BOOKCOMP, Inc. — John Wiley & Sons / Page 1207 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSPORT LIMITATIONS 1207 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 [1207], (27) Lines: 766 to 791 ——— 1.02614pt PgVar ——— Normal Page * PgEnds: Eject [1207], (27) inertial vapor effects. Curve A represents the temperature distribution typical of a heat pipe with subsonic flow conditions and partial pressure recovery (some inertial effects are present in the vapor flow). Increasing the heat rejection rate and lowering the condenser temperature will decrease the evaporator temperature as shown in curve B. Continued increases in the heat flux and reductions in the condenser temperature will result in a decrease in the overall average vapor temperature. However, eventually the vapor velocity at the condenser inlet approaches the sonic velocity and a critical or choked flow condition exists. For this situation, continued reductions in the condenser temperature only serve to decrease the temperature in the condenser region and have no effect on the vapor temperature in the evaporator. Unlike the heat transport limits discussed previously, the sonic limitation actually serves as an upper bound to the axial heat transport capacity and does not neces- sarily result in dryout of the evaporator wick or total heat pipe failure. Attempts to exceed the sonic limit result in increasing both the evaporator temperature and the axial temperature gradient along the heat pipe, thus reducing further the isothermal characteristics typically found in the vapor flow region. Levy (1968) developed a closed-form expression for the sonic limit derived from one-dimensional vapor flow theory. This analysis assumed that the frictional effects may be neglected; thus inertial effects dominate, and the vapor behaves as a perfect gas. Combining these assumptions with the energy and momentum equations results in expressions for the temperature and pressure ratios. Substituting the local Mach number and relating the axial heat flux to the density and velocity, the relationship between the static and stagnation temperatures and pressure can be rewritten as T o T v = 1 + γ v − 1 2 Ma 2 v (16.42) P o P v = 1 +γ v Ma 2 v (16.43) where the subscripts o and v indicate the stagnation and static states of the vapor, respectively. Combining the temperature and pressure ratios with the ideal gas law yields an expression for the density ratio, which when combined with the two rela- tionships, yields an equation for the axial heat flux in terms of the physical properties, geometrical dimensions, and Mach number. For choked flow, the Mach number will equal unity, which yields an expression for the maximum axial heat transport: Q s = A v ρ o λ γ v R v T o 2(γ v + 1) 1/2 (16.44) Busse (1973) presented an alternative approach by assuming that only inertial effects are present in one-dimensional flow. In this case the momentum equation yields dP dx =− d dx ρv 2 (16.45) BOOKCOMP, Inc. — John Wiley & Sons / Page 1208 / 2nd Proofs / Heat Transfer Handbook / Bejan 1208 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 [1208], (28) Lines: 791 to 822 ——— 1.07602pt PgVar ——— Short Page PgEnds: T E X [1208], (28) Integration of this expression, combining it with the continuity equation, and assum- ing that the vapor behaves as an ideal gas yields an expression for the maximum heat transport capacity as a function of the thermophysical and geometric properties Q = λ ρ v P v A 1/2 P P v 1 − P P v 1/2 (16.46) A determination of the point where the first derivative, dQ/dP , vanishes yields a relationship for the sonic limit Q s = 0.474λA v (ρ v P v ) 1/2 (16.47) where ρ v and P v are the vapor density and pressure at the evaporator exit. The greatest difficulty in determining the sonic limit is determining these two quantities along with the inlet pressure to the condenser. In addition, fractional effects have been included by Levy and Chou (1973) to improve correlation with experimental data. Other experimental and theoretical investigations of the sonic limitation have been performed by Kemme (1969) and Deverall et al. (1970). Several attempts to describe the sonic limit from solutions to the two-dimensional Navier–Stokes equations have been developed. Bankston and Smith (1971) and Ro- hani and Tien (1974) all used numerical methods. The former study indicated that axial flow reversal occurred for high condensation rates at the end of the condenser. Comparison with the predicted results of a one-dimensional model developed by Busse (1973) indicated good agreement for high condensation rates in the condenser region despite this flow reversal (Rohani and Tien, 1974). It is interesting to compare the results for the viscous limit and sonic limit where a relationship between the two exist with respect to the quantity P v ρ v . Inertial effects are found to vary with the product (P v ρ v ) 1/2 , while the viscous effects vary linearly with respect to P v ρ v . As a result, when this product is small, the transport capacity is typically limited by viscous effects but with increasing P v ρ v , inertial effects begin to dominate and a transition occurs from the viscous to the sonic limit. The boundary be- tween these two limits can be determined by setting these two equations equal to each other and solving for the combined terms as a function of temperature (Ivanovskii et al., 1982; Busse, 1973). The results indicate that the transition temperature is depen- dent on the thermophysical properties of the working fluid, the geometry of the heat pipe, and the length of the evaporator and condenser regions. Experimental work by Vinz and Busse (1973) verified that this transition compared quite well with predicted values. 16.2.7 Condenser Limit The heat transfer rate in the condenser section is governed by the coupling of the condenser with the system heat sink. At steady state, the heat rejection rate in the condenser must equal the heat addition rate in the evaporator. Typically, the condenser BOOKCOMP, Inc. — John Wiley & Sons / Page 1209 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT PIPE THERMAL RESISTANCE 1209 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 [1209], (29) Lines: 822 to 852 ——— 8.504pt PgVar ——— Short Page PgEnds: T E X [1209], (29) coupling is either by convection and/or radiation. In this case the heat transfer from the condenser, assuming a constant condenser temperature, is determined from Q c = Q conv + Q rad = hA(T c − T ∞ ) + εAσ T 4 c − T 4 surr (16.48) where h is the convective heat transfer coefficient, A the condenser surface area, and the emissivity of the surface. If either convective or radiation effects are negligible, the first or second term, respectively, in eq. (2.6.1) is neglected. Examples of reaching the condenser limit can be low convective heat transfer coefficients (e.g., natural con- vection), low surface emissivity, or limited surface area. In these cases, increased heat addition to the heat pipe results in an increased temperature of the heat pipe because an increased temperature difference between the heat pipe condenser and heat sink is required. Methods to improve the condenser limit may include augmentation of the heat transfer coefficient (e.g., forced convection), increasing the surface emissivity (e.g., surface coatings), and increasing the condenser surface area (e.g., fins). For cases where the condenser surface temperature is not uniform, integration of the dif- ferential heat transfer along the condenser surface is required. Moreover, if the heat pipe condenser is in an enclosure, radiation view factors between the heat pipe and surroundings need to be determined. For both of the latter two cases, information can be used from any standard heat transfer textbook (e.g., Incropera and DeWitt, 2002). 16.3 HEAT PIPE THERMAL RESISTANCE The overall temperature difference between the heat sink and the heat source is an important characteristic for thermal control systems utilizing heat pipes. As the heat pipe is typically referred to as an overall structure of very high effective thermal con- ductivity, an electrical resistance analogy similar to that found in conduction heat transfer analysis is used. As the heat transfer occurs from the heat source to the heat sink, each part of the heat pipe can be separated into an individual thermal resistance. The combined resistances provide a mechanism to model the overall thermal resis- tance and the temperature drop from heat sink to heat source associated with the given heat input. In addition, the resistance analogy provides a means to estimate the mean operating temperature (adiabatic vapor temperature) that is typically needed in deter- mining the transport limit at a given operating condition. The temperature gradient is found utilizing a thermal resistance network, where Fig. 16.13 illustrates the analogy for a simple cylindrical heat pipe. The overall thermal resistance of the heat pipe only is comprised typically of nine resistances arranged in a series–parallel combination. The resistances can be summarized along with estimates of typical magnitudes as follows: 1. R p,e : pipe wall radial resistance, evaporator (∼ 10 −1 °C/W) 2. R w,e : saturated liquid/wick radial resistance, evaporator (∼ 10 +1 °C/W) BOOKCOMP, Inc. — John Wiley & Sons / Page 1210 / 2nd Proofs / Heat Transfer Handbook / Bejan 1210 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 [1210], (30) Lines: 852 to 880 ——— 0.01103pt PgVar ——— Short Page * PgEnds: Eject [1210], (30) Vapor space Wick Wall Heat source T 1 T 2 Heat sink R o,e R p,c R p,a R w,e R w,c R w,a R i,e R i,c R v,a R ext,e R ext,c Figure 16.13 Equivalent thermal resistance of a heat pipe. (From Peterson, 1994, with per- mission.) 3. R i,e : liquid–vapor interface resistance, evaporator (∼ 10 −5 °C/W) 4. R v,a : adiabatic vapor section resistance (∼ 10 −8 °C/W) 5. R p,a : pipe wall axial resistance (∼ 10 +2 °C/W) 6. R w,a : saturated liquid–wick axial resistance (∼ 10 +4 °C/W) 7. R i,c : liquid–vapor interface resistance, condenser (∼ 10 −5 °C/W) 8. R w,c : saturated liquid–wick radial resistance, condenser (∼ 10 +1 °C/W) 9. R p,c : pipe wall radial resistance, condenser (∼ 10 −1 °C/W) By examination of the typical range of resistance values, several simplifications are possible. First, due to comparative magnitudes of the resistance of the vapor space and the axial resistances of the pipe wall and liquid–wick combinations, the axial resistance of the pipe wall, R p,a , and the liquid–wick combination, R w,a , may be treated as open circuits and neglected. Second, the liquid–vapor interface resistances and the axial vapor resistance (in most situations) can be assumed negligible. Thus, the primary resistances of the heat pipe are the pipe wall radial resistances and the liquid–wick resistances in the evaporator and condenser. The pipe wall resistances are found using for flat plates R p,e = δ k p A e (16.49) where δ is the plate thickness and A e is the evaporator surface area, and for cylindrical pipes as BOOKCOMP, Inc. — John Wiley & Sons / Page 1211 / 2nd Proofs / Heat Transfer Handbook / Bejan FIGURES OF MERIT 1211 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 [1211], (31) Lines: 880 to 901 ——— -1.94284pt PgVar ——— Short Page * PgEnds: Eject [1211], (31) R p,e = ln(d o /d i ) 2πL e k p (16.50) where L e is the evaporator length (or is replaced by the condenser length when evaluating R p,c ). The resistance of the liquid–wick combination is also found from eq. (16.50), where the effective conductivity k eff is used instead of the pipe wall value k p . Relationships for calculating k eff are given in Table 16.3. Two other resistances shown in Fig. 16.13 have a significant role in the design of heat pipe thermal control systems. These are the external resistances occurring between the heat source and heat pipe evaporator and the heat pipe sink and heat pipe condenser, R ext,e and R ext,c , respectively. The external resistances are found by using information related to contact resistances and convective resistance where informa- tion on these can be found in most heat transfer textbooks. In many applications, these two resistances combined are greater than the overall heat pipe resistance, thus, these are typically the controlling resistances in applications. One additional important observation from the resistance analogy can be made. This is the case where the heat pipe reaches a dryout condition, such as exceeding the capillary limit. In the case of dryout, the vapor flow from the evaporator to the condenser will be discontinued and the resistance R v,a will increase significantly, such that this circuit now may be considered as an open circuit. Thus, any heat input to the system must be transported along the heat pipe wall, R p,a , and the wick structure combination, R w,a . As the difference between the axial resistances is several orders of magnitude, the temperature drop along the heat pipe will correspond to an increase of several orders of magnitude. This is as expected since the heat must now be transferred by conduction instead of using the latent heat of vaporization of the working fluid. 16.4 FIGURES OF MERIT As a first step in the steady-state modeling of the transport capacity of a heat pipe, it can be useful initially to select a working fluid and/or to compare one fluid with another regardless of the heat pipe geometry. To maximize heat transport, the ideal fluid has a high surface tension to increase capillary pumping, high fluid density, and latent heat of vaporization to reduce mass flow rates (and thus frictional losses) and low viscosity. These properties are combined to form a figure of merit (FM) for a given fluid in terms of fluid properties. In conventional heat pipes, the greatest pres- sure loss is typically associated with the liquid flow in the wick structure. Equating the Young–Laplace equation for capillary pressure to the wick pressure drop predicted by Darcy’s law, Chi (1976) was able to separate the fluid terms from the geometric and produce a figure of merit (liquid based) FM = ρ l σh fg µ l (16.51) BOOKCOMP, Inc. — John Wiley & Sons / Page 1212 / 2nd Proofs / Heat Transfer Handbook / Bejan 1212 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 [1212], (32) Lines: 901 to 924 ——— 0.7831pt PgVar ——— Normal Page PgEnds: T E X [1212], (32) Figure 16.14 Heat pipe liquid-based figure of merit as a function of operating temperature for several working fluids. which has units of heat flux (W/m 2 ). Working fluid candidates with high figures of merit are considered to have better performance characteristics. The figure of merit (liquid based) as a function of operating temperature for several working fluids is shown in Fig. 16.14. For a given heat pipe geometry, this provides a good estimation of the working fluid potentially providing the greatest transport capacity at a specific temperature. In cases where the pressure drop due to flow through the wick structure is not the dominant pressure drop (e.g., the loop heat pipe, discussed later), Dunbar and Cadell (1998) developed a figure of merit under the assumption that the pressure drop in the vapor flow is the dominate pressure loss term. The analysis equated the frictional pressure drop due to turbulent flow in the vapor channel to the Young– Laplace equation for capillary pressure and separated the fluid terms from geometric terms to yield the figure of merit (vapor based) FM = ρ v σh 1.75 fg µ 0.25 v (16.52) 16.5 TRANSIENT OPERATION 16.5.1 Continuum Vapor and Liquid-Saturated Wick The most common transient occurring for a heat pipe is when the vapor is in a continuum flow condition and the wick is saturated with liquid. As the effective BOOKCOMP, Inc. — John Wiley & Sons / Page 1213 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSIENT OPERATION 1213 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 [1213], (33) Lines: 924 to 939 ——— 6.16003pt PgVar ——— Normal Page PgEnds: T E X [1213], (33) thermal conductivity of the heat pipe is very high and the overall mass is rela- tively low, a heat pipe typically has a low thermal capacitance and a low over- all thermal resistance compared to the overall thermal control systems. In these situations, the heat pipe can be modeled using a lumped capacitance method as- suming uniform heat pipe temperature at any given time. This approach may also be applicable in numerous situations, such as when free convection is used as the heat addition or heat rejection mechanism due to large convective resistances be- tween the heat pipe and the airstreams. Applying an energy balance to the entire heat pipe, the general energy equation for using a lumped capacitance approach is given by Q e (t) − Q c (t) = (ρc p ) eff V dT dt (16.53) where Q e (t) is the evaporator heat input, Q c (t) the condenser heat rejection, and (ρc p ) eff V the effective thermal mass of the heat pipe and must include the specific heats of the case, wick, and fluid. Solutions to eq. (16.53) depend directly on the boundary conditions (convection, radiation, conduction) corresponding to the heat input and heat rejection for the given system. In some cases it will be required to know more accurate axial temperatures and pressures occurring in the heat pipe; thus a more detailed spatial-temporal model may be required. For these situations, numerical techniques commonly are utilized, even for cases of one-dimensional ap- proximations. Information regarding numerical analyses of heat pipe transients can be found in Colwell and Modlin (1992), Bowman (1991), and Faghri (1995). 16.5.2 Wick Depriming and Rewetting Heat pipes utilized in spacecraft thermal management systems may be subjected to orbital attitude adjustments, reboosts, and docking maneuvers, which may result in accelerations sufficient to cause redistribution of the liquid within the heat pipes. When the magnitude, direction, and/or duration of the acceleration is sufficient, liquid will flow out of the wick and accumulate in the ends of the liquid and vapor channels opposite to the direction of acceleration. For accelerations in the direction of an evap- orator, dryout may easily occur. To predict the performance of a thermal management system following an evaporator dryout, the process of repriming/rewetting the heat pipe evaporator must be evaluated to determine the time required and conditions for the heat pipe to become operational once again. Additionally, evaporator dryout will occur if the thermal loading on a heat pipe exceeds the heat pipe transport capability. Typical scenarios where this condition may occur are excess power input in electronic chip cooling, cold plate/heat sink cooling in electronics or spacecraft, initial clamp- ing of a heat pipe to a heat source, and heat pipe heat exchangers. In the event of evaporator dryout, the power or thermal loading to the heat pipe must be decreased. Investigations examining the time required to rewet/reprime the wick structure have been conducted by Ambrose et al. (1987), Ivanovskii et al. (1982), and Ochterbeck et al. (1995). BOOKCOMP, Inc. — John Wiley & Sons / Page 1214 / 2nd Proofs / Heat Transfer Handbook / Bejan 1214 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 [1214], (34) Lines: 939 to 974 ——— -3.92393pt PgVar ——— Long Page PgEnds: T E X [1214], (34) 16.5.3 Freeze–Thaw Issues Heat pipe startup from the frozen state involves the preliminary phase-change process of melting the working fluid before the heat pipe becomes fully functional. The heat input must effectively melt the working fluid in the wick structure and allow for liquid return to the evaporator prior to evaporator dryout. The presence of all three phases of the working fluid during the melting process significantly affects the startup characteristics and complicates the investigation of frozen startup. Heat pipes experiencing frozen startup can be divided distinctly into two regimes (liquid metal and room temperature). The primary difference between startup of liquid metal heat pipes and room-temperature heat pipes is the absence of the conduc- tion-dominated regime, which exists in early periods with free molecular vapor flow conditions, common to liquid metals. In room-temperature heat pipe frozen startup, a saturated solid–vapor phase and continuum flow conditions exist; thus sublimation of the working fluid is possible, which results in heat and mass transport by the working fluid vapor in addition to heat diffusion in the solid regions. The sublimation and corresponding loss of mass in the evaporator is a direct disadvantage to the restart process for room-temperature heat pipes, since there is no mass return to the evaporator. Previous investigations of liquid metal and room-temperature heat pipes have demonstrated that prior to continuum vapor flow conditions, heat and mass transfer in the vapor channel are essentially negligible under free molecular flow regimes. Neg- ligible evaporation results from an insufficient vapor pressure P v to sustain vaporiza- tion or from the collision of vapor molecules with the physical boundaries instead of other vapor molecules. Transition criteria, usually based on a transition temperature T ∗ , is evaluated where the mean free molecular path of the vapor molecules becomes much less than the minimum vapor space dimension. This criterion is represented mathematically by the Knudsen number Kn ≡ λ D v (16.54) where λ is the mean free molecular path. Specific ranges regarded to exist for the various flow regimes have been given as (Zucrow and Hoffman, 1976) • Continuum vapor flow: Kn < 0.01 • Free molecular flow: Kn > 3.0 • Transition region: 0.01 < Kn < 3.0 In terms of the vapor temperature and pressure, an expression for the transition temperature T ∗ is given by T ∗ = 2 π Kn 2 R P v D v µ v 2 (16.55) where R is the working fluid gas constant. The transition corresponds to the specified Knudsen number, generally 0.01, at which continuum flow conditions will exist for BOOKCOMP, Inc. — John Wiley & Sons / Page 1215 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSIENT OPERATION 1215 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 [1215], (35) Lines: 974 to 983 ——— 0.781pt PgVar ——— Long Page PgEnds: T E X [1215], (35) temperatures greater than T ∗ and for a specific vapor channel diameter D v . Equation (16.55) provides a relationship for the transition temperature expressed in terms of the fluid properties and the vapor channel diameter. However, there is a difficulty in evaluating eq. (16.55) because of the temperature dependence of the vapor viscosity; thus an iterative solution using individual fluid properties is required. Results of the transition temperature versus vapor channel diameter are shown in Fig. 16.15. For room-temperature working fluids (water and ammonia), continuum flow con- ditions exist for temperatures below the melting point, T mp , where the fluid exists in a saturated solid–vapor phase, while continuum flow conditions do not exist until approximately 2T mp for liquid metal working fluids. In this case, due to a saturated solid–vapor phase in the frozen state, and due to the existence of continuum flow conditions, sublimation of the working fluid is therefore possible. This results in heat and mass transport by the working fluid vapor in addition to heat diffusion in the solid regions. Because the free molecular regime is no longer present, the condition of melting a significant portion of the working fluid by conduction prior to large- scale vaporization is not found in room-temperature heat pipes. The sublimation and corresponding loss of mass in the evaporator is a direct disadvantage to the restart process for room-temperature heat pipes. Thus, the entire thermal history of room- temperature heat pipes that may undergo freeze-thaw cycles is important. Although frozen startup is always required at least once for liquid metal heat pipes, conditions occur for other working fluids where the heat pipe or heat sink is at sub- freezing temperatures. In liquid metal heat pipe startup, it is often possible to melt a significant amount of the working fluid before evaporation begins. This is not the 10 Ϫ2 10 Ϫ1 10 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 2223334445566 Vapor Channel Diameter, (cm)D TT * / mp Sodium ( = 370.5 K)T mp Lithium ( = 459 K)T mp Methanol ( = 175.5 K)T mp Water ( = 273 K)T mp Ammonia ( = 195.5 K)T mp Figure 16.15 Transition temperature as a function of vapor channel diameter for several fluids. BOOKCOMP, Inc. — John Wiley & Sons / Page 1216 / 2nd Proofs / Heat Transfer Handbook / Bejan 1216 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 [1216], (36) Lines: 983 to 993 ——— 0.0pt PgVar ——— Normal Page PgEnds: T E X [1216], (36) case with room-temperature heat pipes, especially those without a noncondensable gas charge. For room-temperature heat pipes, the frozen-state temperature is gener- ally not far below that of the triple point. For heat pipe fluids such as water, the vapor pressure may be great enough to allow migration of the working fluid toward the colder regions of the heat pipe (condenser). Also, relatively higher vapor pressures at the melting point can lead to significant evaporation immediately after the working fluid has melted, for which the condenser may remain in a frozen state, thus hindering the return of sufficient liquid. In the analysis of frozen startup for room-temperature heat pipes (122 to 630 K) (Ochterbeck, 1997), all reported experiments, not including noncondensible gas charging, have resulted in evaporator dryout. Although several tested frozen heat pipes have been restarted successfully, these instances of startup have involved evap- orator dryout followed by evaporator rewetting. Increases in the noncondensible gas level resulted in aiding restart by effectively blocking the sublimation–migration pro- cess of working fluid, and the heat pipe restarted in a gas-controlled mode. During startup of a gas-loaded liquid metal heat pipe, the noncondensible gas initially occupies the entire vapor space under frozen conditions, with this condition prevailing until the vapor pressure P(T e ) is equivalent in magnitude to the initial gas pressure, P i . Also, this condition of P v (T e ) = P i typically occurs at temperatures well above the working fluid melting point. The movement of the vapor–gas front and melt front essentially propagate at the same rate; thus freezeout of the working fluid is hindered by the noncondensible gas. However, in a room-temperature heat pipe, the vapor–gas front and melt front do not necessarily coincide. Increases in the evaporator temperature correspond to an increased evaporator vapor pressure, which drives the movement of the vapor–gas front, even with a frozen working fluid. The initiation of a melt front within the wicking structure does not occur until the evaporator temperature is greater than the working fluid triple state (Faghri, 1995; Ivanovskii et al., 1982). 16.5.4 Supercritical Startup In heat pipes for cryogenic applications, a gaseous or supercritical state prevails for the working fluids (e.g., hydrogen, nitrogen, oxygen) at room-temperature conditions. During startup, the heat pipe must be cooled until the fluid temperature is below the critical point, requiring condensation of the working fluid and wetting of the wicking structure before the heat pipe becomes operational (Chang and Colwell, 1985). The investigation by Chang and Colwell (1985) of supercritical startup in cryogenic heat pipes utilized a finite-difference numerical analysis and gave good insight into the problem. However, as finite-difference techniques may be difficult to utilize, an analytical model was presented by Yan and Ochterbeck (1999). The basics of this model utilized two underlying assumptions in the analysis: that the condenser temperature is uniform and that condensation within the heat pipe begins at T = T c . For the system, the analysis is separated into two regions. First, the condenser temperature remains above the critical point temperature, T cond >T cr , such that condensation within the heat pipe, and phase-change heat transfer, do not yet . analogy similar to that found in conduction heat transfer analysis is used. As the heat transfer occurs from the heat source to the heat sink, each part of the heat pipe can be separated into an individual. the heat pipe because an increased temperature difference between the heat pipe condenser and heat sink is required. Methods to improve the condenser limit may include augmentation of the heat transfer. from any standard heat transfer textbook (e.g., Incropera and DeWitt, 2002). 16.3 HEAT PIPE THERMAL RESISTANCE The overall temperature difference between the heat sink and the heat source is an important