Capillary Limit

During heat pipe operation, the working fluid evaporates in the evaporator and condenses in the condenser, transferring the latent heat from one end of the heat pipe to the other.  The liquid condensate is passively returned to the evaporator by capillary forces in the wick.  The maximum power that the heat pipe can carry and still return the condensate by capillary forces is the capillary limit:

heat pipe capillary limit equation

 

where:

ΔPc      Capillary force generated in the wick, Pa
ΔPg      Pressure drop due to gravitation and acceleration, Pa
ΔPL      Liquid pressure drop in the wick, Pa
ΔPV      Vapor pressure drop in the heat pipe, Pa

The capillary limit states that the capillary force generated by the wick must be larger than sum of the pressure drops in the wick.

 

ΔPc, Capillary Force

The capillary pumping capability depends on surface tension and two radii of curvature of the liquid/vapor interface, measured perpendicular to each other:

heat pump capillary pumping capability equation

 

 

where:

σ          Surface tension, N/m
r1 and r2, are the radii of curvature (m)

For sintered and screen wicks, the two radii are identical, so the equation reduces to:

sintered and screen wicks equations

 

 

where:

rc is the pore radius

One of the radii is infinite for grooves, so the equation becomes:

infinite grooves equation

 

 

 

ΔPg, Gravitational Pressure Drop

The gravitational pressure drop is:

gravitational pressure drop equation

 


where:

ρL         Liquid density, kg/m3
ρV        Vapor density, kg/m3
g          gravity or acceleration, m/s2
h          adverse heat pipe elevation, m;  See Figure 4.

Since the vapor density is typically much less than the liquid density, this reduces to:

 

 

 

The adverse elevation, h, is the height of the evaporator over the condenser.

Figure 1. The adverse elevation, h, is the height of the evaporator over the condenser.

 

ΔPL and ΔPV, Liquid and Vapor Pressure Drops

The mass flow rate circulating through the heat pipe is directly proportional to the power:

mass flow rate equation

 

where:

QHeatPipe            heat pipe power, W
mDot     liquid mass flow, kg/s
λ          latent heat, J/kg

With the exception of grooved wicks, the liquid pressure drop in the wick, ΔPL, is calculated with Darcy’s law for fluid flow through a porous media:

Darcy's law equation

 

where:

μL        Liquid viscosity, kg/(m s)
k          Wick permeability, an intrinsic property of the wick, m2.
AWick    Wick area, measured perpendicular to the liquid flow direction, m2
LEffective Effective length of the heat pipe, defined below, m

Solving for ΔPL, the equation becomes:

 

 

For a grooved wick, ΔPL is calculated with the standard pressure drop equations, found in any fluid mechanics textbook.  Similarly, ΔPV for all heat pipes is calculated using the standard pressure drop equations.

 

Effective Length

As discussed above, the capillary limit is calculated using simple, one-dimensional equations.  An effective length is used in the pressure drop equations to account for the variation in velocities along the heat pipe.  As shown in Figure 5, the vapor and liquid velocities at the start of the evaporator are zero.  They increase linearly due to evaporation to a maximum at the start of the adiabatic section, and then are constant in the adiabatic section.  In the condenser, condensation causes the vapor and liquid velocities to decrease linearly to zero at the end of the condenser.

The full velocity and half of the evaporator and condenser length is used for the effective length, to compensate for the changing velocity.

Figure 2. The full velocity and half of the evaporator and condenser length is used for the effective length, to compensate for the changing velocity.

 

To account for the varying velocity, an effective length is used to calculate the vapor and liquid pressure drops.

varying velocity equation

 

 

 

Capillary Limit Example

Heat pipe capillary limit versus temperature for water heat pipes with various diameters.

Figure 3. Heat pipe capillary limit versus temperature for water heat pipes with various diameters.

 

Figure 6 shows typical capillary limits as a function of temperature for several different heat pipe diameters, calculated using ACT’s heat pipe calculator. The heat pipe limit generally peaks somewhere in the middle of the working fluid temperature range:  At low temperatures, the capillary limit is restricted by high liquid viscosity and low vapor pressure (low vapor density → high vapor velocities).  At high temperatures (approaching the critical point), the maximum power drops off, since the surface tension and latent heat of vaporization go to zero.

 

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