Prévia do material em texto
HEAT PIPE
SCIENCE AND
TECHNOLOGY
Amir Faghri
Professor and Head
Department of Mechanical Engineering
University of Connecticut, Storrs, Connecticut
Taylor&Kkancis
Chapter 4
HEAT TRANSFER
LIMITATIONS
4.1 INTRODUCTION
Although heat pipes are very efficient heat transfer devices, they are subject
to a number of heat transfer limitations. These limitations determine the
maximum heat transfer rate a particular heat pipe can achieve under certain
working conditions. The type of limitation that restricts the operation
of the heat pipo is determined by which limitation lias the lowest value
at a specffic heat pipe working temperature. The possible limitations ou
maximum axial heat transfer rate are shown sehematically in Fig. 4.1 as a
function of heat pipe working temperature and are brietly described below:
Continuum flow limit: For small heat pipes, such as micro heat pipes,
and for heat pipes with very low operating temperatures, the vapor
flow itl the heat pipe may be in the free molecular or rarefied condition.
The heat transport capability under this eondition is limited because
the coutinuum vapor state has not been reached.
Frozetx startup limit: During the startup process from the frozen state,
vapor from the evaporation zone may be refrozen in the adiabatic
or condensation zones. This may deplete the working fluid from the
evaporation zone and cause dry out of the evaporator.
Viscous 'imiti When the viscous forces dominate the vapor flow, as for a
liquid-metal heat pipe, the vapor pressure at the condenser end may
reduce to zero. Under this condition the heat transport of the heat
221
222 HEAT PIRE SCIENCE AND TECHNOLOGY
pipe may be limited. A heat pipe operating at temperatures below . its
normal operating range can encounter this limit, which is ais° know
as the vapor pressure limit.
Sonic limit: For some heat pipes, especially those operating with liquid
metal working fluids, the vapor velocity may reach sonic or supersonic
values during the startup or steady state operation. This choked
worlçing condition is called the sonic limit.
Entrainment limit: When the vapor velocity in the heat pipe is suffi-
ciently high, the shear force eacisting at the liquid-vapor interface
may tear the liquid from the wick surface and entram n it into the
vapor flow stream. This phenomenon reduces the condensate return
to the evaporator and limits the heat transport capability.
Capillary limit: For a given capillary wick structure and working fluid
combination, the pumping ability of the capillary structure to provide
the circulation for a given working medium is limited. This limit is
usually called the capillary or hydrodynamic limit.
Condenser limit: The maximum heat rate capable of being transported
by a heat pipe may be limited by the cooling ability of the condenser.
The presence of noncondensible gases can reduce the effectiveness of
the condenser.
Boiling limit: If the radial heat flux or the heat pipe wall temperature
becomes excessively high, boiling of the working fluid in the wick
may severely affect the circulation of the working fluld and lead to
the boiling limit.
Limitations to heat transport arise mainly from the ability of the wick
to return condensate to the evaporator, and from thermodynamic barriers
encountered in the flow of the vapor. A schematic of the limitations to
heat transport in a heat pipe is aLso shown in Fig. 4.2, which is a graph
of the axial heat flux versus the overall temperature drop rather than the
operating temperature as shown in Fig. 4.1. When both ends of the heat
pipe are at the same temperature, T o --= Tia , where To and TL, are the
evaporator and condenser end cap temperatures, no heat is transported.
As the temperature drop is increased, the heat transport quickly increases,
since the effective thermal conductivity of heat pipes is very high. At point
1, the heat transport can drop suddenly to the value which is transferred
by axial conduction in the pipe wall. Point 1 is either the capillary limit,
where the wick structure faiLs to maintain a sufficient amount of condensate
return to the evaporator, or the boiling limit, where the pipe wall overheats
HEAT TRANSFER LIMITATIONS 223
1 a h
j
Condenaer limit
.p) Rolling
e limit s.
-a
it
o
a
el %coxia O ce limit
.h Capillary
limit
0
Sonie limit
o
'R Frozen atartup limit
e Continuum flow limit
Operating temperature. fr
Figure 4.1: Maximum heat transfer limitations for a heat pipo as a function
of operating temperature.
duo to the formation of vapor bubbles ir' the wick structure. Assumir% that
the wic,k does not reach the capillary or boiling limits, the heat transport
continues to increase until it leveis off at point 2. At point 3, the temper-
ature drop is large enough that the working fluid ir' the condenser section
becomes frozen, and the heat pipo evaporator dries out. This again resulta
in a sudden drop in the heat transport to the levei of axial conduction
the wick structure and the pipo wall.
The levei portion of the curve between points 2 and 3 is ralled the vapor
flow limit. The vapor flow limit is characterized by the relative magnitude
of the viscous and inertia forces on the vapor fiow. In the limiting case
of negligible viscous forces (inertia ilow regime), the vapor flow is lirnited
by the choking phenomenon, where tire vapor pressure at tire end of the
evaporator section is approximately half of that at tire upstream end of tire
evaporator. Tire vapor flow rate (axial heat flux) can only be increased
by increasing tire pressure (and therefore temperature) in tire evaporator
section. This situation is referred to as tire sonic limit, since tire vapor
reaches tire local speed of sound at tire end of the evaporator.
• Tire otlaer limiting case is when inertia forces are negligible (viscous
Entrainrnent
limit
224 HEAT PUJE SCIENCE AND TECHNOLOGY
Vapor flow linkit
Solidification of
working fluid
Liquid tlow limit
To = const.
Temperature drop, Tc, —T L
Figure 4.2: Schematic of the heat transport limitations of a heat pipe
(Busse, 1973).
flow regime). In this situation, the axial heat flux increases with increasing
overail pressure drop until the pressure at the condenser end cap reaches
essentially zero. At this point, the axial heat flux cannot be increased
further, which is called the viscous limit. When both inertia and viscous
forces are present but the inertia forces are dominant, choking again occurs
in the heat pipe, but it occurs at the beginning of the condenser section in
this esse. This is due to the fact that the viscous forces increase the vapor
velocity from subsonic at the end of the evaporator to the sonic velocity at
the beginning of the condenser (Levy, 1971).
Among the heat pipe limitations mentioned above, the capillary, sonic
and boiling limits are most commonly encountered during normal heat pipe
operation. These limitations will be described in the next sections.
4.2 CAPILLARY LIMITATION
4.2.1 PRESSURE BALANCE IN A HEAT PIPE
The stable working fluid circulation in a heat pipe is achieved through the
capillary pressure head developed by the wick structure. The maximum
HEAT TRANSFER LIMITATIONS 225
heat transfer attainable in a heat pipe is achieved under conditions such
that the capillary pressure head is greater than or equal to the sum of
pressure losses along the vapor-liquid path. Figure 4,3 shows schematically
the pressure distributions in the vapor and liquid along tire heat pipe length,
where tire wet point, at which tire liquid and vapor pre,ssures are equal, was
at tire condenser end cap. In order for the heat pipe to operate properly,
tire following pressure balance must be satisfied
APeap,max APe APv 4- Po Apc , 5 + àpg (4.1)
Tire maximum capillary head cari be calculated by
1»cap,max = —
reff
2a:
(4.2)
where reff is tire effective pore radius correctedby the factor // cos 0„„„ ) ,„„„.
The vapor pressure drop along the heat pipe, Ap y , has been described
ir detail in Chap. 3. It should be pointed out that Ap, in eqn. (4.1) should
be interpreted to be the absolute vapor pressure drop along tire heat pipe.
49 is tire pressure drop in tire liquid due to the effect of tire gravitational
force in tire direction of tire heat pipe axis, and can be expressed as
Ap =pgL sinçb
(4.3)
—4-
LiPc, d
Lip
ap
g
Adverse gravity
force
fra-Evaporator-v-H- Adiabatic Condenserl
section section section
Figure 4.3: Pressure balance In a heat pipe.
226 HEAT PIPE SCIENCE AND TECHNOLOGY
where cP is the inclination angle of the heat pipo from horizontal. Ap e,6 and.
Ap c,6 are the pressure drops duo to the evaporation and condensation at
the liquid-vapor interface, respectively, and usually can be neglected. Ape
in eqn. (4.1) is the pressure drop of the liquid flow in a wick structure due
to the frictional drag. This pressure drop is discussed in Chap. 3 and is
one of the major factors which cause the capillary limit.
4.2.2 CAPILLA_RY LIMIT EXPRESSION FOR CON-
VENTIONAL HEAT PIPES
By neglecting the pressure drops due to the evaporation and condensation
at the liquid-vapor interface in eqn. (4.1), the general expression for the
capillary limit is
—
2a
> Ape + Ap t, + pegLe sin eb (4.4)
reff
For laminar incompressible vapor flow when the wet point is at the be-
ginning of the cooling zone, eqn. (3.73) is applicable for the vapor pressure
drop and eqn. (3.7) is applicable for the liquid pressure drop. Therefore,
for a cylindrical heat pipe, eqn. (4.4) can be written as
(L• L• e) • Ptrne 4p„Q,
7f peA„K
dz +
irp„R`,1,h f 9
[L e (1 + FRer) + 2L a]
+pip(Le + La)sinip (4.5)
where F is given by eqn. (3.74). The positive z direction is opposite
to the liquid flow direction in eqn. (4.5). Since at steady state rhe
= Q(z)/h1 9„ if a heat pipe has a uniform heating distribution along
its evaporator and condenser sections, the axial mass flux distribution will
be described by eqn. (3.62). In addition, if both A.„ and K are constant
along the heat pipe length, the pressure drop in terms of the axial heat flow
is
fo (L e +La ) . (Le +L.)
girnt ite dz — Q(z) dz
peA„K peke K h fg
A heat transport factor (QL) eap,,„a„ is defmed as
(Q ncap,max — fo
(L e + L•
Q(z) dz = (0.5L + La)Q, (4.6)
Therefore, we have
HEAT TRANSFER LIMITATIONS 227
2cr > Pt(QL)cap
'
max 4tivQe
[Le (1 FQ° ) 241
reff peile,Khf 9
+
pe lit,hf g 271- p„Le h f g
+peg(Le + La) sin (4.7)
where Qe is the total heat input to the evaporator.
For laminar incompressible vapor flow when the wet point is near the
condenser end cap, eqn. (3.76) is applicable. Noting that for laminar fiow
in a circular pipe, fRe z ,„ = 16, eqn. (3.76) can be written as
(f
aPv —
Rez '
r )ito
Qe(0.5.Le + La + 0.5L, )
211Ple pahi9
The heat transport factor (QL)cap,mex in this case is
(QL)cap,max = j Q(z)dz (0.5Le + 0.5L e)Q
o
Or
(C/neaP,max
Qg =
0.5Le L a + O.5Le
Equation (4.4) can be expressed as
2a
± p
z,r)Pv ) f nr N
> \ ••-i-dicap,maic oLt sin 0
reip — C Me (fRe
op.A.K h f g + 210,AvPvh f g
Therefore, we have
2o/reff — Pe9Lt sin
(QL)cap,max 5_ +
where the vapor and liquid frictional coefflcients are defined as
At Fp — (4.11)
pgA w IChf g
(fRe z ,„)p„
= (4.12)
2R.,?,A e pv hfg
The above equation was derived based on a circular heat pipe. For noncir-
cular heat pipes, eqn. (4.10) is still appficable by introducing a hydraulic
radius in eqn. (4.12).
(4.8)
(4.9)
(4.10)
(4.13) Fe — (fRer,ii)Pv
2R2e,hAePv
Expressions of fReh for noncircular ducts have been given previously in
relations such as eqns. (3.15), (3.17) and (3.19). Although these equations
228 HEAT PIPE SCIENCE AND TECHNOLOGY
were focused on liquid flow ir, noncircular ducts, they are applicable to
laminar incompressible vapor flow as well.
The closed-form solutions described above represent a large number of
practical conventional heat pipe operations. However, when vapor com-
pressibility has a strong influence ou the vapor flow, or the liquid perme-
ability K or flow area A v, are not constant along the heat pipe length, the
closed-form solution for capillary limit can no longer be obtained. In this
case, a numerical procedure for evaluating the capillary limit is needed.
Equation (4.4) can be written as
zmin z. peke ( z)
—
2a
— f peg sin0 dz fo
nn
Aw Kdz + Ap, (4.14)
reff c) Pehf 9
where ;ain is the location of the wet point.
The vapor pressure drop Ap should be obtained by numerically or ana-
lytically solving the conservation equations presented in Sec. 3.3. The local
liquid mass flow rate fne(z) can be obtained by numerically integrating
the radial mass flux along the heat pipe length at the liquid-vapor inter-
face, rim = v6p5, which cai' be calculated from the conjugate temperature
solution in the wick.
rits(z)= f v6p6W dr, (4.15)
o
where W is the circumferential length of the liquid-vapor interface. Once
chi(z) is obtained, the first term ou tire right-hand side of eqn. (4.14) can
be integrated. Tire capillary limit thus obtained should be more accurate
and reliable.
Tire location of tire wet point, z„,;„, in eqn. (4.14) should be determined
before the integrations are carried out. The location of z nik, cai' usually be
considered to be at the condenser end cap. However, if the heat transport
rate is high, and the vapor pressure recovery in tire condenser region is
greater than tire liquid pressure drop in tire wick, tire location of z min shifts
to the beginning of tire cooling zone (see Fig. 1.4).
4.2.3 CAPILLARY LIIVLIT EX_PRESSION FOR AN-
NULAR HEAT PIF'ES
In this section a simple analysis for predicting the capillary heat transport
limitation of tire concentric annular heat pipe (Faghri and Thomas, 1989)
will be presented. This is done to show the generalization of previous re-
sults as well as the improvement of tire heat transport capability of tire
annular design cnrer a conventional heat pipe with the same outer dimen-
HEAT TRANSFER L/MITATIONS 229
sions. This comparison is valid because the same assumptions were marte
for the prediction of the capillary limit of the conventional heat pipe.
Consider an annular heat pipe, as shown previously in Fig. 3.14, oper-
ating under steady state conditions. The sum of the pressure changes in
the closed-cycle system may be described by the following mathematical
relation.
2 Ep„(zref) — Pe(z)] + [73,,(z) — p e , i (z)] + [p(z) —
+ [Pi,i(z) — Pe,i(zref)] + [Pe,i(zref) — Pv(zref)]
+ [14,0(z.f) — Pv(zref)1 + — Pt ,0( ;tf )1 = O
where the subscripts i and o refer to the inner and outer walls, respectively.
Introducing into the above equation the capillary pressure p eep , defined as
the pressure at the vapor side of the liquid interface minus that of the liquid
side, results in
peep ,i (z) +pesp,e(z) =Peap,i(Zref) ±Pcap,o(Zref)
(4.17)
+24ápv (z — z„f) + Apt, z (zref — z) + Arn, e (zref — z)
where in general the notation Ap(z — z ref) means that Ap is evaluated over
the distance (z — z„f). Assuming that the z„f is such that it is located at
zrei„ where the capillary pressure is minimum and equal to zero results in
the following equation.
Pcap,i(Z) ±Pcap,o(Z) = 2APe(Z — Zmin) ák,i(Zmin 4" APP,o(Zmin — z)
(4.18)
In conventiona1 heat pipes, there exists a maximum capillary pressure that
can be developed for a liquid-wick pair. However, for the concentric annular
heat pine there exist two different maximum capillary pressure forces for
the inner and outer tubes. If a heat pipe is to operate continuously without
drying out the wick, the required capillary pressure for each wall should
not exceed the maximum possible capillary pressure. The magnitude of
the capillary pressure may bedetermined by balancing the forces at the
liquid-vapor interface. This requires that the maximum capillary pressure
for each wall should be
Pcap,m
2a
reff,,
(4.19)
2o-
Pcap,max,o = (4.20)
Teff,o
where reff is the effective capillary radius of the wick pores at the liquid-
(4.16)
230 HEAT PIFE SCIENCE AND TECHNOLOGY
vapor interface. Upon the assumption of laminar flow, incompressible fluid,
and neglecting the inertia terms and the shear stress at the liquid-vapor in-
terface in the conservation of momentum equation, one can get the following
force balances for the liquid flow ia the inner and outer wall wick structures
dpe,e
peg sin (4.21)
dz
dpe, 0
pgg sin (4.22)
dz
where re is the viscous stress at the liquid-solid interface and Dh,e is the
hydraulic diameter for the wick. Equations (4.21) and (4 22) can be rep-
resented in terms of the local axial heat flow rate Q, and Qo for the inner
and outer walls, respectively.
= sin (4.23)
dz
Fe,0 Q0 ± pg g sinel, (4.24)
dz
The frictional coefficients, F, are deflned as
Ft,t = (4.26)
KiA o,,eh f g
4° = KO A„,,„hf9
(4.26)
where K is the wick permeability, A„, the wick cross-sectional area, so the
porosity, f the friction drag coeflicient, and Re p the wick-liquid Reynolds
number. These properties are defuied by the following relationships.
Rei,
Dh,e,iwe,e ph,t,02714,0 Ree,0 =-
vi vi
fes
t,ti
1 2
fe st -=
Te i a
ptílT,L0/2
(4.27)
Qi = ruesspiAwsPehfg
(Pipi, e e
2 (fItei,i)
Qo = We,o <Po Atv,o n h f 9
WoDi, e o
2 (fRee 0)
KJ = Ko =
HEAT TRANSFER LIMITATIONS 231
The total local average axial heat flow rate Q is the sum of the inner wall
local axial heat flow rate Q., and the outer wall local axial heat flow rate
Qo . For open rectangular passages, laminar incompressible fully developed
fluid flow analysis shows that fRe is only a function of the geometry and
dimensions of the wick. Therefore Q., and Qo are the only variables in eqns.
(4.23) and (4.24) that change with axial distance along the heat pipe. The
value of f Re needed for the evaluation of F can be obtained from the
discussion of wick structures in Chap. 3.
The problem of calculating the vapor pressure drop in the annulus is
complicated in the evaporating and condensing regions by the radial vapor
flow due to evaporation and condensation. It is convenient to neglect the
effect of blowing and suction on the vapor pressure drop to obtain a closed
form solution. For a more accurate prediction, one should use an analysis
that includes the effect of blowing and suction on the vapor pressure drop as
given in Sec. 3.3. Upon application of the conservation of axial momentum
to the vapor flow between the concentric pipes, one obtains the following
relationship provided that lhe flow is laminar
dp„r,v 2
dz
. „
A,—
dp,
= (71-D,)-1-„,,oerDo)—
dz +
+ m0) 27,„ (4.28)
Since the mass flux of the vapor is related to lhe axial heat flow rate at the
same z hf9 ), eqn. (4.28) can be presented in lhe following
form when the last term on lhe right-hand side of the above equation is
neglected
(4.29)
2 (7,,Rez4 v„
A.„D7, ,v hf 9
7 - _ + DiTv
v P/2 (Do + .D4) p»/2
ft—e, Pvwv-Dh.,,
itv
= Do —
1
E„ —
,n,p„hL
Substituting Aps(z — and apv (z — za,,,,) from eqns. (4.23), (4.24) and
(4.29) into eqn. (4.18) and neglecting the effect of gravity re,sults in the
following equation.
dp„ d ■rj2
= — F,,Q — E,
dz dz
where
(4.30)
232 HEAT PIFE SCIENCE AND TECHNOLOGY
2o-
(
--
1
+
1
= 2rt, dz + + .F,„Q„)dz (4.31)
reff,i refl,o O
The above relation simplifies, if one assumes that the geometry and dimen-
sions of the wick structures of the inner pipe are the same as the outer
pipe, as well as the same heat input to the inner and outer walls, i.e.,
= .F£,„ = .F£ and (2,, = Qa = Q/2.
1 1
Li
— —
,i
(QL)cap,mas Q dz —
reff reff, o
)
o 2-.F, +7 e
(4.32)
The maximum heat transport for a conventional heat pipe is given by the
following equation.
fo t
Qdz
, 2o-/reff
(4.33) (Qncap,max = F'í + F,
It should be noted that eqn. (4.33) is the same as eqn. (4.10) with (/) = 0.
A comparison of the maximum heat transport capillary limit of the axially-
grooved concentric annular heat pipo to a conventional heat pipe with the
same outer diameter using the above analysis (Faghri and Thomas, 1989)
shows an incresse of 80% using water as the working fluid at 100°C. Faehri
and Thomas (1989) also presented tilt tests on conventional and concentric
annular heat pipes at tilt angles from —1° to 00 (Fig. 4.4), where the
maximum power increases as the evaporator section is lowered with respect
to the condenser section of the heat pipo.
Faghri and Buchko (1991) experimentally optimized the heat input dis-
tribution for a water heat pipe with multiple heat sources. It was concluded
that by redistributing the heat load, the maximum heat transport capacity
can be increased. Schmalhofer and Faghri (1993) developed a generalized
analytical methodology for the prediction of the capillary limit of both
circumferentially-heateÁl and block-heated heat pipos. The experimental
and analytical capillary limit was compared (Fig. 4.5) for both modos of
heating for a copper-water heat pipe.
4.3 BOILING LIMITATION
4.3.1 BOILING PHENOMENA
The boiling limit is directly related to bubble formation in the liquid. In
order that a bubble can exist and grow in a liquid, a certain liquid superheat
Concentric Annular Heat Pipe
1 1
-0.8 -0.6 -0.4
Tilt Ang1e, Arc Degrees
-02
o
800
Conventional Heat Pipe
600-
g 400 -
a
200 -
233 HEAT TRANSFER LIMITATIONS
(a)
1500
o
(h)
I I
-0.8 -0.6 -0.4 -0.2 O
Tilt Ang1e, Arc Degrees
Figure 4.4: Maximum power versus tilt angle for the eonventional and
coneentrie annular heat pipe (Faghri and Thomas, 1989).
is required. Ou the other hand, in order that a bubble can mdst and grow
in a superheated liquid, its size must be larger than a eritica1 value. Figure
4.6 shows sehematically a bubble or nueleus formed in a superheated liquid.
The whole system is at a uniform temperature T. The liquid and vapor
30 40 50
R
o 100-
75-
50-
25-
O
o
a
300
10 20
O Experimental Circumferential Heating
Analytical Block Heatint
o Experimental Block Heating
Analytical Circumferential Heating
o
o
1 60 70 á 40 tOo do 120
o
o
234 HEAT PIPE SCIENCE AND TECHNOLOGY
Vapor Temperature (°C)
Figure 4.5: Analytical and experimental capillary limit versus vapor tem-
perature for circumferential and block heating modes (Schmalhofer and
Faghri, 1993).
is at an equilibrium state with pressure Pv in the vapor and pt in the
liquid surrounding the bubble. The pressure p oo at the flat interface is
the saturation pressure corresponding to the system temperature T. The
saturation temperature corresponding to the liquid pressure adjacent to
the bubble is denoted by n at. The mechanical equilibrium of the spherical
vapor bubble in the liquid requires that
2a
Pv = Ti7;
Pe ;T
Figure 4.6: Bubble formation in a liquid.
275-
250-
225-
200-
175-
150-
125-
(4.34)
HEAT1 RANSFER UMITATIONS 235
where a is the liquid-vapor interface surface tension and Rb the radius of
the bubble. The relation between p, and Na , according to multiphase
thermodynamic theory, is given by (Colher, 1972)
( 2a ps
PouRbPe)
Since 2o pa pasRbpe « 1, eqn. (4.35) can be written as
p„ poo 20-P-
PooRbPi
Combining eqns. (4.34) and (4.36), we obtain
2a
Poo 12=)
vLb pe
To obtain the liquid superheat T — Tsat corresponding to the pressure dif-
ference pas — pÉ, the Clausius-Clapeyron equation and the perfect gas law
are employed
(4.38)
For p1» ps, and Pv = PvRgT, eqn. (4.38) can be written as
dp _ hi g dr
(4.39)
p RgT2
Integrating eqn. (4.39) from pi to Na andfrom Tsat to T, we obtain
In (pc°
P1) Rgnat T
(4.40)
h g (T — Tsat)
Substituting eqn. (4.37) into eqn. (4.40), one obtains
T nat Rg TsatT ± 2u (1 4, e2IM
nig I PiRt, P1)]
For most cases, p„ pie, and if 2cr/PeRa << 1, eqn. (4.41) can be simplified
to
RgTsatT 2o RgTa2at 2o- Tsat 2a
T — Tsat = AT
hf g peRb h f g PeRb hfgPv,sat Rb
ps = pas exp (4.35)
(4.36)
(4.37)
dp h f g
dT ( 1 _
Pv Pt )
(4.41)
(4.42)
236 HEAT PIRE SCIENCE AND TECHNOLOGY
where ,o,,,sat is the vapor density corresponding to pe and the saturation
temperature Tsat. Equation (4.42) can be written as
2aTbab
h f gpv ,sas AT
In reality, bubble formation is always a non-equilibrium process. In arder
for a bubble to grow, eqn. (4.34) must be replaced by
2a
Pv - PE a —
Rb
AT >
20"Tbat
h f g p,,,sas-kb
14 >
- h f gp,,,BabakT
Equation (4.46) gives the relation for the criticai size of a bubble or nucleus
under a certain liquid superheat and physical properties. If the nucleus is
smaller than the criticai size, it is unstable and the nucleus will disappear
atter a short period of time, The nucleus formation in a homogeneous liquid
is of a statistical nature and is related to energy fluctuations of molecule
clusters In the liquid itself. From eqn. (4.45) we learn that AT oo when
Rb -4 O. Therefore, it is very difficult to form a vapor nucleus or embryo
hl a homogeneous liquid at the molecular levei. The superheating required
for boiling in a bulk of degassed pure water, for example, could reach as
high as 300°C at atmospheric pressure. For liquid metais, the superheat
required is much larger than that for water, since the term 2onat/hf gpv,sat
for liquid metais is much larger than that for nonmetallic fluids. In practice,
there are always other factors which promote nucleation. The liquid cannot
be theoretically pure; iinpurities always exist in the liquid suei as dust,
noncondensible gases or small vapor bubbles which act as vapor-forming
centers or seeds. Therefore, the initial embryo size is always larger than
that predicted theoretically, and the liquid superheat required is much lower
than the theoretical value. This is especially true for boiling at a solid wall.
Cavities always exist in tire wall surface. Even for a very smooth surface
this is still unavoidable from tire microscopic point of view. Some of these
pits or cavities are capable of trapping noncondensible gas ar vapor even
under subcooled conditions. When a liquid superheat is imposed, these
cavities may act as nucleation sites. Bubbles generated from these sites
require a much lower superheat than in the liquid. Some of these cavities
are schematically shown in Fig. 4.7. Not ali cavities are ame to trap vapor.
(4.43)
Therefore, we have
ar
2aTsab
(4.44)
(4.45)
(4.46)
HEAT TRANSFER LIMITATIONS 237
Only those cavities whose walls are poorly wetted by the liquid (Fig. 4.7(c))
or whose shapes are irregular, such as re-entrant (Fig. 4.7(b)) cavities can
effectively trap vapor and work as active nucleation sites.
Boiling from a plane heated surface submerged in a large volume of
stagnant liquid is a common boiling heat transfer example and is usually
referred to as pool boiling. The results of investigations on heat transfer
rates in pool boiling are usually plotted on a graph of surface heat flux
versus wall superheat T1 — Tsat , where T,„ is the heated wall surface tem-
perature and Tsat is the saturation temperature corresponding to the liquid
pool pressure. The graph was first obtained by Nalciyama (1934) for water,
and it was found that ali liquids behave in a similar trend. The component
parts of the boiling curve are well known and are indicated in Fig. 4.8.
Natural convection: The region AH corresponds to the natural convec-
tion region, where temperature gradients are established in the pool,
and heat is removed by natural convection to the free surface and
then by evaporation to the vapor space.
Nucleate boiling: The region BC corresponds to the nucleate boiling
region. As the heat flux is increased, bubbles form at the heating
surface and very high heat transfer rates can be reached with a rela-
tively small temperature difference T. The bubbles transport heat
both by latent heat and by agitating the thermal boundary layer in
the liquid dose to the wall.
Critical heat flux: At point C, the bubble population becomes so high
that the interaction of the liquid and vapor streams causes a restric-
tion of the liquid supply to the heating surface and a continuous va-
por film forms adjacent to the wall. The wall temperature increases
rapidly and the criticai heat flux is reached.
(a) Wetting
cavity
(b) Re—entrant (c) Non—wetting
cavity cavity
Figure 4.7: Different cavities in the wall surface.
natural
convection
region
nucleate partial
boiling film
region boiling
region
C
stable
film
bolling
reglon
E
critica!
heat
flux
13
238 HEAT PIPE SCIENCE AND TECHNOLOGY
Tw—Teat
Figure 4.8: Pool boiling curve for water at a plane surface.
Partial fim boiling: Region CD is known as partia' fim boiling and is
characterized by the edstence of an unstable vapor blanket over the
heating surface that releases laxge patches of vapor at more or less
regular intervals. Intermittent wetting of the surface is believed to
occur.
Stable fim boiling: Region DE is called stable film boiling. Heat trans-
fer is principally by conduction and convection through the vapor fim
with radiation becoming significant as the surface temperature is in-
creased. The point E is determined by the melting temperature of
the heating material.
4.3.2 BOILING HEAT TRANSFER AND VAPOR
FORNIATION MODELS IN HEAT PIPE
WICKS
Heat transfer from wicked surfaces is more complicated than that from plain
surfaces due to the existence of capillary structures. Also, heat transfer
and vapor formation may change with different types of wick structures
and working fluids. As a result, different heat transfer modes may eadst in
heat pipe wicks. Four basic modes of heat transfer and vapor formation are
shown schematically in Fig. 4.9.
Mode 1, Conduction-convection: The whole wick is filled with liquid,
where conduction occurs across the liquid layer and vaporization takes
rietemtâilid
HEAT TRANSFER LIMITATIONS 239
place from its surface. No boiling occurs within the wick. However,
natural convection may take place within the wick for some thick
wicks under a gravitational field. This is the situation for non-metallic
working fluids under a low heat flux and metallic worlcing flui& under
a low or moderately high heat flux. The heat transfer across the wick
can be calculated by a conduction model with sufficient accuracy.
Heat transfer under this mode is normal and no heat transfer limits
are encountered.
Mode 2, Receding liquid: As heat flux is increased, the evaporation at
the liquid surface is intensified. The capillary or body forces available
in the heat pipe may not be enough to drive a sufficient amount of
liquid back to the evaporator zone. As a result, the liquid layer begins
to recede into the wick structure. If the heat flux at the evaporator is
reduced, the receding liquid layer will stop, and refilling of the capil-
lary structure with liquid will begin. However, if the receding of liquid
continues, the liquid at the evaporator may be completely depleted,
and heat pipe wick may be burned out. This heat pipe limit encoun-
vapor
liquid
wick
qfiwIng,SWID
WeineXtibN
UI) III) III geneatutio
qe e
Mode 1
Moda 2
Conduction—Convection
Receding liquid
ge
Mode 3
Nucleate boiling
(n.o
onosesit
Moda 4
Film boiling
(Rolling lima)
Figure 4.9: Modes of heat transfer and vapor formation in wicks.
240 HEAT PIPE SCIENCE AND TECHNOLOGY
tered is actually the capillary limit, not the boiling limit Before the.
liquid is completelydepleted, the heat transfer across the liquid layer
is still by conduction, and the liquid vaporization takes place at the
liquid-vapor interface. No boiling occurs within the wick structure.
This is applicable to both non-metallic and metallic working fluids.
Mode 3, Nucleate boiling: For some non-metallic working fluids, when
the temperature difference across the wick is large, nucleate boiling
may take place within the wick. Bubbles grow at the heated wall,
escape to the liquid surface and burst rapidly. The nucleate boiling
within the wick does not necessarily represent a heat transfer limit
unless bubbles cannot escape from the wick. This is especially true
for gravity-assisted heat pipes or thermosyphons. In thermosyphons,
nucleate boiling in the liquid pool is a normal working condition, with
the condensate being fed back to the evaporator by gravity. However,
for wicked heat pipes whose liquid circulation is brought about only
by the capillary force, nucleate boiling in the wick indeed represents a
heat transfer limit for the following reasons: Large bubbles bursting
at the liquid surface may disrupt the menisci established at the liquid-
vapor interface and eliminate the capillary force circulating the liquid
condensate; vapor bubbles generated at the evaporator section may
block the liquid return from the condenser section.
Mode 4, Filia boiling: As the temperature difference across the wick is
increased, a large quantity of bubbles are generated at the heated wall.
These bubbles coalesce together before escaping to the surface, form-
ing a layer of vapor adjacent to the heated wall, which prevents the
liquid from reaching the wall surface. As a result, the heat pipe wall
temperature will increase rapidly and the heat pipe may be burned
out. This heat transfer lirnit is similar to fllm boiling in pool boiling
heat transfer, and is a common heat transfer limit for wicked heat
pipes.
4.3.3 EXPERIMENTAL STUDIES OF BOILING
HEAT TRANSFER FROM SURFACES COV-
ERED BY POROUS MEDIA
Heat transfer from wicked surfaces lias been extensively studied in recent
years. Since the wick provides additional sites for nucleation, the heat
transfer in the wick is more complicated than the boilhig heat transfer
from plain surfaces. In addition, the superheat AT for boiling incipience in
a wick should be lower than that from a smooth surface. Marto and Lepere
1 1 1 1 1 111
Freon-113 —
16 111111, 1 1 1 1 1 II
"e
1 05
106
104
102
01 1.0 10.0
10 3
100.0
• I 1 1 1 1 1 111 1 1 1 1 1 1 111 1 1 1 1 1 1::
- : O • Run 4, surface aging A
A • Run 5, surface aging B
_ O • Run 8, ~face aging C
V • Rim 9, surface aging D
r - - - puiu tube data
Incipient boiling
V
Nueleate _
boiling
from
plane
surface
0 41—
4—
- =
•
A
HEAT TR,ANSFER LIMITATIONS 241
(1982) studied the pool boiling heat transfer from various porous metallic
coatings and enhanced surfaces experimentally. Their experimental resulta
for a porous coating called the High Flux surface are shown in Fig. 4.10. It
can be seen from the figure that boiling heat transfer from a surface covered
by a porous coating requires a much lower wall superheat due to additional
nucle,ation sites.
Ferrell and Alleavitch (1970) experimentally studied the heat transfer
from a horizontal surface covered with beds of Monel beads. The bed
depth ranged from 3.2 mm to 25.4 mm using water as the working fluid
at atmospheric pressure. The experimental resulta are shown in Fig. 4.11.
They concluded that the heat transfer mechanism at lower AT was conduc-
tion through the saturated wick-liquid matrix to the liquid-vapor interface,
which corresponda to Mode 1 in Fig. 4.9. At higher superheats, nucleate
boiling occurs and the experimental data deviates from the curve predicted
T, - T, ÇC)
Figure 4.10: Comparison of high flux surface to plain tube for F'reon-113
(Marto and Lepere, 1982).
Bed depths
O 3.2 oun
Cl 12.7 min
zh 25.4 mm
Curve preclicted by
conduction modo!
n II
O
o
1
1
Robsenbow correlation
(pool boiling)
Normal pool boding
(without wick)
Water at atmosphetic pressure
242 HEAT PIPE SCIENCE AND TECHNOLOGY
100
1.0
01 1.0 10
AT (t)
Figure 4.11: Heat transfer from a submerged wick (Ferrell and Alleavitch,
1970).
by the conduction model. The experimental data intercepts the boiling
curves from the plain surface and Rohsenow's (1985) pool boiling corre-
lation at a higher T. The nucleate boiling incipience occurs at a lower
superheat than that of pool boiling.
Philips and Hinderman (1967) conducted experiments using a wick of
nickel foam 0.14 em thick and one layer of stainless steel screen attached
to a horizontal plate, with distilled water as the worldrig fluid. Their ex-
perimental resulte are shown in Fig. 4.12. The heat transfer is primarily
by conduction through the flooded wick. This heat transfer mode also cor-
responds to Mode 1 as shown in Fig. 4.9. At higher Ai, nucleate boiling
was observed and the liquid superheat AT began to decrease. The heat
transfer mode at this point corresponds to Mode 3. It should be pointed
out that boiling heat transfer in porou,s media ou plaM surfaces has a dif-
ferent consequence from boiling in a heat pipe porous wick. The capillary
menisci in the heat pipe may be disrupted due to the nucleate boiling and
consequently the liquid circulation in the heat pipe wick may be stopped.
The boifing limit may be reached during heat pipe operation insteaLl of a
continuous drop in wall temperature for boiling in porous media on a plain
surface as shown in Fig. 4.12.
Reiss et al. (1968) measured the limiting heat flux at the evaporator for
Liquid head
O otom
O (-12.7) mm
• (-25.4)mm
Nickel-feam wick
water at 28 kN/m 2
Nueleate boiling
¡first observed
-
•
•
• iffi Caleulated assuming heat
conduction through fim
of water equal to thickness
of wick
HEAT TRANSFER LIMITATIONS 243
sodium in grooved heat pipes. Very high radial heat fluxes up to 2 x 10 3
W/cm2 were achieved at wall temperatures higher than 800°C. The heat
flux achieved is much higher than those for sodium pool boiling as shown
in Fig. 4.13. The heat transfer from the sodium wick is by conduction
corresponding to Modes 1 and 2. Since the sodium liquid layer in grooves is
thin and the thermal conductivity is very high, a high evaporation heat flux
was possible from the sodium surface. During the experiment, the authors
observed dark stripes ou the outside of the heat pipe and concluded that
evaporation was taking place only from the grooves. The limiting heat flux
was brought about by depletion of the liquid from the wick neer the heated
surface due to the onset of the capillary limit corresponding to Mode 2.
Ivanovskii et al. (1982) conducted experimento on the evaporation of
sodium from a compound wick in a vapor chamber. The experimental heat
flux data obtained are shown in Fig. 4.14 together with the curve represent-
ing the criticai heat fluxes of sodium pool boiling. The experimental data
shown in the figure are not the critical heat fluxes of evaporation from the
wick. The values were not limited by the heat removal from the wick but
by the melting temperature of the copper lining the heating section of the
experimental apparatus. The heat filmes at the heated wall are as high at
901 W/cm2 , and are more than three times greater than the criticai heat
fluxes under pool boiling conditions. In summary, the boiling limit rarely
occurs in a heat pipe wick with liquid metal working fluids. The dryout of
100
10
1.0
0.1
00 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0
(t)
Figure 4.12: Heat transfer from a submerged wick (Philips and Hinderman,
1967).
• Noyes
O Carbon
Subbotin
o Subbotin
X Subbotin
Caswell
• Heat pipo
A'
244 HEAT PIPE SCIENCE AND TECHNOLOGY
1o3
to`
lo'
1 0
102p (Torr)
1 0
1 0 io'
1 0 4
500 600 700 800 900 1000 1200
T (°C)
Figure 4.13: Criticai heat flux for sodium pool boiling compared to the
limiting heat flux in grooved sodium heat pipe (Reiss et al., 1968).
the wick at the evaporator is mainly caused by other factors such as the
capillary limit.
4.3.4 CRITICAL NUCLEATION SUPERHEAT AND
HEAT FLUX IN POROUS WICKS
Nucleation within flue wick is undesirable for wicked heat pipe operation
because the bubbles can obstruct the liquid circulation and hence cause hot
spots ou the heated wall. Analysis of the boiling limit involves the theory
of nucleate boiling, which is concerned with two processes: the formation
of bubbles (nucleation), and the subsequent growth and motion of these
bubbles. There always edsts some vapor iiuclei or srnall bubbles within the
wick structure as mentioned before, but the superheat required is essential
for these bubbles to grow. A small trapped hemispherica1 vapor bubble
with effective radius R6 in the vicinity of the wall-wick interface is shown
in Fig. 4.15. The bubble as well as the liquid adjacent to the wall are at the
wall temperature T. The vapor pressure Inside the vapor bubble is pt,v,
while the liquid pressure adjacent to the bubble is pe. The temperature
S
/1,
Figure 4.15: Bubble formation at the wall-wick interface.
HEAT TRANSFER LIMITATIONS 245
600
— 400
200
6 8 104 2 4
p,(ahn)
Figure 4.14: Heat flux for sodium evaporation Irem a compound wick com-
pared to criticai heat flux for sodium pool boiling (Ivanovskii et al., 1982).
and pressure of lixe heat pipo vapor space adjacent to the meniscus are T„
and py . At equilibrium
P6,v — pe = — (4.47)
2a
-Rb
Neglecting the gravity influence, the relation between the liquid pressure
lie and the vapor space pressure pz, can be written as
Pv Pe Rrnei,
(4.48)
where Rmen is the radius of the liquid-vapor meniscus. Thus, combining
eqns. (4.47) and (4.48) gives
1 1 )
Pb,v Pv Za
Rmen (4.49)
Wick—wall-
interface
R Porous !fiel(
246 HEAT PIPE SCIENCE AND TECHNOLOGY
If we denote the saturation pressure corresponding to Tu, by psat , the rela-
tion between psat and No according to eqn. (4.36) is
Pb,v Ft% Psat (1
PsatRbPS
Combining eqns. (4.49) and (4.50), we have
2o-
Psat — Pv = 20. (—p
1 1 )p„
(4.51)
amen IlbPe
Assuming that the vapor adjacent to the meniscus is at the saturation state,
and applying the Clausius-Clapeyron equation between (pv , Tv ) and (psat,
Ta) ), we obtain
RgT„T„ ± 2o ( 1 1 ± 2a p„,
iáTsát = T,„ –2; (4.52)
tbf g Pv Rb Ruim I Pv Rbpt
If
2rr ( 1 1 ) 2o p„
,
Vi, \ Rb Rmen Pvnbfte
eqn. (4.52) can be written as
RgT,T,„ pa ( 1 1 2crpl
afcrit =
r_
h fg /37, Ftõ w- ) men PuRbn
(4.53)
In addition, neglecting the term 2apv/pabpe and letting Tor,„ T, we
have
ATent n"- (4.54) hfg pvv
2aT (1
Rb Rinen )
1
Equation (4.54) is a commonly used relation for criticai superheat in the
literature. Based on the ATcrit obtained above, the critical heat input rate
related to the boiling limit for a cylindrical heat pipe is given by
awLe kaa Micra
(4.50)
Q6 =
in(Rab)
where R is the inner radius of the pipe wall, and R„ is the vapor space
radius. Equations (4.52) and (4.54) represent the temperature drop across
the wick structure to maintain vapor bubbles of radius Rb at the wick-
wall interface. As can be seen, the boiling limit is very sensitive to Rb,
the effective initial radius of the vapor bubble at its formation, which is
(4.55)
- 14) \12crTeatkt(v!„,
Rb =
h f gqr
(4.56)
HEAT TRANSFER LIMITATIONS 247
dependent ou the wall surface conditions and is affected by the presence of
dissolved gas in the liquid.
The term 1/Renee in eqn. (4.52) or eqn. (4.54) has a significant ef-
fect ou the criticai superheat, which is substantially reduced when 1?,,,, e
is very small and has the same magnitude as Rb. This means that the
existence of the porous structure on the wall prometes boiling incipience in
the wick structure and reduces the MaXiMUM radial heat flux attainable at
the evaporator. Since both Rmen and keff in eqn. (4.55) are dependent ou
wick properties such as the wick geometry, porosity and material, different
types of heat pipe wick structures have different boiling limitations.
While Rmen can be taken to be approximately the effective pore radius,
reff, in the calculation of the boiling limit the values of Rb are very difficult
to estimate accurately due to the extremely complicated surface condition
(Cao and Faghri, 1991). For the approximate calculation, the relation from
the nucleation theory of Bergles and Rohsenow (1964) can be used
where q, is the radial heat flux, kÉ is the liquid thermaI conductivity and 14
and /4 are the specific volume of saturated vapor and liquid, respectively.
The experiment conducted by Ponnappan et al. (1988) for the double wall
artery water heat pipe showed that q, could reach 22 W/cm 2 without a
boiling problem. The Rb estimated from eqn. (4.56) was ou the order of
10-5 m. Woloshun et al. (1990) studied the boiling limitation in heat
pipes with annular gap wick structures. They found a nucleation radius
Rb --- 3 x 10 -6 m predicted their lower bound of the boiling limit data
with acceptable accuracy using their analyticaI model. They also pointed
out that the effective nucleation radius can vary by as much as two orders
of magnitude in experiments with the same wall materiais, and it may
also change with time due to the volatile nature of the boiling process and
changes iii surface conditions and wetting between successive runs.
Experimental data for the boiling limit should be obtained whenever
possible. In the absence of sufficient experimental data, the values of Re
can be taken to be 10 m and 10 -7 m for the gas-loaded and conventional
heat pipes, respectively. It can be seen from eqn. (4.52) or eqn. (4.54)
that a smailer Rb requires a larger AT, making boiling incipience within
the wick more difficult. To reduce Rb and obtain a laxger heat load at the
evaporator without reaching the boiling limit, attention should be paid to
the following points:
1. Good wetting of the heating wall by the liquid.
248 HEAT PIPE SCIENCE AND TECHNOLOGY
2. Achieving good thermal contact of the wick with the wall.
3. Completely degassing and distilling of the liquid and using a clen,
filtered working fluid.
4. Using a smooth heating wall with a high thermal conductivity and
heat capacity.
5. Clean both the container and the wick as completely as possible.
4.4 SONIC LIMITATION
In Chap. 3, the one-dimensional incompressible vapor flow has been dis-
cussed. However, when the vapor Mach number is high, especially when the
vapor flow approached the sonic limitation, vapor compressibility must be
taken into account. A closed-form relation for the sonhe limit was first de-
rived by Levy (1968). By applying the principies of conservation of mass,
momentum, and energy for a one-dimensional vapor control volume, the
following equations were obtained
Av—dz (PW)= Thie
dp d
piTjAcd —2 —2 2
dz
w (w + h _
2 ate 2 2 as
where A„ is the cros,s-sectional area of the vapor flow passage, til is the
average axial vapor velocity over the cross section, rh'e is the mass injection
rate per unit length of evaporator, h is the enthalpy of vapor, hg is the
enthalpy of saturated vapor, and v6 is the normal velocity of the injected
mass Frictional effects have been neglected in the above equations. The
radial mass injection rate Thei and the local axial heat rate per unit length
Q' are related by
The (hf g + 4) (4.60)
Considering a perfect gas, the equation of state is given by
dp _ dp dT
p p T
(4.61)
By using the definition of the Mach number, the equation of state can also
(4.57)
(4.58)
(4.59)
HEAT TRANSFER LIMITATIONS249
be written as
dMa2 cliD 2 CU
ma2 ,t-u2 T (4.62)
From eqns. (4.57)—(4.59), (4.61) and (4.62), the following equation for the
axial gradient of vapor Mach number is obtained
c/Ma2
dz --= Ma
2 [K'Ma2 + 11
1 — Ma2 j
{ ( x (Ma2 (K' — 1) + 2) 2 ( fis) ± w
2cp,„T
2
rh
(4.63)
where K' is the ratio of specific heats and ih is the total axial vapor flow
rate
=
/
2 ifee (4.64)
o
Since the normal velocity of injected mass vs is relatively small and
h9 , the second terms in eqns. (4.60) and (4.63) can be neglected. The
maximum Mach number which can be obtained in the evaporator section is
unity from compressible fluid mechanics. In this case, there is a maximum
axial heat transport rate due to the choked flow corresponding to the axial
temperature drop along the evanorator. For the case of a uniform mass
injection rate, eqn. (4.63) can be integrated to obtain the maximum
mass flow rate and thus the maxinmm heat flow rate Q, corresponding to
Ma = 1
Q,— = Aypohfg[
KR 9 T° 11/2
(4.65)
pocohf g At,
,V2(K' + 1) 2(K' + 1)
where Po, co, and To are density, speed of sound and temperature at the
evaporator end cap defined as the operating condition In heat pipe.
In the above analytical solution, the influence of friction ou lhe operation
of lhe heat pipe has been neglected. Sufficiently simple closed-form relations
describing the effect of friction ou lhe sonic limit have not been obtained.
Levy and Chou (1973) also considered the vapor dissociation-recombination
and homogeneous vapor condensation phenomena in a sodium heat pipo,
and found that neither the dissociation-recombination reaction nor the va-
por condensation has a large influence on the sonic-limit heat transfer rate.
Faghri (1989) developed a generalized one-dimensional compressible vapor
flow analysis including friction, as discussed in Sec. 3.3.2. The single most
. important factor is the wall friction. The friction effects control the location
of the sonic point, determine if the flow in the condenser section is subsonic
250 HEAT PIPE SCIENCE AND TECHNOLOGY
or supersonic, and substantially decrease the sonic-limit heat transfer rate.
compared to those predicted from the inviscid analyses.
The numerical one-dimensional solutions including friction and the ana-
lytical closed-form solution with no friction are compared with experimental
data from Dzakowic et al. (1969) in Figs. 4.16 and 4.17. At higher working
temperatures, the frictionless analytical model agrees relatively well with
the experimental data. At lower working temperatures, however, the an-
alytical results have a large deviation from the experimental data. For a
sodium heat pipe at a working temperature near 500°C, the deviation is
about 20%.
More generally, the solution of the sonic limit should be obtained by
solving the complete two-dimensional compressible mass, momentum and
energy conservation equations including the equation of state in the vapor
core. The governing equations for transient, compressible, laminar vapor
flow with constant viscosity are as follows (Chaps. 3 and 5)
Dt +p • 17) r= O
DV
p
yt
= —Vp + —
1
ity V(V - V) + ji.,,V 2 -17 (4.67)
D 3
DT Dp
Pve" TD—t- = V . kvVT ± I5t -E liv(*
(4.68)
where G is the viscous chosipation. The equation of state is given by
P = py RgT (4.69)
Cao and Faghri (1990) solved the above conservation equations with ap-
propriate boundary conditions using the control volume funte difference
approach. Figures 4.18 and 4.19 show the numerical axial pressure and
Mach number profiles compared with the experimental data by Bowman
(1989). The experimental data were obtained by simulating the vapor flow
of a cylindrical heat pipe with a porous pipe that has an inside diameter of
16.51 mm and a length of 0.61 m. The evaporator and condenser sections
have equal lengths and were simulated by the injection and suction of air
at the interface. The agreement between the experimentai data and the
numericaI solution in Fig. 4.18 is generally good. For the two subsonic flow
cases in Fig. 4.19, the axial velocity increased until it decelerated in the con-
denser due to mass removal. For the supersonic case, mass removal caused
a further acceleration in the condenser until a shock was encountered.
In summary, the approximate analytical solutions for vapor flow (eqns.
(4.57)—(4.59)) and the sonic limit (eqn. 4.65) can be recommended for
Dp,
(4.66)
Frictionless
perfect gas
Reacting flow
with friction
Los Alamos heat pipa
251
102
500 600
700
(z=0) (°C)
Figure 4.16: Sonic limit solutions compared with the Los Alamos heat pipo
data (Levy and Chou, 1973).
approximate calculations of the characteristics of heat pipes. For more
general and accurate results, the numerical solutions of the two-dimensional
compressible conservation equations as described by eqns. (4.66)—(4.69) are
recommended (Chap. 3 presents the steady state models and Chap. 5 gives
the transient models).
The sonic limit does not necessarily represent an operational failure.
When the sonic limit is reached, a further increase in the heat flow can be
achieved by increasing the evaporator temperature. This is especially true
for high-temperature heat pipe startup from the frozen state, where sonic
or supersonic vapor flow will a1ways be encountered due to the low vapor
density. In most cases, sonic vapor flow in a heat pipe is temporary and will
disappear when the heat pipe working temperature rises to a sufficiently
HEAT TRANSFER LIMITATIONS
3 x101
lOs
2 x10'
Westinghouse heat pipe
Frictionless
prefect gas
Reading tlow
with friction
1
252 HEAT PIFE SCIENCE AND TECHNOLOGY
4x10 3
io3
10 2
3x10'
400 500
(z=0) (°C)
Figure 4.17: Sonic limit compared with the Westinghouse heat pipe data
(Levy and Chou, 1973).
high levei. However, for some heat pipes, when the heat transfer coefficient
at the condenser is high or the heat input at lhe evaporator is low, the sonic
or supersonic vapor flow will not disappear when the heat pipe reaches the
steady state. A heat pipe operating at or near the sonic limit will experience
large axial temperature and pressure differences along the heat pipe length,
as shown by Kemme (1969). More importantly, the heat pipo operating at
or near the sonic limit will also experience a large axial pressure difference
along the heat pipe as shown in Fig. 4.18. This large vapor pressure drop
will reduce lhe capillary potentia1 of the heat pipo and consequently reduce
the heat transport capacity described in the next section. Although the
sonic limit is not as catastrophic as other heat pipe limits, it still needs to
be avoided.
600
0.5—
1 ill —Ç r II
-°- -11---0-4-0"---e, la,
.. --n-
N ''\ ..‘%.
-
‘9 O ./.7
\ /
O Experimental data, :á =0.40 Ibmrs
/ \
O /
&Numerical resuLM =0.4016m/s \
/X Experimental data, .h= 0.15 linn/a .5.
a. Numelical result., 41 = O • I 5 To r et / s. li N,,,, /
• Expelimental data, ira =0-.03 Ibmis ° ‘"--- „/
V Numerical reaulkrii = 0.03 Ilaints tf--
a.
D
O ..-"""
Ag
1, e
1
01.2 0.4 0.6
z/L
0.8
Le
o fit = 0.4 lbmis
A ?ft =0.15 Itnnis
e. tu = 0.03 lbads
1 .5 -
0.4 0.6 à o
2
0.5-
HEAT TRANSFER LIMITATIONS 253
Figure 4.18: Axial pressure profiles along the centerline of the simulated
heat pipe (Cao and Faghri, 1990).
4.5 ENTRAINMENT LIMITATION
The vapor and liquid flows within a heat pipe are in direct contact with
each other and flow in opposite directions. Depending ou the properties
of the vapor and liquid phases, at high vapor velocities the shear stress at
the liquid-vapor interface can cause waves to appear ou the free surface of
the liquid. As the vapor velocity increases, the interaction between the two
ilL
Figure 4.19: Axial Macia number profiles along the centerline of the simu-
latedheat pipe (Cao and Faghri, 1990).
254 HEAT PIPE SCIENCE AND TECHNOLOGY
phases also increases, and the amplitude of the waves on the free surface
become larger. Eventually, droplets form on the wave crests, which are
entrained into the countercurrent vapor flow. The droplets are carried
back toward the condenser end cap, having never reached the evaporator
section. This essentially short-circuits the liquid return path and causes
the onset of evaporator dryout, due to the decrease in condensate reaching
the evaporator section. Shortly after the invention of the heat pipe, Cotter
(1967) identifled entrainment as one of the limitations in operating heat
pipes, and developed a criteria for the occurrence based on a Weber number
of unity.
Since the vapor velocity is generally several orders of magnitude larger
than that of the liquid, the shear stress at the liquid-vapor interface is
mainly due to the vapor flow. The shear force imposed by the vapor on the
liquid-vapor interface is proportional to the dynamic pressure of the vapor
flow, and the area of the liquid exposed to the vapor flow
pv -,02 Aev
Fe, = cl (4.70)
2
where C1 is a proportionality constant, Pv is the vapor density, D is the
mean axial vapor velocity, and At, is the area of the individual surface pores
of the wick. The surface tension force maintaining flue liquid in the wick
is proportional to the surface tension coefflcient and the wetted perimeter
around the area of free surface exposed to the vapor flow
F = C2 ceP (4.71)
where C2 is the proportionality constant, a is the coefficient of surface
tension, and P is the wetted perimeter of the individual surface pores.
When the ratio of these two forces, the Weber number, We, is on the order
of unity, the entrainment limit is reached.
CV1,1-52 Ae, We = — 1 (4.72)
2C2oP
Kemme (1967) and Wright (1970) give experimental data on the value of
the ratio of the proportionality constants, e1/C2, which is approximately
8. By deflning the hydraulic radius of the wick surface pore as
2A.e,
eqn. (4.72) can be written as
We —
2Rh
'
wp,7,12
= 1
a
(4.73)
(4.74)
HEAT TRANSFER LIAHTATIONS 255
For screen wicks, Rh, w =- 0.5W, for axial grooves, Rh,v,= W, and for
packed sphere wicks, Rh,„, =- 0.205D, where W is the wire spacing for
screen wicks and the groove width for axial grooves, and D is the sphere
diameter for packed sphere wicks. In terms of the axial heat flow rate, the
entrainment limit is given by
1/2
Qent Anhfg CYPv (4.75)
2Rh,10
This phenomenon was first described experimentally for capillary-driven
heat pipes by Kemme (1967), where the sound of liquid droplets were heard
from the condenser end cap of the heat pipe. The evaporator section quickly
overheated after the droplets began to impinge ou the condenser end cap.
However, it was shown later by Kemme (1976) that the performance in
these experiments was probably limited by another vapor flow effect. Thus
lar, the only direct evidenc,e of an entrainment limitation was obtained
in vertical performance tests with gravity-assisted heat pipes, which are
discussed in Chap. 6. In contrast to the evidence of many experimental
data ou entrainment in gravity-assisted heat pipes, the study of entrainment
in capillary-driven heat pipes is not observed experimentally. It is doubtful
that the entrainment really occurs in conventional capillary-driven heat
pipes because the capillary structure would most likely retard the growth
of any surface waves. Several investigators (Tien and Chung, 1978; Rice and
Fulford, 1987) later developed new equations for the entrainment limit, but
none have been confirmed with experimental data on conventional capillary-
driven heat pipes.
4.6 VISCOUS LIMITATION
The sonic and viscous limas to axial heat transport are described by an-
alyzing the conservation equations and the appropriate equation of state,
where the vapor is assumed to be a perfect gas These two limits are at
opposite ends of the spectrum which includes the effects of viscous and iner-
tia forces on the vapor flow. In the viscous regime, where inertia forces are
negligible, the limitation to the heat transport is due to the fact that the
vapor pressure at the cnndenser end cap cannot be less than zero (viscous
limit). In the inertia regime, where viscous forces are small compared to
inertia forces, the heat transfer is limited by the choking phenomena (sonic
limit).
The sonic limitation is discussed in Sec. 4.4. Some analysis of the sonic
limitation is also given here in conjunction with the viscous limitation. The
analysis (Busse, 1973) is carried out for laminar flow of a isothermal perfect
256 HEAT PIPE SCIENCE AND TECHNOLOGY
gas within a cylindrical heat pipe, which has a length that is large compared
to its diameter. Also, the heat input and output is uniform in the azimuthal
direction. The equation of state of an isothermal perfect gas at a constant
temperature of To is
P Po HuTo - — - (416)
P Po
where Ra is the universal gas constant, M is the average molar mass, and
the subscript O refers to the evaporator end cap It is assumed that the
amount of heat required to increase the temperature of the vapor is small
compared to that needed to vaporize the condensate in the wick, and can be
neglected. h is siso assumed that the temperature dependence of the heat
of vaporization, hf g, is negligible. The average axial heat flux transported
by the heat pipe can be expressed as follows
= prnvhfg (4.77)
where the overbar denotes the average over the cross section. The Navier-
Stokes equations can be simplified by assumptions similar to Prandtl's
boundary layer theory. From the solution to these approximate conser-
vation equations, the following results can be obtained (Busse, 1973):
1. The radial pressure distribution is constant, and from eqn. (4.76), so
is the radial density profile.
aP aP (4.78)
2. The axial velocity profde is parabolic ia the viscous flow regime (w
1- 4r2 /D), where D„ is the diameter of the vapor core. In the inertia
flow regime, the velocity profile is proportional to cos(27rr 2 //n), and
becomes steadily flatter towards the evaporator exit, where the sonic
limit occurs.
3. In the viscous flow regime, the pressure gradient is
dp 32g121(z)
dz
In the inertia flow regime, the pressure drop is
(4.79)
P o - P( z) = p(z)w2 (z) (4.80)
The occurrence of w 2 in eqn. (4.80) requires the determination of the
ratio A' w 2 /T72 , which was found by Busse (1973) to be a weak function
of the pressure ratio p o /p• From eqns. (4.76)-(4.80), the heat transport
limits in the viscous and inertia flow regimes can be found.
HEAT TRANSFER LIMITATIONS 257
VISCOUS HEAT TRANSFER LIMIT
The pressure gradient in the axial direction can be cast in terms of the
heat transport by combining eqns. (4.76), (4.77), and (4.79) and taking
eqn. (4.78) into account
dp 32p po ri
P dz D.?, h fg P0
Integrating along the length of the heat pipe gives
voz _ p2L, 6D4:: h fP:so o ifo Lt 4(z) dz (4.82)
where L refers to the total length of the heat pipe. The effective length of
the heat pipe is defined as
1
Lt
Leff Ez) dz
qmax o
where 0,,,,a„ is the maximum value of the axial heat flux. The effective
length can be calculated for a given heat input and output distribution.
With this definition, eqn. (4.82) can be solved for flia.„•
_
=
Mhfoopo ( 1 pit )
grila% 2 64gLe ff Po
The viscous he,at transport limit is reached when the pressure at the con-
denser end cap approaches zero, i.e., pts /po = O.
D2„//fg pepe
4we
64pLeff
SONIC LIMIT
Combining eqns. (4.76), (4.77) and 4.80), the axial heat flux can be de-
termined.
(4.81)
(4.83)
(4.84)
(4.85)
ri= hf
--2 1/2
PoPow (1 P
w2 po 1)0
(4.86)
It can be seen that has a maximum value at the point where choking
occurs. This maximum, which isthe sonic limit, can be found by setting
258 HEAT RIPE SCIENCE AND TECHNOLOGY
cki/dp = O.
1[1 (po cilln(r7/W2 )11
Po) s 2 1_ ) d(PoiP)
The relationship between w 2 /rd2 and po/p is given be Busse (1973) by inte-
grating the axial momentum equation over the r-direction. An approximate
solution was derived for compressible flow.
—
w2
= 1 234 – 0.358g + 0.20692 (4.88)
{x – 0.310 –
(4.89)
– 0.6691
= 214 . (4.89) g
x – 0.765
(Po/P -1) x (4.90)
ln(po/p)
Equation (4.87) can be solved iteratively using eqns. (4.88)–(4.90)
(Po —) = 2.08 (4.91)
P s
( w2
=3 = 1.11
w
(4.92)
s
Substituting these values into eqn. (4.86) gives the sonic heat transport
limit.
= 0.4741ti gS, (4.93)
The dependence of the viscous and sonic limits ou temperature is due to
the factor popo, which increases dramatieally with temperature. The vis-
cous and sonic limits are therefore proportional to T and IT. At low
temperaturas, the heat transport will be limited by the viseous effects,
while at higher temperatures the heat transfer is limited by the choking
phenomenon. A transition from the viscous limit to the sonic limit occurs
when the maximum heat transport described by eqns (4 85) and (4.93)
are of the same magnitude, which is dependent ou temperature.
(Ler ) VPoPo
– 0.033 (4.94)
tr 1-1 )tr
Using eqn. (4.76) and solving for the transition temperature gives
2 ,
(TA, = 918.2—
M
(. 1.i) efeff ) 2
po
(4.87) •
and
(4.95)
c e e II 1 1 le 1 1 1 I
1 Sonic heat transfer limitation
IViscous heat transfer limitativa
I 1/1 L 1 I II 1 1 II
HEAT TRANSFER LIMITATIONS 259
Figure 4.20 shows the transition temperature (To)t,. versus Leff irn for Na,
K, and Cs (Busse, 1973). In Fig. 4.21, the above analysis is compared to the
experimental resuhs obtained by Kemme (1969) with Na, K, and Cs heat
pipes. The experimental data agree closely with the theoretical sonic limit
except at low temperatures, especially with Na. The curve intersecting
lhe sonic limit is the viscous limit with the assumption of a transition
temperature of (To)tr = 525°C. While a direct calculation of L ett is not
possible from the data given by Kemme (1969), it is obvious that another
limitation is occurring at lower temperatures. More experimental data is
needed in the lower temperature range to verify the presence of the viscous
limit.
4.7 CONDENSER LIMITATION
Condenser heat transfer limitations are directly associated with the con-
denser heat dissipation capabilities. When a heat pipo reaches the steady
state, the heat input in lhe evaporator section is equal to the heat output
in lhe condenser section. For a high temperature heat pipe with radiation
as lhe major mede of heat transfer, this requires that
800
700
o
600
1 500
400
o
'rol 300
200
10 102 1 o3
L eo- /A? (em1 )
Figure 4.20: Transition temperature from viscous to sonic limitation (Busse,
1973).
100
260 HEAT PIRE SCIENCE AND TECHNOLOGY
o'
10
200 500
1000
Temperature ai the heginning
of the heating zone, 1 (°C)
Figure 4.21: Comparison of theoretical heat transport limits with experi-
mental data (Busse, 1973).
Q.= ff ea(T4 dSc (4.96)
where Q, is the total heat input in the evaporator section, S, is the con-
denser heat transfer arca, E is the emissivity of condenser outer surface, a
is the Stefan-Boltzmann constant, and T the ambient temperature. For
a cylindrica1 heat pipe, the above equation can be reduced to
Q. = 2/rR0Lcea (T4 ito (4.97)
where Ro and Le are the outer pipe wall radius and length of the condenser
section, respectively, and T is the average temperature. For a high temper-
ature test system with a concentric calorimeter, eqn. (4.96) can be written
as
Qe
—
_1 + ( 1 à
E I, D2 k, C2 ,/
Sco- (T4 —
(4.98)
HEAT TRANSFER LIMITATIONS 261
where T2 is the calorimeter wall temperature, El is the condenser wall emis-
sivity, D1 is the heat pipe diameter, D2 is the calorimeter inner diameter,
and E2 is the calorimeter wall emissivity.
According to the eguations above, the heat transfer rate is highly de-
pendent ou the heat transfer surface arca and the operating temperature,
22. In practical applications, the condenser heat transfer surface area and
the heat pipe operating temperature are often subject to many design coa-
straints, suei' as the operating environment and the maximum allowable
temperature of the heat pipe materiais. As a result, the heat transport
capacity of a heat pipe may be limited by the heat dissipation capacity
of the condenser. An example is given by the high temperature heat pipe
experiment conducted by Buchko (1990), and is shown in Fig. 4.22.
The condenser heat transfer limit is not only related to the high tem-
perature heat pipe with radiation as major heat transfer mode. For a low
temperature heat pipe with convection as the major heat transfer mode,
the energy balance at the steady state reguires that
Qe Se h(T, — Too ) (4.99)
where h is the heat transfer coefficient between the condenser outer surface
and the cooling fluid, and no is the temperature of cooling fluid. When
the heat transfer coefficient ft is low, such as the case when a heat pipe
dissipates heat to the ambient by natural convection, the heat transfer
800
Test environrnent
o Vacuom
• Air
600-
200 -
/ Sonie limit r - - - - - .,, . Capifiaty
limit
I Entrainment Condenser
r— hunt . radiation
....
-- \ ,
. / uniu \..-
.• .0 , o
/ ...# /lo ....-
." ..
/.. .' ...... 1.0-•
... .......
V
,„.•
....... :.$'' ...... .... ..-
0
300 350 400 450 500 550
Transport vapor temperature,T('C)
Figure 4.22: Axial heat transfer limits versus transport vapor temperature
for operation in air and vacuurn.
600
262 HEAT PIPE SCIENCE AND TECHNOLOGY
capacity of the heat is also limited by the condenser heat transfer limit.
Faghri (1987) showed that a much better performance could be obtained
by installing circular fins along the condenser section of a copper-water heat
pipe compared to a conventional cooling jacket under positive and negative
tilt angjes.
4.8 CONTINUUM FLOW LIMITATION
The vapor flow in the heat pipe core is usually in the continuum state for
conventional heat pipes. However, as the size of the heat pipe decreases,
the vapor in the heat pipe may lose its continuum characteristics. The heat
transport capability of a heat pipe operating under non-continuum vapor
flow conditions is very limited, and large temperature gradients exist along
the heat pipe length. As a result, the heat pipe will lose its advantages as
an effective heat transfer device (Cao and Faghri, 1993). This is especially
true for a miniature or micro heat pipe, whose dimensions may be extremely
small.
The continuum criterion is usually expressed in terms of the Knudsen
number
À 0.01 continuum vapor flow
Kn = h =
{ <
-> 0.01 ratefied or free molecular flow
(4.100)
where À is the mean free path of the vapor molecule, and D is the minimum
dimension of the vapor flow passage. For a circular vapor space, D is the
vapor core diameter. The mean free path based on the kinetic theory of
dilute gases is
1.051KT
À (4.101)
V2itn2p
where tç is the Boltzmann constant, a is the collision diameter, and p is the
vapor pressure. By combining eqns. (4.100) and (4.101) and applying the
equation of state, p = pRgT, one can obtain the transition density from
continuum vapor flow to rarefied or free molecular vapor flow as
1.051K
P —
-‘51ra 2 RgDKn
(4.102)
Assuming that the vapor is in the saturation state, the transition vapor
temperature corresponding to the transition density can be obtained by
using the Clausius-Clapeyron equation combined with the equation of state
(Cao and Faghri, 1992)
HEAT TRANSFER LIMITATIONS 263
psat exp hfg 1 1 \
(4.103)
pR9Eg Tsat )1
where psat and Tsat are the saturation pressure and temperature, h jg is the
latent heat of evaporation, and the vapor density p is given by eqn. (4.102).
Equation (4.103) can be rewritten as
(TtrpR9 ) hfs, 1 1 )
ln
Rg Tsat
(4.104)
Psat
and solved iteratively for Ter using the Newton-Raphson/sec,ant method.
For a heat pipe with a very smafl minimum dimension D, the transition
temperature Ti,. is rather high. A heat pipe working below this transition
temperature will be subject to the continuum flow limitation. The temper-
ature gradient along the heat pipe in this case is very large, and the heat
pipe will lose its isothermal characteristics. Applications of eqn. (4.104) to
micro/miniature heat pipes are discussed in Chap. 10.
4.9 FROZEN STARTUP LIMITATION
Under normal working conditions, the working fluid in the wick structure
of a heat pipe is in the liquid state, and flows from the condenser section
to the evaporator section due to the capillary pumping action in the heat
pipe. However, if a heat pipe is started from the ambient temperature, the
working fluid in the wick structure may be in the solid state, depending ou
the speciflc working fluid used. For low or mediam temperature heat pipes,
the worlcing fluid is usually in the liquid state at the ambient temperature.
For high temperature heat pipes on the other hand, the working fluid ia the
heat pipe is usually ia the solid state at ambient temperature, due to the
high melting temperature of the working fluid. Therefore, frozen startup is
a routine occurrence during the high temperature heat pipe operation.
Prior to heat pipe startup, the working fluid In the wick structure is
ia the solid state, and the heat pipe core is nearly evacuated. After the
application of heat to the evaporator section, the temperature of the evap-
orator begins to increase. However, the temperature in the rest of the heat
pipe may still be at the ambient temperature. When the temperature
ia the evaporator section exceeds the melting temperature of the working
fluid, the working fluid liquefles and evaporation begins to take place at the
wick-vapor interface. The vapor flows from the evaporator to the adiabatic
and condenser sections and is condensed at the wick-vapor interface, re-
leasing its latent heat energy. The vapor condensed at the liquid-saturated
wick surface may flow back to the evaporator section due to the capillary
264 HEAT PIFE SCIENCE AND TECHNOLOGY
pumping action of the liquid-saturated wick structure. However, the vapor
condensed onto the frozen wick structure may be frozen out, and not be
able to flow back to the evaporator section. At the same time, the working
fluid in the wick structure dose to the evaporator liquefies due to axial heat
conduction, and may flow back to the evaporator section, which increases
the amount of liquid available for vaporization. These two processes deter-
mine if a particular heat pipe can start successfully. A mass balance over
the liquid-saturated wick region results in the following equality (Cao and
Faghri, 1992)
wpeA'w hfg >1
C(Trnei - no) -
where cp is the porosity of the wick structure, pt is the liquid density of
the working fluid, 24',, is the cross-sectional area of the working fluid in the
wick, hf g is the latent heat of evaporation, C is the heat capacity per unit
length of heat pipe wall and wick, Tmei is the melting temperature of the
working fluid, and To.o is the initial or ambient temperature of the heat pipe.
When the above equation is not satisfied, the amount of liquid in the liquid-
saturated region begins to reduce, and eventuaIly the liquid in this region
may be depleted. In this case, dryout will occur in the evaporator region,
and the frozen startup limit will be reached. A more detailed discussion of
the frozen startup limitation is given in Sec. 5.3.6.
REFERENCES
A.E. Bergles and W M. Rohsenow, 1964, ASME J. Heat Transfer, Vol.
86, p. 365.
W.J. Bowman, 1987, "Simulated Heat Pipe Vapor Dynamics," Ph.D.
Dissertation, Air Force Institute of Technology, Dayton, OH.
M.T. Buchko, 1990, "Experimental and Numerical Investigations of Low
and High Temperature Heat Pipes with Multiple Heat Source and Sinks,"
Master's Thesis, Wright State University, Dayton, OH.
C.A. Busse, 1973, "Theory of the Ultimate Heat Transfer Limit of Cylin-
drical Heat Pipes," Int. .1. Heat Mass Transfer, Vol. 16, pp. 169-186.
Y. Cao and A. Faghri, 1990, "Transient Two-Dimensional Compress-
ible Analysis for High-Temperature Heat Pipes with Pulsed Heat Input,"
Numerical Heat Transfer, Part A., Vol. 18, pp. 483-502.
Y. Cao and A. Faghri, 1991, "Transient Multidimensional Analysis of
Nonconventional Heat Pipes with Uniform and Nonuniform Heat Distribu-
tions," ASME J. Heat Transfer, Vol. 113, pp. 995-1002.
Y. CEIA and A. Faghri, 1992, "Closed-Form Analytical Solutions of
High-Temperature Heat Pipe Startup and Frozen Startup Limitation,"
ASME J. Heat Transfer, Vol. 114, pp. 1028-1035.
(4.105)
HEAT TRANSFER LBIITATIONS 265
Y. Cao, and A. Faghri, 1993, "Micro/Miniature Heat Pipes and Oper-
ating Limitations," Proc. ASME National Heat Transfer Conf., Atlanta,
Georgia, ASME HTD-Vol. 236, pp. 55-62.
S.W. Chi, 1976, Heat Pipe Theory and Practice, Hemisphere Publishing
Corporation, Washington, D.C., p. 91.
J.G. Colher, 1972, Convective Boiling and Condensation, Second Edi-
tion, McGraw-Hill Book Company.
T.P. Cotter, 1967, "Heat Pipe Startup Dynamics," Proc. 1967 Thermi-
onic Conversion Specialist Conf., Paio Alto, CA., pp. 344-348.
G. Dzakowic, Y. Tang, and F. Arcella, 1969, "Experiment Study of
Vapor Velocity Limit in a Sodium Heat Pipe," ASME Paper No. 69-HT-
21.
A. Faghri, 1987, "Condenser Performance of an Axially-Grooved Heat
Pipe," Proc. 6th Int. Heat Pipe Conf., Grenoble, France, Vol. 1, pp.
142-145.
A. Faghri and S. Thomas, 1989, "Performance Characteristics of a Con-
centric Annular Heat Pipe: Part 1—Experimental Prediction and Analysis
of the Capillary Limit," ASME J. Heat Transfer, Vol. 111, pp. 844-850.
A. Faghri, 1989, "Performance Characteristics of a Concentric Annular
Heat Pipe: Part II—Vapor Flow Analysis," ASME J. Heat Transfer, Vol.
111, pp. 851-857.
A. Faghri and M. Buchko, 1991, "Experimental and Numerical Analysis
of Low-Temperature Heat Pipes with Multiple Heat Sources," ASME J.
Heat Transfer, Vol. 113, pp. 728-734.
J.K. Ferrei', and J. Alleavitch, 1970, "Vaporization Heat Transfer in
Capillary Wick Structures," Citem. Eng. Symp. Series, No. 66, Vol. 2.
M.N. Ivanovskii, V.P. Sorolçin, and IN. Yagodkin, 1982, The Physical
Principies of Heat Pipes, Clarendon, Press, Oxford, p. 130.
J.E. Kemme, 1967, "High Performance Heat Pipes," Proc. 1967 Ther-
mionic Conversion Specialist Conf., Falo Alto, CA.
J.E. Kemme, 1969, "Ultimato Heat Pipe Performance," Trans. Electron
Devices, ED-16, p.717.
J.E. Kemme, 1976, "Vapor Flow Considerations in Conventional and
Gravity-Assisted Heat Pipes," Proc. 2cd Intl. Heat Pipe Conf., Bologna,
Italy, pp. 11-22.
E.K. Levy, 1968, "Theoretical Investigation of Heat Pipes Operating
at Low Vapor Pressures," ASME J. Engineering Industry, Vol. 90, pp.
547-552.
E.K. Levy, 1971, "Effects of Friction on the Sonic Velocity Limit in
Sodium Heat Pipes," Proc. 6th AIAA Therrnophysics Conf.
E.K. Levy, and S.F. Chou, 1973, "The Sonic Limit in Sodium Heat
Pipes," ASME Heat Transfer, pp. 218-223.
266 HEAT PIPE SCIENCE AND TECHNOLOGY
P.J. Marto and V.J. Lepere, 1982, "Pool Boiling Heat Transfer from
Enhanced Surfaces to Dielectric Fluids," ASME J. Heat Transfer,Vol. 104,
pp. 292-299.
S. Nalciyama, 1934, "Maximum and Minimum Vaiues of Heat Trans-
mitted from Metal to Boiling Water Under Atmospheric Pressure," J. Soc.
Mech. Eng., Japan, Vol. 37, p. 367.
E.C. Philips and J.D. Hinderman, 1967, "Determination of Capillary
Properties Useful in Heat Pipe Design," Proc. ASME-AIChE Heat Transfer
Conf., Minneapolis,Minnesota.
Ponnappan, M.L. Ramalingam, J.E. Johnson, and E.T. Mahefkey,
1988, "Analysis of Criticai Heat Flux in the Double Wall Artery Heat Pipe
Evaporator," 261h Aerospace Sei ente Meeting, Reno, Nevada, AIAA Paper
No. AIAA-88-0356.
F. Reiss et ai., 1968, "Pressure Balance and Maximum Power Density at
the Evaporator Gained from Heat Pipe Experimenta," 2nd Int Syraposium
Ort Thermionic Electrical Power Generation, Stresa, Italy.
G. Rice and D. Fulford, 1987, "Influente of a Fine Mesh Screen on
Entrainment in Heat Pipes," Proc. 6th HW Heat Pipe Conf., Grenoble,
France, pp. 243-247.
W.M. Rohsenow, J.P. Hartnett, and E.N. Cante, Eds., 1985, Handbook
of Heat Transfer Fundamentais, McGraw-Hill, New York, p. 12.31.
J. Schmalhofer and A Faghri, 1993, "A Study of Circumferentially-
Heated and Block-Heated Heat Pipes: Part 1—Experimental Analysis and
Generalized Analytical Prediction of the Capillary Limit," Int. J. Heat
Mass Transfer, Vol. 36, pp. 201-212.
C.L. Tien and K S Chung, 1978, "Entrainment Limits in Heat Pipes,"
AIAA Paper No. 78-382.
K. Woloshun, J. Sena, E. Keddy, and M Merrigan, 1990, "Boiling Limits
in Heat Pipe with Annular Gap Wick Structures," Proc. AIA A/ASME 5th
Joint Therrnophysics and Heat Transfer Conf., AIAA 90-1793.
P.E. Wright, 1970, "ICICLE Feasibility Study," Final Report NASA
Contract No. NAS5-21039, RCA, Camden, N.J.Mais conteúdos dessa disciplina
Associação de Situações com Leis de Newton- conteudo-programatico-tecnico-judiciario-2025
- texto motores
- Sistemas Estruturais De Concreto Q04
- ANEXO I CONTEÚDO PROGRAMÁTICO
- ATIVIDADE 1 - EELE - GERAÇÃO DE ENERGIA ELÉTRICA - 54_2025
- c) Vantagens diretas para o usuário do uso deste combustível
- Determine a temperatura do gás nas saídas do compressor e da turbina sendo que a entalpia do gás na saída do compressor é de 544,35 kJ kg
- Imagine que você atuando como engenheiro e dando consultoria sobre geração de energia, quando chega um cliente querendo
- Para o processo de compressão Para o processo de adição de calor Para o processo de expansão
- Para o processo de rejeição de calor O trabalho do ciclo calcula-se como Para o ciclo, a eficiência é determinada a partir de Razão de pressão
- Você encontrou os seguintes parâmetros no livro e quer descobrir outros parâmetros Uma usina a gás que opera em um ciclo Brayton
- ual lei da termodinâmica fundamenta a operação dos equipamentos de refrigeração? A Lei Zero B 1a lei C 2a lei D 3a lei
- Com base nos motores de ciclo Otto
e de ciclo diesel, associe
- s motores podem ser classificados com base em várias das suas características, como o combustível utilizado e a forma de ignição. Um desses tipos d...
- Aparelhos de ar-condicionado dão 0 conforto térmico tão necessário durante períodos mais quentes do clima. O resfriamento por meio dos condicionadores
- Uma máquina térmica opera nas temperaturas 340 K e 580 K. Para determinar a efici- ência dela, de acordo com O ciclo de Carnot, só precisamos Pesquisa
- compressores de ar condicionado residencial quais são
- Segundo a NR 13, caldeiras da categoria B têm pressão de operação superior a 60 kPa (0,61 kgf/cm2) e inferior a 1.960 kPa (19,98 kgf/cm2), com volume
- I) Curve A describes the behavior of a non-azeotropic mixture in which component 1 is more volatile than component 2. True, this statement is correct,
- Considere as afirmações abaixo e indique as assertivas corretas: Quando ocorre uma transferência de calor para o sistema, o calor Q adquire valor nega
- Sabendo que, 1kg de água a 100°C foi convertido em vapor à pressão atmosférica padrão (1,01 x 105 Pa). A variação do volume de água (líquida) foi de 1
- Existe uma tendência dos gases circulantes em depositar cinzas e fuligem. Tais partículas se acumulam em anteparos e nas superfícies de troca de calor
- Separadores e purificadores de vapor impedem que água líquida se encaminhe junto com o vapor extraído. Sobre esses componentes, podemos afirmar que: A
- LUVA SEGURANÇA, MATERIAL: LÁTEX NATURAL, TAMANHO: ÚNICO, APLICAÇÃO: MANUSEIO DE AGENTES ABRASIVOS E ESCORIANTES, CARACTERÍSTICAS ADICIONAIS: SUPERFÍCI
- Anexo 10 (PDF) RT05 - MEDIÇÃO DE RESISTÊNCIA DE ISOLAÇÃO EM MOTORES DE INDUÇÃO
- Anexo 7 (PDF) RT02 - ENSAIO EM CURTO-CIRCUITO DE UM TRANSFORMADOR TRIFÁSICO
