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:

*where:*

ΔP_{c} Capillary force generated in the wick, Pa

ΔP_{g} Pressure drop due to gravitation and acceleration, Pa

ΔP_{L} Liquid pressure drop in the wick, Pa

ΔP_{V} 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.

## ΔP_{c}, 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:

*where:*

σ Surface tension, N/m

r_{1} and r_{2}, are the radii of curvature (m)

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

*where:*

r_{c} is the pore radius

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

## ΔP_{g}, Gravitational Pressure Drop

The gravitational pressure drop is:

where:

ρ_{L} Liquid density, kg/m^{3
}ρ_{V} Vapor density, kg/m^{3
}g gravity or acceleration, m/s^{2
}h adverse heat pipe elevation, m; See Figure 4.

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

## ΔP_{L} and ΔP_{V}, Liquid and Vapor Pressure Drops

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

*where:*

Q_{HeatPipe} heat pipe power, W

m_{Dot} liquid mass flow, kg/s

λ latent heat, J/kg

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

*where:*

μ_{L} Liquid viscosity, kg/(m s)

k Wick permeability, an intrinsic property of the wick, m^{2}.

A_{Wick} Wick area, measured perpendicular to the liquid flow direction, m^{2
}L_{Effective} Effective length of the heat pipe, defined below, m

Solving for ΔP_{L}, the equation becomes:

For a grooved wick, ΔP_{L} is calculated with the standard pressure drop equations, found in any fluid mechanics textbook. Similarly, ΔP_{V} 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.

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

## Capillary Limit Example

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.