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 f · Reµ v
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TABLE 16.2 Expressions for the Effective Capillary Radius r cfor Several Wick Structures
Circular cylinder (artery
or tunnel wicks)
reff = r r = radius ofliquid flow passage
Rectangular groove reff = w w = groove width
Triangular groove reff = w
cosβ w = groove width
β = half-included angle
Wire screens reff = w + d w
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
whereK represents the wick permeability The wick permeability is related directly
to the porosity ofthe wick structure, which is defined as the ratio ofpore volume to total volume, orε = Vpore/Vtot, and is given by
K = f2ε(r h )2
As the hydraulic radius ofthe 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· Rel) 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
(f · Reµ v ˙m v (x)
2A v r2
h,vρv −
2˙m v
A2
vρv
d ˙m v (x)
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 ofthe heat transport rate and the latent heat ofvaporization ofthe fluid, or
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TABLE 16.3 Expressions of Wick Permeability K for Several Wick Structures
8 Open rectangular grooves K = 2
2
h,l
f l· Rel
w s
s = groove pitch
r h,l= w + 2δ2wδ
w = groove width
δ = groove depth
(f l· Rel ) from (a) below
Circular annular wick K = 2r
2
h,l
f l· Rel r h,l = r1− r2
(f l· Rel ) from (b) below
Wrapped screen wick K = d2 3
122 2 d = wire diameter
1.05πNd
4
N = mesh number
37 2 r s= sphere radius
on packing mode)
0
12
16 14
18 16
20 18
22 20
24 22
26 24
r r2 1/
a
r2r1
␦
w
a⬅ ␦— w
Source: Chi (1976), with permission.
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The effective length of the heat pipe,Leff, is used to represent the average distance that the liquid and vapor must travel along the heat pipe
Leff = 1
˙m
0
Q
0
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
Leff = L e
2 + L a+L c
and the total liquid pressure drop can then be expressed as
0
dP l
KA wρl h fg (16.18)
Assuming that the dynamic pressure drop is fully recovered in the condenser region, the vapor pressure drop will be
0
dP v
(f · Re) vµv QLeff
2A v r2
h,vρv 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 ofthe 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) vmust be evaluated from a friction factor correlation for turbulent flow
To determine the friction factorf v, the vapor flow regime must be evaluated
Expressing the Reynolds number in terms ofthe heat inputQ, the flow regime is
determined from
Rev =A2r v Q
For laminar flow (Re < 2300) in a circular cross section, the term (f · Re) v is a constant
while for turbulent flow(Re > 2300) in a circular cross section the Blasius
correla-tion can be used
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f v = 0.038
Values forf ·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 oflaminar vapor flow, the substitution ofthe individual pressure-drop terms into the capillary limit, eq (16.9), results in an algebraic expression that can be solved directly forQ 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 ofan iterative solution This procedure begins with
an initial estimation ofthe capillary limit where the solution first assumes laminar, incompressible vapor flow Using these assumptions, the maximum heat transport capacityQ 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 ofthe 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 ofthe fluid properties
For a more direct solution ofthe 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 inputQ(x) and length
x into friction coefficients F vandF l, the capillary limit can be expressed as
(∆P c )max− ∆P⊥− ∆P=
0
(F l + F v )Qdx (16.23) where the liquid frictional coefficientF lis given by
and the vapor frictional coefficientF vis evaluated from the expression
2r2
For the case ofturbulent 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
r2
v A vρv h fg
2r v Q
A vµv h fg
(16.26)
Substitution ofthis expression and combining with those discussed previously results
in a general pressure balance relationship for turbulent vapor flow which takes the form
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(∆P c )max− ∆P⊥− ∆P= 0.019µ v
r2
v A vρv h fg
2r v
A vµ v h fg
0
Q7/4 dx + F l
0
Q dx
(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 ofthe 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 105the fric-tion factor becomes constant as the flow enters the fully turbulent region By assuming
a friction factor for Re> 105, 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 ofthe heat flux or boiling limit is based on nucleate boiling theory and is comprised oftwo separate phenomena: (1) bubble formation and (2) subse-quent growth or collapse ofthe bubbles Bubble formation is governed by the size (and number) ofnucleation 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 = keff
wherekeff is the effective thermal conductivity of the liquid–wick combination The critical superheat∆Tcis defined by Marcus (1965) as
h fgρv
2σ
(16.29)
whereTsatis the saturation temperature ofthe fluid andr nis the critical nucleation
site radius, which according to Dunn and Reay (1982) ranges from 0.1 to 25.0µm for conventional metallic heat pipe case materials As discussed by Brennan and Kroliczek (1979), this model yields a very conservative estimate ofthe amount of
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TABLE 16.4 Effective Thermal Conductivity k efor Liquid-Saturated Wick Structures
Wick and liquid in series
k e= ek k l k w
w + k l k e= effectivethermal
conductivity
k l= liquid thermal
conductivity
k w= wick material
thermal conductivity
Wick and liquid in parallel
k e = ek l + k w
Wrapped screen k e= k l[(k l + k w l − k w )]
(k l + k w l − k w )
Packed spheres k e= k l[(2k (2k l + k w l − k w )]
Rectangular grooves k e= (w f k (w + w l k w δ) + wk l (0.185w f k w + δk l )
f )(0.185w f k f + δk l ) w f = groove finthickness
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 ofheat pipes
The growth or collapse ofa 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 ofthe 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= 2πLeffkeffT v
A v h fgρv ln(r i /r v )
2σ
(16.30)
wherer i is the inner pipe wall radius andr v is the vapor core radius Relationships
to determine the effective conductivity,keff, ofthe liquid saturated wick are given in Table 16.4
16.2.4 Entrainment Limit
Examination ofthe 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,
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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 Ofall the limits for heat pipes, the entrainment limit has produced one ofthe largest amounts ofwork, 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 ofany surface waves In a majority ofcases stud-ied, the wick structure ofthe heat pipe was flooded (i.e., excess liquid), which al-lowed entrainment to occur Additionally, much ofthe work has been an adaptation
to work conducted in the study ofannular two-phase flow, where the onset ofdroplet 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 ofthe 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,wρv V h2
By relating the vapor velocity and the heat transport capacity to the axial heat flux as
and assuming that to prevent entrainment ofliquid droplets in the vapor flow, the Weber number must be less than unity, the maximum transport capacity based on entrainment can be written as
v
2r c
(16.33)
wherer c is the capillary radius ofthe wick structure However, this assumption typically results in an overestimation ofthe entrainment limit since the axial critical wavelength may be much greater than the width ofthe 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
(16.34)
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and another by Rice and Fulford (1987)
U c=
8σ
P v d
(16.35)
These relations can be converted into the heat transport limitation due to entrainment
by combining with the continuity equation, which yields
2πσ
(16.36) or
Q e = A vρvλ
8σ
(16.37)
respectively, whered 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 ofbuoyancy, or
v
A∗
2πσ
(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
Dpipe
lσ
(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 ofavailable 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
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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 ofthe 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 ofthe fluid is low
Busse (1973) provided an analytical investigation ofthe 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 ofthe vapor flow as-suming laminar flow conditions was developed and expressed as
Q v= A v r v2h fgρv P v
16µv Leff
(16.40)
whereP v andρv are the vapor pressure and density at the evaporator end ofthe heat pipe The values predicted by this expression were compared with the results ofprevious 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 v2h fgρv P v
16µv Leff
1−p P v,c22
v
(16.41)
whereP v,cis the vapor pressure in the condenser Busse (1973) noted that the viscous limit could be reached in many cases whenP 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
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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 ofvery small vapor channel diameters With the increased va-por velocities, inertial, or dynamic, pressure effects must be included It is imva-portant
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 ofheat 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 to the 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 ofheat 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 ofchoked 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.)