BOOKCOMP, Inc. — John Wiley & Sons / Page 1197 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSPORT LIMITATIONS 1197 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 [1197], (17) Lines: 410 to 475 ——— 2.23228pt PgVar ——— Normal Page * PgEnds: Eject [1197], (17) TABLE 16.2 Expressions for the Effective Capillary Radius r c for Several Wick Structures Structure r c Data Circular cylinder (artery or tunnel wicks) r eff = rr= radius of liquid flow passage Rectangular groove r eff = ww= groove width Triangular groove r eff = w cos β w = groove width β = half-included angle Parallel wires r eff = ww= wire spacing Wire screens r eff = w + d w 2 = 1 2N N = screen mesh number w = wire spacing d w = wire diameter Packed spheres r c = 0.41r s r s = sphere radius Source: Chi (1976), with permission. The differential liquid pressure drop in the wick structure assuming one-dimen- sional laminar flow can be expressed as dP l dx =− µ l ˙m l (x) KA w ρ l (16.12) where K represents the wick permeability. The wick permeability is related directly to the porosity of the wick structure, which is defined as the ratio of pore volume to total volume, or ε = V pore /V tot , and is given by K = 2ε(r h ) 2 f l · Re l (16.13) As the hydraulic radius of the porous structure is typically small and the liquid flow velocity is low, the liquid flow can be assumed laminar. Thus, the values of (f l ·Re l ) can be assumed constant and depend only on the flow passage shape, where typical values of the permeability for different wick structures are given in Table 16.3. Assuming one-dimensional vapor flow, the differential vapor pressure drop can be expressed in terms of the pressure drops due to frictional forces and dynamic pressure, or dP v dx = (f · Reµ v ˙m v (x) 2A v r 2 h,v ρ v − 2 ˙m v A 2 v ρ v d ˙m v (x) dx (16.14) Recognizing that the total vapor mass flow rate and liquid mass flow rate are equal at steady-state conditions, the mass flow rate through the system can be expressed in terms of the heat transport rate and the latent heat of vaporization of the fluid, or BOOKCOMP, Inc. — John Wiley & Sons / Page 1198 / 2nd Proofs / Heat Transfer Handbook / Bejan 1198 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 [1198], (18) Lines: 475 to 475 ——— 0.8034pt PgVar ——— Normal Page PgEnds: T E X [1198], (18) TABLE 16.3 Expressions of Wick Permeability K for Several Wick Structures Wick Structure K Expression Circular artery K = r 2 8 Open rectangular grooves K = 2r 2 h,l f l · Re l = porosity = w s s = groove pitch r h,l = 2wδ w + 2δ w = groove width δ = groove depth (f l · Re l ) from (a) below Circular annular wick K = 2r 2 h,l f l · Re l r h,l = r 1 − r 2 (f l · Re l ) from (b) below Wrapped screen wick K = d 2 3 122(1 −) 2 d = wire diameter = 1 − 1.05πNd 4 N = mesh number Packed sphere K = r 2 s 3 37.5(1 − ) 2 r s = sphere radius = porosity (value depends on packing mode) 00 0.20.2 0.40.4 0.60.6 0.80.8 1.01.0 1412 1614 1816 2018 2220 2422 2624 rr 21 /a r 2 r 1 ( Re)f . ( Re)f . ␦ w a w — ϵ ␦ ()a ()b Source: Chi (1976), with permission. BOOKCOMP, Inc. — John Wiley & Sons / Page 1199 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSPORT LIMITATIONS 1199 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 [1199], (19) Lines: 475 to 531 ——— 0.13634pt PgVar ——— Normal Page * PgEnds: Eject [1199], (19) ˙m l =˙m v = Q h fg (16.15) The effective length of the heat pipe, L eff , is used to represent the average distance that the liquid and vapor must travel along the heat pipe L eff = 1 ˙m L 0 ˙m(x) dx = 1 Q L 0 Q(x) dx (16.16) Assuming uniform evaporation and condensation in the evaporator and condenser regions, the mass flow rates in the evaporator and condenser vary linearly and the effective length of a heat pipe becomes L eff = L e 2 + L a + L c 2 (16.17) and the total liquid pressure drop can then be expressed as ∆P l = L 0 dP l dx dx = µ l QL eff KA w ρ l h fg (16.18) Assuming that the dynamic pressure drop is fully recovered in the condenser region, the vapor pressure drop will be ∆P v = L 0 dP v dx dx = (f · Re) v µ v QL eff 2A v r 2 h,v ρ v h fg (16.19) For cases where compressibility effects must be included and the dynamic pressure is not fully recovered, see Busse (1973) or Ivanovskii et al. (1982). Due to low liquid velocities and the small characteristic dimensions of the wick structure, the liquid flow is always generally assumed to be laminar. However, the vapor flow velocities may be sufficient to correspond to turbulent flow. In this case, the heat transfer rate Q and the Reynolds number Re are related, and the term (f ·Re) v must be evaluated from a friction factor correlation for turbulent flow. To determine the friction factor f v , the vapor flow regime must be evaluated. Expressing the Reynolds number in terms of the heat input Q, the flow regime is determined from Re v = 2r v Q A v µ v h fg (16.20) For laminar flow (Re < 2300) in a circular cross section, the term (f · Re) v is a constant f v · Re v = 16 (16.21) while for turbulent flow (Re > 2300) in a circular cross section the Blasius correla- tion can be used BOOKCOMP, Inc. — John Wiley & Sons / Page 1200 / 2nd Proofs / Heat Transfer Handbook / Bejan 1200 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 [1200], (20) Lines: 531 to 577 ——— 0.08427pt PgVar ——— Long Page * PgEnds: Eject [1200], (20) f v = 0.038 Re 0.25 v (16.22) Values for f ·Re for noncircular cross sections can be obtained from most convective heat transfer textbooks as a function of the cross-sectional geometry. In the case of laminar vapor flow, the substitution of the individual pressure- drop terms into the capillary limit, eq. (16.9), results in an algebraic expression that can be solved directly for Q. For turbulent vapor flow, different methods exist for solving for Q. The first method for determining the capillary limit when the vapor flow is turbulent is that of an iterative solution. This procedure begins with an initial estimation of the capillary limit where the solution first assumes laminar, incompressible vapor flow. Using these assumptions, the maximum heat transport capacity Q can be determined by substituting the values for individual pressure drops and solving for the axial heat transfer. Once this value has been obtained, the axial heat transfer can be substituted into expressions for the vapor Reynolds number to determine the accuracy of the original assumptions. Using this iterative approach, accurate values for the capillary limitation as a function of the operating temperature can be determined where the operating temperature effects the capillary limit due to the temperature dependence of the fluid properties. For a more direct solution of the capillary limit when turbulent flow is present in the vapor channel, it is possible to substitute the Blausius correlation into the vapor pressure-drop term. Then, separating all terms other than heat input Q(x) and length x into friction coefficients F v and F l , the capillary limit can be expressed as (∆P c ) max − ∆P ⊥ − ∆P = L 0 (F l + F v )Qdx (16.23) where the liquid frictional coefficient F l is given by F l = µ l KA w ρ l h fg (16.24) and the vapor frictional coefficient F v is evaluated from the expression F v = (f v · Re v )µ v 2r 2 v A v ρ v h fg (16.25) For the case of turbulent vapor flow, the vapor friction coefficient was modified by substituting eq. (16.22) for (f · Re) v which results in an expression for the vapor friction coefficient of F v = 0.038 2 µ v r 2 v A v ρ v h fg Re 0.75 v = 0.019 µ v r 2 v A v ρ v h fg 2r v Q A v µ v h fg 0.75 (16.26) Substitution of this expression and combining with those discussed previously results in a general pressure balance relationship for turbulent vapor flow which takes the form BOOKCOMP, Inc. — John Wiley & Sons / Page 1201 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSPORT LIMITATIONS 1201 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 [1201], (21) Lines: 577 to 639 ——— -0.09183pt PgVar ——— Long Page PgEnds: T E X [1201], (21) (∆P c ) max − ∆P ⊥ − ∆P = 0.019µ v r 2 v A v ρ v h fg 2r v A v µ v h fg 0.75 L 0 Q 7/4 dx +F l L 0 Qdx (16.27) This expression and a Newton–Raphson method to determine the roots from the re- sulting polynomial equation, the maximum heat transport capacity (i.e., the capillary limit) for a given heat pipe can be determined as a function of the evaporator and condenser lengths and the operating temperature. To solve for the capillary limit without iteration or numerical integration, an esti- mation of the capillary limit may be obtained where the friction factor is estimated and assumed constant for the entire operating range. Inspection of a traditional Moody (1944) friction factor diagram reveals that beyond a Reynolds number of 10 5 the fric- tion factor becomes constant as the flow enters the fully turbulent region. By assuming a friction factor for Re > 10 5 , the capillary limit results in a quadratic equation for Q and a much easier solution. This method typically produces reasonable results that tend more on the conservative side. 16.2.3 Boiling Limit At higher heat fluxes, nucleate boiling may occur in the wick structure, which may allow vapor to become trapped in the wick, thus blocking liquid return and resulting in evaporator dryout. This phenomenon, referred to as the boiling limit, differs from the other limitations discussed previously, as it depends on the radial or circumferential heat flux applied to the evaporator, as opposed to the axial heat flux or total thermal power transported by the heat pipe. Determination of the heat flux or boiling limit is based on nucleate boiling theory and is comprised of two separate phenomena: (1) bubble formation and (2) subse- quent growth or collapse of the bubbles. Bubble formation is governed by the size (and number) of nucleation sites on a solid surface and the temperature difference between the heat pipe wall and the working fluid. The temperature difference, or superheat, governs the formation of bubbles and can typically be defined in terms of the maximum heat flux as Q = k eff T w ∆T c (16.28) where k eff is the effective thermal conductivity of the liquid–wick combination. The critical superheat ∆T c is defined by Marcus (1965) as ∆T c = T sat h fg ρ v 2σ r n − (∆P c ) max (16.29) where T sat is the saturation temperature of the fluid and r n is the critical nucleation site radius, which according to Dunn and Reay (1982) ranges from 0.1 to 25.0 µm for conventional metallic heat pipe case materials. As discussed by Brennan and Kroliczek (1979), this model yields a very conservative estimate of the amount of BOOKCOMP, Inc. — John Wiley & Sons / Page 1202 / 2nd Proofs / Heat Transfer Handbook / Bejan 1202 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 [1202], (22) Lines: 639 to 654 ——— 2.83023pt PgVar ——— Normal Page PgEnds: T E X [1202], (22) TABLE 16.4 Effective Thermal Conductivity k e for Liquid-Saturated Wick Structures Wick Structure k e Expression Wick and liquid in series k e = k l k w ek w + k l (1 − ) k e = effective thermal conductivity k l = liquid thermal conductivity k w = wick material thermal conductivity = wick porosity Wick and liquid in parallel k e = ek l + k w (1 − ) Wrapped screen k e = k l [ (k l + k w ) − (1 −)(k l − k w ) ] (k l + k w ) + (1 −)(k l − k w ) Packed spheres k e = k l [ (2k l + k w ) − 2(1 −)(k l − k w ) ] (2k l + k w ) + (1 −)(k l − k w ) Rectangular grooves k e = (w f k l k w δ) + wk l (0.185w f k w + δk l ) (w + w f )(0.185w f k f + δk l ) w f = groove fin thickness w = wick thickness δ = groove depth Source: Chi (1976), with permission. superheat required for bubble formation and is true even when using the lower bound for the critical nucleation site radius. This is attributed to the absence of adsorbed gases on the surface of the nucleation sites caused by the degassing and cleaning procedures used in the preparation and charging of heat pipes. The growth or collapse of a given bubble once established on a flat or planar surface is dependent on the liquid temperature and corresponding pressure difference across the liquid–vapor interface caused by the vapor pressure and surface tension of the liquid. By performing a pressure balance on any given bubble and using the Clausius–Clapeyron equation to relate the temperature and pressure, an expression for the heat flux beyond which bubble growth will occur may be developed (Chi, 1976) and expressed as Q b = 2πL eff k eff T v A v h fg ρ v ln(r i /r v ) 2σ r n − (∆P c ) max (16.30) where r i is the inner pipe wall radius and r v is the vapor core radius. Relationships to determine the effective conductivity, k eff , of the liquid saturated wick are given in Table 16.4. 16.2.4 Entrainment Limit Examination of the basic flow conditions in a heat pipe shows that the liquid and vapor flow in opposite directions. The interaction between the countercurrent liquid and vapor flow results in viscous shear forces occurring at the liquid–vapor interface, BOOKCOMP, Inc. — John Wiley & Sons / Page 1203 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSPORT LIMITATIONS 1203 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 [1203], (23) Lines: 654 to 689 ——— 3.7872pt PgVar ——— Normal Page * PgEnds: Eject [1203], (23) which may inhibit liquid return to the evaporator. In the most severe cases, waves may form and the interfacial shear forces may become greater than the liquid surface- tension forces, resulting in liquid droplets being picked up or entrained in the vapor flow and carried to the condenser. The majority of previous work has been for thermosyphons or for gravity-assisted heat pipes. Of all the limits for heat pipes, the entrainment limit has produced one of the largest amounts of work, even though much is debated about when, and if, this limit occurs. Busse and Kemme (1980) expressed doubt as to whether entrain- ment actually occurs in a capillary-driven heat pipe because the capillary structure would probably retard the growth of any surface waves. In a majority of cases stud- ied, the wick structure of the heat pipe was flooded (i.e., excess liquid), which al- lowed entrainment to occur. Additionally, much of the work has been an adaptation to work conducted in the study of annular two-phase flow, where the onset of droplet formation, the rates of entrainment, and the contribution to momentum transfer by the entrained droplets have been investigated in much detail (Langer and Mayinger, 1979; Hewitt, 1979; Nguyen-Chi and Groll, 1981). It is important to note that for ther- mosyphons, the entrainment and flooding limitation is typically the most important factor limiting heat transport (Faghri, 1995). The most common approach to estimating the entrainment limit in heat pipies is to use a Weber number criterion. Cotter (1967) presented one of the first methods to determine the entrainment limit. This method utilized the Weber number, defined as the ratio of the viscous shear force to the forces resulting from the surface tension, or We = 2r h,w ρ v V 2 h σ (16.31) By relating the vapor velocity and the heat transport capacity to the axial heat flux as ν v = Q A v ρ v h fg (16.32) and assuming that to prevent entrainment of liquid droplets in the vapor flow, the Weber number must be less than unity, the maximum transport capacity based on entrainment can be written as Q e = A v λ σρ v 2r c 0.5 (16.33) where r c is the capillary radius of the wick structure. However, this assumption typically results in an overestimation of the entrainment limit since the axial critical wavelength may be much greater than the width of the capillary structure. In addition to the Weber number criterion, several different onset velocity criteria have been proposed for use with this expression. These include one by Busse (1973) U c = 2πσ ρ v d 1/2 (16.34) BOOKCOMP, Inc. — John Wiley & Sons / Page 1204 / 2nd Proofs / Heat Transfer Handbook / Bejan 1204 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 [1204], (24) Lines: 689 to 736 ——— 0.73114pt PgVar ——— Normal Page PgEnds: T E X [1204], (24) and another by Rice and Fulford (1987) U c = 8σ P v d 1/2 (16.35) These relations can be converted into the heat transport limitation due to entrainment by combining with the continuity equation, which yields Q e = A v ρ v λ 2πσ ρ v d 1/2 (16.36) or Q e = A v ρ v λ 8σ ρ v d 1/2 (16.37) respectively, where d is the wire spacing for screen wicks or the groove width for grooved wicks. However, as mentioned earlier, these criteria may overestimate the entrainment limit, due to problems associated with the characteristic dimensions. Tien and Chung (1979) presented correlations for vertical (gravity-assisted) heat pipes. This correlation was applied to data reported by Kemme (1976), who expanded the Weber number criterion suggested by Cotter (1967), to include the balancing force term of buoyancy, or Q e = A v λ ρ v A ∗ 2πσ ι + ρ l gD 0.5 (16.38) Prenger (1984) developed a correlation for textured wall, gravity-assisted heat pipes. A model was presented which included both liquid and vapor inertia terms. It was found that for textured wall heat pipes, the liquid inertia term was dominant because the liquid was partially shielded from the vapor flow. This fact allowed the vapor inertia term to be neglected and the model to be reduced to Q e = 2A v λ D wire D pipe ρ l σ πD pipe 0.5 (16.39) which correlated well with previous data taken by Prenger and Kemme (1981). While the model presented by Tien and Chung (1979) found that heat flux limited by entrainment or flooding (as a function of the capillary structure), Prenger (1984) found the entrainment limit to be a function of the depth of the liquid layer or flow channel. A review paper by Peterson and Bage (1991) provides a full description of work in the entrainment area as well as comparisons of available models. An interesting item in this comparison is that the models resulted in values for the heat pipe entrain- ment limit which varied by as much as a factor of 27 and demonstrate the level of unresolved issues in predicting the entrainment limit. BOOKCOMP, Inc. — John Wiley & Sons / Page 1205 / 2nd Proofs / Heat Transfer Handbook / Bejan TRANSPORT LIMITATIONS 1205 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 [1205], (25) Lines: 736 to 758 ——— 5.51013pt PgVar ——— Normal Page PgEnds: T E X [1205], (25) 16.2.5 Viscous Limit When operating at low temperatures, the available vapor (saturation) pressure in the evaporator region may be very small and be of the same magnitude as the required pressure gradient to drive the vapor from the evaporator to the condenser. In this case, the total vapor pressure will be balanced by opposing viscous forces in the vapor channel. Thus, the total vapor pressure within the vapor region may be insufficient to sustain an increased flow. This low-flow condition in the vapor region is referred to as the viscous limit. As the viscous limit occurs at very low vapor pressures, the viscous limit is most often observed in longer heat pipes when the working fluid used is near the melting temperature (or during frozen startup conditions) as the saturation pressure of the fluid is low. Busse (1973) provided an analytical investigation of the viscous limit. The model first assumed an isothermal ideal gas for the vapor and that the vapor pressure at the condenser end was equal to zero, which provides the absolute limit for the condenser pressure. Using these assumptions, a one-dimensional model of the vapor flow as- suming laminar flow conditions was developed and expressed as Q v = A v r 2 v h fg ρ v P v 16µ v L eff (16.40) where P v and ρ v are the vapor pressure and density at the evaporator end of the heat pipe. The values predicted by this expression were compared with the results of previous experimental investigations and were shown to agree well (Busse, 1973). For cases where the condenser pressure is not selected to be zero, as could be the case when the viscous limit is reached for many conditions, the following expression is used Q v = A v r 2 v h fg ρ v P v 16µ v L eff 1 − p 2 v,c P 2 v (16.41) where P v,c is the vapor pressure in the condenser. Busse (1973) noted that the viscous limit could be reached in many cases when P v,c /P v ∼ 0.3. To determine whether the viscous limit should be considered as a possible limiting condition, the vapor pressure gradient relative to the vapor pressure in the evaporator may be evaluated. In this case, when the pressure gradient is less than one-tenth of the vapor pressure, or ∆P v /P v < 0.1, the viscous limit can be assumed not to be a factor. Although this condition can be used to determine the viscous limit during normal operating conditions, during startup conditions from a cold state, the viscous limit given by Busse (1973) will probably remain the limiting condition. As noted earlier, the viscous limit does not represent a failure condition. In the case where the heat input exceeds the heat input determined from the viscous limit, this results in the heat pipe operating at a higher temperature with a corresponding increase in the saturation vapor pressure. However, this condition typically is associated with the heat pipe transitioning to being sonic limited, as discussed in the following section. BOOKCOMP, Inc. — John Wiley & Sons / Page 1206 / 2nd Proofs / Heat Transfer Handbook / Bejan 1206 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 [1206], (26) Lines: 758 to 766 ——— 1.927pt PgVar ——— Normal Page PgEnds: T E X [1206], (26) 16.2.6 Sonic Limit The sonic limit is typically experienced in liquid metal heat pipes during startup or low-temperature operation due to the associated very low vapor densities in this con- dition. This may result in choked, or sonic, vapor flow. For most heat pipes operating at room temperature or cryogenic temperatures, the sonic limit is typically not a fac- tor, except in the case of very small vapor channel diameters. With the increased va- por velocities, inertial, or dynamic, pressure effects must be included. It is important to note that in cases where inertial effect of the vapor flow are significant, the heat pipe may no longer operate in a nearly isothermal case, resulting in a significantly increased temperature gradient along the heat pipe. In cases of heat pipe operation where the inertial effects of the vapor flow must be included, an analogy between heat pipe operation and compressible flow in a converging–diverging nozzle can be made. In a converging–diverging nozzle, the mass flow rate is constant and the vapor velocity varies due tothe varying cross-sectional area. However, in heat pipes, the area is typically constant and the vapor velocity varies due to mass addition (evaporation) and mass rejection (condensation) along the heat pipe. As in nozzle flow, decreased outlet (back) pressure, or in the case of heat pipes, condenser temperatures, results in a decrease in the evaporator temperature until the sonic limit is reached. Any further increase in the heat rejection rate does not reduce the evaporator temperature or the maximum heat transfer capability but only reduces the condenser temperature due to the existence of choked flow. Figure 16.12 illustrates the relationship between the vapor temperature along a heat pipe with Figure 16.12 Temperature as a function of axial position. (From Dunn and Reay, 1982, with permission.) . in terms of the heat transport rate and the latent heat of vaporization of the fluid, or BOOKCOMP, Inc. — John Wiley & Sons / Page 1198 / 2nd Proofs / Heat Transfer Handbook / Bejan 1198 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 [1198],. the heat pipe transitioning to being sonic limited, as discussed in the following section. BOOKCOMP, Inc. — John Wiley & Sons / Page 1206 / 2nd Proofs / Heat Transfer Handbook / Bejan 1206 HEAT. maximum heat transport capacity Q can be determined by substituting the values for individual pressure drops and solving for the axial heat transfer. Once this value has been obtained, the axial heat