Calculating Transport Capacity
The most important heat pipe design consideration is the amount of power a heat pipe is capable of transferring. Heat pipes can transfer much higher powers for a given temperature gradient than the best metallic conductors. The maximum power that the heat pipe can carry can be set either by the heat source and heat sink conditions or by internal heat pipe limits (can be minimized through proper design).
The temperature drops across a heat pipe are:
 Conduction through the envelope wall and wick
 Evaporation
 Vapor Space Temperature Drop
 is usually small compared to the temperature drops required to conduct heat into and out of the heat pipe.
 Condensation
 Conduction through the envelope wick and wall
In addition, there are further temperature drops to bring the heat to the heat pipe evaporator, and reject the heat from the condenser, for example, using a finned heat sink and forced convection at the condenser.
Critical design point: ensuring that the heat pipes meet your maximum system requirements. If operating conditions cause the heat pipe to exceed its power capacity, the effective conductivity of the heat pipe will be significantly reduced.
Heat Pipe Limits
There are five primary heat pipe transport limitations that must be considered during design: viscous, sonic, capillary, entrainment/flooding and boiling; see table below. These limits are a function of many variables including operating temperature, wick selection, and fluid properties. The most common limit for terrestrial applications is the capillary limit. ACT developed a heat pipe calculator for terrestrial copperwater heat pipes to help customers design accordingly.
Heat Pipe and Thermosyphon Limits Table
Heat Pipe Limit  Description  Cause 
Viscous (Vapor Pressure)  Viscous forces prevent vapor flow within the heat pipe.  Heat pipe operating near triple point with a very low vapor pressure – need to use a different working fluid. 
Sonic  Vapor flow reaches sonic velocity when leaving the evaporator, choking the flow.  Too much power at lower operating temperature. Typically this is seen at startup and will selfcorrect. 
Heat Pipe Entrainment  High velocity vapor flow strips liquid from the wick.  Not enough vapor space for the given power requirement. Occurs at low temperatures. 
Thermosyphon Flooding  High velocity vapor flow prevents liquid return in a gravity aided thermosyphon.  Not enough vapor space for the given power requirement. Occurs at low temperatures. 
Capillary  The capillary action of the wick structure cannot overcome gravitational, liquid, and vapor flow pressure drops.  Power input too high. Wick structure not designed appropriately for power and orientation. 
Boiling  Boiling occurs in the wick which prevents liquid return  High radial heat flux into the heat pipe evaporator. 
Viscous (Vapor Pressure)
Sonic
Note: as the vapor temperature in the heat pipe is lowered, the vapor pressure drops. To carry a given amount of heat, the vapor velocity must increase, which in turn increases the pressure drop from the evaporator to the condenser. At these low vapor pressures, compressible flow effects become important.
Design rule of thumb, always operate with powers less than onehalf of the sonic limit.
The sonic limit is reached when the Mach number reaches 1 at the beginning of the heat pipe adiabatic section. There are a number of ways to reach the sonic limit in compressible flow, such as decreasing the flow area, adding heat, and frictional effects.
Sonic Limit Caclulation
The heat pipe sonic limit is approached by mass addition, where mass is added to the flow rate from the start to the end of the heat pipe evaporator:
 where:
m_{Dot_Sonic }Mass flow at the entrance to the adiabatic section, when the flow is choked, kg/s
ρ_{V} Vapor density, kg/m^{3 }V_{Sound_V} Speed of sound in the vapor, m/s
A_{Vapor} Heat pipe vapor space area, measured perpendicular to the flow, m^{2 }γ Ratio of specific heats, c_{p}/c_{v}  The sonic limit is then:
 where:
q_{Sonic} Sonic limit, W
λfg Latent heat, liquid to vapor, J/kg
 where:
Sonic Limit Example
The chart to the right compares heat pipe power versus temperature for identical heat pipes using either cesium or potassium as the working fluid. The sonic and capillary limits were calculated, and then curves using the lower of the two limits were plotted in the figure.
On the left side of the graph, the maximum heat pipe power is set by the sonic limit (the roughly parabolic part of the curve), while on the right side of the graph, the maximum power is set by the capillary limit (the roughly flat part of the curve).
At lower temperatures, more power can be carried with cesium, since it has a higher vapor density (and higher sonic limit) at any given temperature. Once the temperature is increased above roughly 500°C, the potassium heat pipe carries more power (for this particular design). This is the reason that cesium is normally used at temperatures below 600°C, and is replaced by potassium and then sodium at higher temperatures.
Heat Pipe Entrainment/Thermosyphon Flooding
Like the viscous and sonic limits, the entrainment or flooding limit is related to the vapor velocity and is more significant at lower temperatures. The reason is that the vapor pressure and vapor density decrease as the temperature is lowered, so the vapor velocity must increase to carry the same power.
Design rule of thumb, always operate with powers less than 0.75 of the entrainment or flooding limit.
Heat Pipe Entrainment
In a heat pipe, the vapor flows from the evaporator to the condenser, with countercurrent liquid flow from the condenser to the evaporator. The entrainment limit is reached when the vapor velocity in the heat pipe is high enough to shear liquid from the wick.
The entrainment limit is given by:
 where:
q_{Entrain} Entrainment limit, W
A_{Vapor} Heat pipe vapor space area, measured perpendicular to the flow, m^{2 }λ_{fg} Latent heat, liquid to vapor, J/kg
σ Surface tension, N/m
ρ_{V} Vapor density, kg/m^{3 }r_{c} Wick pore radius, m
Thermosyphon Flooding Limit
In a thermosyphon, the vapor flows from the evaporator at the bottom of the thermosyphon to the condenser at the top of the thermosyphon. At the same time, the condensed liquid drains back to the evaporator along the wall (driven by gravity). The flooding limit is reached when the vapor velocity in the thermosyphon is high enough that shear stresses prevent liquid from returning to the evaporator.
 Note: loop thermosyphons are sometimes used to eliminate the flooding limit since the vapor and liquid flows are separated from each other.
One flooding correlation that is often used was developed by A. Faghri and published in the book, Heat Pipe Science and Technology, CRC Press, pp. 387397, 1995.
The first step is to define the Bond number, a dimensional number that measures the importance of surface tension forces versus body forces:
 Where:
ID Internal Diameter, m
ρL Liquid density, kg/m3
ρV Vapor density, kg/m3
g gravity or acceleration, m/s2
σ Surface tension, N/m  The flooding constant, KFlooding, is defined as:
 The Flooding Limit, qFlooding, is then:
AVapor Heat pipe vapor space area, measured perpendicular to the flow, m2
λfg Latent heat, liquid to vapor, J/kg
θ Tilt from vertical (valid for small tilts)
Capillary
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.
Design rule of thumb: the capillary limit states that the capillary force generated by the wick must be larger than the sum of the pressure drops in the wick.
Capillary Limit Calculation
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 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/mr_{1} and r_{2}, are the radii of curvature (m)
 where:
 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:
 where:
 The capillary pumping capability depends on surface tension and two radii of curvature of the liquid/vapor interface, measured perpendicular to each other:
 ΔP_{G}, Gravitational Pressure DropThe 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:
 where:
 P_{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
 where:
 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.
 The mass flow rate circulating through the heat pipe is directly proportional to the power:where:
 ΔP_{C}, Capillary Force
Effective Length
As discussed above, the capillary limit is calculated using simple, onedimensional 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 the figure at the right, the full velocity and half of the evaporator and condenser length are used for the effective length, to compensate for the changing velocity. 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
The calculation shown in the chart to the right 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.
Boiling
When a low heat flux is applied to a heat pipe evaporator, the heat is conducted through the wick, and liquid vaporizes on the inner surface of the wick, into the vapor chamber. As the heat flux increases, the temperature difference across the wick increase linearly. The boiling limit or heat flux limit takes place when the transverse heat flux into the evaporator is enough to create nucleate boiling in the wick of the evaporator section. This generates vapor bubbles, which can become trapped in the wick, blocking the liquid coming back, which can result in evaporator wick dryout.
Boiling Limit Calculation
The boiling limit can be calculated by applying nucleate boiling theory:
 where:
Q_{Boil} Boiling limit, W
L_{Evap } Evaporator length, m
k_{eff} Effective thermal conductivity of the liquidwick combination, W/(m K)
T Vapor temperature, K
λ Latent heat of vaporization, J/kg
ρ_{Vapor} Vapor density, kg/m3
r_{IDWall } Inner radius of the heat pipe wall, m
r_{ODVapor} Radius of the vapor core radius, m
σ Surface tension, N/m
PCapillary Capillary pressure of the wick structure, Pa
r_{nucleate } Nucleation site radius, which can be [2.54 x 105 m to 2.54 x 107 m] for conventional heat pipes.
Experimental Boiling Limits
 Rules of thumb for the boiling limit in some typical heat pipe wicks:
 Sintered Wicks with Water: ~ 75 W/cm^{2}
 Screen Wicks with Water: ~ 75 Wcm^{2}
 Grooved, Aluminum Wicks with Ammonia: ~ 15 W/cm^{2}
 In special cases, wicks can be designed with much higher boiling limits. The photo at the right shows a specially designed copper/water vapor chamber wick, which can remove 750 W/cm^{2} over a 1 cm^{2} area, shown in the center of the figure.
Case Study: Increasing the boiling limit with a hybrid Wick Heat Pipe
Grooved Constant Conductance Heat Pipes (CCHPs) transport heat from a heat source to a heat sink with a very small temperature difference. Aluminum/ammonia CCHPs are used for transferring the thermal loads onorbit due to their high wick permeability and associated low liquid pressure drop, resulting in the ability to transfer large amounts of power over long distances in microg environment. The maximum heat flux into a CCHP is set by the boiling limit, which is roughly 5 to 15 W/cm^{2} for typical grooves.
In order to increase the heat flux limit to more than 50 W/cm^{2}, ACT developed heat pipes with a hybrid wick that contains screen mesh, metal foam, or sintered evaporator wicks for the evaporator region, which can sustain high heat fluxes, where the axial grooves in the adiabatic and condenser sections can transfer large amounts of power over long distances due to their high wick permeability and associated low liquid pressure drop as shown in the figure to the right.

For 0.5” OD aluminum/ammonia hybrid heat pipe, boiling and capillary limits are shown in the chart to the right as a function of the evaporator’s sintered wick thickness in the CCHPs performance.
 The boiling limit can be improved by minimizing the wick thickness in the evaporator, but the capillary limit will be reduced. As the boiling limit is more sensitive and important than the capillary limit in hybrid CCHPs, the 0.06 in. (1.5 mm) wick should be selected.
Heat Pipe Performance Limit
To calculate the heat pipe performance limit, the different heat pipe limits are plotted as a function of temperature; see Top Graph at right that shows that the entrainment and capillary limits are controlling over certain temp ranges.
 Note: the viscous limit is not shown, since it is not relevant in the normal operating temperature range.
The lowest limit at each temperature is then the heat pipe performance limit curve, which is calculated by taking the lowest limit at each temp.; see Bottom Graph at right.
The viscous, sonic, and entrainment/flooding limits are all related to the vapor velocity and are more significant at lower temperatures. The reason is that the vapor pressure and vapor density decrease as the temperature is lowered, so the vapor velocity must increase to carry the same power.
Interesting in working with ACT’s experienced engineers on your heat pipe design?
Contact ACT today
Water Heat Pipe Parameters and Limitations
Heat Pipes are passive twophase heat transfer devices that transfer heat by evaporation and condensation. The heat pipes used in electronics thermal management, Spot Cooling Heat Pipes, HiK™ Plates., and Vapor Chambers, all typically use water as the working fluid.
 Note: This section pertains specifically to water heat pipes; heat pipes with other working fluids have different limitations, one reason being that their operating temperature ranges are different.
Maximum power for a given heat pipe geometry drops off at high and low temperatures, and as the adverse elevation increases.
There are three general limitations for passive twophase devices, which include heat pipes, HiK™ plates, and vapor chambers:
Operating Temperature
All heat pipes have a temperature range over which they have the best performance. Maximum power versus operating temperature for typical water heat pipes is shown in the Copper Water Heat Pipe Calculation output above. The maximum power is high from roughly 60 to 200°C, falling off gradually at lower temperatures (the power also falls off at higher temperatures, but this is typically not a concern for electronics cooling).
The dropoff in power as the temperature is reduced is set by the fluid properties of the working fluid. As the temperature (and associated saturation pressure) of the water is reduced, the water vapor density is also reduced. To carry a given amount of power, the vapor velocity in the heat pipe must increase, which in turn increases the pressure drop in the heat pipe while reducing the power that can be carried.
ACT generally designs heat pipes to operate at temperatures above ~25°C. At temperatures below 0°C, the water is frozen in the heat pipe, and conduction through the heat pipe wall is the primary method of heat removal. It is important to note that this is not generally a problem for electronics cooling, since the primary concern is to maintain the electronics below a maximum temperature. When the system starts up from a colder condition, say 40°C, the electronics will warm up until the temperature is around 25°C, and the heat pipe starts operating. Properly designed heat pipes can operate after thousands of freeze/thaw cycles, see testing data graphed at right.
If operation below 25°C is required, then the thermal designer can switch to a different working fluid such as methanol, or use an encapsulated conduction card.
Adverse Vertical Height
Heat pipes return liquid from the condenser to the evaporator through a wick, which allows them to operate in any orientation. During operation, capillary forces in the wick must overcome the sum of liquid and vapor pressure drops as well as the adverse gravity head and acceleration. As the adverse elevation is increased (evaporator located above condenser), more of the wick pumping capability is used to counteract the adverse gravity head, and the maximum heat pipe power is reduced. This can be seen when comparing the Copper Water Heat Pipe Calculation output from above, sides (a) and (b), which shows power with a level heat pipe, and a heat pipe with an adverse 4 inch (10 cm) elevation, respectively. It can be seen that the maximum power is reduced significantly. Other adverse elevations can be examined through calculations with ACT’s heat pipe calculator.
In general, water heat pipes can operate with the evaporator elevated a maximum of 910 inches (2325 cm) above the condenser. This sets the maximum elevation for spot cooling heat pipes and vapor chambers. This elevation is doubled for HiK™ plates to 1820 inches (4650 cm), when the HiK™ plates are cooled on both the top and the bottom, and a double set of heat pipes is embedded. Note that if the electronics and sink can be arranged so that the electronics are lower than the heat sink, then the heat pipe behaves as a thermosyphon. In this orientation, gravity returns the liquid to the evaporator rather than capillary forces, and the heat pipe length is essentially unlimited.
Acceleration
During operation, capillary forces in the wick must overcome the sum of liquid and vapor pressure drops as well as the adverse gravity head and acceleration. Water heat pipes will stop operating under high adverse acceleration when the wick can no longer return condensate to the evaporator, and the heat pipe deprimes, or dries out. The wick will quickly reprime after the acceleration stops. Heat is stored by a rise in temperature during acceleration. Most accelerations are relatively short, and this temperature rise is acceptable.
If the acceleration is of longer duration, then the thermal designer has three choices:
 Design the heat pipes “gravity aided” under acceleration if the axis and direction of the acceleration are known.
 Arrange heat pipes in pairs, so that one pipe is always “gravity aided”; see image at right.
 Use an encapsulated conduction card.