Cap 2_Solid_Liquid_Vapor Phenomena do livro HEAT PIPE SCIENCE AND TECHNOLOGY, do autor Amir Faghri

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 2 
SOLID-LIQUID-VAPOR 
PHENOMENA, 
DRIVING FORCES 
AND INTERFACIAL 
HEAT AND MASS 
TRANSFER 
2.1 INTRODUCTION 
The purpose of this chapter is to discuss various interfacial phenomena, such 
as capillarity and disjoining pressure, and their impact on the pressure and 
temperature differences generated in a heat pipe wick structure. Capillarity 
can be defined as the macroscopic motion or flow of a liquid resulting from 
the surface free energy and forces generated within the pore structures of 
the wick at the surface of the liquid. These driving forces are manifested 
by a pressure differential across the length of the heat pipe wick with the 
direction of flow determined by the direction of decreasing capillary pressure 
potential. The pressure difference which causes the capillary flow is due to 
the variations in curvature and/or the surface tension at the liquid interfaces 
in the different regions (i.e., evaporator and condenser) of the heat pipe. 
The pressure difference is also due to the body forces, hydrodynamic forces, 
61 
62 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
phase-change interactions, and disjoining pressure losses. The basic factors 
which define the capillary driving potential are the forces acting ou the 
system momentum, and the contact angle. Details of these factors as well 
as the interfacial momentum, heat, and mass transfer during evaporation 
and condensation ia heat pipes are also discussed. The significance of the 
original literature on capillarity and the disjoining effect is more related 
to driving forces or potentiaLs and equilibrium shapes than with the fluid 
dynamics of the phenomena. An effort is made in this chapter to cover 
both aspects equally in reference to the operation of heat pipes. 
2.2 PHYSICAL SURFACE PHENOMENA, 
CAPILLARY AND DISJOINING 
FORCES 
2.2.1 SURFACE TENSION 
When a liquid is in contact with another medium, be it liquid, vapor, 
or solid, a force imbalance occurs at the boundaries between the differ-
ent phases. For example, a liquid molecule surrounded by other liquid 
molecules will not experience any resultant force since it will be attracted 
ia ali directions equally. However, if the same liquid molecule is at or near 
a liquid-vapor interface, then the resultant molecular attraction of a liquid 
molecule on the surface would be in the direction of the liquid, since forces 
between the interacting gas and liquid molecules are less than the forces 
between the liquid molecules. It is basically due to the asymmetry of the 
force field acting ou a molecule on the surface tending to pull it back to 
a higher density region or phase. If a liquid is bounded by its own vapor, 
then the force in the surface layer is directed into liquid because, in gen-
eral, the liquid is more dense than the vapor. As a result of this effect, 
the liquid will tend toward the shape of minimum area and behave like a 
rubber membrane under tension. In this context, if the surface area of the 
liquid is to be increased, then negative work must be done on the liquid 
against the liquid-to-liquid molecular forces. Any increase in the surface 
area will require movement of molecules from the interior of the liquid out 
to the surface. The work or energy required to increase the surface area 
can be obtained from the following relation, which is also the definition of 
surface tension 
(a.9Es 
a — (2.1) 
SOLID-LIQUID-VAPOR PHENOMENA 	 63 
where E is the surface fite energy, S is the surface arca, and ri, is the num-
ber of moles for the ith component for multicomponent systems. Equation 
(2.1) is valid for solid-liquid, solid-vapor, liquid-vapor, and liquid-liquid 
interfaces. Liquid-liquid interfaces are present between two immiscible liq-
uids, such as oil and water. 
From this expression, the surface tension can be described as a funda-
mental quantity which characterizes the surface properties of a given liquid. 
Additionally, the surface tension is referred to as free energy per unit area 
or as force per unit length. Surface tension exists at all phase interfaces; 
i.e., solid, liquid and vapor. Therefore, the shape that the liquid assumes is 
determined by the combination of the interfacial forces of the three phases. 
The surface in which interfacial tension exista is not two-dimensional, but 
three-dimensional with very small thickness. In this very thin region, prop-
erties differ from the bordering bulk phases. 
2.2.2 ANGLE OF CONTACT 
The angle of contact of a liquid interface is dependent only upon the physical 
properties of the three contacting media (solid, liquid and fluid), and is 
independent of the container shape and gravity. The fiuid can be another 
liquid or a vapor. For the general discussion in this chapter, we refer to 
the second fiuid as the vapor phase unless it is specified otherwise. The 
three surface tension forces are applied to the lime of contact of the three 
phases. Each of these forces is directed tangentially along the surface of 
contact of the two respective media. Figure 2.1(a) shows a concave (acute 
contact angle) surface which results from a wetting fluid. Figure 2.1( b) 
shows a convex (obtuse contact angle) surface which results from a non-
wetting fiuid. The contact angle is measured through the liquid. The 
surface tensions resulting from the media interactions axe denoted by 
as ,,, and cra,,, where s, E, and v correspond to the solid, liquid, and vapor 
phases, respectively. Mathematically, they exist along a une. Physically, 
they exist in each phase within a few molecular diameters of the other two 
phases. The surface forces act tangentially at the interface. It should be 
noted that surface tension is a scalar quantity. Like pressure, however, a 
direction is usually associated with surface tension. The contact angle can 
be determined by considering the force balance of the surface tensions at 
the three-phase boundary lime. 
To provide a clearer picture of the interfacial forces and the relationship 
of these forces to wetting and non-wetting liquids, consider a drop of liquid 
on a plane surface (Fig. 2.2(a)). The angle between the solid-liquid (s,P) 
interface and the liquid-vapor (1,v) interface is denoted as O. If the drop is 
allowed to move through a small virtual displacement (from point a to point 
64 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
Key 
= Solid 
ti -= Vapor 
L = Liquid 
e = Contact Angle 
Figure 2.1: Meniscus shape at a solid wall: (a) Wetting liquids; (b) Non-
wetting liquids 
b), then the v interface caia be considered to move parallel to itself (Fig. 
2.2(b)). The increase of the v interface ia the arca per unit distance along 
the boundary une where the three plisses meet is bc = ab cos B. The s,,e 
and s, v interfaces are increased and decreased, respectively, by the distance 
ab. For equilibrium conditions to hold, the total change In the forces must 
be zero. Taking the summation of forces at the three-phase boundary une 
gives 
— ct si„AS + cr s,e s' + 	AS;,„ = O 	 (2.2) 
where AS's , is, for example, flue change in surface area per unit length along 
the solid-vapor interface. Applying the results of the above diocussion gives 
Vapor 
Liquid 
e 
Solid 	b a Solid 
(a) 	 (b) 
Figure 2.2: Drop of liquid on a plane surface: (a) Virtual displacement; (b) 
Magnifiecl view of the liquid and solid boundaries 
(a) (b) 
SOLID-LIQUID-VAPOR PHENOMENA 	 65 
as,v =- as,e + us, v cos 	 (2.3) 
The angle of contact is then defined as 
= 005-1 ( Crs 'v 	— Caie ) 	 (2.4) 
Cr e,v 
If crs ,,, > as , e , then the angle of contact will be acute; i.e., a wetting condi-
tion (Fig. 2.1(a)). For as , v <a8 , , then lhe angle of contact is obtuse and 
the liquid is called non-wetting (Fig. 2.1(b)). The best lmown examples of 
wetting and non-wetting liquids, respectively, are water andmercury ia a 
glass container. For the situation in which complete wetting occurs, O = O, 
and the liquid spreads over the surface without reaching an equilibrium con-
dition. Similarly, for a complete non-wetting case, O = ir, and equilibrium 
is again not reached. The angle of contact or capillary behavior will be af-
fected by surface-active agents, surface roughness and electrostatic charges. 
The wetting characteristics of a liquid with any given solid is an important 
consideration for heat pipe design. It will be shown in the next section why 
wetting liquids are always used in heat pipes. While the minimum wetting 
contact angles can be predicted, experimental data are often used in heat 
pipe analyses. In Table 2.1, some data on the minimum wetting contact 
angles for different solidfliquid combinations obtained by Stepanov et al. 
(1977) are reproduced. 
2.2.3 CAPILLARY PRESSURE 
The term capillarity as related to heat pipes is defined as the flow of a liquid 
under the influence of its own surface and interfacial forces. The flow is ia 
the direction of the decrease of capillary pressure. The capillary pressure 
difference causing flow is generated by the differences ia curvature along 
Table 2.1: Minimum wetting contact angles O rne„,„, ft, (are degrees) (The 
upper value is for advancing and the lower is for receding liquid fronts) 
(Stepanov et al., 1977) 
Acetone Water Ethanol R-113 
Aluminum 73/34 
Beryllium 25/11 63/7 0/0 
Brass 82/35 18/8 
Copper 84/33 15/7 
Nickel 16/7 79/34 16/7 
Silver 63/38 14/7 
Steel 14/6 72/40 19/8 16/5 
Titanium 73/40 18/8 
66 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
the liquid-vapor interface and the existence of the surface tension. The 
three basic factors that determine the driving potential are surface tension, 
the contact angle and the geometry of the solid surface at the three-phase 
boundary une. 
The basic mode of operation in flue heat pipe is through a cycle of 
evaporation and condensation. This process allows for extremely high heat 
transfer rates. To sustain this evaporation/condensation cycle, the liquid 
in the heat pipe must be continuously supplied to the evaporator section. 
This liquid supply is provided by a porous wick structure of various forms: 
gauzes, sintered porous materiais, grooves ou the interior heat pipe wall, 
or any other material capable of transporting the liquid to the evaporator 
section. Operation of the heat pipe in its simplest form involves flue evap-
°ration of the liquid in the heated end of the heat pipe. This evaporative 
process leads to the formation of or increase in the curvature of the concave 
menisci in the wick pores. As a result of surface tension forces, a capillary 
pressure pcap develops in the menisci which a,cts against the surface tension 
forces. Thus, the capillary pressure can be determined by examining the 
radii of curvature of the menisci. 
In general, it is necessary to specify two radii of curvature to describe an 
arbitrarily curved surface, Ri and R/1, as shown in Fig. 2.3. The surface 
section is taken to be small enough such that R/ and R11 are approximately 
constant. If the surface is now displaced outward by a sma11 distance, the 
change in Bica is 
aS= (x + dx)(y+dy)—xy 
	
(2.5) 
If dxdy O, then 
LSS=y dx+xdy 	 (2.6) 
The energy required to displace the surface is 
dE a(x dy + y dx) 	 (2.7) 
There is also a pressure difference L‘p across the surface due to the dis-
placement which acts ou the area xy over the distance dz. The work or 
energy attributed to generating this pressure difference is 
dE = Lip xy dz = pcapxy dz 	 (2.8) 
From the geometry of Fig. 2.3, it follows that 
x + dx 	x 
(2.9) 
+ dz 
Of 
(2.10) 
SOLID-LIQUID-VAPOR PHENOMENA 	 67 
and similarly, 
= y dz 
dy 	 (2.11) 
For the two surfaces to be in equilibrium, the two expressions for energy E 
must be equal. 
cr(x dy + y dx) = aip xy dz 	 (2.12) 
a xy dz xy dz) 
— ap xy dz 	 (2.13) 
k 	RH 
1 	1 )P"P AP = a (—Ri +) 
 
This expression is called the Young-Laplace equation, and is the fundamen-
tal equation for capillary pressure. In systems such as heat pipes, where 
is nearly constant along the liquid-vapor interface (assuming no significant 
temperature change), p cap is a function of curvature only. Therefore, p cap 
depends mainly on the geometry of the interface. The best known example 
of the Young-Laplace equation is the rise of liquid in capillary tubes, as 
Figure 2.3: Arbitrarily curved surface with two radii of curvature Ri and 
RH 
68 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
shown in Fig. 2.4. In this case, there is a constant opposing force of gravity 
to balance the capillaxy force. The pressure difference across the interface 
is given by eqn. (2.14). It should be noted that the radli of curvature (with 
both having the same sign) always lie ou that side of the interface having 
a higher pressure. If the radius of the tube in Fig. 2.4 is small, 
r / cos O 	 (2.15) 
then 
(Pv 	)c= Pcap,c = 	cos O 
	
(2.16) 
If H denotes the height of the meniscus above the flat surface a (p cap,„ = O) 
Pa = Pv + Pv911 
	
(2.17) 
Pb = Pa = Pv Pai/ — Pcap,c 	 (2.18) 
Combining eqns. (2.16), (2.17) and (2 18) gives 
2 	, 
	
Pcap,c =
a cos O 
— 	
Pv )gH 	 (2.19) 
For a liquid to completely wet the tube, O = O. When it falis to wet the 
wall, O> ir/2. In this case, there is a capillaxy depression as shown in Fig. 
2.4(b), where the meniscus is convex and H is now the depression depth. 
The saturated vapor pressure for a flat interface between the liquid and 
a) Capillary rise under 	b) Capillary depreasion under 
wetting condition 	non—wetting condition 
Figure 2.4: Capillary phenomena in an open tubo 
SOLID-LIQUID-VAPOR PHENOMENA 	 69 
vapor is a function of temperature only. This is not the case when there is 
a curved interface due to the capillary effect. From Fig. 2.5, it is seen that 
the vapor pressure at point c (concave surface) is 
Pa,v = Pc,v Pv9H 	 (2.20) 
Substituting H from eqn. (2.19) into the above equation gives 
Pa,v Pc,v 
	2o- cos 	p, 	
(2.21) 
Pe Pv 
Therefore, the vapor pressure over a concave interface is less than that for 
a flat interface by 
(2o- cos B .\ 	p„ 
,()¢ — pv ) 
which is also a function of the curvature of the surface. In general, for heat 
pipes with capillary pore radii of more than 10 pm, this effect is small and 
can be neglected. 
In the condenser section of the heat pipe, condensation returns liquid 
to the wick. The curvature of the menisci in the condenser is considerable 
smaller than in the evaporator. Thus, the capillary pressures of the two 
Figure 2.5: Capillary phenomena in a tube in a closed system 
70 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
regions produce a pressure drop and correspondingly, the driving potential 
for lhe liquid transfer through the wick. 
As mentioned previously, wick structures and geometries are varied. 
The angle of contact derived in Sec. 2.2.2 (or eqn. (2.4)) is based ou 
lhe smooth-surface model, which is not always equal to the real meniscus 
contact angle. For example, for an open rectangular channel, the meniscus 
contact angle is greater than that given by eqn. (2.4). The capillarity of a 
given wick, in most cases, must be determined experimentally. However, for 
some geometries it is possible to derive the theoretical maximum capillary 
pressure, peep,„,ex . These speciflc cases are characterized by a constant 
cross-sectional flow area. As an example, in the case of a circular capillary 
(Fig. 2.6(a)), the minimum radii are 
	
(Raiar' = (R1.1)mia = 	cos 	o 	 (2.22) 
Substituting this expression into the Yotmg-Laplace equation, the capillary 
pressure is 
Pcap = r 	
	
cos O 	
(2.23) 
For this expression to be a maximum value, the contact angle must be zero; 
i.e., a perfectly wetting fluid. Thus 
2cr 
	
Pcap,max = — 	 (2.24) 
Similarly, for a rectangular channel, as shown in Fig. 2.6(b), 	= ao and 
(EM= = 2 cos 
Thus, 
2cr cos O 
Pc" — V/ 
For this expressionto be a maximum value, the contact angle must be zero. 
2a 
Pcap,max = vv 
(2.27) 
From these and other cases, lhe Young-Laplace equation can be generalized 
as 
2a 
Pcap,max = — 
reg 
where reg is the effeetive pore ra,dius. Table 3.1 lists various common wick 
types and corresponding effective 
(2.25) 
(2.26) 
(2.28) 
SOLID-LIQUID-VAPOR PHENOMENA 	 71 
Vapor! 
7/17~ 
Liquidi 
a) Cylindrical capinar), 	b) Rectangular capillary 
Figure 2.6: Effective pumping radii 
It is important to note that in the above expressions, the maximum 
capillary pressure is reached when the angle of contact is zero for a wetting 
fiuid. Conversely, for a non-wetting fluid the expression for the maximum 
capillary pressure will be negative. This negative sign indicates that the 
high pressure region resides on the liquid side of the liquid-vapor interface. 
Thus, for effective capillary pumping through the heat pipe wick, wetting 
fluids must be used. 
2.2.4 DISJOINING PRESSURE 
The disjoining pressure introduced by Derjaguin (1955) represents the pres-
sure losses due to the attraction of the liquid phase by the solid. This pres-
sure gradient is generated within the thin layer of liquid which covers the 
solid section in contact with the vapor (Israelachvili, 1985; Ivanov, 1988). 
The properties of a liquid in a very thin fim are significantly different from 
the properties of the bulk liquid. The disjoining pressure is a product of 
long range intermolecular forces composed of molecular and electrostatic 
interactions. Since the properties and chemical potentials of the bulk liq-
uid and liquid thin fim are not the same, an additiona1 pressure difference 
arises. This pressure, the disjoining pressure, is described by 
A B 
Pd = 	 (2.29) 
where A and B are constants that characterize the molecular and electro- 
static interactions, and b is the fim thickness. The nature of this phe- 
nomenon produces increasing negative pressure with decreasing film thick- 
72 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
ness. The pressure is considered to be positive for repulsion and negative 
for attraction of the surface fim. Due to the extremely high values of pd in 
ultra-thin films, the transport of liquid in thin films can be significant and 
their role in evaporation can be essential, especially for low-temperature 
fluids. The disjoining pressure is one of the fundamental phenomena which 
affect the formation of the thin evaporating films and the magnitude of the 
contact angles. 
In summary, the maximum capillary pressure can be estimated by the 
Young-Laplace equation for a specific wick structure. However, to obtain a 
complete expression for the capillary pressure, the process becomes much 
more complex. 
2.3 CHANGE IN SATURATED VAPOR 
PRESSURE OVER THE LIQUID FILM 
As discussed before, for very thin films, the repulsion of the vapor phase 
by the solid and liquid produces a pressure difference (disjoining pressure) 
across the liquid-vapor interface in addition to the capillary effect. These 
two effects reduce the saturated vapor pressure over a thin film compared 
to the normal saturated condition. Consider a thin liquid film with liq-
uid thickness 6 over a substrate with liquid interface temperature 716, and 
normal saturation vapor pressure N et (Tb) corresponding to 115. Under equi-
librium, the chemica1 potential in the two phases must be equal. 
Pv 
	 (2.30) 
Integrating the Gibbs-Duhem equation 
dbt = —s dT + v dp 
	
(2.31) 
at constant temperature from the normal saturated pressure p eat (T5) to an 
arbitrary pressure gives 
tp 
P — Psat = 	v dp 
	
(2.32) 
Nat(T6) 
Using the ideal gas law (v v = R9 715 ipv ) for the vapor phase, and the incom-
pressible assumption (p = pe) for the liquid phase, one obtains the following 
relations upon integration of eqn. (2.32) for the vapor and liquid chemical 
potentia1s, respectively. 
/4,6 = Psat,v R9T5 ln P psatvç,',,i6 8) (2.33) 
SOLID-LIQUID-VAPOR PHENOMENA 	 73 
1405 = Psat,1 Vf [Pe p„t (Ta )1 	 (2.34) 
Since Peat,t = gsat,v, substituting eqns. (2.33) and (2.34) into eqn. (2.30) 
yields 
Pv.d = Psat (TA eXP 	
[Pe — Psat (T6 )] } 	 (2.35) 
R9T6 
The pressure difference in the vapor phase ps ,a and liquid phase pe due to 
capillary and disjoining effects are related as follows. 
2o 
Pv,ê — Pe = — Pd 	 (2.36) 
reff 
Equation (2.36) can be used to eliminate pe in eqn. (2.35). 
[p.„ — ps,(T6) —2a/reff Pcd } 
Pv,6. Psat(n) exP 	 (2.37) 
piRg T6 
When the interfarP is flat and pd = O, N Ó = psat (Tb). For a curved interface 
and pd = 0, eqn. (2.37) coincides with the Kelvin equation. 
2.4 INTERFACIAL RESISTANCE IN VA-
PORIZATION AND CONDENSATION 
PROCESSES 
The high heat transfer coefficients typically associated with evaporation 
and condensation processes in a heat pipe make it possible to transfer a 
high heat transfer rate with a relatively low driving temperature difference. 
The discussion in this section is related to the thermal resistance due to 
these phase change problems. 
When condensation takes place at the interface, the flux of vapor mole-
cules into the liquid must exceed the fax of liquid molecules escaping to the 
vapor phase. The opposite occurs when evaporation takes place. Schrage 
(1953) used the kinetic theory of gases to describe the condensation and 
evaporation processes, and considered the fluxes of condensing and vapor-
izing molecules for each direction separately. Furthermore, it was assumed 
that the interaction between the molecules leaving the interface and those 
approaching the interface were under equilibrium to obtain the following 
relation for the net mass flux at the interface 
fM rp„ 
(2.38) 
27r.R. 	irk) 
where Ra is the universal gas constant, Mv is the molecular weight of the 
vapor, and the function 1' is given by 
74 	 HEM' PIPE SCIENCE AND TECHNOLOGY 
r (a) exp (a 2) + a fïr [1 + erf (a)] 
	
(2.39) 
r (—a) exp (a2 ) — 	— erf (a)] 
	
(2.40) 
	
1 	
js 
pvhfg\217.„T„ 
a — 	 (2.41) 
The heat flux to the interface is equal to the net mass flux multiplied by 
the latent heat (q5 = Thithig ). The empirical term a in eqn. (2.38) is the 
accommodation coefficient Since r is a function of qs, eqn. (2.38) does not 
provide an explicit relation for the interfacial heat flux. Assuming that pe 
and pv are the saturation pressures corresponding to Te and Te , eqn. (2.38) 
can be represented in the following form. 
q's ah!' .5/27rR„ 	Nfiv.v 
Mv [rpsat (Tv) Psat (Te)] 
For evaporation and condensation processes of working fluids at moderate 
and high temperatures, a is usually very small by examining the definition. 
In such a case, eqn. (2.39) can be approximated by 
= 1 + a NFr 	 (2.43) 
An explicit relation for q,5 and ?til; was obtained by Silver and Simpson 
(1961) by substituting eqn. (2.43) into eqn. (2.38), and using pv = 
Pv . Mv /R.Tv. 
r.4 =_ 	( 2a \ I Mv ( Pv 	Pe (2.44) 
hfg k 2 — a.; V 27rR„, kfz," „fru 
The above equation has been referred to as the Kucherov-Ifikenglaz equa-
tion (1960) in the Soviet literature. 
Carey (1992) developed an alternative form to eqn. (2.38) for small a 
by assuming that (pv — pe) /pv c 1, - T R VT„ c 1, and by using the 
Clausius-Clapeyron relation. 
( 2a .\ 	fi 2f g 	Mv (1 pvvig ) 
— a ) T g V 27r.R„T, 	2hfg 
(Te —Te) 	(2.45) 
Using the above relation, the heat transfer coefflcient at the interface ha is 
obtained from the following equation. 
(Tv — Te) ( 	 ( 	 2 — a 	Tv v fg) 
/ 	 Mv 	
(246 
V 27r11„Tv 	2h19 
h6 	 
	
( 	Pvuf g --) .) 
The interfacial resistance is of particular importance in low-temperature 
Mv 
(2.42) 
SOLID-LIQUID-VAPOR PHENOMENA 	 75 
heat pipes. In general, this interfacial resistance should be included in 
detailed simulations of heat pipes and thermosyphons. The above equa-
tions should definitely be used to provide an estimate of h6 in order to be 
compared with other heat transfer coefficients associated with other mech-
adorno in the heat pipe. If itsis of the same order of magnitude of the 
other h values, the effect of interfacial resistances should be accounted for. 
It should be noted that the above equations relating q6, /tê , and (Tu — Te ) 
presented in this section apply equally well to both evaporation and conden-
sation in heat pipes with the convention that q6 is positive for condensation 
and negative for evaporation. 
It is clear that prediction of the interface resistance using any of the 
above equations depends on the value of the accommodation coefficient a, 
which varies widely in the literature. Paul (1962) compiled the accommoda-
tion coefficients for evaporation for a large number of working fluido. Mills 
(1965) recommended that a should be lesa than unity when the working 
fluid or the interface is contaminated. This can also be due to a deviation 
from the assumptions used to develop these equations. Fortunately, due 
to careful processing procedures in heat pipes having pure fluids and clean 
components, the value of a dose to unity may be appropriate. Cao and 
Faghri (1993b) successfully used a -= 1 in a detailed numerica1 simulation 
which predicted the frozen startup of liquid metal heat pipes. However, for 
low-temperature heat pipes, the predicted resulto can be sensitive to the 
value of a because of the more significant role of the ultra-thin films in the 
evaporative heat transfer process. 
2.5 INTERFACIAL MASS, MOMENTUM 
AND ENERGY BALANCES 
The physical understanding and mathematical modeling of the phenom-
ena at the liquid-vapor interface is important for predicting the capillary 
limit as well as specifying the boundary conditions that various hydrody-
namic and transport equations must satisfy to properly simulate the ther-
mal characteristics of heat pipes. In fluid mechanics and heat transfer, 
the conservation laws are reduced to local partial differential equations if 
they are considered to be at a point which does not belong to a surface 
of discontinuity, such as an interface. When considering a discontinuous 
point, appropriate jump considerations which relate the values of the fun-
damental qua.ntities on both sides of the interface should be considered. 
The jump conditions are traditionally derived from the global laws written 
for the conservation of mass, linear momentum, angular momentum, total 
Energy, and entropy. In this section, the general jump conditions in the 
Vapor 
76 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
case of two-phase flows involving surface tension and surface properties will 
be presented. The general relations are also simplified for heat pipe appli-
cations. The reader is siso directed to the works reported by Burelbach et 
al. (1988), and Delhaye (1974, 1976). 
The ba,sic assumption made for the jump conditions at the liquid-vapor 
interface is that the liquid and vapor phases are at the steady state. The 
jump balance at the liquid-vapor interface satisfies the continuity of mass 
flux normal to the interface (Fig. 2.7) 
Thit = PECite — VIS) • h = Pvcrft, 175) • h 
	
(2.47) 
where V is the resultant fluid velocity and frtit represents a flux of mass 
crossing lhe liquid-vapor interface normally into lhe vapor space. Sub-
scripts f and v indicate the liquid and vapor phases, respectively, and 6 
indicates the condition at the interface. Here and t are unit normal and 
tangential vectors directed into the vapor space. If the interface is not 
moving, V6 = O. 
The jump energy balance including ali of the important terms is 
rhit — h f 9 + —12 17£ 	— Ve • 17, — 	• i7„ + • 17,5 ) — kiVre ft 
+1c,VT, • ft — 2p e(f e • h) • (17 e — fl6) + 2p„ei-„ • 	—176) = O (2.48) 
z, w 
Solid 
Figure 2.7: Physica1 configuration describing lhe liquid-vapor interfacial 
phenomena 
SOLID-LIQUID-VAPOR PHENOMENA 	 77 
where Tale and Ç are the rate of deformation tensors in the liquid and vapor, 
respectively. It should be noted that in eqn. (2.48) the surface-entropy 
effect has not been included since this term in most cases is negligible 
compared to other effects. 
The conservation of normal and tangential momentum at the liquid-
vapor interface can be represented by the following relations, respectively. 
2a(T) 
(fte — -Vv) — e — Sv) • ft = V a(T)+ D — 	+ D (2.49) 
R 
74(fle — fiv) • — (Se — v) • fx • i = —Va(T) • E 	(2.50) 
D is the contribution due to the disjoining pressure. This effect has tradi-
tionally been neglected in heat pipe analyses, but it may have a significant 
effect on the analysis of micro heat pipes and miniature thermosyphons. 
is the stress tensor defined by the following relation for newtonian fluids 
S= —p 1 +2 p (2.51) 
where / is the identity tensor. .11 is the mean radius of curvature of the 
interface, and a is the surface tension, which is a linear function of temper-
ature. 
ao — -y(T6 — Tb) 	 (2.52) 
For common liquids, ey = — da/dT > 0. Therefore, there is a surface flow 
from the hot end toward the cold end. The bulk of the fluid is dragged along 
since the fluid is viscous. This phenomenon is called the thermocapillary 
effect (Levich, 1962; Davis, 1987). ao is the surface tension at the reference 
temperature. The term on the right-hand side of eqn. (2.50) is the axial 
gradient of the surface tension due to surface temperature variation, which 
is called the Marangoni effect. This effect can also induce fluid motion. 
The following assumptions are made to simplify eqns. (2.47)—(2.50): 
1. A flat interface is assumed except for the calculation of the cap-
illarity, 2a (T)/R. 
2. The effects of disjoining pressure are neglected. 
Equations (2.47)—(2.50) can be simplified in the 2-D cylindrical coordinate 
system in the following forms. 
r- — ,90p —pti Vv 	 (2.53) 
78 	 HEAT PIFE SCIENCE AND TECHNOLOGY 
avi aWg\ w 	11 011v + 0211v \ w 
	
7. 19T1) 	aTe 
	
Or 	Dr 	az 	Or } 1 	k, az 	ar ) v 
+frill {–hf g + 	
Pe Pv 
+2fit'Á — 	– — -w-) = O 
Pv vr 	pe r 
ge ave 
(pe pv ± ± 04)2 (1 1) = 2 (pte w:Ove pv wavv ) 
	
' R 	 Pv 
avt atue \ 	(avv awv ) Ou „ 
7"isi (" wv) –14 ç-0.7 U 
Equations (2.54)–(2.56) can be further simplified by assuming that the 
phases are inviseid, the interfacial mass flux is small, and the surface tension 
gradient is negligible. 
 
— 	 — Ths f g — 
n 
 
ar 	ar 
(2.57) 
(2.58) 
Wg = Wv 
	 (2.59) 
Equations (2.53), (2.57) and (2.58) are the forms which have been used in 
most conventional heat pipe models due to their simplicity. 
Aceording to the discussion in fixe previous section, the interfacial mass 
flux fiei can be calculated as 
ifilt =2 2—act) ,I2irmitav CA—}P — Á,P 
Therefore, the interfacial velocity v5 can be eakulated as 
V6 
Pv 
On the other hand, an energy balance at the interface gives 
r_ 9a 
° 	hig 
By neglecting the liquid flow in the wick and the interfacial resistance in 
(2.54) 
(2.55) 
(2.56) 
(2.60) 
(2.61) 
(2 .62) 
SOLID-LIQUID-VAPOR PHENOMENA 	 79 
comparison with that of the wick, the interfacial heat flux can be calculated 
by q = —k eff0T/Or, using the temperature distribution in the wick. There-
fore, instead of directly calculating Mit by eqn. (2.60), we can calculate the 
heat flux q6 in the wick to obtain the mass flux rhit. This method lias been 
used by Chen and Faghri (1990) for a steady, 2-D heat pipe model, and by 
Cao and Faghri (1990) for a continuum transient simulation of conventional 
heat pipes. One can also use eqn. (2.60) to obtain fri; directly. However, 
this method involves the uncertainty of the evaporation/condensation co-
efficient a, which was discussed in the previous section. Cao and Faghri 
(1993a, 1993b) used an expression similar to eqn. (2.60) for a 2-D simu-
lation of high-temperature frozen startup of heat pipes, and showed that 
the variation of a does not affect the final result when the wall, wick and 
vapor are coupled as a single domain by including the effects of conjugate 
heat transfer. However, Khrustalev and Faghri (1994) showed a marked 
variation of the final results with a for thecase of axially-grooved low-
temperature heat pipes. 
Since evaporation and condensation is a very complicated process with 
various parameters, there are no exact relations available in the literature 
for all working fluids and operating conditions. Ou the other hand, the 
calculation of q6 only involves the temperature distribution in the wick, 
and proves to be more practical. It is therefore recommended that Thg be 
calculated via q5 whenever possible. 
2.6 INTERFACIAL PRESSURE BALA-
NCE AND MAXIMUM INTERFAC-
IAL PRESSURE DIFFERENCE 
The heat transport capability of a heat pipe can be affected by one or more 
limiting factors such as the capillary limit, the sonic limit, the entrainment 
limit, and the boiling limit. For most applications, however, the capillary 
fina determines the pumping capacity of the heat pipe. The capillary limit 
manifests itself in the form of a wick dryout condition. This results from 
the required interfacial pressure exceeding the maximum capillary pressure 
that the wick is capable of sustaining. When the pumping rate of the wick 
is insufficient, more liquid is evaporated from the wick than is supplied. 
For a heat pipe, the pressure distribution in the liquid and vapor can be 
determined by integrating the differential equations for the axial pressure 
gradient. 
p(z) =Vpe, z dz +M0) 	 (2.63) 
o 
80 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
py (z) = f Vp,, z dz +p,(0) 	 (2.64) 
o 
The liquid-vapor pressure difference at the interface is a function of position 
along the heat pipe axis. 
fps (z) = (z) — p(z) 
	
(2.65) 
Substituting eqns. (2.63) and (2 64) into the expression for Aps gives 
Ap6(z) -= 	[Vp„,z — Vps, z ]dz +p,(0) — MO) 	(2.66) 
Equation (2.66) is an expression that defines the required interfacial pres-
sure difference for any given location of the heat pipe axis with respect 
to the pressure difference at z = 0, which is given by [p 2 (0) — p6(0)]. To 
determine the interfacial pressure difference with respect to the minimum 
pressure point in the heat pipe, it is assumed that at a specific location, zi, 
the pressure difference is zero. 
p6(z1) = O 	 (2.67) 
Then z2 
p„ (0) — MO) = — 	[vpv,. - Vn,2]dz 
o 
(2.68) 
Substituting eqn. (2.68) into eqn. (2.66), we get 
AP6(z) 	N7/4,z — 	 (2.69) 
Just as there is a point of minimum pressure difference, there is also a point 
where the interfacial pressure reaches a maximum. Once this location (22) is 
found, it can be used to calculate the maximum pressure difference relative 
to the zero pressure location by evaluating eqn. (2.69) with the appropriate 
limits of integration. 
2 
(APs)ma,2 = Ap5(z2) = 
z
[VPv,z — .1/47Pe,21dz (2.70) 
There is always at least one axial location 2 2 where the interfacial pressure 
difference Ap6(z) is maximum. The location of this point as well as the 
value of this maximum interfacial pressure difference can be cakulated us-
ing the above equations, either by a numerical or a closed-form analytical 
solution of the momentum equations in the liquid and vapor regions with 
SOLID-LIQUID-VAPOR PHENOMENA 	 81 
the appropriate boundary conditions. In a general case, numerical methods 
need to be employed when multiple evaporators or condensers are involved, 
or the geometric shape is nonconventional. For proper fluid circulation to 
be maintained in the heat pipe, this maximum interfacial pressure difference 
should be less than or equal to the maximum capillary pressure difference, 
á/kap ,mar 
The above discussion dealt with a specific case in which the capillary 
forces due to surfa,ce tension are the only forces which balance the interfacial 
liquid and vapor pressure drops in the heat pipe. In general, however, as 
discussed in the previous section, other forces impact the total force balance 
in a heat pipe. The generalized expression at any given axial location 
including all of the important effects for proper fluid circulation in the wick 
of a heat pipe is 
+ áPph áPb < aPcap,max 
	 (2.71) 
Ap6 represents the liquid and vapor pressure drops, Apph is the pressure 
loss that occurs due to phase transition, and Apb is the pressure drop in 
the vapor and liquid regions as a result of body forces, such as gravity, cen-
trifugai, electromagnetic, etc. It should be noted that in some cases, Apb is 
included in Ap6 when a complete solution of the Navier-Stokes equations 
is obtained including body forces. The disjoining pressure is mainly im-
portant in the analysis of the thin films in micro heat pipes and capillary 
grooved heat pipes, and Appb is considered under high condensation or 
evaporation rates. 
2.7 INTERFACIAL PHENOMENA IN 
CAPILLARY GROOVED STRUCT-
URES 
A detailed mathematical model is presented in this section which describes 
heat transfer through thin liquid films in the evaporator and condenser of 
heat pipes with capillary grooves. The model accounts for the effects of 
interfacial thermal resistance, disjoining pressure and surface roughness for 
a given meniscus contact angle. The free surface temperature of the liquid 
film is determined using the extended Kelvin equation and the expression 
for interfacial resistance given by the kinetic theory. The numerical results 
obtained are compared to existing experimental data. The importance of 
the surface roughness and interfacial thermal resistance in predicting the 
heat transfer coefficient in the grooved evaporator is demonstrated. 
• The model presented in this section is that of Khrustalev and Faghri 
82 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
(1994) which is a significant contribution over the previous investigators' 
attempts (Kamotani, 1976a, 1978; Vasiliev et al., 1981; and Stephan and 
Busse, 1992). The discussion in this section is developed for rectangular, 
triangular and trapezoidal grooves in a circular tube, but flat evaporators 
and condensers can also be described by the presented equations. Heat 
transfer processes in the heat pipe container and working fluid was con-
sidered to be one-dimensional in the radial direction, such that axial heat 
conduction was neglected. The emphasis has been placed on the formation 
of the thin liquid films affected by the operational conditions. During the 
condensation process, liquid in the subcooled thin fim flows towards the 
meniscus region along the s-coordinate, as shown in Fig. 2.8(a). During 
evaporation, liquid in the superheated thin film flows from the meniscus 
region in the opposite direction, as presented in Figs. 2.8(6) and 2.9. Fi-
naIly, the numerical results were obtained using an iterative mathematical 
procedure which involved the following boundary-value problems (except 
the first): 
1. Formation of and heat transfer through thin liquid films. 
2. Heat transfer in the evaporating fim ou a rough surface. 
3. Heat transfer in the condensate fim on the fim top surface. 
4. Heat conduction in a metallic fim and liquid meniscus. 
These problems are described in detail in the following subsections. 
2.7.1 FORMATION OF AND HEAT TRANSFER 
THROUGH THIN LIQUID FILMS 
The thermal resistance of a low-temperature axially-grooved heat pipe 
(AGHP) depends mostly ou the thickness of the thin films in the con-
denser and evaporator sections. Since the heat transfer and fluid dynamics 
processes in a thin fihn are similar in both sections, it is possible to describe 
the formation of the films by the same equations, but taking into account 
the different directions of the temperature potential. In this section a thin 
evaporating film on a heat-loaded surface with curvature K u, is considered, 
as shown in Fig. 2.9(6). The local heat flux through the film due to heat 
conduction is 
(2.72) 
where the local thickness of the liquid layer 6 and the temperature of the 
free liquid film surface 7:5 are functions of the s-coordinate. For small 
SOLID-LIQUID-VAPOR PHENOMENA 	 83 
 
 
 
 
 
 
(a) 
(b) 
Figure 2.8: Cross-sections of the characteristic elements of an axially-
groovédheat pipe: (a) Condenser; (b) Evaporator 
eguilibrium film 
microfilm region 
transition region 
meniscus 
region 
RMen 
84 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
Figure 2.9: Thin evaporating fim on a fragment of the rough solid surface 
Reynolds numbers, an assumption of a fully developed laminar liquid flow 
velocity profile is valid 
1 dpe 
'tt = — 4,£W (2776 -712) 
where ii is the coordinate normal to the solid-liquid interface. The vapor 
(2.73) 
	
SOLID-LIQUID-VAPOR PHENOMENA 	 85 
pressure is assumed to be constant along the s-coordinate, and the liquid 
flow is driven mainly by the surface tension and the adhesion forces. 
dpe 	dK dpd 	do dTe d , 22 ( 1 	1 
— — 	 — kp,1) 6) 	 (2.74) 
ds 
= 	
ds 	ds 	dT6 ds ds 	 pe 
K is the local interface curvature, pd is the disjoining pressure (Derjàguin, 
1955) and the last terra is the kinetic reaction of the evaporating fluid 
pressure. The impact of the last two terras ou the results was found to be 
negligible in the present analysis, therefore they are omitted in following 
equations. 
The continuity equation for the evaporating liquid layer is 
ds o 	hfgfte 
d 6 
ue 	— 	 (2.75) 
Substituting eqns. (2.72)—(2.74) into eqn. (2.75) gives the following relation 
for the thickness of the evaporating fim, 6(s). 
314 cTã ds ujd 	 hf g pe6 
1 d rs3 d 	aK)1_ Ice(Ty„ — Te) 	
(2.76) 
The fim surface curvature K is expressed in terms of the solid surface 
curvature K,„ and fim thickness as 
d2 5 	d6) 21 —3/2 
	
H- 
	 (2.77) 
Following Potash and Wayner (1972), a power-law dependence of Pd ou 
is given for non-polar liquids. 
pd = —A 18— B 
	
(2.78) 
For water, however, the logarithmic dependence is preferable (Holm and 
Goplen, 1979). 
It is assumed that the absolute value of the vapor core pressure at any z-
location along the groove is related to vapor temperature by the saturation 
conditions 
Pv = Psat(Tv) 
	
(2.79) 
and therefore can be defined for a given Ti, using the saturation tables. 
. The temperature of the interface T6 is affected by the disjoining and 
capillary pressures, and siso depends on the value of the interfacial resis- 
86 	 HEAT PIPE SCIENCE A1VD TECHNOLOGY 
tance, which is defined for the case of a comparatively small heat flux by 
the following relation. 
= ( 
2 2—aa 	1/2f7rg 	Pv Rg 	v 	(PsaTt 6)6] 
(2.80) 
/4 and (peat )6 are the saturation pressures corresponding to Tv in the bulk 
vapor and at the thin liquid film interface, respectively. 
While eqn. (2.80) is used in the present analysis, it seems useful to 
mention that for the case of extremely high heat fluxes during intensive 
evaporation in thin films , Solov'ev and Kovalev (1984) have approximated 
the interfacial heat flux by the following expression. 
	
q = 3.2N, / RgTv[(Peat)6 — Pui 
	
(2.81) 
Equation (2.81) was derived with the assumption that the accommodation 
coefficient a r- 1 from the expressions given by Labuntsov and Krukov 
(1977). 
The relation between the vapor pressure over the thin evaporating fim, 
(Psat),5, affected by the disjoining pressure, and the saturation pressure 
corresponding to Tb? Psat(TÕ is given by the extended Kelvin equation. 
(Psat).5 = Pest(Ts)eXP 
[ (Psat )5 Psat (T6) pd — 
psTI9TÕ 	
(2.82) 
Equation (2.82) refiects the fact that under the infiuence of the disjoining 
and capillary pressures, the liquid free surface saturation pressure (p sat )6 is 
different from the normal saturation pressure peat (T6) and varies along the 
thin fim (or s-coordinate), while p, and Tv are the same for any value of s 
at a given z-location. This is also due to the fact that 716 changes along s. 
For a thin evaporating fim, the difference between (p s„t )6 given by eqn. 
(2.82) and that for a given using the saturation curve table is larger. 
This difference is the reason for the existence of the thin non-evaporating 
superheated fim, which is in the equilibrium state in spite of the fact that 
716 > T. 
Under steady state conditions the right-hand sides of eqns. (2.72) and 
(2.80) can be equated. 
( 2a \ 	hf9 [ Pv 	(Psat),51 
ke 	— a) .5/271-Rg N/ri, 	N/76 J 
Equations (2.82) and (2.83) determine the interfacial temperature, 7'6, and 
pressure, (psat )6. Tv, has to be provided as an input to the solution pro- 
cedure, resulting from the solution of the heat conduction problem in the 
(2.83) 
(2.85) 
For water the following equation for the disjoining pressure was used (Holm 
and Goplen, 1979) 
—1/13 
{ 
1 
X.,{ 
	T,,, 
Pv .‘ 	— Psat(Tvt) — peRgn 	
(T) 
, ln ( 	 .‘ 
T, Psatw ,, 
SOLID-LIQUID-VAPOR PHENOMENA 	 87 
fim between the grooves. The four boundary conditions for eqns. (2.76) 
and (2.77) must be developed taking the physical situation into account, as 
shown In the following sections. 
As the liquid film thins, the disjoining pressure, pd, and the interfacial 
temperature, Ts, increase. Under specific conditions, a non-evaporating fim 
thickness is present which gives the equality of the liquid-vapor interface 
and the solid surface temperatures, 716 = T. This is the thickness of the 
equilibrium non-evaporating fim 60 , which can be determined from eqns. 
(2.82) and (2 83) For a non-evaporating equilibrium fim (q = 0), it follows 
from eqn. (2.83) that 
(Psat)5 = Pv 
7',„ 	
(2.84) 
Substitution of eqns. (2.78) and (2.84) into eqn. (2.82) gives 
= 
Pd = PeR9T5 ln [a (mi] 	(2.86) 
where a = 1.5336 and b = 0.0243. The thicluiess of the equilibrium film is 
given for water by 
= 3.3 
1/6 
(2.87) 
2.7.2 HEAT TRANSFER IN THE THIN-FILM RE-
GION OF THE EVAPORATOR 
This problem has been analyzed by several investigators with various 
apprcrximations. Kamotani (1978), Holm and Goplen (1979), Stephan 
(1992), and Khrustalev and Faghri (1994) modeled an evaporating 
x —
a 
exp 
1 	hat (7"; ) — p„ IT,IT, + 0- 1( 	Pv 	I-Tw 
peRgn, 	 psat(Tw) 
\ 
Tv 
88 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
extended meniscus in a capillary groove (Fig. 2.8). In all of the above 
papers, it is emphasized that most of the heat is transferred through the 
region where the thickness of the liquid layer is extremely small. The sig-
nificance of the temperature difference between the saturated vapor core 
and the interface has been stressed by Khrustalev and Faghri (1994) and 
Stephan and Busse (1992). In ali of the mathematical models except that of 
Khrustalev and Faghri (1994), the sofid surface was assumed to be smooth. 
The model by Khrustalev and Faghri (1994) will be presented here since 
the difference between the saturated vapor temperature and that of the free 
liquid surface wa.s considered, and the existence of the surface roughness and 
its infiuence on evaporative heat transfer was taken into consideration. In 
general, manufacturing processes always leave some degree of roughness ou 
the metallic surface. Alloys of copper, brags, steel and aluminum invariably 
have some distinct grain structure, resulting from processing the materiais. 
In addition, corrosion and deposition of some substances on the surface can 
infiuence its microrelief. This means the solid surface is totally covered 
with microroughnesses, where the characteristic size may vary from, for 
example, R,. =10-8 to 10' m. Apparently, the thin liquid fim formation 
can be affected by some of these microroughnesses. It can be as.sumed that 
at least some part of a single roughness fragment, ou which the thin fim 
formation takes place, has a circular cross section and is extended in the 
z-direction due to manufacturing the axial grooves (Fig. 2.9). 
The free liquid surface is divided into four regions (Fig. 2.9). The first 
region is the equilibrium non-evaporating fim. The second (microfilm) 
region ranges in the interval 50 <8 < 61 , where the increase of the liquid 
fim thickness up to the value Si is described by eqns. (2.76) and (2.77). 
In this region, the generalized capillary pressure /k ap aK — pd (here peap 
was defined so that its value is positive) is changing drastically along thes-
coordinate from the initial value up to an almost constant value at point 
where the fim thickness, 8 1 , is large enough to neglect the capillary pressure 
gradient. It is useful to mention that some investigators have denoted this 
microfilm region as the "interline region." The third (transition) region, 
where the liquid-vapor interface curvature is constant, is bounded by õ< 
5 < R,. + 60 , and the local fim thickness is determined by the geometry of 
the solid surface relief and the value of the meniscus radius Rmen . In the 
fourth (meniscus) region, where by definition 6 > R,. + 80, the local fim 
thickness can be considered independent of the solid surface microrelief. 
In the third and fourth regions, the heat transfer is determined by heat 
conduction in the meniscus liquid fim and the metallic fim between the 
grooves. However, in the second region, the temperature gradient in the 
solid body can be neglected in comparison to that in liquid duo to the 
extremely small size of this region. 
SOLID-LIQUID-VAPOR PHENOMENA 	 89 
The total heat flow rate per unit g-roove length in the microfilm region 
is defined as 
17,-„ic(81)= 
81 
(2.88) fo 	.5/k 
 
da _ 	q da 
Equations (2.76)—(2.79), (2.82) and (2.83) must be solved for four variables: 
5, 5', peap and Q'Tnic (s) in the interval from $ = 0 to the point s = 81, where 
Pcap can be considered to be constant. Now, instead of the two second-order 
equations (2.76) and (2.77), the following four first-order equations should 
be considered with their respective boundary conditions. 
dS 
= 6' 	 (2.89) 
da 
	
de = (1 + 812
) 3/2 ( pcap — A15 -13 	1 
± 	 (2.90) 
da a 	R, 
	
dpeap _ 	3vg , 
(2.91) 
da 	hf9t 3QmI c(s) 
	
dC2',.. 1, 	Ty, — T6 
(2.92) 
da — (5/kt 
	
5Is=o = 6o 	 (2.93) 
	
518=0=-0 	 (2.94) 
	
Peapis=o — 	a 	+ A'6 8 
Rr + fio 
	
Qtrnicis=o = o 	 (2.96) 
The value of 60 is found from eqn. (2.85), where K = 
Though the initial-value problem, eqns. (2.89)—(2.96), is completely 
determined, its solution must satisfy one more condition. 
a Pcap1.9=- 	in 91 	Ren 
Since the only parameter which is not fixed in this problem is connected 
. with the surface roughness characteristics, the boundary condition (2.97) 
(2.97) 
90 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
can be satisfied by the choice of R„. Physically, it means that the beginning 
of the evaporating fim is shifted along the rough surface depending on the 
situation so as to satisfy the conservation laws. However, in a smooth 
surface model (R, — ■ oo) the solution will probably not satisfy eqn. (2.97). 
As a result of this problem, the values of b i and gime (si) can be determined 
and the transition region can be considered, provided that 61 <14, where 
the free iiquid surface curvature is constant and its radius R,,, en is many 
times larger than 14. Based ou the geometry shown in Fig. 2.9, the 
following approximation for the liquid filia thickness in the transition region 
xf < x < xt, is given. 
6 = 60 Rr fEr 2 xy _Rmen (Rrnen 2 ± x2 
+ 2Rmenx sin Of )
1/2 
(2.98) 
Equation (2.98) is valid for the rough surface model (Of can be set equal to 
zero for very small Ri.) and also the smooth surface model in the meniscus 
region (R, co and Of is given as a result of the microfilm problem 
solution). For the smooth surface model, 9 f is the angle between the solid-
liquid and liquid-vapor interfaces at point s i , where the capillary pressure 
becomes constant. 
The heat flow rate per unit groove length in the transition region is 
(ft, _ f x " T„, — T6 dx 
(2.99) 
ixt 	6/kt 
where xf and xt, are obtained from eqn. (2.98) provided 6 = 6 1 and 
6 = 	+ bo, respectively. 
Now, the connecting point between the transition and meniscus regions 
must be considered. At this point, the fim thickness, the free surface cur-
vature, and the liquid surface slope angle must coincide from both sides. In 
the rough surface model, the last condition is a1ways satisfied because the 
length of the microfilm region is smaller than R„, and the rough fragment 
with the fim can be "turned" around its center in the needed clirection (see 
Fig. 2.9). In other words, because of the circular geometry of the rough 
fragment and the constant temperature of the solid surface in the microfilm 
region, the slope of the fim free surface is not fixed in the mathematical 
model. Ou the contrary, in the smooth surface model the numerical results 
give Of which is generally not equal to 0,„,„ determined by the fluid flow 
along the groove. Stephan (1992) seems to have answered this contradiction 
using a rounded fim comer, however, this explanation is not completely 
satisfactory. Note that ir' the situation when Of 0,,, e,,, the smooth surface 
model can be used a1ong with the rounded fim comer, where the radius is 
Rfin. In this case eqn. (2.98) can also be used provided R, Is changed 
to Ran. 
SOLID-LIQUID-VAPOR PHENOMENA 	 91 
It is useful to mention here that the values of Rinen and 0men are con-
nected by the geometric relation O rnen = arccos (W/2Rmen) — ry and should 
be given as a result of the solution of the problem for the fluid transport 
along the groove. The fim top temperature T u, should be defined from the 
consideration of the heat conduction problem in the fim between grooves 
and in the meniscus liquid fim discussed below. 
The free liquid surface curvature K in the microfilm region varies from 
the initial value to that in the meniscus region. Its variation is described 
by eqns. (2.89)—(2.97) with respect to the p eap and pd definitions. In spite 
of a sharp maidmum which the K function has in the microfilm region, its 
variation only slightly affects the total heat transfer coefficient. To check 
this hypothesis numerically, a simplified version of the heat transfer model 
of the microfilm region was developed by Khrusta1ev and Faghri (1994), 
where it was assumed that the microfilm free surface curvature is equal to 
that in the meniscus region. Therefore, instead of solving eqns. (2.89)— 
(2.97), the microfilm thickness in this region (and also in the transition 
region) can be given by eqn. (2.98) for the interval O < x < x tr . In this 
case, the heat flow rate per unit groove length in both the microfilm and 
transition regions is 
rtr T — T6 
climic+Q1
r — t 	 ti:51kt dx 
(2.100) 
2.7.3 HEAT TRANSFER IN THE THIN-FILM RE-
GION OF THE CONDENSER 
Heat transfer during condensation on a grooved surface has been considered 
by Kamotani (1976b), Babenko et al. (1981), and Khrustalev and Faghri 
(1994). Analyzing their results, the following conclusions are made, which 
lead to the simplification of eqns. (2.76) and (2.77): 
1. The surface of the liquid fim is smooth and the fim thicicness 
variation along the s-coordinate is weak (see Fig. 2.8). 
db) 2 
«1 ' 
C1.9 
2. The disjoining pressure gradient along the fim flow can be ne-
glected in comparison to that of the capillary pressure due to 
the surface tension force because of the large fim thickness. 
Taking the above points into consideration, and substituting eqn. (2.77) 
92 	 HEAT PIPE SCIENCE AND TECHNOLOGY ' 
into eqn. (2.76) gives the following differential equation for the fim thick-
ness at the top of the fim between grooves. 
6— 
d [
53 	
u,
)] 
(d36 dK\1 _ 	
(T6 T 
31.telcs 
(2.101) 
ds 	ds3 	ds 	ahfgpt
n,) 
where To, is the temperature of the top of the fin. The boundary conditions 
for eqn. (2.101) at s = O are 
d6 
	
 
= 	
das 
, 
	
ds3 	
(2.102) 
These conditions imply that the thickness and curvature of the fim are 
symmetric around $ = 0. For small N (see Fig. 2.8(a)) at s = L2, the 
curvature of the fim and its surface slope angle are determined by the 
radius of the meniscus in the groove 
	
d2 5 	1 
	
ds2 	R irei, 
—
de 
= tan 	- arcsin 	 + arcsin Ll 	(2.104) 
ds 	 2R,,,,„ 	2R, 
where Ly is the half-length of the fim, which is equal to L 1 /2 in the case 
of a flat fim top geometry (Fig. 2.8). The boundary valueproblem, eqns. 
(2.101)—(2.104), is solved approximately by introducing the following poly-
nomial function for the fim thiclmess. 
6(s) = Co ± 	- L2)± C2(8 - L4 2 +G3(8 - L2)3 +C4(8 - L2) 4 (2.105) 
From the boundary conditions (2.102)—(2.104) the values of the coefficients 
are 
( 	
W 	. Li 	 1 
	
Cl = tan is — arcsin 	 + arcsin — 	C2 
	
2R. 	2R) ' 	
—
2R„„„ 
2C2L2 - Cl 	 C3 
C3 = 	 = 
4L2 
At the point s = L2, where the thickness of the fim is usually at a minimum, 
eqn. (2.101) must be satisfied exactly, and the total mass flow rate of the 
condensate due to the surface tension force must be equal to the total 
amount of fluid condensed in the region O < s < L2. Thus, integrating eqn. 
(2.101) we have 
(2.103) 
SOLID-LIQUID-VAPOR PHENOMENA 	 93 
ahfot í63 ( d36 dlfw \ 1 
314 
8=4,2 
= ki 
fo L2 T5 Tiv d 
6 	s 
( 2.1 06) 
Substituting eqn. (2.105) into eqn. (2.106) and solving numerically for Co, 
the heat flow rate per unit groove length through the thin fim region is 
L2 	 ke (T6 Tw ) 
ds 
Q16 -= fo Co + Oi(s — L2) + C2(s — L2) 2 + C3(8 — L2) 3 ± C4(8 — L2) 4 
(2.107) 
The fim top temperature, Tio , is given from the results of the heat conduction 
problem in the fim and the meniscus region. Equation (2.106) must aiso be 
solved within the following iterative procedure because of the influence of 
the fim surface curvature and the disjoining pressure on T5. In the first 
iteration, Tb is defined from eqns. (2.82) and (2 83) assuming that K = 
and Pd = 0. Note that for the case of condensation, the plus sign after 	in 
eqn. (2.83) should be replaced by a minus sign. In the second and following 
steps 
d26 
ds2 
where the last term is calculated using the solution of the previous step 
for 6(s) and pd = pd(7) (here the bar denotes an average value). While 
the influence of K ou the presented results was negligible in comparison 
with the effect of the meniscus radius variation, it can be important for 
extremely thin films of condensate with large free surface curvatures, in 
which case the problem should be treated numerically in the frames of a 
more complicated analysis. 
Now, the consideration of the meniscus region gives the opportunity to 
obtain the heat transfer coefficients. 
2.7.4 HEAT CONDUCTION IN THE METALLIC 
FIN AND MENISCUS REGION FILM 
For low-temperature heat pipes, the thermal conductivity of the metallic 
casing is several hundred times higher than that of the liquid working fluid. 
Nearly ali of the heat is transferred from the metallic fim between grooves to 
the saturated vapor or vice versa through a thin liquid fim in the vicinity of 
the fim top. The temperature drop in the metallic fim is many times smaller 
than in the liquid film (Schneider et al., 1976; Stephan and Busse, 1992). 
Therefore, Khrustalev and Faghri (1994) assumed that the temperature 
94 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
gradient ia the metallic fim in the direction transverse to the x-coordinatd 
can be neglected (Fig. 2.8). The heat conduction in the metallic fim and 
meniscus liquid film is described by the following equation, which was ob-
tained as a result of an energy balance over a differential element (Vasiliev 
et al., 1981). 
d2T dT tan('y + x) 	 ke 
+ = 0 	(2.108) 
dx 2 	dx L(x) 
+ (T6 T) 
lc5(x)Lfiri(x) 
where x = TrIN for the circular geometry, x -= 0 for the plain grooved 
surface, and N is the number of grooves. The fim thickness variation is due 
to its wall inclination angle and the circular tube geometry 
Lft(x) = Li /2 + x tan(Y + X) 
and the liquid fim thickness is 
(2. 109) 
2R,,,enx 	. 	
1/2 
x2 
= - Rmen + [Rnaen 2 + 
cos2 (7 + x) 
+ 
cose)
, 
 x) 
5111 amen] 	(2.110) 
where 6 is measured perpendicularly from the liquid-vapor interface. It 
should be noted that the last term ia the left-hand side of eqn. (2.108) lias 
been corrected by the correction factor cos('y+ x) in order to compensate for 
the two-dimensional nature of the heat conduction in the liquid fim and the 
fim, which is only significant for large values of (ry + x)• Equations (2.108)- 
(2.110) are valid for the evaporator and condenser sections. However, the 
boundary conditions for eqn. (2.108) in these two sections are different, 
a.nd the value of 62 should be chosen as follows: 
62 = R + 60 ia the rough surface evaporation model, 
62 = Si in the smooth surface evaporation model, 
62 = 61s_-L 2 for the condenser heat transfer model. 
The boundary conditions for eqn. (2.108) ia the circular evaporator are 
dT 
= 	
(2.112) 
dx x=pg _ tt kID NEL1/2 ± (D9 - tt)tallet + X)I 
where boundary condition (2.111) is written with the assumption that 
Dg — te. For the simplified model, eqn. (2.108) was solved abo 
dT 
dx 
_ (2,„lic + 
kL 2 /2 
q€ 7i 
SOLID-LIQUID-VAPOR PHENOMENA 	 95 
in the microfilm and transition regions, where 6 was given by eqn. (2.98) 
and the right-hand side of eqn. (2.111) was set equal to zero. 
The boundary conditions for eqn. (2.108) in the circular condenser are 
dT 
dx 
While the 
dT 
dx 
I 
Cfg. 
(2.113) 
(2.114) 
T1 1 =0, which is 
x=o 	ko,L i / 2 
= I x.D9—tt 	kN[L 1 /2 + (Dg — tt)tan(ry + x)] 
values of Q'oilo , Cit, and Q 16 depend ou T„, 
obtained from the solution of eqns. (2.108)—(2.114), this problem is to be 
solved in conjunction with those concerning heat transfer in the thin film 
regions. 
The local heat transfer coefficient (for a given z) in the evaporator from 
the bottom of the groove surface to the vapor is 
he,bot 	
R qe 	o 
Vi r=pg _tt — To] R + Dg 
The local heat transfer coefficient from the externai surface of the evapo-
rator to the vapor is 
R 
— [ R° ln 	° 	
1 	R0 1-1 
(2.116) 
ka, R + Dg he,bot R + D9 
where the thermal resistance of the circular tube wall is aken into account 
For the condenser region, the heat transfer coefficients are defined in a 
similar manner. 
Tic,bat 
- 
nr- 
ln 
(lx 	Ro 
(2.117) 
(2.118) 
[71 1,=Da —t, — Tv] R,, + D9 
1-1 
R° 	
R, ± _1 
[ R° 
+ Dg 	hc,bot Rv 	Dg ] 
2.7.5 NUMERICAL PREDICTION 
Khrustalev and Faghri (1994) solved eqns (2 82) and (2.83) simultane-
ously for Tg (absolute error a a = 0.0001 K) and (Psat)6 (aa = 1 Pa) for 
every point ou $ by means of Wegstein's iteration method (Lance, 1960). 
The system of the four first-order ordinary differential equations with four 
initial conditions and one constitutive condition describing the evaporating 
(2.115) 
96 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
microfilm region, eqns. (2.89)-(2.97), were solved using the fourth-order 
Runge-Kutta procedure and the shooting method (on parameter 117%). The 
controlled relative error was less than 0.001% for each of the variables. The 
results obtained for comparatively small temperature drops through the 
thin fim were compared with those from the simplified model. Since the 
agreement was good, the simplified model was used further in the predic-
tion of the AGHP characteristics. Equation (2.106) was solved for Ca by 
means of Muller's iteration method (A a = 10-11 m), and the integration 
in eqn. (2.107) was made using Simpson's method. The heat conduction 
problem, eqns. (2.108)-(2.114), was also solved by Lhe standard Runge-
Kutta method (A a = 0.0001 K and Ar = 0.001% for the functions T and 
d'I/dx, respectively) along with eqns. (2.106) and (2.107) within the iter-
ative procedure to find Te, (A g = 0.0001 K). 
Khrustalev and Faghri (1994) compared their numerical results with 
Lhe experimental data provided by Schlitt et al. (1974). Therefore, the 
results presented in this section mostly refer to the AGHP with the following 
geometry: L t = 0.914 m, La = 0.152 m, 0.15 < L e < 0.343 m, W = 0.61 
mm, Dg = 1.02 mm, L1 = 043 mm, R„ = 4.43 mm, Ha r- 7.95 mm, 
= 0° , N = 27, N = 0°, tt = 0. The working fluids were ammonia and 
ethane, the casing material thermal conductivity was assumed to be ke, =- 
170 W/(m-K), a = 1, dispersion constant A' = 10'1 J and B = 3. 
The data in Figs. 2.10-2.12 wereobtained for ammonia with a vapor 
temperature in the evaporator of T„ = 250 K and a = 1. The solid surface 
superheat is AT = ITa - Te I, and the results obtained using the simplified 
model for evaporating fim are denoted as SIMPL. 
Figure 2.10(a) shows the variations of the free liquid surface temperature 
along the evaporating film for AT = 0.047 K, 0.070 K and 0.120 K, which 
are from the solutions of eqns. (2.78), (2.82)-(2.84), and (2.89)-(2.97) in 
the microfilm region. These results are compared to those obtained by the 
simplified model, where eqns. (2.78) and (2.82)-(2.84) were solved a1ong 
with eqns. (2.98), (2.108) and (2 109) with the boundary conditions 
Tia-0 = Tw, 
dT 
Tx 
=0 
x=o 
 
in the microfilm and transition regions for the same values of the roughness 
characteristic sizes (11,, = 0.33 pm, 1.0 pm and R, -4 oo). It should be 
noted that the temperature drop in the solid body in these regions was 
negligible in the results of the simplified model in comparison to AT, and 
the equilibrium fim thickness was defined within Lhe assumption that its 
free surface curvature is equal to 1/R man . In the simplified model for the 
case of a smooth surface, the value of the contact angle in the microfilm 
250.15 
250.1- 
g 	 ‘NN 
250.05- 	 
SOLID-LIQUID-VAPOR PHENOMENA 	 97 
--(a) 
250 
00 	2.0 	4.0 	6.0 	8.0 	10.0 
•10-7 
1-t 10.0 • 
()) 
2.0 	4.0 	6.0 	8.0 
Er = 0.39 iam, AT = 0.047 K 
Rr = 1.0_apkAT = 0.070 K 
Smount eu:Tece AT w. 0.120 K 
Rr = 0s33 ami AT = 0,047 IC. SIMPL 
RrLOjim,AT=O.OIOIÇ SIIdPL. 
Smooth eurface AT r. 0.120 ksiMpy 
(c) 
00 
	
2.0 	4.0 	6.0 	8.0 
	
10.0 
s(m) 
	
*10-7 
Figure 2.10: Characteristics of the evaporating film along the solid-liquid 
interface (ammonia, T = 250 K): (a) Free liquid surface temperature; (b) 
Thickness of the fim; (c) Generalized capillary pressure (Khrustalev and 
Faghri, 1994) 
0.0 
00 10.0 
•10-7 
Rr 0.99 um, AT = 0.047 X 
Ike 1.0 ma AT se 0.070 X 
Smoo_tkourfoosa=9: 120 X _ 
Rr 0.39 um. AT e 0.047_ X. 9114.1 — 
92a14bon,. AT = 0.970 K, SpOL 
Smooth ourfase, ATOKStMPL 
10 3 
o 
• 
(a) 
o 
o 
4.0 	6.0 
s (m) 
= 20°, eqn. (2,80) 
A,,seitrieg !Leal 
S ri0 eqa2(2.80) 
= 60°, eqn. 
( 3) 
8.0 10.0 2.0 00 
98 	 HEM' PIPE SCIENCE AND TECHNOLOGY 
00 	 5.0 	10.0 
	
15.0 
x (m) 
	
*10-1 
Figure 2.11: Heat flux through the evaporating film (ammonia, T„, =- 250 
K, a = 1): (a) Along the solid-liquid interface (microfilm region); (b) Along 
the fir. axis = 0.33 pra, AT = 1 K) (Khrustalev and Faghri, 1994) 
region wa.s 	= 7°, which was given by the numerical solution of eqns. 
(2.89)—(2.97). 
The corresponding variations of the fim thickness 8 and generalized 
capillary pressure p eap are shown in the Figs. 2.10(b) and 2.10(c). The 
results obtained by the simplified model have been artificially shifted along 
the s-coordinate in these figures (and siso in Fig. 2.11(a)) to make the com-
parison more understandable. Also, it should be noted that there is some 
difference between the s-coordinate and the x-coordinate used in the sim-
plified model. The following relation lias been used in the present analysis: 
15000 
a 10000- 
"E 
Ia 5000 - 
	 o 
20°, 91MPL 
efin= 40117MPL 
ijamel.91•' MN- 
fflaita8_02aPI. 
0,,,,n 0° = 20°, 40°. 8. 80°.-AT = 0.0;7 X 
(a) o e„,. 20°, 40°, 80°, 80°, AT = 0.070 IC 
1 	1 	1 	t 	1 
4.0 	6.0 	8.0 	10.0 	12.0 
Rr (m) 
00 	2.0 
12000 
O 
O 
(b) 
Nino. 20°. SIMPL 
e IN 40° S1MPL -A.- 
O 0,„„,=20°, 40°, AT = 0.07K 
1 	1 
O 	5000 	10000 
ge (%Wn12) 
Figure 2.12: Local heat transfer coefficient in the evaporator of the 
ammonia-Al heat pipe (T = 250 K): (a) Versus roughness size; (6) Versus 
heat flux (R,. = 1 pm) (Khrustalev and Faghri, 1994) 
4000 
15000 20000 
SOLID-LIQUID-VAPOR PHENOMENA 	 99 
s =14 arcsin (x/R,.). 
In Fig. 2.10(a), the interval of 1:5 variation along the evaporating film 
from the value of Tu, to approximately T was more prolonged in com-
parison to the results by Stephan and Busse (1992), and the interfacial 
thermal resistance was still significant even when the fim thickness was 
larger than 0.1 pm. For a smaller characteristic size Rr , the fim thickness 
increased more sharply along the solid surface (Fig. 2.10(b)), which is in 
agreement with eqn. (2.98). It should be mentioned that for the problem, 
eqns. (2.89)—(2.97) (unfike for the simplified model) 14 is not a parameter 
but the result of the numerical solution. The values of the maximum heat 
100 	 HEAT PIPE SCIENCE AND TECHNOLOGY ' 
20000 
= 0.05, Rr = 0.02 ism 
= 0.05_, .1_2rIam 
\'‘'.‘ • 	aSpr = 1 min , 
Gaiata.= 41.9_2S31___ 
N 
15000- 
e. 
10000 - 
Lor 
5000- 
(a) 
o 
o 20 40 60 80 100 
60000 
40000 
há6 20000 
20 	40 	60 	80 
1 	 1 
Ornee (ars degrees) 
Figure 2.13: Effect of lhe meniscus contact angle ou the local evaporative 
heat transfer coefficients (AT .= 1 K) : (a) Ammonia-Al heat pipe by Schlitt 
et al. (1974), (Tu = 250K); (6) Ethane-Al heat pipe by Schlitt et al. (1974), 
(Ti, = 200 K); (c) Water-copper evaporator by Ivanovskii et al. (1984), (T, 
= 300 K) (Khrustalev and Faghri, 1994) 
o 
100 
SOLID-LIQUID-VAPOR PHENOMENA 	 101 
flux in the microfilm region were extremely high in comparison to those In 
the meniscus region (Fig. 2.11). For AT = 0.120 K, the generalized cap-
illary pressure peap decreased from the initia1 value to an almost constant 
value by approximately 5000 times (Fig. 2.10(c)). For a larger AT, this 
sharp decrease can cause some difficulties in the numerical treatment while 
solving eqns. (2.89)—(2.97); that is why the simplified model is useful. The 
simplified model has given the variation of p cap along the fim which is even 
more drastic because of the surface tension term is absent in the capillary 
pressure gradient (Fig. 2.10(c)). However, the decrease of the total heat 
flow rate in the microfilm region caused by this assumption was compara-
tively small, which is illustrated by Fig. 2.11(a). The distributions of the 
heat flux in the microfilm, transition and beginning of meniscus regions for 
different meniscus contact angles B ruen as predicted by the simplified model 
are presented in Fig. 2.11(b). The total heat flow through the meniscus 
region was significantly larger in comparison to that through the microfilm 
region. This means that while estimating the heat transfer coefficient for 
an evaporator element, shown in Fig. 2.8, the simplified model should pro-
vide the accuracy needed. To verify this, the numerical results for the local 
heat transfer coefficient h in Fig. 2.12(a) have been obtained. The sim-
plified model underestimated h 6 by only 5%, which enables its use when it 
is necessary to avoid the numerical difficulties mentioned above. The local 
evaporative heat transfer coefficient h, depends upon the meniscus contact 
angle 0„,,„, especially for small ()men , and is practically independent of the 
heat flux on the externai wall surface of the evaporator and also of AT, as 
shown in Fig. 2.12(b). The characteristic roughness size affected the value 
of T.t e , decreasing it up to 30% for a = 1 in comparison to the value obtained 
for the smooth solid surface. For large meniscus contact angle the influence 
of the roughness size on the heat transfer coefficient is at lhe maximum 
when R, is dose to lhe length of the microfilm region. For small values of 
the accommodation coefficient (for example for a = 0.05) the effect of the 
surface roughness on the heat transfer is insignificant because the heat flux 
in the microfilm region in this case is comparatively smaIl (Fig. 2.13). 
The results of the present model were compared with lhe experimental 
data by Schlitt et al. (1974) and Ivanovskii et al. (1984) for the case of 
a small heat load applied to lhe AGHP (or evaporator). For a small heat 
load (Q, < Qmar) the vamos of the meniscus angle in both evaporator 
and condenser of the AGHP under consideration are comparatively large: 
Ora,. > 60°in the evaporator and O mer, > 800 in the condenser. This is 
valid because in the case without a heat load the grooves of an AGHP in 
the horizontal position are completely filled with liquid (i.e., the meniscus 
angle is dose to 90°). For O men > 60 0 the local evaporative heat transfer 
coefficients are practically independent on Orne., as shown in Fig. 2.13. 
102 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
The values of the evaporative heat transfer coefficients (based ou the outei 
tube diameter) obtained experimentally by Schlitt et ai. (1974) and those 
reported by Ivanovskii et al. (1984) were also found to be independent 
of heat load, which resulted in a valid comparison, as given in Table 2.2. 
The agreement of the results for ammonia, ethane and water is good for 
,€ 1 since it was mentioned by Carey (1992) that, for some substances 
(ethanol, methanol, water, etc.), the accommodation coefficient had been 
found to have very small values (0.02 to 0.04) in the experiments by Paul 
(1962). The physical reason for low a values in the microfilm region of 
the evaporator can be the concentration of the contaminants which usually 
exist in a heat pipe in this region. For the case of a = 1, the prediction gave 
significant (up to 100%) overestimations of Ti e even for a rough surface, as 
can be seen from Fig. 2.13. The experimental data by Ivanovskii et al. 
(1984) correspond to the case of evaporation of water from a copper plate 
with rectangular grooves for heat fluxes on the wall up to 20 W/cm 2 (W 
= 0.34 mm, Dg = 0.8 mm, L1 = 0.5 mm, L e = 100 mm, Te -= 300 K). 
A comparison with the numerical data reported by Stephan and Busse 
(1992) has also been made for ammonia with: T = 300 1{, k e, = 221 
W/(91-K), A' = 2 x 10 -21 J, a -=- 1, L1 = 10 -3 m, W =10-3 m, = 450 , 
Dg = 0.5 x 10 -3 m, = 10-3 m, Of = Omen = 19.7°, AT -= 1.31 K. Since 
Stephan and Busse used a flat plate for their experiments, the values for 
the vapor space radius and the outer pipe radius were set to Te t, = 1 m 
and Re = 1.0015 m to approximate a planar geometry. The results of the 
comparison are listed in Table 2.3. T6„ is the temperature of the vapor 
side of the interface and Q'rnie is the heat flow rate per unit groove length 
in the region O < x < 1 pm. The value of the heat transfer coefficient 
by Stephan and Busse (1992) was 23,000 W/(m 2-K), while the result of 
the present numerical analysis is 17,385 W/(m 2-K) for a rough surface (for 
R,. = 0.02 pm) and 23,900 W/(m 2-K) for a smooth surface, which validates 
the present analysis. 
The influence of the meniscus contact angle on the local heat transfer 
coefficient in the condenser (configuration by Schlitt et al. (1974)) is demon-
strated in Fig. 2.14. The results agree qualitatively with those obtained 
by Babenko et al. (1981). The increase of the liquid surface curvature 
causes the strong decrease of the heat transfer coefficient, where a sharp 
maximum occurs in the vicinity of maximum O men • In this location, the 
heat transfer coefficient is also dependent on the temperature drop ST. 
In the numerical experiments the liquid fim thickness was comparatively 
large (Fig. 2.14(a)) and the interfacial thermal resistance was negligible in 
comparison with that of the fim. The values of the heat transfer coefficient 
in the condenser based ou the outer tube diameter for the ammonia (z, 
250 K) and ethane (T = 200 K) heat pipes reported by Schlitt et al. (1974) 
103 
15000 
AT = 0.6 K 
AT = 1.0 K 
AT 4; 2.9 
AT = 4.0 K 
AT = 8.0 K 
10000— 
5000— 
o 
80 	82 	84 	86 	818 
Or„,„ (ara degrees) 
90 
K 
4121.0K 
AT ac 2.0 K 
AT = 4.0 K 
AT = 8.0 K 
4000 
3000 
2000 
1000 
o 
104 	 HEAT PIRE SCIENCE AND TECHNOLOGY 
P. 10.0 
	
ez.o° 	i3 	= 87.7" 
	
em.= 8t 0° 	Omon 
5 .0 - 
to 
------- 
-- 
0.0 
1 O 0 	5:0 	10.0 	1.0 	20.0 	25.0 
$ (m) 	 •10-5 
80 	82 	84 	86 	88 
Ornr, (aro degrees) 
Figure 2.14: Effect of the meniscus contact angle in the heat pipe condenser 
ou; (a) Liquid fim thickness variation along the surface of the fim top 
(ammonia, AT = 1 K, Ti, = 250 K); (b) Local heat transfer coefficient for 
ammonia (T, = 250 K); (c) Local heat transfer coefficient for ethane (T„ 
200 K) (Khrustalev and Faghri, 1994) 
90 
SOLID-LIQUID-VAPOR PHENOMENA 	 105 
are 7600 and 3300 W/(m 2-K), respectively. The numerical predictions were 
of the same order of magnitude as that reported by Schlitt et al. (1974). 
2.8 HEAT TRANSFER IN WAVY THIN 
LIQUID FILMS 
it is often difficult to establish a smooth fim of liquid ou the inner surface 
of thermosyphons, except when the liquid Reynolds number is mal. 
Ree — 	 G 12 	 (2.119) 
porDhf g 
Visual observations of ripples and waves developing in the liquid fim in the 
condenser section of the thermosyphon have been reported. The waves in-
crease the heat transfer coefficient compared to a smooth surface due to an 
increase in the interfacial surface area and the mixing action. Hirschburg 
and Florschuetz (1982) calculated, for approximately sinusoidal and for 
more complicated wave forms, the heat transfer coefficient for evaporation 
or condensation based ou unidimensional conduction as the only heat trans-
fer mechanism and have shown a favorable comparison of the prediction to 
experimental data. These data are of course also predicted fairly well by 
the empirical specification of Kutateladze (1982) and of Zazuli, as given by 
Kutateladze (1963). A theoretical analysis based on solving the convective 
diffusion equation, using the velocity distribution predicted from the hy-
drodynamic analysis of the wavy fiow, is the only way to understand the 
mechanism whereby the waves increase the rate of heat and mass transfer. 
Faghri and Seban (1985) solved the transient convective energy equation 
in a laminar falling liquid layer with a sinusoidally varying thickness for 
Reynolds numbers of 35 and 472. The system analyzed by Faghri and Se-
ban (1985) is a liquid fim at zero temperature initially, which flows down 
a vertical wall and evaporates at the free surface. The energy equation is 
07' 	OT OT 	702T 02T) 
(2.120) 
at 	az 	ay 	822 	,9y2 
In this analysis, the form of the w and v velocities are those predicted by 
linear hydrodynamic analysis, but the wave properties are those of Kapitza 
and Kapitza (1975) and Rogovan et al. (1969). 
Following common practice, the velocity in the z-direction is assumed 
to be quasi-parabolic 
w = 3(z, (77 - 5) 	(2.121) 
106 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
where rn is the local mean velocity, ij = y/6, and 6 is the local fim thickness. 
The y component of velocity is defined by the continuity equation. 
V = – f OW 
dy = –3 
oz 
aw) 
[—.5 
az 
( 77 2 
— 
2 
1,73 ) 
– — – 
6 
(36 C2 
az 
Ti3 
– (2.122) 
Consider the macroscopic mass balance, which may be written in the form 
as 	a is 	a _ 
i = — L 
W dy = – wz (tv6) 	 (2.123) 
Now, express the fim thickness in terms of the average fim thickness and 
the local amplitude O. 
(2.124) 
For a wave having a periodic character, one may write the following relation 
aro 	aro 
= 
(2.125) 
at –c az 
06 	06 
= 	 (2.126) 
where c is the wave velocity. Upon combining eqns. (2.123)–(2.126) and 
integrating, one obtains 
(c — rg)(1 + 0 ) = (c — .(T)o) 
	
(2.127) 
where ruo is the average velocity for the average film thickness 75. For O < 1, 
te) may be approximated by the following relation upon neglecting the third-
order terms. 
(2.128) 
Substituting eqn. (2.128) for iT; and eqn. (2.124) for 6 into eqn. (2.122) 
gives 
–30 
v = –ruo b {– 	– 1) (1– 204 ( if – 
az 	Wo 	 6 2 	6 
+ [1 + 	– 1) 0 – 	– 1) 021 C/22 – 	(2.129) 7-7;) 
WO 
With the assumption of a sinusoidal wave, Ø = A sin[(2/r/A)(z – ct)]. The 
independent variables of the energy equation are transformed from (t, z, y) 
to (e, where e = (2ir/À)(z – cl), to give 
OT 	OT 	021. 	827, 82T „, 
u l — + — C3 	4- C4 — ± U5 an 	N2 	onae 	ae2 (2.130) 
SOLID-LIQUID-VAPOR PHENOMENA 	 107 
where= -1-(w – c) 	 (2.131) 
27rn 
C2 = 	— (C — w)Acose – 1=a7/Asine 
À 	 À 5 
9 	2 	x 2 	 (2.132) 
–2 (-±r ) a e) yA2 cos2 + –v 
6. 
n 	2 -Á- 2 
C3 = +a ( r) ei) 	n2 A2 cos2 e 	(2.133) 
21 2 3 
C4 = –2a (—
A n
-A cos e 	 (2.134) 
(27r 2 
C5 = 	 (2.135) 
) 
These coefficients are evaluated with w from eqns. (2.121) and (2.128), and 
v from eqn. (2.129). 
In the evaporation problem, with a fluid of high latent heat of vaporiza-
tion, the average fim thicicness over a wavelength, 8, will not vary substan-
tially with distance z, and far from the location at which heating ar cooling 
begins, the temporal average temperature profile will be invariable with z. 
For condensation, the average thickness varies more, but the assumption 
about the temperature profiles is still justifiable. Thus, these cases are 
approximated by the following boundary conditions in terms of the nor-
malized temperature, where the z-axis is along the wall and the y-axis is 
perpendicular to it, with y = 0 coinciding on the wall. 
	
T(0, 77) = T(27r, 77) 	 (2.136) 
T(e,0) = 1 	 (2.137) 
T(e,1)= O 	 (2.138) 
Given the solution of eqn. (2.130) for these conditions, the heat flux at the 
wall is 
–
k 
—
aT 
1 and 1 i,„ (2.139) 
The heat flux at y = 1 is the flux normal to the wavy surface, T. ft, and 
this gives 
qks 	[27; 
(cose) é
,z l é OT 
6 	an 
(2.140) 
 
"Cr and êv are unit vectors along the z- and y-axes, respectively. The average 
heat.flow in O < e < 27r is obtained by integration. For the wavy layer this 
108 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
is with respect to the actual surface length. For small amplitudes and large 
wavelengths, the average flux for the surface (77 =1) is approximately 
and for the wall it is 
	
1 ( 21r 1 _OT 	I 	( 06) 2] 1/2 
fo 	anil 	(9,Z 
de 	(2.141) 
4„ = 1 f 2 w ar 
k 	27r jo b 
 
o 
(2.142) 
 
The calculations were made using 40 increments in both C and n , for the 
values of 2/r/À, A, and c/27,0, listed in Table 2.4, which are experimental 
determinations made by the cited authors. In Table 2.4, Columns 1-7 
give the experimental conditions and measurements. Column 8 gives the 
Prandll numbers for which the calculations are made. One, of the order of 7, 
corresponds to the experimental conditions, and the other, 1.7, was selected 
for a comparison to show the effect of Prandtl number. Column 9 gives the 
calculated average Nusselt number at the wall, and Column 10 gives the 
average Nusselt number at the outer edge of the layer. These should be the 
sa,me, and the difference indicates the failure of the calculation to satisfy 
the energy balance. This difference is small; it increases with Fteynolds 
number, reflecting some increase in truncation error. Column 11 gives the 
average Nusselt number as evaluated for unidimensional conduction. On 
this basis, the local Nusselt number is h8/k = 1 and then 
For the sinusoidal wave 
Ti 	1 	f21T 
h 	1 
k 	27r Ia 
de1 I- ( 1` 
ifo 
de 
(2.143) 
(2.144) 
k 	27r6 ia 	1 + A sin e 	2r8 
r 
fc,7 
	de 
1 — A sinel 
1 + A sin 
1 
\/(1 — A2 ) 
The difference between the calculated average Nusselt numbers is due to 
the contribution of convection and to the two-dimensional nature of the 
conduction that exists because of the variation of the layer thickness. It is 
of interest to investigate the result for w = v = e = 0, which corresponds to 
the two-dimensional conduction solution for the wave. The Nusselt num-
bers for such a calculation are shown in Columns 12 and 13. These should 
15- t- 	 15- t5- 
05 05 	05 Cr) 05 
0 .0 
en 	Cd Cd 	ac 
OC CO CO Cl O 0 
.— 
T—I 	 O O ri 1-1 
1-1 1-1'-1 1-1 
CO CC 
50 ":1 5 
O O 
r". 
CJC 
IO 
▪ •-■ 
CO CO 
11, 
O O 
1-1 
CO C- 
1.0 
R 
N 
tc: 
o 
on 
o 
o 
M 
1.
9
3
 0
.
52
 
st 	F2 
o 
lo 
e 
o 
IN 
-0(#© 
I, O C, O CO Cl 
00 	 c-1 C- 
Á 1-1 
I 
ti-o, 	ci,tl- 
o d. 	0) C- Cd GIC 
1.0 	CO Mt- 
•-■ 	 121 
011- 	Cl 	O C- 
IC 1-1 	C.: 	C-: 
a 
1-1 
o 
CO 	cd 
o 	t- 
Cd 
o 	o 
Cl 	•x 
Cd 
1-1 
o 
1-1 
co 
t- 
o 
Lo 
xr 
cx 
▪ O 	te, 
Li, 	 LO 
d 
Li' 	 ài 
109 
.25 
.20 
.05 
110 	 HEAT PIPE SCIENCE AND TECHNOLOGY 
be equal, and the difference between them indicates the effect of lhe trun-
cation error in lhe cakulation, made in the same way as for the cases for 
which there is fluid motion. These values are essentially the same as that of 
Column 11, for unidimensional conduction, and this correspondence shows 
that two-dimensiona1 conduction effects are negligible. Therefore, the dif-
ference between Nusselt numbers in Columns 9 and 11 reflect only the effect 
of convection. 
Figure 2.15 shows by circles the values of the Nusselt number, (Ti TV/c) 
as given by Column 9 of Table 2.4 for a Prandtl number of about 7. The 
value for (4F/it) of 472 is shown also for an arbitrarily higher value of A = 
0.55, because the value of A probably should not decrease as the Reynolds 
number increases. This, and a1so Column 11, shows how important the 
amplitude is. 
Figure 2.15 also contains lines, A, to show the specification of Zazuli 
given by Kutateladze (1963), and, B, to show that of Kutateladze (1982). 
Curve C is that of Hirschburg and Florschuetz (1982) for the intermediate 
wave solution designated by them as f+ = 0.65, which fits better lhe average 
of the data. Three data points from Chun and Seban (1971) for evaporation 
of water with a Prandtl number of about 5.6 are shown by plus symbols, 
four numerical results of Faghri and Seban (1985) are shown by circles in 
the same figure. 
2 
	
3 
log10 4.#," 
Figure 2.15: The average Nusselt number as a function of the Reynolds 
number: Curve A, Kutateladze (1982); Curve B,Kutateladze (1963); Curve 
C, Hirschburg and Florschuetz (1982); for f+ = 0.65 (Faglui and Seban, 
1985) 
4F/g via 
- - - 35 7.2 
— 472 7.2 
SOLID-LIQUID-VAPOR PHENOMENA 	 111 
0.5 
4/27c 
Figure 2.16: The local Nusselt number (Faghri and Seban, 1985) 
Figure 2.16 shows the variation of Lhe local heat transfer coefficient, 
normalized with respect to the average value over a period, as a function 
of the results being for the Prandtl number of 7.2 and the Reynolds 
numbers of 35 and 472. (The same portrayal for the Prandtl number of 1.7 
is not much different.) The ratio h/Tt, for the wall, varies considerably 
for the low Reynolds number, but not very much for the high Reynolds 
number. For the surface, the ratio h6/Ti varies substantially for both cases. 
The numerical results for the Nusselt numbers of Faghri and Seban 
(1985) are of the order of, but tending to be higher than, Lhe correlation 
equations of Kutateladze (1963, 1982). Comparison of Columns 9 and 11 in 
Table 2.4 indicates a considerable effect of convection and two-dimensional 
conduction. 
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