Advances in Heat Flux Measurements

Publisher Summary The decade of the 1950s saw great advances in heat-transfer measurement techniques. Optical methods became popular, along with several new heat-flux gages that are still in wide use today, as evidenced by the commercial heat-flux-gage manufacturers. In the last 10 to 20 years, a number of new techniques have been developed and applied, which have greatly increased the resolution and operating range of heat-flux instrumentation. Although the older methods are discussed as the background, the focus of this chapter is on the newer techniques and novel uses of the older techniques. In addition, an effort is made to bring together information from a variety of fields that deal with heat transfer, but that often isn't communicated. Three areas of new capability with important applications are time-resolved heat-flux measurements, simultaneous measurement of spatially distributed heat flux, and heat-flux measurement at high-temperature conditions. Examples of recent advances in these areas are discussed. Such new capabilities, when applied to real-world problems, make the field of heat transfer exciting. From these reviews, it is clear that no one gage or method is good for every application. The limitations of previous heat-flux-gage performance are highlighted and the need for better gage characteristics is expressed. Recent advances have been able to overcome some of these limitations and provide measurements with improved accuracy under conditions previously not possible. The discussion of measurement methods is organized by the first three categories of heat-transfer measurement categories, given in the chapter. Recent measurements using the methods discussed are showcased to demonstrate the advances made in measurement capability. Calibration is briefly addressed in the chapter.

[1]  H. G. Groehn.  INTEGRAL AND LOCAL HEAT TRANSFER OF A YAWED SINGLE CIRCULAR CYLINDER UP TO SUPERCRITICAL REYNOLDS NUMBERS , 1986 .

[2]  C. G. Miller,et al.  Comparison of thin-film resistance heat-transfer gages with thin-skin transient calorimeter gages in conventional hypersonic wind tunnels , 1981 .

[3]  T. V. Jones,et al.  Heat-transfer measurements in short-duration hypersonic facilities , 1973 .

[4]  R. Gardon.  A Segmented‐Mirror Solar Furnace for High‐Intensity Thermal Radiation Studies , 1954 .

[5]  B. Bellhouse,et al.  Determination of mean and dynamic skin friction, separation and transition in low-speed flow with a thin-film heated element , 1966, Journal of Fluid Mechanics.

[6]  Bryan E. Richards,et al.  Short Duration Measurements of Heat-Transfer Rate to a Gas Turbine Rotor Blade , 1982 .

[7]  Curt H. Liebert,et al.  Heat flux measurements , 1989 .

[8]  J. Moreno,et al.  New experimental apparatus for the study of the Bénard–Rayleigh problem , 1980 .

[9]  M. Hirata,et al.  Characteristics of dynamic behavior and local heat transfer around single row tubes immersed in floating particles , 1990 .

[10]  James W. Baughn,et al.  An Experimental Study of Entrainment Effects on the Heat Transfer From a Flat Surface to a Heated Circular Impinging Jet , 1991 .

[11]  R. Simoneau,et al.  Unsteady heat transfer and direct comparison to steady-state measurements in a rotor-wake experiment , 1986 .

[12]  A. C. Alkidas,et al.  Transient Heat Flux Measurements in a Divided-Chamber Diesel Engine , 1985 .

[13]  J. Hayes,et al.  Unsteady heat transfer coefficient estimation for long duration , 1986 .

[14]  W. George,et al.  An Evaluation of Analog and Numerical Techniques for Unsteady Heat Transfer Measurement with Thin-Film Gauges in Transient Facilities , 1991 .

[15]  Mohammad E. Taslim,et al.  An Experimental Investigation of Heat Transfer in an Orthogonally Rotating Channel Roughened With 45 deg Criss-Cross Ribs on Two Opposite Walls , 1991 .

[16]  D. E. Metzger,et al.  Periodic Streamwise Variations of Heat Transfer Coefficients for Inline and Staggered Arrays of Circular Jets with Crossflow of Spent Air , 1980 .

[17]  Maher I. Boulos,et al.  Dynamics of heat transfer from cylinders in a turbulent air stream , 1974 .

[18]  A heat flux meter to determine the local boiling heat flux density during a quenching experiment , 1979 .

[19]  Thomas J. Horvath,et al.  Aerothermodynamic measurements on a proposed assured crew return vehicle (ACRV) lifting-body configuration at Mach 6 and 10 in air , 1990 .

[20]  Denis J. Doorly,et al.  Simulation of the Effects of Shock Wave Passing on a Turbine Rotor Blade , 1985 .

[21]  Theo G. Keith,et al.  A laser-induced heat flux technique for convective heat transfer measurements in high speed flows , 1991, ICIASF '91 Record., International Congress on Instrumentation in Aerospace Simulation Facilities.

[22]  J. Louis,et al.  High temperature heat transfer and skin friction measurements in turbulent flows along highly cooled walls , 1986 .

[23]  Tony Arts,et al.  Short-Duration Measurements and Numerical Simulation of Heat Transfer Along the Suction Side of a Film-Cooled Gas Turbine Blade , 1985 .

[24]  J. E. Doorly.  Procedures for Determining Surface Heat Flux Using Thin Film Gages on a Coated Metal Model in a Transient Test Facility , 1987 .

[25]  G. M. Buck,et al.  Surface temperature/heat transfer measurement using a quantitative phosphor thermography system , 1991 .

[26]  C. Liebert.  Measurement of local high-level, transient surface heat flux , 1988 .

[27]  Carl T. Kidd.  Coaxial surface thermocouples - Analytical and experimental considerations for aerothermal heat-flux measurement applications , 1990 .

[28]  Stanisław Łopata.  Détermination du flux de chaleur instationnaire sur les surfaces cylindriques , 1990 .

[29]  C. Hoogendoorn,et al.  LOCAL COHVECTIVE-HEAT-TRANSFER COEFFICIENTS FOR JETS IMPINGING ON A PLATE; EXPERIMENTS USING A LIQUID-CRYSTAL TECHNIQUE , 1974 .

[30]  A. C. Alkidas.  Heat Release Studies in a Divided-Chamber Diesel Engine , 1987 .

[31]  S. Polat,et al.  Sensor for transient heat flux at a surface with throughflow , 1991 .

[32]  A. C. Alkidas,et al.  An experimental investigation of surface thermometry and heat flux , 1989 .

[33]  C. G. Miller.  Experimental and predicted heating distributions for biconics at incidence in air at Mach 10 , 1984 .

[34]  Hans-Jörg Bauer,et al.  Effects of wakes on the heat transfer in gas turbine cascades , 1985 .

[35]  C. Hoogendoorn.  The effect of turbulence on heat transfer at a stagnation point , 1977 .

[36]  M. Zamora,et al.  Design and construction of precision heat fluxmeters , 1984 .

[37]  D. François,et al.  Thin film temperature heat fluxmeters , 1990 .

[38]  R. J. Moffat,et al.  Measurement of temperature and heat transfer coefficient distributions in a complex flow using liquid crystal thermography and true-color image processing , 1989 .

[39]  Denis J. Doorly.  Modeling the unsteady flow in a turbine rotor passage , 1988 .

[40]  A. Mujumdar,et al.  On the use of hot and cold-film sensors for skin friction and heat transfer measurements in impingement flows , 1976 .

[41]  Thomas E. Diller,et al.  A Convection Calibration Method for Local Heat Flux Gages , 1984 .

[42]  K.G.T. Hollands,et al.  The Measurement of Surface Heat Flux Using the Peltier Effect , 1989 .

[43]  D. François,et al.  Sputtered alumina layers and platinel thermocouples for high temperature surface thermometers - Evaluation of their electrical and mechanical characteristics , 1987 .

[44]  B. R. Hollworth,et al.  Heat Transfer to Arrays of Impinging Jets in a Crossflow , 1987 .

[45]  K. Winter.  An outline of the techniques available for the measurement of skin friction in turbulent boundary layers , 1979 .

[46]  Thomas E. Diller,et al.  Experimental Performance of a Heat Flux Microsensor , 1990 .

[47]  Akira Sakurai,et al.  STUDY OF A MULTI-LAYERED THIN FILM HEAT TRANSFER GAUGE AND A NEW METHOD OF MEASURING HEAT TRANSFER RATE WITH IT. , 1987 .

[48]  Green's function partitioning procedure applied to foil heat flux gages , 1987 .

[49]  E. Achenbach,et al.  Total and local heat transfer from a smooth circular cylinder in cross-flow at high reynolds number , 1975 .

[50]  L. Matthews,et al.  Radiative flux measurements in a sooty pool fire , 1989 .

[51]  C. Kidd.  High heat flux measurements and experimental calibrations/characterizations , 1992 .

[52]  M. F. Young.  Effect of Free-Stream Velocity Vector and Aspect Ratio on the Output of a Free-Standing Circular Disk Heat Flux Gage , 1984 .

[53]  Robert C. Hendricks,et al.  Heat transfer in aerospace propulsion , 1988 .

[54]  W. Cook,et al.  Determination of heat-transfer rates from transient surface temperature measurements , 1970 .

[55]  M. L. G. Oldfield,et al.  Heat Transfer Optimized Turbine Rotor Blades—An Experimental Study Using Transient Techniques , 1984 .

[56]  Curt H. Liebert.  Heat flux measurement in SSME turbine blade tester , 1990 .

[57]  G. J. Vanfossen,et al.  A Study of the Relationship Between Free-Stream Turbulence and Stagnation Region Heat Transfer , 1985 .

[58]  S. Saxena,et al.  Experimental techniques for the measurement of radiative and total heat transfer in gas fluidized beds: A review , 1989 .

[59]  M. L. G. Oldfield,et al.  The theory of advanced multi-layer thin film heat transfer gauges , 1987 .

[60]  Klaus Dullenkopf,et al.  The Effect of Incident Wake Conditions on the Mean Heat Transfer of an Airfoil , 1991 .

[61]  R. J. Field,et al.  Liquid Crystal Thermography and Its Application to the Study of Convective Heat Transfer , 1975 .

[62]  D. L. Schultz,et al.  Unsteady Aerodynamic and Heat Transfer Processes in a Transonic Turbine Stage , 1985 .

[63]  R. Gardon.  A Transducer for the Measurement of Heat-Flow Rate , 1960 .

[64]  Alan H. Epstein,et al.  High frequency response heat flux gauge for metal blading , 1985 .

[65]  R. Moffat,et al.  Development of porous plug radiometers for use in advanced gas turbine engine programs , 1988 .

[66]  S. Kline,et al.  A SUMMARY OF EXPERIMENTS ON TURBULENT HEAT TRANSFER FROM A NONISOTHERMAL FLAT PLATE , 1960 .

[67]  T. Diller,et al.  Design and Calibration of a Local Heat-Flux Measurement System for Unsteady Flows , 1989 .

[68]  J. Beck,et al.  Rectangular Heat Source on a Semi-infinite Solid—An Analysis for a Thin Film Heat Flux Gage Calibration , 1988 .

[69]  D. E. Beasley,et al.  A generalised analysis of a local heat flux probe , 1988 .

[70]  Alan H. Epstein,et al.  High-frequency response heat-flux gauge , 1986 .

[71]  M. L. G. Oldfield,et al.  New Heat Transfer Gages for Use on Multilayered Substrates , 1986 .

[72]  N. Afgan,et al.  A heat flux-meter for ash deposit monitoring systems—I. Ash deposit prevention , 1991 .

[73]  Terry V. Jones,et al.  The measurement of local heat transfer coefficients in blade cooling geometries , 1985 .

[74]  L. M. Russell,et al.  High-resolution liquid-crystal heat-transfer measurements on the endwall of a turbine passage with variations in Reynolds number , 1988 .

[75]  C T Kidd.  A Durable, Intermediate Temperature, Direct Reading Heat Flux Transducer for Measurements in Continuous Wind Tunnels , 1981 .

[76]  N. R. Keltner,et al.  Transient response of circular foil heat‐flux gauges to radiative fluxes , 1975 .

[77]  J. Watts,et al.  A TECHNIQUE FOR THE MEASUREMENT OF LOCAL HEAT TRANSFER COEFFICIENTS USING COPPER FOIL , 1984 .

[78]  Richard M. White,et al.  Microsensors for heat transfer and fluid flow measurements , 1990 .

[79]  Michael G. Dunn,et al.  Heat-Flux Measurements for the Rotor of a Full-Stage Turbine: Part II—Description of Analysis Technique and Typical Time-Resolved Measurements , 1986 .

[80]  John R. Grace,et al.  The measurement of instantaneous local heat transfer coefficients in a circulating fluidized bed , 1989 .

[81]  D. J. Doorly,et al.  Wake-passing in a turbine rotor cascade , 1985 .

[82]  L. M. Russell,et al.  Local heat-transfer measurements on a large scale-model turbine blade airfoil using a composite of a heater element and liquid crystals , 1985 .

[83]  G. Woschni,et al.  Heat Transfer With Insulated Combustion Chamber Walls and Its Influence on the Performance of Diesel Engines , 1988 .

[84]  T. V. Jones,et al.  Detailed measurements of heat transfer on and around a pedestal in fully developed passage flow , 1986 .

[85]  J. Grace,et al.  Instantaneous local heat transfer and hydrodynamics in a circulating fluidized bed , 1991 .

[86]  N. Afgan,et al.  A heat flux-meter for ash deposit monitoring systems—II. ‘Clean’ heat flux-meter characteristics , 1991 .

[87]  Influence of thermal boundary conditions on heat transfer from a cylinder in cross flow , 1981 .

[88]  M. Chyu.  On the boundary condition and data reduction of heat transfer experiment , 1987 .

[89]  L. M. Russell,et al.  Evaluation of a Method for Heat Transfer Measurements and Thermal Visualization Using a Composite of a Heater Element and Liquid Crystals , 1981 .

[90]  T. Diller,et al.  SIMULTANEOUS MEASUREMENTS OF TIME-RESOLVED SURFACE HEAT FLUX AND FREESTREAM TURBULENCE AT A STAGNATION POINT , 1990 .

[91]  S. Aso,et al.  Measurement of heat-transfer coefficients in shock wave-turbulent boundary layer interaction regions with a multi-layered thin film heat transfer gauge , 1986 .

[92]  D. Dewitt,et al.  Heat Transfer in a Cooled and an Insulated Diesel Engine , 1989 .

[93]  Heat-transfer distribution around a cylinder in pulsating crossflow , 1985 .

[94]  Charles G. Miller,et al.  Refinement of an 'alternate' method for measuring heating rates in hypersonic wind tunnels , 1985 .

[95]  B. Barry,et al.  Measurements of Heat Transfer Coefficients on Gas Turbine Components: Part I — Description, Analysis and Experimental Verification of a Technique for Use in Hostile Environments , 1982 .

[96]  Roger W. Ainsworth,et al.  Surface Heat Transfer Fluctuations on a Turbine Rotor Blade Due to Upstream Shock Wave Passing , 1989 .

[97]  R. W. Graham.  Impact of new instrumentation on advanced turbine research , 1980 .

[98]  C. H. Liebert,et al.  High-temperature thermocouple and heat flux gauge using a unique thin film-hardware hot junction , 1985 .

[99]  K. E. Starner.  USE OF THIN-SKINNED CALORIMETERS FOR HIGH HEAT FLUX ARC JET MEASUREMENTS , 1967 .

[100]  Roger W. Ainsworth,et al.  Developments in Instrumentation and Processing for Transient Heat Transfer Measurement in a Full-Stage Model Turbine , 1989 .

[101]  G. Simeonides,et al.  Infrared thermography in blowdown and intermittent hypersonic facilities , 1989 .

[102]  J. E. LaGraff,et al.  Unsteady Interaction Effects on a Transitional Turbine Blade Boundary Layer , 1989 .

[103]  Robert Gardon,et al.  An Instrument for the Direct Measurement of Intense Thermal Radiation , 1953 .

[104]  W. H. Atkinson,et al.  Turbine blade and vane heat flux sensor development, phase 1 , 1984 .

[105]  S. Gordon.  On the Limits on Wall-Thickness in the Thin-Wall Calorimeter Heat Flux Gauge , 1964 .

[106]  R. M. C. So,et al.  Short Duration Heat Transfer Studies at High Free-Stream Temperatures , 1982 .

[107]  Terry V. Jones,et al.  High-resolution heat-transfer-coefficient maps applicable to compound-curve surfaces using liquid crystals in a transient wind tunnel , 1987 .

[108]  I. Mabuchi,et al.  TEMPORAL AND SPATIAL CHARACTERISTICS OF HEAT TRANSFER AT THE REATTACHMENT REGION OF A BACKWARD-FACING STEP , 1987 .

[109]  Anil K. Kulkarni,et al.  Analysis of heat flux measurement by circular foil gages in a mixed convection/radiation environment , 1991 .

[110]  T. Diller,et al.  Analysis and design of experimental systems for heat transfer measurement from constant-temperature surfaces , 1989 .

[111]  Steven A. Hippensteele,et al.  Use of a liquid-crystal and heater-element composite for quantitative, high-resolution heat-transfer coefficients on a turbine airfoil including turbulence and surface-roughness effects , 1987 .

[112]  Michael G. Dunn,et al.  Measurement and Analyses of Heat Flux Data in a Turbine Stage: Part II—Discussion of Results and Comparison With Predictions , 1984 .

[113]  L. Hearne,et al.  Interpretation of asymptotic calorimeter measurements. , 1967 .

[114]  G. Jovanovic,et al.  Instrumented cylinder for studying heat transfer to immersed tubes in fluidized beds , 1981 .

[115]  Measurement of the local convection coefficient by pulsed photothermal radiometry , 1991 .

[116]  J. Baughn,et al.  An Instrument for the Measurement of Heat Flux from a Surface with Uniform Temperature , 1980 .

[117]  H. Grönig,et al.  A new method for manufacture of thin film heat flux gauges , 1991 .

[118]  R. Taylor,et al.  Heat Transfer Measurements in Incompressible Turbulent Flat Plate Boundary Layers With Step Wall Temperature Boundary Conditions , 1990 .

[119]  A high sensitivity, fast response heat flux sensor , 1985 .

[120]  Michael G. Dunn,et al.  Measurement of Heat Flux and Pressure in a Turbine Stage , 1982 .

[121]  Miniature high temperature plug-type heat flux gauges , 1992 .

[122]  W. H. Atkinson,et al.  Development of advanced high-temperature heat flux sensors. Phase 2: verification testing. Final Report , 1982 .

[123]  W. Cook,et al.  Reduction of data from thin-film heat-transfer gages - A concise numerical technique. , 1966 .

[124]  A comparison of the transient and heated-coating methods for the measurement of local heat transfer coefficients on a pin fin , 1989 .

[125]  Terry V. Jones,et al.  A Technique for Measuring Convective Heat Transfer at Rough Surfaces , 1990 .

[126]  James W. Baughn,et al.  Heat Transfer Measurements From a Surface With Uniform Heat Flux and an Impinging Jet , 1989 .

[127]  D. E. Metzger,et al.  Use of Melting Point Surface Coatings for Local Convection Heat Transfer Measurements in Rectangular Channel Flows With 90-deg Turns , 1984 .

[128]  Michael G. Dunn.  Phase and Time-Resolved Measurements of Unsteady Heat Transfer and Pressure in a Full-Stage Rotating Turbine , 1990 .

[129]  Michael G. Dunn,et al.  Phase-resolved heat-flux measurements on the blade of a full-scale rotating turbine , 1988 .

[130]  Herbert J. Gladden,et al.  Experience with advanced instrumentation in a hot section cascade , 1989 .

[131]  R. J. Simoneau,et al.  Effect of a rotor wake on heat transfer from a circular cylinder , 1984 .

[132]  G. Hayward,et al.  Local heat transfer from a single sphere to a turbulent air stream , 1978 .

[133]  James W. Baughn,et al.  Heat transfer to air from a yawed cylinder , 1982 .

[134]  D. E. Metzger,et al.  Transient Liquid Crystal Measurement of Local Heat Transfer on a Rotating Disk With Jet Impingement , 1991 .

[135]  J. E. LaGraff,et al.  Measurement and Modeling of the Gas Turbine Blade Transition Process as Disturbed by Wakes , 1989 .

[136]  D. Telionis,et al.  Time-Resolved Heat Transfer and Skin Friction Measurements in Unsteady Flow , 1989 .

[137]  J. H. Klems,et al.  Large‐area, high‐sensitivity heat‐flow sensor , 1982 .

[138]  W. H. Bennethum,et al.  Sensors for ceramic components in advanced propulsion systems: Summary of literature survey and concept analysis, task 3 report , 1988 .

[139]  T. Diller,et al.  Heat Flux Microsensor Measurements and Calibrations , 1992 .

[140]  R S Longenbaugh,et al.  EXPERIMENTAL AND THEORETICAL ANALYSIS OF THE RADIATIVE TRANSFER WITHIN A SOOTY POOL FIRE , 1988 .

[141]  G. K. Steenvoorden,et al.  Thick film heat flux sensor , 1983 .

[142]  B. Launder,et al.  Heat Transfer, Temperature, and Velocity Measurements Downstream of an Abrupt Expansion in a Circular Tube at a Uniform Wall Temperature , 1989 .

[143]  M. G. Dunn,et al.  Heat-Flux Measurements for the Rotor of a Full-Stage Turbine: Part I—Time-Averaged Results , 1986 .

[144]  T. Baba,et al.  Analysis of the operational error of heat flux transducers placed on wall surfaces , 1985 .

[145]  David R. Englund,et al.  Recent advances in high temperature instrumentation for hot section applications , 1988 .

[146]  A. E. Forest,et al.  Heat Transfer Measurements With Film Cooling on a Turbine Blade Profile in Cascade , 1985 .

[147]  A. C. Alkidas,et al.  Heat Transfer Characteristics of a Spark-Ignition Engine , 1980 .

[148]  Richard J Goldstein,et al.  Visualization of heat transfer from arrays of impinging jets , 1982 .

[149]  Shoichi Furuhama,et al.  Study on Thin Film Thermocouple Measuring Instantaneous Temperature on Surface of Combustion Chamber Wall In Internal Combustion Engine : 2nd Report, Study on Thin Film Thermocouples Embedded in Combustion Chamber Wall , 1986 .

[150]  J. Hayes,et al.  The application of numerical techniques to model the response and integration of thermal sensors in wind tunnel models , 1991 .

[151]  Peter Erbland,et al.  Instrumentation of hypersonic structures - A review of past applications and needs for the future , 1988 .

[152]  Terry V. Jones,et al.  The response time of a surface thermometer employing encapsulated thermochromic liquid crystals , 1987 .

[153]  R. P. Benedict,et al.  Fundamentals of temperature, pressure, and flow measurements , 1977 .

[154]  B. B. Mikic,et al.  An experimental investigation into the effect of surface thermal conductivity on the rate of heat transfer in dropwise condensation , 1976 .

[155]  Michael G. Dunn,et al.  Measurement of heat-transfer rate to a gas turbine stator , 1979 .

[156]  J. C. Jaeger,et al.  Conduction of Heat in Solids , 1952 .

[157]  T. Simon,et al.  Development of a special-purpose test surface guided by uncertainty analysis , 1989 .

[158]  P. A. Gnoffo,et al.  Laminar heat-transfer distributions on biconics at incidence in hypersonic-hypervelocity flows , 1984 .

[159]  F. L. Test,et al.  The Effect of Free-Stream Turbulence on Heat Transfer From a Rectangular Prism , 1984 .

[160]  Richard B. Rivir,et al.  Investigation of Transient Technique for Turbine Vane Heat Transfer Using a Shock Tube , 1985 .

[161]  L. W. Florschuetz,et al.  Effects of crossflow temperature on heat transfer within an array of impinging jets , 1986 .

[162]  C. Subramanian,et al.  Effects of Vortices With Different Circulations on Heat Transfer and Injectant Downstream of a Single Film-Cooling Hole in a Turbulent Boundary Layer , 1991 .

[163]  R. J. Moffat,et al.  A review of turbulent-boundary-layer heat transfer research at Stanford, 1958-1983 , 1984 .

[164]  J. Simonich,et al.  Liquid Crystal Visualization of Surface Heat Transfer on a Concavely Curved Turbulent Boundary Layer , 1984 .

[165]  R. D. Jackson,et al.  Heat Transfer 1 , 1965 .

[166]  R. Takahashi,et al.  Local Heat Transfer Measurements Using an Electrically Heated Thin Gold-Coated Plastic Sheet , 1985 .

[167]  Michael G. Dunn,et al.  Heat-Flux and Pressure Measurements and Comparison With Prediction for a Low-Aspect-Ratio Turbine Stage , 1986 .

[168]  M. A. Hilditch,et al.  Unsteady Heat Transfer Measurements on a Rotating Gas Turbine Blade , 1990 .

[169]  D. Kenning.  Wall temperature patterns in nucleate boiling , 1991 .

[170]  Robert Haimes,et al.  Fully Scaled Transonic Turbine Rotor Heat Transfer Measurements , 1988 .

[171]  G. A. Dreitser,et al.  Unsteady Convective Heat Transfer and Hydrodynamics in Channels , 1970 .