BOOKCOMP, Inc. — John Wiley & Sons / Page 1227 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 1227 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1227], (47) Lines: 1233 to 1250 ——— 0.99185pt PgVar ——— Short Page PgEnds: T E X [1227], (47) cr critical d droplet dry dryout e entrainment evaporator section eff effective ext external f frictional fin g gravitational h hydraulic I inner inertial interface l liquid m maximum n nucleation o outer p pipe s sonic surface t total v vapor w wire spacing wick wetting axial hydrostatic pressure + normal hydrostatic pressure REFERENCES Akachi, H., and Polasek, F. (1995). Pulsating Heat Pipe Review of the Present State of the Art, Technical Report ITRI-ERL, Chutung, Taiwan, May. Ambrose, J. H., Chow, L. C., and Beam, J. E. (1987). Transient Heat Pipe Response and Rewetting Behavior, J. Thermophys. Heat Transfer, 1(3), 222–227. Babin, B. R., Peterson, G. P., and Wu, D. (1990). Experimental Investigation of a Flexible Bellows Heat Pipe for Cooling Discrete Heat Sources, J. Heat Transfer, 112(3), 602–607. Bankston, C. A., and Smith, J. H. (1971). Incompressible Laminar Vapor Flow in Cylindrical Heat Pipes, ASME-71-WA/HT-15, ASME, New York. Bowman, W. J. (1991). Numerical Modeling of Heat-Pipe Transients, J. Thermophys. Heat Transfer, 5(3), 374–379. BOOKCOMP, Inc. — John Wiley & Sons / Page 1228 / 2nd Proofs / Heat Transfer Handbook / Bejan 1228 HEAT PIPES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1228], (48) Lines: 1250 to 1298 ——— 9.0pt PgVar ——— Custom Page (-4.0pt) PgEnds: T E X [1228], (48) Brennan, P. J., and Kroliczek, E. J. (1979). Heat Pipe Design Handbook, NASA Contract Report NAS5-23406. Busse, C. A. (1973). Theory of the Ultimate Heat Transfer of Cylindrical Heat Pipes, Int. J. Heat Mass Transfer, 16, 169–186. Busse, C. A., and Kemme, J. E. (1980). Dry-out Phenomena in Gravity-Assist Heat Pipes with Capillary Flow, Int. J. Heat Mass Transfer, 23, 643–654. Carey, V. P. (1992). Liquid–Vapor Phase-Change Phenomena, Taylor & Francis, Washing- ton, DC. Chang, W. S., and Colwell, G. T. (1985). Mathematical Modeling of the Transient Operating Characteristics of a Low-Temperature Heat Pipe, Numer. Heat Transfer, 8, 169–186. Chi, S. W. (1976). Heat Pipe Theory and Practice, Hemisphere Publishing, Washington, DC. Colwell, G. T., and Modlin, J. M. (1992). Mathematical Heat Pipe Models, Proc. 8th Interna- tional Heat Pipe Conference, Vol. 1, pp. 162–166. Cotter, T. P. (1967). Heat Pipe Startup Dynamics, Proc. SAE Thermionic Conversion Specialist Conference, Palo Alto, CA. Delhaye, J. M. (1981). Basic Equations for Two-Phase Flow Modeling, in Two-Phase Flow and Heat Transfer in the Power and Process Industries, A. E. Bergles, J. G. Collier, and J. M. Delhaye, eds., Hemisphere Publishing, Washington, DC. Deverall, J. E., Kemme, J. E., and Florschuetz, L. W. (1970). Sonic Limitations and Startup Problems of Heat Pipes, Report LA-4578, Los Alamos Scientific Laboratory, Los Ala- mos, NM. Dunbar, N., and Cadell, P. (1998). Working Fluids and Figures of Merit for CPL/LHP Appli- cations, CPL-98 Workshop Proc., Aerospace Corporation, El Segundo, CA, Mar. 2–3. Dunbar, N., and Supper, W. (1997). Spacecraft Capillary Pumped Loop Technology towards a Qualified Thermal Control Tool, Proc. 10th International Heat Pipe Conference, Stuttgart, Germany. Dunn, P. D., and Reay, D. A. (1982). Heat Pipes, 3rd ed., Pergamon Press, New York. Faghri, A. (1995). Heat Pipe Science and Technology, Taylor & Francis, Washington, DC. Gaugler, R. S. (1944). Heat Transfer Devices, U.S. patent 2,350,348. Grover, G. M., Cotter, T. P., and Erikson, G. F. (1964). Structures of Very High Thermal Conductivity, J. Appl. Phys., 218, 1190–1191. Hewitt, G. F. (1979). Liquid Mass Transport in Annular Two-Phase Flow, Two-Phase Momen- tum, Heat and Mass Transfer in Chemical, Process, and Energy Engineering Systems, Vol. 1, Hemisphere Publishing, New York, pp. 273–302. Incropera, F. P., and DeWitt, D. P. (1996). Fundamentals of Heat and Mass Transfer, 4th ed., Wiley, New York. Ivanovskii, M. N., Sorokin, V. P., and Yagodkin, I. V. (1982). The Physical Properties of Heat Pipes, Clarendon Press, Oxford. Kemme, J. E. (1969). Ultimate Heat Pipe Performance, IEEE Trans. Electron Devices, 16, 717–723. Kemme, J. E. (1976). Vapor Flow Consideration in Conventional and Gravity-Assist Heat Pipes, Proc. 2nd International Heat Pipe Conference, pp. 11–21. Ku, J. (1997). RecentAdvances in Capillary Pumped Loop Technology, AIAA-97-3870, AIAA, New York. BOOKCOMP, Inc. — John Wiley & Sons / Page 1229 / 2nd Proofs / Heat Transfer Handbook / Bejan REFERENCES 1229 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1229], (49) Lines: 1298 to 1339 ——— 13.0pt PgVar ——— Custom Page (-4.0pt) PgEnds: T E X [1229], (49) Langer, H., and Mayinger, F. (1979). Entrainment in Annular Two-Phase Flow under Steady and Transient Flow Conditions, Two-Phase Momentum, Heat and Mass Transfer in Chem- ical, Process, and Energy Engineering Systems, Vol. 2, Hemisphere Publishing, New York, pp. 695–706. Levy, E. K. (1968). Theoretical Investigation of Heat Pipes Operating at Low Vapor Pressure, J. Eng. Ind., 90, 547–552. Levy, E. K., and Chou, S. F. (1973). The Sonic Limit in Sodium Heat Pipes, J. Heat Transfer, 95, 218–223. Maidanik, Y. F., and Fershtater, Y. G. (1997). Theoretical Basis and Classification of Loop Heat Pipes and Capillary Pumped Loops, Proc. 10th International Heat Pipe Conference, Stuttgart, Germany, Sept. Marcus, B. D. (1965). On the Operation of Heat Pipes, Report 9895-6001-TU-000, TRW, Redondo Beach, CA. Moody, L. F. (1944). Friction Factors for Pipe Flow, Trans. ASME, 66, 671–684. Nguyen-Chi, H., and Groll, M. (1981). Entrainment or Flooding Limit in a Closed Two-Phase Thermosyphon, Proc. 4th International Heat Pipe Conference, pp. 147–162. Ochterbeck, J. M. (1997). Modeling of Room-Temperature Heat Pipe Startup from the Frozen State, J. Thermophys. Heat Transfer, 11(2), 165–172. Ochterbeck, J. M., Peterson, G. P., and Ungar, E. (1995). Depriming/Rewetting of External Artery Heat Pipes: Comparison with SHARE-II Flight Experiment, J. Thermophys. Heat Transfer, 9(1), 101–108. Peterson, G. P. (1994). An Introduction to Heat Pipe, Wiley, New York. Peterson, G. P., and Bage, B. (1991). Entrainment Limitations in Thermosyphons and Heat Pipes, J. Energy Resour. Technol., 113(3), 147–154. Prenger, F. C. (1984). Performance Limits of Gravity-Assist Heat Pipes, Proc. 5th International Heat Pipe Conference, pp. 1–5. Prenger, F. C., and Kemme, J. E. (1981). Performance Limits of Gravity-Assist Heat Pipes with Simple Wick Structures, Proc. 4th International Heat Pipe Conference, pp. 137–146. Rice, G., and Fulford, D. (1987). Influence of a Fine Mesh Screen on Entrainment in Heat Pipes, Proc. 6th International Heat Pipe Conference, pp. 168–172. Rohani, A. R., and Tien, C. L. (1974). Analysis of the Effects of Vapor Pressure Drop on Heat Pipe Performance, Int. J. Heat Mass Transfer, 17, 61–67. Stenger, F. J. (1966). Experimental Feasibility Study of Water-Filled Capillary Pumped Heat Transfer Loops, NASA X-1310, NASA LeRC Report. Tien, C. L., and Chung, K. S. (1979). Entrainment Limits in Heat Pipes, AIAA J., 17(6), 643– 646. Trefethen, L. (1962). On the Surface Tension Pumping of Liquids or a Possible Role of the Candlewick in Space Exploration, GE Tech. Int. Ser. No. G15-D114, General Electric Co., Schenectady, NY. Vinz, P., and Busse, C. A. (1973). Axial Heat Transfer Limits of Cylindrical Sodium Heat Pipes between 25 W-cm −2 and 15.5 kW-cm −2 , Proc. 1st International Heat Pipe Conference, Stuttgart, Germany, Paper 2-1. Wayner, P. C., Jr. (1999). Long Range Intermolecular Forces in Change-of-Phase Heat Transfer, Proc. 33rd National Heat Transfer Conference, Albuquerque, NM, Aug. 15–17. BOOKCOMP, Inc. — John Wiley & Sons / Page 1230 / 2nd Proofs / Heat Transfer Handbook / Bejan 1230 HEAT PIPES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [Last Page] [1230], (50) Lines: 1339 to 1347 ——— 443.04701pt PgVar ——— Normal Page PgEnds: T E X [1230], (50) Wu, D., Peterson, G. P., and Chang, W. S. (1991). Transient Experimental Investigation of Micro Heat Pipes, J. Thermophys. Heat Transfer, 5(4), 539–545. Wulz, H., and Embacher, E. (1990). Capillary Pumped Loops for Space Applications: Ex- perimental and Theoretical Studies on the Performance of Capillary Evaporator Designs, AIAA-90-1739, AIAA, New York. Yan, Y., and Ochterbeck, J. M. (1999). Analysis of the Supercritical Startup Behavior for Cryogenic Heat Pipes, J. Thermophys. Heat Transfer, 13(1), 140–145. Zucrow, M. J., and Hoffman, J. D. (1976). Gas Dynamics, Wiley, New York. BOOKCOMP, Inc. — John Wiley & Sons / Page 1231 / 2nd Proofs / Heat Transfer Handbook / Bejan 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [First Page] [1231], (1) Lines: 0 to 82 ——— 5.18208pt PgVar ——— Normal Page PgEnds: T E X [1231], (1) CHAPTER 17 Heat Transfer in Manufacturing and Materials Processing RICHARD N. SMITH Department of Mechanical Engineering, Aeronautical Engineering and Mechanics Rensselaer Polytechnic Institute Tr oy, New York C. HARIS DOUMANIDIS Department of Mechanical Engineering Tufts University Medford, Massachusetts RANGA PITCHUMANI Department of Mechanical Engineering University of Connecticut Storrs, Connecticut 17.1 Introduction 17.2 Heat transfer to moving materials undergoing thermal processing 17.2.1 Uniform thermal environment Thin solid model Two-dimensional workpieces 17.2.2 Interaction between a discrete heat source and a continuously moving work- piece Thin plate or rod with a moving planar heat source Thin plate with a moving line heat source Semi-infinite solid with a moving point source Semi-infinite plane with finite size moving heat source 17.3 Thermal issues in heat treatment of solids 17.4 Machining processes: metal cutting 17.4.1 Background 17.4.2 Thermal analysis Tool–chip interface temperature rise Energy generation at the shear plane Assessment of steady-state metal cutting temperature models 17.5 Machining processes: grinding 17.5.1 Background 17.5.2 Workpiece temperatures during grinding 17.6 Thermal–fluid effects in continuous metal forming processes 17.6.1 Background 1231 BOOKCOMP, Inc. — John Wiley & Sons / Page 1232 / 2nd Proofs / Heat Transfer Handbook / Bejan 1232 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1232], (2) Lines: 82 to 157 ——— -2.0pt PgVar ——— Normal Page PgEnds: T E X [1232], (2) 17.6.2 Considerations for thermal–fluid modeling in extrusion and drawing Deformation heating considerations Frictional heating considerations 17.7 Processing of polymer-matrix composite materials 17.7.1 Introduction 17.7.2 Processing of thermosetting-matrix composites Thermal model Kinetics model Laminate consolidation model 17.7.3 Processing of thermoplastic-matrix composites Heat transfer Void dynamics Interlaminar bonding Polymer degradation Solidification (crystallization) 17.8 Thermal process control for manufacturing 17.8.1 Control of SISO thermal systems Thermostatic (on–off) control Proportional–integral–derivative (PID) control Software implementation of SISO controllers 17.8.2 Control of MIMO thermal systems State controllers by pole placement State observers by pole placement 17.8.3 Optimal formulation: linear quadratic Gaussian Optimal control: linear quadratic regulator (LQR) Optimal observation: Kalman–Bucy filter 17.8.4 Smith prediction 17.8.5 Sliding mode control 17.8.6 Adaptive control Model reference adaptive control (MRAC) Self-tuning regulation 17.8.7 Parameter identification Orthogonal projection Least squares Nomenclature References 17.1 INTRODUCTION The last two decades of the twentieth century have witnessed a significant move in the thermal-fluid sciences toward studying fundamental problems motivated by applications in manufacturing and materials processing. Establishment of a number of industrially supported research centers within universities had the express purpose of developing advances that could directly affect U.S. industrial competitiveness in manufacturing and that could train a new generation of technical specialists. BOOKCOMP, Inc. — John Wiley & Sons / Page 1233 / 2nd Proofs / Heat Transfer Handbook / Bejan INTRODUCTION 1233 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1233], (3) Lines: 157 to 161 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [1233], (3) Recognition that many processes, such as casting, welding, spray deposition, quenching, crystal growth, extrusion and drawing, rolling, and metal cutting, are enabled and/or controlled by heat transfer and fluid mechanics phenomena has led heat transfer researchers to turn their attention to these types of problems. The report of the National Science Foundation (NSF)-sponsored workshop on critical technolo- gies in the thermal systems (Jacobs and Hartnett, 1992) emphasized the importance of work in this area. The proceedings of several recent international symposia (Shah et al., 1992; Tanasawa and Lior, 1992; Guceri, 1993) have provided significant col- lections of review articles and of recent research results which can be of great utility to engineers and researchers. In 1988, the Heat Transfer Division of the American Society of Mechanical Engineers (ASME) initiated a new technical committee (K- 15, Heat and Mass Transfer in Manufacturing and Materials Processing, referred to subsequently as MMP) to coordinate the orderly presentation of research results at division-sponsored meetings. These activities were recently celebrated with a more focused NSF-sponsored workshop (Prasad et al., 1998). Kitto et al. (1995) highlighted critical technologies associated with thermal phenomena, with particular emphasis on MMP. A new journal devoted to this subject was also initiated (Guceri, 1992), and two other handbook chapters have appeared (Radford and Tong, 2000; Viskanta and Bergman, 2000). Necessarily, the scope of this presentation must be limited. First, MMP is an ex- tremely broad subject, and it is difficult to identify common technological founda- tions among such disparate topics as, for example, machining and injection molding and chemical vapor deposition, even with a focus strictly on thermal-fluid aspects. Furthermore, the thermal-fluid elements of even a single process cannot be separated fully from other fundamental engineering elements, such as material behavior, me- chanics, and control. Indeed, entire books have been devoted even to a single type of manufacturing process (e.g., DeVries, 1992; Steen, 1991). Nonetheless, the goal of the present chapter is to characterize the important transport phenomena, particu- larly heat transfer, that are associated with a number of manufacturing and materials processing operations. The general approach will be to introduce and review the es- sential features of a particular manufacturing process, to identify appropriate physical models of the process that are useful for describing the thermal and transport fields, to show how the physical model leads to a mathematical model of the process, and to interpret these models in light of experimental results and in terms of the expected performance and operation of the actual process. The particular subjects have been chosen largely from segments of three graduate courses taught by the authors, developed independently between 1995 and 2000. The focus is more on manufacturing operations than on materials processes (as much as those two can be distinguished), and there is limited overlap with similar monographs (Radford and Tong, 2000; Viskanta and Bergman, 2000), so that no specific reference will be made to them. Examples of topics that have reluctantly been omitted are most phases of solidification processing, chemical and vapor deposition processes, poly- mer processes (except for composites processing), and others that the reader will note. Some texts that may serve as appropriate background in some of these areas are Poirier and Poirier (1992), Guthrie (1989), Gaskell (1992), Flemings (1974), Poirier BOOKCOMP, Inc. — John Wiley & Sons / Page 1234 / 2nd Proofs / Heat Transfer Handbook / Bejan 1234 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1234], (4) Lines: 161 to 178 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [1234], (4) and Geiger (1994), Yang et al. (1994), and Kou (1996). For background on general manufacturing and materials processes, the reader is referred to classical texts (Schey, 2000; Kalpakjian, 1996; DeGarmo et al., 1997). The authors have chosen not to at- tempt a complete literature review of the particular subjects. Rather, important refer- ences that comprise the essential focus will be cited, in addition to broad background references. Most of the necessary background in conduction, convection, and radiation phe- nomena has been discussed in earlier chapters of this book. Unique features of MMP that must be considered carefully in developing appropriate physical models are un- usual boundary conditions, conjugate heat transfer problems, in which two adjoining media are thermally coupled; moving heat sources or sinks; multimode heat transfer (convection and radiation); and phase change. Some background material on con- duction heat transfer to objects in motion and on the thermal response of a solid to a moving, local, or distributed heat source is offered first. These elements are com- mon to several of the process descriptions that follow. Section 17.3 contains a very brief review of some of the issues in heat treatment. In Sections 17.4 to 17.6 some material removal and metal forming processes are discussed. In Section 17.7 some thermal characteristics of composite materials manufacturing are introduced. Finally, in Section 17.8 we present an introduction to analysis for thermal control, which has common application in almost any manufacturing operation involving an applied heat source. In all cases, the heat transfer elements are emphasized over considerations of mechanical behavior and material property response. 17.2 HEAT TRANSFER TO MOVING MATERIALS UNDERGOING THERMAL PROCESSING A large variety of processes involve the continuous movement of a workpiece or man- ufactured part through a thermal environment. These can include rolling, extrusion and drawing, continuous casting, crystal growth, welding, and heat treatment. The thermal environment can be localized, such as the forming zone of a rolling process, or continuous, such as quenching of a wire following a drawing process. When a heat source is localized, it may be that the source itself is moving and the workpiece stationary, although it is often convenient to consider the thermal field from the point of view of a stationary source. Processing speeds can vary from a few centimeters per hour (Czochralski crystal growth) to several meters per second (wire drawing). 17.2.1 Uniform Thermal Environment Consider first the case of a continuously moving workpiece exposed to a uniform thermal environment. Jaluria (1993) has discussed three basic approaches in the math- ematical modeling of such processes. In the first approach, only conduction heat transfer in the solid material (workpiece), including the advection associated with the workpiece motion, is considered. The external heating or cooling is accommo- dated by a known surface heat transfer coefficient and/or radiative environment or BOOKCOMP, Inc. — John Wiley & Sons / Page 1235 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER TO MOVING MATERIALS UNDERGOING THERMAL PROCESSING 1235 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1235], (5) Lines: 178 to 203 ——— -3.4389pt PgVar ——— Normal Page * PgEnds: Eject [1235], (5) Figure 17.1 Conjugate temperature and velocity fields in a moving material undergoing convective cooling. by a specified distribution of surface heat flux. The second approach focuses on the convective flow field in the surroundings, which is induced, at least partially, by the workpiece movement. One attempts to obtain a distribution of the local surface heat transfer coefficient from the thermal field calculated in the fluid. The thermal condi- tions in the solid workpiece are prescribed as boundary conditions. (This would be considered a classical convection problem.) The third approach is a combination of the first two, called a conjugate problem of heat transfer, as illustrated schematically in Fig. 17.1. The thermal fields in the workpiece and in the surroundings are both determined, with a coupling of the thermal and fluid conditions at the solid–fluid interface. The concentration here is only on the first approach, because attention is necessarily focused on the workpiece thermal response. However, it should be rec- ognized that a determination of surface heat transfer coefficients, presumed as being known and uniform in most of what follows, is seldom a straightforward task. Thin Solid Model Consider the case in which the temperature variation through the cross section of the workpiece is negligible compared to that along its length. For a workpiece moving at a uniform velocity (implying that the cross section of the workpiece is uniform in the direction of motion), an energy balance on a differential control volume of size dA c × dx yields the following equation for the temperature distribution as a function of time and distance x along the workpiece (Fig. 17.2): 1 α ∂T ∂t = ∂ 2 T ∂x 2 − V α ∂T ∂x − hP kA c (T −T ∞ ) (17.1a) with some very simple boundary conditions: T = T o at x = 0 ∂T ∂x = 0atx = L (17.1b) BOOKCOMP, Inc. — John Wiley & Sons / Page 1236 / 2nd Proofs / Heat Transfer Handbook / Bejan 1236 HEAT TRANSFER IN MANUFACTURING AND MATERIALS PROCESSING 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [1236], (6) Lines: 203 to 248 ——— 2.08276pt PgVar ——— Long Page * PgEnds: Eject [1236], (6) Figure 17.2 Continuously moving material whose surface is exposed to a convective heat transfer environment. Under steady-state conditions, the following dimensionless parameters may conve- niently be introduced: x ∗ = x γ θ = T −T ∞ T o − T ∞ Pe = V γ α L ∗ = L γ Bi = hγ k (17.2) where γ ≡ A c /P (P is the perimeter). The dimensionless form of eq. (17.1a) is d 2 θ dx ∗2 − Pe dθ dx ∗ − Bi · θ = 0 (17.3a) subject to θ = θ o at x ∗ = 0 ∂θ ∂x ∗ = 0atx ∗ = L ∗ (17.3b) The solution to eq. (17.3) is straightforward: θ(x) θ o = m 1 e m 1 L ∗ e m 2 x ∗ − m 2 e m 2 L ∗ e m 1 x ∗ m 1 e m 1 L ∗ − m 2 e m 2 L ∗ (17.4) where m 1 and m 2 = Pe ± Pe 2 + 4Bi 2 (17.5) . E. J. (1979). Heat Pipe Design Handbook, NASA Contract Report NAS5-23406. Busse, C. A. (1973). Theory of the Ultimate Heat Transfer of Cylindrical Heat Pipes, Int. J. Heat Mass Transfer, 16,. Change-of-Phase Heat Transfer, Proc. 33rd National Heat Transfer Conference, Albuquerque, NM, Aug. 15–17. BOOKCOMP, Inc. — John Wiley & Sons / Page 1230 / 2nd Proofs / Heat Transfer Handbook /. a known surface heat transfer coefficient and/or radiative environment or BOOKCOMP, Inc. — John Wiley & Sons / Page 1235 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER TO MOVING