57 1 mc. rgnge 0.00 0.328 0.0 0.01 0.007 0.05 0.051 0.71 10. 11. 12. REFERENCES Andrade, E. N. da C. (1931). On the groupings and general behavior of solid particles under the influence of air vibrations in tubes. Phil. 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'A Textbook 9; Sound. 3rd Edition, G. Bell & Sons, London. 323. 111.: (M 5?: ‘41!“ ”71117111711111 1293 flflthQIiHTUijflifiLilllflfilfl'“