Heat Pipe Limits

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.

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

A Heat Pipe Assembly used to move heat away from the hot side of TEC to an external heat sink.

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 copper-water heat pipes to help customers design accordingly.

Heat Pipe and Thermosyphon Limits Table

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 one-half 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:

1. 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/m³
   V_{Sound_V}       Speed of sound in the vapor, m/s
   A_Vapor            Heat pipe vapor space area, measured perpendicular to the flow, m²
   γ                      Ratio of specific heats, c_p/c_v
2. The sonic limit is then:
   1. where:
      q_Sonic       Sonic limit, W
      λfg            Latent heat, liquid to vapor, J/kg

Sonic Limit Example

Sonic and Wicking Limits: The sonic limit controls the power below 400ºC for cesium, and below 500ºC for potassium.

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:

1. where:
   q_Entrain    Entrainment limit, W
   A_Vapor      Heat pipe vapor space area, measured perpendicular to the flow, m²
   λ_fg             Latent heat, liquid to vapor, J/kg
   σ                Surface tension, N/m
   ρ_V             Vapor density, kg/m³
   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.

1. 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. 387-397, 1995.

The first step is to define the Bond number, a dimensional number that measures the importance of surface tension forces versus body forces:

1. 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
2. The flooding constant, KFlooding, is defined as:
3. 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:

1. 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
   1. ΔP_C, Capillary Force
      1. The capillary pumping capability depends on surface tension and two radii of curvature of the liquid/vapor interface, measured perpendicular to each other:
         1. where:
            σ          Surface tension, N/m
            

            Radii of Curvature

            r₁ and r₂, are the radii of curvature (m)

      2. For sintered and screen wicks, the two radii are identical, so the equation reduces to:
         1. where:
            r_c is the pore radius
         2. One of the radii is infinite for grooves, so the equation becomes:
   2. ΔP_G, Gravitational Pressure DropThe gravitational pressure drop is:
      1. where:
         

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

         ρ_L   Liquid density, kg/m³
         ρ_V   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:

   3. P_L AND ΔP_V, Liquid and Vapor Pressure Drops
      1. 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
      2. 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:
         1. where:
            μ_L             Liquid viscosity, kg/(m s)
            k               Wick permeability, an intrinsic property of the wick, m².
            A_Wick       Wick area, measured perpendicular to the liquid flow direction, m²
            L_Effective  Effective length of the heat pipe, defined below, m
      3. Solving for ΔP_L, the equation becomes:
      4. 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

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 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 increases 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 from coming back, which can result in evaporator wick dry-out.

Boiling Limit Calculation

The boiling limit can be calculated by applying the nucleate boiling theory:

1. where:
   Q_Boil            Boiling limit, W
   L_Evap           Evaporator length, m
   k_eff               Effective thermal conductivity of the liquid-wick 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 10-5 m to 2.54 x 10-7 m] for conventional heat pipes.

Experimental Boiling Limits

1. Rules of thumb for the boiling limit in some typical heat pipe wicks:
   1. Sintered Wicks with Water: ~ 75 W/cm²
   2. Screen Wicks with Water: ~ 75 Wcm²
   3. Grooved, Aluminum Wicks with Ammonia: ~ 15 W/cm²
   4. 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/cm² over a 1 cm² area, shown in the center of the figure.

Case Study: Increasing the boiling limit with a hybrid Wick Heat Pipe

Hybrid CCHPs: axial grooved adiabatic & condenser sections; screen mesh or sintered evaporator wick

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 on-orbit 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 a micro-g environment. The maximum heat flux into a CCHP is set by the boiling limit, which is roughly 5 to 15 W/cm² for typical grooves.

In order to increase the heat flux limit to more than 50 W/cm², 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.

1. 

   Evaporator wick thickness effect, CCHP, Fx of temp

   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.

2. 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

Heat pipe performance limits

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 controlled over specific temperature 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 two-phase 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.

Copper Water Heat Pipe Limits: Left: Horizontal heat pipe; Right: Vertical heat pipe. Calculated with ACT’s heat pipe calculator.

There are three general limitations for passive two-phase devices, which include heat pipes, HiK™ plates, and vapor chambers:

• Temperature
• Vertical Height
• Acceleration

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 drop-off 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 9-10 inches (23-25 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 18-20 inches (46-50 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 de-primes, or dries out. The wick will quickly re-prime 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.

Dual heat pipes for high accelerations. One set of the heat pipes always operates, since the acceleration returns the condensate to the evaporator.

If the acceleration is of longer duration, then the thermal designer has three choices:

1. Design the heat pipes “gravity aided” under acceleration if the axis and direction of the acceleration are known.
2. Arrange heat pipes in pairs, so that one pipe is always “gravity aided”; see image at right.
3. Use an encapsulated conduction card.

Heat Pipe FAQs