The heat pipe is a sealed system containing a liquid, which when vaporized transfers heat under isothermal conditions. The temperature of the vapor corresponds to the vapor pressure, and any temperature variation throughout the system is related directly to vapor pressure drop. The choice of liquid charge is related to the required operating temperature range of the heat pipe. This may vary from cryogenic conditions (well below 0°C) to high temperature operation (above 600°C), in which case liquid metals are used (e.g., potassium, sodium or lithium).

The heat pipe has three major operating zones, namely evaporator, adiabatic section and condenser, see Figure 1. In the case of the elementary pipe design, liquid returns from the condenser via a wick structure. The wick is designed to provide a capillary pumping action, as described below. The heat pipe is a development of the *thermosyphon*, in which there is no wick structure and liquid is returned to the evaporator by gravity. Thus in the case of the thermosyphon the condenser region must be above the evaporator region, angle in Figure 1 being negative.

The heat pipe as we now know was originated by Grover in Los Alamos for use in thermionic direct conversion devices. One of its main features, namely isothermalization, is of major significance in this application. It is further possible to control the temperature of operation of the pipe by introducing a controlled pressure of inert gas, such as helium or argon. The vapor pressure of the liquid charge will be equal to that of the gas, provided operation is ensured to be of the nature illustrated by Figure 2. This pipe is referred to by Dunn and Reay as "gas-buffered" or "variable conductance" design.

The heat pipe has four major operating regimes, each of which sets a limit of performance in either heat transfer rate (axial or radial) or temperature drop. The limit for each regime is presented below for a simple cylindrical geometry heat pipe, as illustrated in Figure 3. These limits were catogorized by Busse and are as follows.

At low temperature range of operation of the working fluid, especially at start-up of the heat pipe, the minimum pressure at the condenser end of the pipe can be very small. The vapor pressure drop between the extreme end of the evaporator and the end of the condenser, represents a restriction in operation. The maximum rate of heat transfer under this restricted vapor pressure drop limit is given by:

where D_{v} is the diameter of the vapor passageway, h_{lg} is the enthalpy of vaporization, p_{v} is the pressure, ρ_{v} the vapor density, and η_{v} the vapor dynamic viscosity.

NOTE. All vapor properties in Eq. (1) refer to conditions at the closed end of the evaporator.

where l_{e} is the length of the evaporator region, l_{a} the length of the adiabatic region and l_{c} the length of the condenser region.

At a temperature above the vapor pressure limit, the vapor velocity can be comparable with sonic velocity and the vapor flow becomes "choked". The recommended maximum rate of heat transfer, to avoid choked flow conditions (i.e., sonic limit) is given by

where A_{v} is the area of the vapor passageway.

The vapor velocity increases with temperature and may be sufficiently high to produce shear force effects on the liquid return flow from the condenser to the evaporator, which cause entrainment of the liquid by the vapor. The restraining force of liquid surface tension is a major parameter in determining the entrainment limit. Entrainment will cause a starvation of fluid flow from the condenser and eventual "*dry out*" of the evaporator.

The entrainment limit is given by

where σ is the surface tension of the liquid, x is the characteristic dimension of the wick surface (≡2r_{σ}, where r_{σ} = effective radius of pore structure).

The driving pressure for liquid circulation within the heat pipe is given by the capillary force established within the wick structure, namely:

Circulation will be maintained provided:

where Δp_{l} is the frictional pressure drop in liquid and Δp_{v} is the factional pressure drop in the vapor.

For laminar flow conditions in the wick structure:

where is the rate of heat transfer, η_{l} the liquid viscosity, A_{w} the cross sectional area within the wick, K the permeability of the wick, and ρ_{l} the liquid density.

The gravitational head (ρ_{l}gl cosΦ) may be positive or negative, depending on whether the pipe is gravity assist or working against gravity (see Figure 1).

In calculating the vapor pressure drop (Δp_{v} ) it is important to ensure that the Mach Number M < 0.2 and incompressible flow conditions are assumed.

For laminar flow condition (i.e., Re_{v} < 2000) the *Hagen-Poiseuille equation* may be applied, thus

where r_{v} is the radius of the vapor passageway.

For turbulent flow, the Fanning equation gives:

where l is the length of the vapor passageway, f is *Fanning friction factor* (0.079/Re_{v}^{1/4} for 2100 < Re_{v} < 10^{5}), V_{v} is the vapor velocity and D_{v} the diameter of the vapor passageway; and where Re_{v} is the Reynolds number for vapor flow.

NOTE: for laminar flow, i.e., Re < 2100 the *Fanning friction factor* quoted above is replaced by the *Hagen-Poiseuille* form, f = 16/Re_{v}. (See Friction Factors for Single Phase Flow.)

The vapor pressure drop over the length of the evaporator plus the adiabatic region, for turbulent flow (Re ≥ 2000) is given in ESDU 79012 as

The temperature drop across the wick structure in the evaporator region increases with evaporator heat flux. A point is reached when temperature difference exceeds the degree of superheat sustainable in relation to nucleate boiling conditions. The onset of Boiling within the wick structure interferes with liquid circulation. This eventually leads to "*dry out*", which in the case of constant heat flux heating can cause "**Burn Out**" of the evaporator containment.

In the event of nucleate boiling the relationship between bubble radius and pressure difference sustainable across the curved surface is given by:

where R is the radius of the bubble.

The degree of superheat ΔT_{s} related to Δ_{p} is given by the *Clausius-Clapeyron equation*

where v_{v} is the specific volume of the vapor; and v_{l} is the specific volume of the liquid.

Since v_{l} << v_{v}, then

Nucleation sites, at which bubbles first form, are provided by scratches or rough surfaces and by the release of absorbed gas. Dunn and Reay give the following impirical equation for the degree of superheat in a wick structure:

where δ is the thermal layer thickness (i.e., characteristic dimension) equal to 2.5 × 10^{−5} m for typical surfaces. Using this characteristic dimension they have produced a table showing the degree of superheat for a range of candidate heat pipe working fluids, including ammonia, water and liquid metals (for high temperature operation). The liquid metals, having much higher surface tension give much higher degrees of superheat (e.g., 10°C ΔT_{s} < 54°C compared to ΔT_{s} ≈ 2°C for NH_{3} and H_{2}O).

The associated evaporator heat flux with nucleate boiling is given by:

where λ_{w} is the effective thermal conductivity of the wick (metal plus liquid) and x is the thickness of wick structure.

where

and λ_{s} is the conductivity of the solid, λ_{l} the conductivity of the liquid and ε the porosity of the wick structure.

An alternative equation for the boiling limit is given in ETSU data sheet 79012 as

where Δp_{σ} is the maximum capillary pressure provided by the wick (see above), r_{n} is the nucleate radius (= 2 × 10^{−6} m) and Z the thermal impedance of the wick.

The above limitations are seen to relate to temperature, according to working fluid, in the manner illustrated by Figure 3. The choice of working fluid must be such that the heat pipe is operated at a temperature well beyond the viscous limit, even at start up.

The *thermosyphon* differs from the heat pipe, in having no wick structure. The device can therefore only operate with the condenser above the evaporator with gravity-assist liquid flow return. Equations relating to the various limits of performance of a two-phase closed thermosyphon are given in ESDU data sheet 81038. The viscous and sonic limits are the same as for wicked heat pipes and the equation for the boiling limit and countercurrent flow limits are summarized below.

The boiling limit occurs when a stable vapor film is formed between the liquid and the evaporator wall. The maximum heat flux as given in ESDU 81038:

This condition relates to entrainment or flooding. The maximum heat transfer under this condition is given by

where f_{1} is a function of the Bond Number, defined as

f_{1} can be found from ESDU 81038, but is seen to have a value of 4 at B_{0} = 1.0 and a value of 8 at B_{0} = 10.0.

The factor f_{1} is a function of a dimensionless parameter K_{p}, which is defined as

and

The factor f_{3} is a function of the inclination of the heat pipe. When vertical f_{3} = 1. The varitation of f_{3} with both angle of inclination of the pipe and Bond number is given in Figure 2 of ESDU 81038.

In selecting the working fluid for a heat pipe or thermosyphon it is necessary to ensure that the device operates within the above defined limits. The choice of working fluid very much depends on the thermophysical properties of the fluid as well as the mode of operation of the device. A figure of merit (Φ) may be used to establish the relative performance of a range of prospective working fluids. Values of Φ are given for a range of fluids in ESDU 80017 and a plot of Φ versus temperature illustrates the influence of the working fluid properties on maximum heat flux.

For capillary driven heat pipes

see Figure 4 for Φ_{1} versus Temperature for a range of fluids.

For a thermosyphon

The heat pipe may be used to transfer heat under near isothermal conditions and may also be used to effect temperature control, as illustrated by Figure 2. The use of a buffer gas to control vapor pressure and hence vapor temperature is seen to be a very effective method of temperature control. Both passive and active techniques are illustrated in Heat Pipes by Dunn and Reay. There is also the potential of enhanced heat pipe performance, when operating in the capillary limit regime, with use of composite wick structure design. (Ref. Dunn and Reay.) A further advantage of the heat pipe is its application as a thermal transformer, see Figure 5.

The concept of vaporization of a fluid in a heated porous element was developed firstly at Harwell by Dunn and Rice in the late 1960's for establishing a nuclear reactor design using this principle, and secondly at the University of Reading, leading to the successful submission of a PhD thesis by Rice (1971). Work at Reading lead to the use of the porous element heater for such applications as a fast response vapor diffusion vacuum pump, jointly developed with AERE Harwell and Edwards High Vacuum Ltd. The short residence time for liquid heating and evaporation was exploited in further work associated with pyrolytic chemical reactions.

The principle of vaporization within a porous element, compared to vaporization from a plain surface, is illustrated in Figure 6 and 7. It is seen that stable boiling can only be achieved in a porous media if a uniform flow regime is established. The means for achieving this condition was brought about by the use of a dispenser region through which the liquid feed was fed into the element, see Figure 8. The dispenser also provided a thermal barrier to prevent subcooled boiling at inlet and porous alumina, with small pore size (typically 1-5 μm diameter pore) and low permeability, and produced high pressure drop compared to pressure drop across the heated porous media. Both flat plate and cylindrical geometry porous element boilers were constructed, with stable boiling and superheat in a single pass, see Figure 8.

The concept was developed using electrically heated porous elements, see Figure 7. Specific power ratings in excess of 1 kW/cm^{3} of element were achieved both when vaporizing water and freon. In the case of freon, evaporation and superheat was achieved uniformly with a porous element in excess of l m long.

It was originally conceived that the porous element boiler could be developed to provide a new concept of boiling water reactor design, see Figure 9. The porous element would consist of packed enriched UO_{2} coated particles contained in a porous ceramic “dispenser” tube. The reactor vessel would be fed with water through porous dispenser tubes. It was conceived that this reactor design would permit both boiling and superheating in a single pass through the porous element "*fuel rods*". An experimental "*in-pile*" steam generator was designed, as illustrated in Figure 8, in the hope that the concept may be demonstrated under nuclear heating conditions.

#### REFERENCES

Busse, C. A. (1973) Theory of ultimate heat transfer limit of cylindrical heat pipes, *Int. J. Heat and Mass Transfer*, 16, 169-186. DOI: 10.1016/0017-9310(73)90260-3

Chisholm, D. (1971) *The Heat Pipe*, Mills and Boon Ltd., London.

Cotter, T. P. (1965) *Theory of Heat Pipes*, LA 3246-MS, 26 March 1965.

Dunn, P. D. and Reay, D. A. (1994) *Heat Pipes*, 4th edn., Pergamon.

Grover, G. M., U.S. Patent 3229759. Filed 1963.

Grover, G. M., Cotter, T. P., and Erickson, G. R, (1964) Structures of very high thermal conductance, *J Appl. Phys.*, 35, p. 1990.

Fulford, D., (1989) *Variable Conductance Heat Pipes*, PhD Thesis, University of Reading, U.K.

*Heat Pipes—General Information in their Use, Operation and Design,*

ESDU data sheet 80013, Aug. 1980.

*Heat Pipes—Performance of Capillary-driven Design*, ESDU data sheet>

79012, Sept. 1979.

*Heat-Pipes—Properties of Common Small-pore Wicks*, ESDU data sheet

79013, Nov. 1979.

*Heat Pipes—Performance of Two-phase Closed Thermosyphons*, ESDU data sheet >

81038, Oct. 1981.

Rice, G., (1971) *Porous Element Boiler*, PhD Thesis, University of Reading, U.K.,

Rice, G., Dunn, P. D., Oswald, R. D., Harris, N. S., Power, B. D., Dennis, H. T. M., and Pollock, J. F. (1977) An industrial vapor vacuum pump employing a porous element boiler, *Proc. 7th Int. Vacuum Conf*, Vienna.

Rice, G., Dunn, P. D., (1992) 'Porous Element Boiling and Superheating', *8th International Heat Pipe Conference*,, Beijing Sept. 1992, *Publ. Advances in Heat Pipe Science and Technology*, Ed. by M. A. Tangze—Int. Academic Publishers, ISBN 7-80003-272 1/T 9.

*Thermophysical properties of heat pipe working fluids:* operating range between -60°C and 300°C, ESDU data sheet 80017, Aug. 1980.

#### References

- Busse, C. A. (1973) Theory of ultimate heat transfer limit of cylindrical heat pipes,
*Int. J. Heat and Mass Transfer*, 16, 169-186. DOI: 10.1016/0017-9310(73)90260-3 - Chisholm, D. (1971)
*The Heat Pipe*, Mills and Boon Ltd., London. - Cotter, T. P. (1965)
*Theory of Heat Pipes*, LA 3246-MS, 26 March 1965. - Dunn, P. D. and Reay, D. A. (1994)
*Heat Pipes*, 4th edn., Pergamon. - Grover, G. M., U.S. Patent 3229759. Filed 1963.
- Grover, G. M., Cotter, T. P., and Erickson, G. R, (1964) Structures of very high thermal conductance,
*J Appl. Phys.*, 35, p. 1990. DOI: 10.1063/1.1713792 - Fulford, D., (1989)
*Variable Conductance Heat Pipes*, PhD Thesis, University of Reading, U.K. *Heat Pipesâ€”General Information in their Use, Operation and Design,*- ESDU data sheet 80013, Aug. 1980.
*Heat Pipesâ€”Performance of Capillary-driven Design*, ESDU data sheet>- Sept. 1979.
*Heat-Pipesâ€”Properties of Common Small-pore Wicks*, ESDU data sheet- Nov. 1979.
*Heat Pipesâ€”Performance of Two-phase Closed Thermosyphons*, ESDU data sheet >- Oct. 1981.
- Rice, G., (1971)
*Porous Element Boiler*, PhD Thesis, University of Reading, U.K., - Rice, G., Dunn, P. D., Oswald, R. D., Harris, N. S., Power, B. D., Dennis, H. T. M., and Pollock, J. F. (1977) An industrial vapor vacuum pump employing a porous element boiler,
*Proc. 7th Int. Vacuum Conf*, Vienna. - Rice, G., Dunn, P. D., (1992) 'Porous Element Boiling and Superheating',
*8th International Heat Pipe Conference*,, Beijing Sept. 1992,*Publ. Advances in Heat Pipe Science and Technology*, Ed. by M. A. Tangzeâ€”Int. Academic Publishers, ISBN 7-80003-272 1/T 9. *Thermophysical properties of heat pipe working fluids:*operating range between -60Â°C and 300Â°C, ESDU data sheet 80017, Aug. 1980.