Page 354 - Mechanical Engineers' Handbook (Volume 4)

3 Heat Transport Limitations 343

2 T (p )
T T (p ) s l (22)
l
l
s
hr
ƒg e
v
where r is the meniscus radius of the vapor bubble formed in the wick structure of the heat
e
pipe, which is directly related to the pore size of the wick structure. According to the theory
presented by Hsu 10 and the derivation presented by Carey, 11 an embryo bubble will grow
and a cavity will become an active nucleation site if the equilibrium superheat becomes
equaled or exceeded around the perimeter of the embryo. To avoid boiling near the base of
the wick structure, the temperature difference between the wall and the saturation temperature
must be less than the superheat required for bubble formation. Using the superheat obtained
above, the critical heat flux related to the boiling limit can be found as
k 2 T (p )
eff
l
s
q k [T T (p )] (23)
s
l
l
eff
hr
v
ƒg e
As can be seen, the boiling limit is sensitive to the effective thermal conductivity, k , and
eff
the meniscus radius of vapor bubble, r , at the wick–wall interface.
e
3.3 Entrainment Limit
In an operating heat pipe, when the vapor flow direction is opposite to the liquid flow
direction, the frictional shear stress occurring at the liquid–vapor interface may slow down
the return of liquid to the evaporator. As the vapor velocity increases, the vapor flow effect
on the liquid–vapor interface increases, depending on surface tension, viscosities, and den-
sities of both the vapor and liquid phases. When the influence caused by the frictional shear
stress acting on the liquid–vapor interface by the frictional vapor flow is large enough, the
liquid flow cannot flow back to the evaporator. When this occurs, the liquid in the evaporator
dries out. At this point the heat pipe reaches a heat-transport limit, which is known as the
entrainment limit. Based on a Weber number equal to one, i.e., We F /F 1, where F lv
lv

is the shear stress at the liquid–vapor interface and F is the surface tension force, Cotter 12

developed an approximation of the entrainment limit as follows:
Ah 0.5

q ent v ƒg v (24)
2r h,w
where r is the hydraulic radius of the wick surface pore.
h,w
3.4 Viscous Limit
When the vapor pressure from the evaporator to the condenser cannot overcome the vapor
pressure drop caused by the viscous forces, the heat pipe reaches a heat-transport limit,
which is called the viscous limit. In particular, the viscous limit is reached when the vapor
pressure in the condenser is equal to zero. Using assumptions of laminar flow, ideal gas, and
zero pressure in the cap end of the condenser, the viscous limit can be determined by
4rh pA v
2
ƒg v,e v,e
v
q vis (25)
ƒ Re L eff
v
v
v
where , p v,e are the vapor density and vapor pressure in the cap end of the evaporator,
v,e
respectively.