Page 351 - Mechanical Engineers' Handbook (Volume 4)

340 Heat Pipes

where r c,e is the effective capillary radius depending on the wick structures. Table 2 lists
effective capillary radii for four typical wicks. For other kinds of wicks, the effective capillary
radii can be readily determined using the Laplace-Young equation.
Liquid Pressure Drop
The liquid pressure drop is the result of the combined effect of both viscous and inertial
forces. If the flow rate in the wick is very small, the effect of inertial force can be neglected,
and the pressure difference in the liquid phase is caused only by the frictional forces at the
liquid–solid interface and the liquid–vapor interface due to the vapor flow effect. The total
liquid pressure drop can be determined by integrating the pressure gradient over the length
of the flow passage, or
p (x) x dp l
l
0 dx dx (5)
where the limits of integration are from the evaporator end (x 0) to the condenser end (x
L), and dp /dx is the gradient of the liquid pressure resulting from the frictional forces
l
from the solid wick, wall, and vapor flow at the liquid–vapor interface, which can be writ-
ten as

dp l l ˙ m (6)
dx KA l
wl
where ˙m l is the local mass flow rate in the wick, and K, the permeability, can be expressed
as
2 r 2
K h,l (7)
ƒ Re l
l
The wick porosity, , in Eq. (7) is defined as the ratio of the pore volume V por to the total
volume V tot of wick structure. The hydraulic radius, r , in Eq. (7) is defined as twice the
h,l
cross-sectional area divided by the wetted perimeter, or r h,l 2A /P. It should be noted that
c
Eq. (7) is valid for both the circular and noncircular channels/grooves. If the Reynolds
number of the working fluid flowing through the wick structure is small and less than the
critical value, then laminar flow is assumed. The product of the friction factor and Reynolds
number, ƒ Re , for laminar flow is constant and depends only on the passage shape. It should
l
l
be noticed that when the liquid–vapor interface is affected by the vapor flow, the friction
factor Reynolds number product, ƒ Re , depends on the vapor flow in addition to the contact
l
l
angle and channel angle. 4
Table 2 Effective Capillary Radii 3
Structures r c,e Note
Rectangular groove w w groove width
cos contact angle
Triangular groove 3 cos w w groove width
4 cos contact angle
half groove angle
Wire screen w d w d w wire diameter
2 cos w mesh spacing
contact angle
Packed or sintered particles 0.41r s r s particle radius