BOOKCOMP, Inc. — John Wiley & Sons / Page 1187 / 2nd Proofs / Heat Transfer Handbook / Bejan INTRODUCTION 1187 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 [1187], (7) Lines: 173 to 215 ——— -2.03pt PgVar ——— Long Page PgEnds: T E X [1187], (7) The axially grooved wick shown in Fig. 16.2c possesses highly conductive metal paths for the minimization of radial temperature drop. Axially grooved heat pipes are most commonly found in space applications. The annular and crescent wicks, shown respectively in Fig. 16.2d and e, have small resistance to liquid flow but are vulnerable to liquids of low thermal conductivity. The artery wick, shown in Fig. 16.2f was developed to reduce the thickness of the radial heat flow path through the structure and to provide a low-resistance path for the liquid flow from the condenser to the evaporator. However, these wicks often lead to operating problems if they are not self-priming, because the arteries must fill automatically at startup or after a dryout. All the composite wicks shown in Fig. 16.3 have a separate structure for develop- ment of the capillary pressure and liquid flow. Notice that in some of the structures in Fig. 16.3, a separation of the heat flow path from the liquid flow path can be pro- vided. For example, the screen-covered groove wick shown in Fig. 16.3b has a fine mesh screen for high capillary pressure, axial grooves to reduce flow resistance, and a metal structure to reduce the radial temperature drop. The slab wick displayed in Fig. 16.3c is inserted into an internally threaded container. High capillary pressure is derived from a layer of fine mesh screen at the surface, and liquid flow is assured by the coarse screen inside the slab. The threaded grooves tend to provide uniform circumferential distribution of liquid and enhance radial heat transfer. 16.1.3 Classification by Type of Control In addition to classification by the temperature range of the working fluid, heat pipes may be classified by the type of control employed. Control is often necessary because a heat pipe without control will self-adjust its operating temperature in accordance with the heat source at the evaporator end and the heat sink at the condenser end. For example, it may be desirable to control the temperature in the range prescribed in the presence of a wide range of variations in heat source and heat sink temperatures. On the other hand, it may be required to permit the passage of heat under one set of conditions and block the heat flow completely under another set of conditions. This leads to a consideration of the performance of heat pipes known as thermal switches and thermal diodes. There are four major control approaches that are illustrated in Fig. 16.4 and de- scribed in what follows. 1. Gas-loaded heat pipe. The presence of a noncondensible gas has a marked effect on the performance of a condenser. This effect can be exploited for heat pipe control. Any noncondensible gas present in the vapor space is swept to the condenser section during operation, and gas will block a portion of the condenser surface. The heat flow at the condenser can be controlled by controlling the volume of the noncondensible gas. Examples of self-controlled devices, those that can be controlled by the vapor pressure of the working fluid, are shown in Fig. 16.4a, b, and c. Examples of feedback-controlled devices are shown Fig. 16.4d and e. 2. Excess-liquid heat pipe. Control can also be attained by condenser flooding with excess working fluid. In the excess-liquid heat pipe, excess working fluid in BOOKCOMP, Inc. — John Wiley & Sons / Page 1188 / 2nd Proofs / Heat Transfer Handbook / Bejan 1188 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 [1188], (8) Lines: 215 to 215 ——— * 70.927pt PgVar ——— Normal Page PgEnds: T E X [1188], (8) Q Q ()a ()b ()c ()d ()e Evaporator Adiabatic section Condenser Gas reservoir T Control fluid Heater controller Heater T Figure 16.4 Representive gas-loaded heat pipes. (From Chi, 1976, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 1189 / 2nd Proofs / Heat Transfer Handbook / Bejan INTRODUCTION 1189 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 [1189], (9) Lines: 215 to 219 ——— -2.903pt PgVar ——— Normal Page PgEnds: T E X [1189], (9) Figure 16.5 Representative excess liquid heat pipes. (From Chi, 1976, with permission.) the liquid phase is swept into the condenser and blocks a portion of the condenser. Observe in Fig. 16.5a that a decrease in vapor temperature will expand the control fluid in the bellows, which forces excess liquid to block a portion of the condenser. An example of a thermal diode is displayed in Fig. 16.5b. 3. Vapor flow–modulated heat pipe. The performance of the heat pipe can be controlled by the vapor flow through the adiabatic section as shown in Fig. 16.6a,an increase in heat input or an increase in heatsourcetemperaturefelt at the surface of the evaporator causes a rise in the temperature and pressure of the vapor in the evaporator section. The flow of this vapor through the throttling valve creates a temperature and pressure drop that results in a reduction in the magnitudes of these quantities in the condenser section. Thus, the condensing temperature and pressure can be held at values that yield the required condenser performance even though the temperature at the heat source has increased. In the event that the heat input increases, the condenser can keep pace with this increase and adjust its performance by means of the throttling valve. Figure 16.6b shows a thermal switch where the flow of vapor through the throttling valve can be cut off entirely. 4. Liquid flow–modulated heat pipe. Liquid flow control is also an effective way of maintaining control over heat pipe performance. One way of controlling liquid flow is through the use of a liquid trap, as shown in Fig. 16.7a. This trap is a wick-lined BOOKCOMP, Inc. — John Wiley & Sons / Page 1190 / 2nd Proofs / Heat Transfer Handbook / Bejan 1190 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 [1190], (10) Lines: 219 to 219 ——— * 40.50198pt PgVar ——— Normal Page PgEnds: T E X [1190], (10) Figure 16.6 Representative vapor flow-modulated heat pipes. (From Chi, 1976, with per- mission.) Figure 16.7 Representative liquid flow-modulated heat pipes. (From Chi, 1976, with per- mission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 1191 / 2nd Proofs / Heat Transfer Handbook / Bejan INTRODUCTION 1191 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 [1191], (11) Lines: 219 to 251 ——— 0.89607pt PgVar ——— Normal Page PgEnds: T E X [1191], (11) reservoir located in the evaporator end. The wick in the trap, referred to as the trap wick, is not connected to the operating wick in the rest of the heat pipe. In the normal mode of operation with the heat pipe operating in the standard fashion, the trap wick is dry. If the heat input increases or the attitude of the heat pipe changes, condensation may occur in the trap and the liquid trap may become an alternate condensing end of the pipe. As liquid accumulates in the trap, the main wick begins dryout which results in operational failure. An example of a heat pipe with the evaporator section below the condenser section is shown in Fig. 16.7b. This, in itself, is a type of control because the heat pipe can function as a thermal diode providing that the wick is designed appropriately. Notice that the condensed liquid is returned to the evaporator section with the assistance of the gravitational force. This type of heat pipe is commonly referred to as a thermosyphon. 16.1.4 Capillary Action In capillary-driven systems, the driving potential for the working fluid circulation is provided by the difference in the curvature of the evaporating and condensing liquid–vapor interfaces. Consequently, determining the maximum pumping capacity, and the corresponding heat transfer performance, of these systems depends strongly on the accuracy of the prediction of the shapes of the evaporating and condensing interfaces. On a microscopic scale, a liquid–vapor interface is a volumetric transition zone across which the molecule number density varies continuously. However, on a macroscopic scale, an interface between a liquid and its vapor is modeled as a surface of discontinuity and characterized by the property of surface tension. The surface tension is defined thermodynamically to be the change in surface excess free energy (or work required) per unit increase in interfacial area σ = ∂E ∂A s T,n (16.1) As the capillary pressure at the liquid–vapor interface is due to the curvature of the menisci and the surface tension of the working fluid and is given by the Young– Laplace equation (see Carey, 1992 for a detailed derivation) ∆P c = σ 1 R 1 + 1 R 2 (16.2) where σ is the surface tension and R 1 and R 2 are the principal radii of the meniscus, as shown in Fig. 16.8. Limitations to use of the Young–Laplace equation are typically that the liquid–vapor interface is static, interfacial mass fluxes (evaporation) are low, and disjoining pressure effects are negligible. For cases of very thin films where dis- joining pressure effects must be included to provide a physically correct and accurate prediction of the capillary pressure across an interface, a review of techniques has been provided by Wayner (1999). In predicting the maximum capillary pressure available for a given heat pipe wick structure, the two principal radii of curvature are typically combined into an effective BOOKCOMP, Inc. — John Wiley & Sons / Page 1192 / 2nd Proofs / Heat Transfer Handbook / Bejan 1192 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 [1192], (12) Lines: 251 to 279 ——— 0.5161pt PgVar ——— Normal Page PgEnds: T E X [1192], (12) Figure 16.8 The radii of the meniscus. radius of curvature for the wick structure. This method produces an effective capillary radius equivalent to the inner radius of a circular tube (R 1 = R 2 = r eff / cos θ) and allows for easy comparison between capillary structures with different structures. In this case, the capillary pressure is expressed as ∆P c = 2σ r eff cos θ (16.3) where θ, the apparent contact angle (Fig. 16.9), is dependent on the fluid–wick pair used. The contact angle is a measure of the degree of wettability of the liquid on the wick structure, where θ = 0° relates to a perfectly wetting system. Carey (1992) provides a detailed discussion on parameters affecting wettability. For this expression to be maximized, the wetting angle must be zero (i.e., the liquid wets the wick perfectly). Thus, the maximum capillary pressure with a perfectly wetting fluid will be (∆P c ) max = 2σ r eff (16.4) where r eff is the effective pores radius of the wick and can be determined for various wick structures. The difference in the curvature of the menisci between the evaporator and the condenser section implies a difference in the capillary pressure at the interface along the length of the heat pipe. The capillary pressure developed by the wick between points 1 and 2 can be expressed as (∆P c ) 1→2 = ∆P c,1 − ∆P c,2 (16.5) BOOKCOMP, Inc. — John Wiley & Sons / Page 1193 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSPORT LIMITATIONS 1193 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 [1193], (13) Lines: 279 to 299 ——— 0.78102pt PgVar ——— Normal Page PgEnds: T E X [1193], (13) Figure 16.9 Meniscus in a cylindrical pore. The maximum capillary pressure developed by the wick between the wet point (de- fined as the point where the meniscus is flat) and the dry point (defined as the point where the curvature of the menisci is maximum) is then (∆P c ) max = 2σ r eff (16.6) This capillary pressure differential circulates the fluid against the liquid and vapor pressure losses and any adverse body forces such as gravity. 16.2 TRANSPORT LIMITATIONS 16.2.1 Introduction Limitations of the maximum heat input that may be transported by a heat pipe can be divided into two primary categories: limits that result in heat pipe failure and limits that do not. For the limitations resulting in heat pipe failure, all are characterized by insufficient liquid flow to the evaporator for a given heat input, thus resulting in dryout of the evaporator wick structure. The heat input to the heat pipe, Q, is related directly to the mass flow rate of the working fluid being circulated and the latent heat, h fg , of the fluid as the heat input is the driving mechanism, or Q =˙mh fg (16.7) BOOKCOMP, Inc. — John Wiley & Sons / Page 1194 / 2nd Proofs / Heat Transfer Handbook / Bejan 1194 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 [1194], (14) Lines: 299 to 340 ——— -1.18001pt PgVar ——— Long Page PgEnds: T E X [1194], (14) However, limitations not resulting in heat pipe failure do require that the heat pipe operate at an increased temperature for an increase in heat input. The two categories and basic phenomena for each limit may be summarized as follows: Limitations (Failure) 1. Capillary limit. The capillary limit relates to the fundamental phenomenon governing heat pipe operation which is development of capillary pressure differences across the liquid–vapor interfaces in the evaporator and condenser. When the driving capillary pressure is insufficient to provide adequate liquid flow from the condenser to the evaporator, dryout of the evaporator wick will occur. Generally, the capillary limit is the primary maximum heat transport limitation of a heat pipe. 2. Boiling limit. The boiling limit occurs when the applied evaporator heat flux is sufficient to cause nucleate boiling in the evaporator wick. This creates vapor bubbles that partially block the liquid return and can lead to evaporator wick dryout. The boiling limit is sometimes referred to as the heat flux limit. 3. Entrainment limit. The entrainment limit refers to the case of high shear forces developed as the vapor passes in the counterflow direction over the liquid saturated wick, where the liquid may be entrained by the vapor and returned to the condenser. This results in insufficient liquid flow to the wick structure. Limitations (Nonfailure): 1. Viscous limit. The viscous limit occurs at low operating temperatures, where the saturation vapor pressure may be of the same order of magnitude as the pressure drop required to drive the vapor flow in the heat pipe. This results in an insufficient pressure available to drive the vapor. The viscous limit is sometimes called the vapor pressure limit. 2. Sonic limit. The sonic limit is due to the fact that at low vapor densities, the corresponding mass flow rate in the heat pipe may result in very high vapor velocities, and the occurrence of choked flow in the vapor passage may be possible. 3. Condenser limit. The condenser limit is based on cooling limitations such as radiation or natural convection at the condenser. For example, in the case of radiative cooling, the heat pipe transport may be governed by the condenser surface area, emmissity, and operating temperature. Additionally, the capillary, viscous, entrainment, and sonic limits are axial heat flux limits, that is, functions of the axial heat transport capacity along the heat pipe. However, the boiling limit is a radial heat flux limit occurring in the evaporator. Using the analysis techniques for each limitation independently, the heat transport capacity as a function of the mean operating temperature (the adiabatic vapor tem- perature) can be determined. This procedure yields a heat pipe performance region similar to that shown in Fig. 16.10. As shown, the separate performance limits define an operational range represented by the region bounded by the combination of the in- dividual limits. In effect, this operational range defines the region or combination of temperatures and maximum transport capacities at which the heat pipe will function. Thus, it is possible to ensure that the heat pipe can transport the required thermal BOOKCOMP, Inc. — John Wiley & Sons / Page 1195 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSPORT LIMITATIONS 1195 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 [1195], (15) Lines: 340 to 359 ——— 6.097pt PgVar ——— Long Page PgEnds: T E X [1195], (15) Figure 16.10 Typical heat pipe performance map. load or to improve the design. It is important to note that in the determination of the heat transport capacity, the mean operating temperature must be identified. However, the operating temperature of a standard heat pipe is a function of the heat input, thus resulting in a mutually dependent case between the heat transport and the operating temperature. In Section 16.3 a method is described by which the operating temper- ature can be estimated based on the heat pipe characteristics, the heat input, and the condenser cooling conditions. 16.2.2 Capillary Limit Because the driving potential for the circulation of the working fluid is the capillary pressure difference, the maximum capillary pressure must be greater than the sum of all pressure losses inside the heat pipe: (∆P c ) max ≥ ∆P tot (16.8) The pressure losses in heat pipes can be separated into the frictional pressure drops along the vapor and liquid paths, the pressure drop in liquid as a result of body forces (e.g., gravity, centrifugal, electromagnetic), and the pressure drop due to phase transition ∆P tot = ∆P ν + ∆P l + ∆P b + ∆P ph (16.9) During heat pipe operation, the menisci naturally adjust the radii of curvature for the capillary pressure differential to balance the pressure losses ∆P tot . However, the maximum radius of curvature is limited to the capillary dimension of the wick structure such that the maximum transport occurs when (∆P c ) max = ∆P tot .Itis BOOKCOMP, Inc. — John Wiley & Sons / Page 1196 / 2nd Proofs / Heat Transfer Handbook / Bejan 1196 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 [1196], (16) Lines: 359 to 410 ——— 0.927pt PgVar ——— Normal Page PgEnds: T E X [1196], (16) important to note that the pressure drop associated with phase transition, ∆P ph ,is significant only under very high condensation or evaporation rates and represents the jump condition associated with the kinetic energy change in the liquid–vapor process. Except for very specific conditions (e.g., liquid metal heat pipes with extremely high evaporation rates), the phase transition pressure drop is typically negligible and will not be considered further in following discussions. However, for further information, Ivanovskii et al. (1982) and Delhaye (1981) provide more details related to the phase transition condition. Figure 16.11 shows a pressure-drop diagram along the length of a heat pipe work- ing under low heat flux. If the total pressure drop exceeds the maximum capillary pressure, the return rate of liquid to the evaporator will be insufficient and the heat pipe will experience dryout of the wick. The maximum capillary pressure ∆P c devel- oped within the heat pipe wick structure is given by the Laplace–Young equation of eq. (16.2). Values of the effective capillary radius r eff for different wick structures are provided in Table 16.2. The body forces result from any gravitational field against which the liquid must be pumped. This includes any inclination of the heat pipe ∆P = ρ l gL sin φ (16.10) as well as any hydrostatic pressure drop related to the drawing of the fluid to the top portion of the heat pipe cross section ∆P ⊥ = ρ l gd ν cos φ (16.11) It is important to note that the inclination of the heat pipe can either be an adverse tilt (evaporator above condenser) or a favorable tilt (condenser above evaporator) such that the hydrostatic pressure either subtracts from, or adds to, the capillary pumping pressure. In cases where the liquid flow to the evaporator becomes dominated by gravitational forces, the system is operating as a thermosyphon as opposed to a traditional heat pipe. For basics of a thermosyphon, Faghri (1995) may be consulted. Figure 16.11 Pressure variation along the length of a heat pipe working under low heat flux. . Adiabatic section Condenser Gas reservoir T Control fluid Heater controller Heater T Figure 16.4 Representive gas-loaded heat pipes. (From Chi, 1976, with permission.) BOOKCOMP, Inc. — John Wiley & Sons / Page 1189 / 2nd Proofs / Heat Transfer Handbook. the latent heat, h fg , of the fluid as the heat input is the driving mechanism, or Q =˙mh fg (16.7) BOOKCOMP, Inc. — John Wiley & Sons / Page 1194 / 2nd Proofs / Heat Transfer Handbook /. heat pipe will function. Thus, it is possible to ensure that the heat pipe can transport the required thermal BOOKCOMP, Inc. — John Wiley & Sons / Page 1195 / 2nd Proofs / Heat Transfer Handbook