16.2f was developed to reduce the thickness ofthe radial heat flow path through the structure and to provide a low-resistance path for the liquid flow from the condenser to the evaporator.
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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 oflow thermal conductivity The artery wick, shown in Fig
16.2f was developed to reduce the thickness ofthe 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 ofthe capillary pressure and liquid flow Notice that in some ofthe structures
in Fig 16.3, a separation ofthe 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 ofthe working fluid, heat pipes may be classified by the type ofcontrol 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 ofa wide range ofvariations in heat source and heat sink temperatures
On the other hand, it may be required to permit the passage ofheat under one set of conditions and block the heat flow completely under another set ofconditions This
leads to a consideration ofthe performance ofheat 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 ofa 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 ofthe condenser surface
The heat flow at the condenser can be controlled by controlling the volume ofthe noncondensible gas Examples ofself-controlled devices, those that can be controlled
by the vapor pressure ofthe working fluid, are shown in Fig 16.4a, b, and c Examples offeedback-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
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Q Q
( )a
( )b
( )c
( )d
( )e
Evaporator Adiabatic
reservoir
T
Control fluid
Heater
T
Figure 16.4 Representive gas-loaded heat pipes (From Chi, 1976, with permission.)
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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 ofthe 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 ofa 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 heat source temperature felt at the surface of the evaporator causes a rise in the temperature and pressure ofthe vapor in the evaporator section The flow ofthis vapor through the throttling valve creates a temperature and pressure drop that results in a reduction in the magnitudes ofthese 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 ofvapor through the
throttling valve can be cut off entirely
4 Liquid flow–modulated heat pipe Liquid flow control is also an effective way
ofmaintaining control over heat pipe performance One way ofcontrolling liquid flow
is through the use ofa liquid trap, as shown in Fig 16.7a This trap is a wick-lined
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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.)
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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 ofthe heat pipe In the normal
mode ofoperation with the heat pipe operating in the standard fashion, the trap wick
is dry Ifthe heat input increases or the attitude ofthe 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 ofthe gravitational force This type ofheat 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 ofthe prediction ofthe shapes ofthe 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 ofdiscontinuity and characterized by the property ofsurface 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
R1
+ 1
R2
(16.2)
whereσ is the surface tension and R1andR2are the principal radii ofthe meniscus,
as shown in Fig 16.8 Limitations to use ofthe 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 ofthe capillary pressure across an interface, a review oftechniques has been provided by Wayner (1999)
In predicting the maximum capillary pressure available for a given heat pipe wick structure, the two principal radii ofcurvature are typically combined into an effective
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Figure 16.8 The radii ofthe meniscus
radius of curvature for the wick structure This method produces an effective capillary radius equivalent to the inner radius ofa circular tube (R1 = R2 = reff/ cos θ) and
allows for easy comparison between capillary structures with different structures In this case, the capillary pressure is expressed as
∆Pc= 2σ
reff
whereθ, the apparent contact angle (Fig 16.9), is dependent on the fluid–wick pair
used The contact angle is a measure ofthe degree ofwettability ofthe liquid on the wick structure, whereθ = 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
reff
(16.4)
wherereffis 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 ofthe 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)
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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 (de(de-fined as the point
where the curvature ofthe menisci is maximum) is then
(∆P c )max= r2σ
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 ofthe 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 ofthe evaporator wick structure The heat input to the heat pipe, Q, is related
directly to the mass flow rate ofthe working fluid being circulated and the latent heat,
h fg, ofthe fluid as the heat input is the driving mechanism, or
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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 ofcapillary 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 ofthe evaporator wick will occur Generally, the capillary limit is the primary maximum heat transport limitation ofa 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 ofthe same order ofmagnitude 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 ofchoked 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 ofradiative 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 ofthe 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
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Figure 16.10 Typical heat pipe performance map
load or to improve the design It is important to note that in the determination ofthe heat transport capacity, the mean operating temperature must be identified However, the operating temperature ofa standard heat pipe is a function ofthe 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:
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 ofbody forces (e.g., gravity, centrifugal, electromagnetic), and the pressure drop due to phase transition
∆Ptot= ∆Pν + ∆P l + ∆Pb + ∆Pph (16.9) During heat pipe operation, the menisci naturally adjust the radii ofcurvature for the capillary pressure differential to balance the pressure losses ∆Ptot However, the maximum radius ofcurvature is limited to the capillary dimension ofthe wick structure such that the maximum transport occurs when(∆P c )max = ∆Ptot It is
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important to note that the pressure drop associated with phase transition,∆Pph, 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 ofa heat pipe work-ing under low heat flux Ifthe total pressure drop exceeds the maximum capillary pressure, the return rate ofliquid to the evaporator will be insufficient and the heat pipe will experience dryout ofthe wick The maximum capillary pressure∆Pc devel-oped within the heat pipe wick structure is given by the Laplace–Young equation of
eq (16.2) Values ofthe effective capillary radiusrefffor 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 ofthe heat pipe
as well as any hydrostatic pressure drop related to the drawing ofthe fluid to the top portion ofthe heat pipe cross section
It is important to note that the inclination ofthe 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 ofa thermosyphon, Faghri (1995) may be consulted
Figure 16.11 Pressure variation along the length ofa heat pipe working under low heat flux