5.2. Heat pipe manufacture

filling pipe is pressed, and after disconnection from the vacuuming pump, filling pipe is sealed

In the chemical part of the production, mechanical impurities and rust from the body of the heat pipe are first removed. This is followed by wet cleaning of the heat pipe components including cleaning with solutions, acids, and basic acids which are precisely determined for each type of material. The ultrasound cleaning, vacuuming, degassing, and passivation are processes that guarantee a high purity of the heat pipe material and thus contribute to longlasting failure free operation. Generally, two important goals are achieved by cleaning. The first goal is to ensure good wetting material of the heat pipe well by working. The second goal is to remove all particles of dirt because the presence of impurities in solid, liquid or gaseous form may have an adverse effect on the heat transfer ability of heat pipe. Small particles can inhibit the formation of capillary pressure in the wick structure. Machining or human hand grease may reduce the wettability of the wick structure. Oxides formed on the walls of the wick structure may also reduce the ability of the working fluid to wet the surface. It is also highly advisable to use an ultrasonic cleaner to clean the heat pipe material, as the ultrasound breaks down impurities firmly absorbed on the surface of metallic particles that cannot be removed in any other way. The cleaning of the heat pipe is repeated immediately before filling with the working fluid, after connecting the body with the end caps and the filling tube. After cleaning, the tube is degassed by heating to a higher temperature and vacuuming the interior. In the case of a wick heat pipe, it is necessary to remove the oxide layers from the wick

The working fluid added into the heat pipe must be completely clean, free from all mechanical impurities and gases, as their trace residues can also react with the body material of the heat pipe and the formation of undesirable elements. Clean substances can be purchased without any problems at special chemical stores. However, even in pure liquids and solids, an incompressible gas may be present. These gases can be removed by repeated freezing and thawing cycles. The working fluid in the filling bottle can freeze using the liquid nitrogen or dry ice.

Filling process of each type of working fluid is happening under other conditions. The characteristic of the filling process depends on the state of the working fluid at ambient temperature. If the working fluid is at the room temperature in the gaseous state (cryogenic), the filling can be carried out via a gas container of high quality. Filling and closing process of liquid-metal

The filling of low-temperature heat pipes can be carried out at room temperature without the use of any protective atmosphere. Before filling the heat pipe, it is advisable sucking the air from it to ensure the removal of undesirable components contained in the materials which could be later shown as non-condensing components. In addition, under pressure, the working fluid naturally enters into the heat pipe, and thus the equilibrium state of the pure vapor and

by soldering.

5.1.2. Chemical part of heat pipe manufacture

162 Porosity - Process, Technologies and Applications

structure by chemical cleaning (e.g., solvents).

5.1.3. Filling the heat pipe with the working fluid

heat pipes is appropriate to do in the vacuum chamber [37].

liquid phases at a lower pressure than atmospheric will be achieved [38].

Although the production of the porous wick structure is most difficult from all types of wick structures, it is one of the three most used wick structures in the heat pipe, because it is able to create a large capillary pressure that allows the heat pipe to transfer a high heat flux in the antigravity position. One method of making a porous wick structure is to sinter a copper powder uniformly poured around a coaxially centered steel mandrel located inside the copper pipe at a temperature close to melting the powder material in a high temperature electric furnace. By sintering copper powders is possible made wick structure with the high thermal conductivity, high wick porosity, small capillary radius, and high wick permeability what are the main which the wick structure have to ensure supplies evaporator with the condensed liquid. The high thermal conductivity of copper ensures that the wick structure will not have high thermal resistance, which is one of the expecting properties of wick structure too. The formation of a suitable porous structure by sintering the metallic powder depends, in addition to the sintering temperature, both on the time of sintering and the grain size of the powder. Copper powders with a particle size of 30–100 μm or copper fibbers of 2–3 mm in length and a diameter of 20–100 μm are used for the production of porous sintering structure.

The most important part of the heat pipe is wick structure. This expersiment deal with heat pipes with sintered wick structure made from copper powder with granularity of 100, 63 and 35 μm by sintering in the high thermal electric oven using powder metallurgy. By sintering the copper powder on the inner wall of the heat pipe container, 1.5 mm thick wick structures were created. The sintering process of the wick structure was approx. at temperature of 1000C and time of 30 min. Seeing that the pore size of the wick structure depends on the grain size of the copper powder, sintering the copper powder of various grain size creates the wick structure of various pore size. The overall length of the heat pipes is 0.5 m.

In Figure 21, copper powders are shown, and in Figure 22, manufactured porous wick structure are shown.

The other important part of the heat pipe design depends on factors related to the properties of the working fluid. The working fluid must have good thermal stability in relation to the specific working temperature and pressure. The most important requirements that the working fluid must have are the following: compatibility with the capillary system and with the material of the pipe, high thermal stability, high state of heat, high thermal conductivity, low

Figure 21. Copper powders (35, 63, and 100 μm).

vacuuming of the pipe, pressure drop occurs and this may cause evaporation of working fluid. As a cooling medium, dry ice or liquid nitrogen may be used. After vacuuming, capillary tube

The main goal of the experiments is the determination of influence of the porous wick structure on the amount of thermal performance transferred by heat pipe. To determinate the amount of thermal performance transferred by heat pipe, measuring unit consisting of measuring apparatus (thermostat, data logger, ultrasonic flowmeter, power supply) shown in Figure 24 was proposed. Evaporator section of heat pipe was electrically heated by connecting to the laboratory power supply. Condensation section of heat pipe is placed into the heat exchanger where transferred heat from the evaporator is dissipated. Heat transferred by heat pipe is evaluated by calorimetric method emanating from calorimetric equation, where known mass flow, specific heat capacity, input and output temperature of cooling medium are flowing

> Q ¼ m :

In Figure 25, results of the experimental determination of influence of porous wick structure and working fluid on the heat pipe heat transfer ability at horizontal position and heat source 80�C are shown. It is seen that the heat pipe with working fluid water is able to transfer highest thermal performance in range 150–200 W. The best working wick structure in the water heat pipe is porous wick structure made from Cu powder with grain size 63 μm. On the other side, the porous wick structure made from Cu powder with grain size 35 μm is better for the heat pipes with working fluids such as acetone and ethanol that are able to transfer thermal performance

]—mass flow of liquid, and c [J kg s�<sup>1</sup>

:c:Δt (15)

]—special thermal capacities of liquid.

�C]—output tempera-

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Δt ¼ t<sup>2</sup> � t<sup>1</sup> (16)

�C]—input temperature, t2 [

connected was cramped, disconnected from vacuuming system and free-end soldered.

5.3. Heat transfer ability of the heat pipe

where Δt [�C]—temperature difference, t1 [

in the heat exchanger.

ture, ṁ[J kg�<sup>1</sup> K�<sup>1</sup>

Figure 24. Scheme of measuring unit.

Figure 22. Sintered porous wick structures.

viscosity of the liquid and vapor phase, high surface tension, and acceptable freezing point. For this experiment, a working fluid water and ethanol were chosen.

The amount of the working fluid in heat pipes is other alchemy of the heat pipe manufacturing. There are some recommendations of working fluid amount in heat pipe. Lack of working fluid may lead to drying of evaporator part of heat pipe. Surplus working fluid can lead to congestion of the condensation part of the heat pipe. One of the recommendation about the working fluid amount in heat pipe is that the working fluid has to fill-up at least 50% of evaporator part of the heat pipe. In general, the quantity of the working fluid is determined in the range of 15–30% of the total heat pipe volume [35]. In this experiment, heat pipes was filled with working fluid of 20% total heat pipe volume.

And finally, the process of the vacuuming, filling and closing the heat pipe are the other important part of the heat pipe manufacture. There are some methods on how to perform this process. Each of these methods has a precise plan of filing and vacuuming processes. In Figure 23, schema of filling and vacuuming process used in heat pipe manufacturing is shown. Working fluid was injected into the pipe via connecting capillary tube by syringe. Heat pipe container with working fluid was connected to vacuum system and by vacuum pump, air from heat pipe container was sucked off. Before connecting pipe to vacuum system, the working fluid was cooled by the immersion of pipe into the cooling medium, because during

Figure 23. Schema of heat pipe filling and vacuuming process.

vacuuming of the pipe, pressure drop occurs and this may cause evaporation of working fluid. As a cooling medium, dry ice or liquid nitrogen may be used. After vacuuming, capillary tube connected was cramped, disconnected from vacuuming system and free-end soldered.

#### 5.3. Heat transfer ability of the heat pipe

viscosity of the liquid and vapor phase, high surface tension, and acceptable freezing point.

The amount of the working fluid in heat pipes is other alchemy of the heat pipe manufacturing. There are some recommendations of working fluid amount in heat pipe. Lack of working fluid may lead to drying of evaporator part of heat pipe. Surplus working fluid can lead to congestion of the condensation part of the heat pipe. One of the recommendation about the working fluid amount in heat pipe is that the working fluid has to fill-up at least 50% of evaporator part of the heat pipe. In general, the quantity of the working fluid is determined in the range of 15–30% of the total heat pipe volume [35]. In this experiment, heat pipes was filled

And finally, the process of the vacuuming, filling and closing the heat pipe are the other important part of the heat pipe manufacture. There are some methods on how to perform this process. Each of these methods has a precise plan of filing and vacuuming processes. In Figure 23, schema of filling and vacuuming process used in heat pipe manufacturing is shown. Working fluid was injected into the pipe via connecting capillary tube by syringe. Heat pipe container with working fluid was connected to vacuum system and by vacuum pump, air from heat pipe container was sucked off. Before connecting pipe to vacuum system, the working fluid was cooled by the immersion of pipe into the cooling medium, because during

For this experiment, a working fluid water and ethanol were chosen.

with working fluid of 20% total heat pipe volume.

Figure 22. Sintered porous wick structures.

164 Porosity - Process, Technologies and Applications

Figure 23. Schema of heat pipe filling and vacuuming process.

The main goal of the experiments is the determination of influence of the porous wick structure on the amount of thermal performance transferred by heat pipe. To determinate the amount of thermal performance transferred by heat pipe, measuring unit consisting of measuring apparatus (thermostat, data logger, ultrasonic flowmeter, power supply) shown in Figure 24 was proposed. Evaporator section of heat pipe was electrically heated by connecting to the laboratory power supply. Condensation section of heat pipe is placed into the heat exchanger where transferred heat from the evaporator is dissipated. Heat transferred by heat pipe is evaluated by calorimetric method emanating from calorimetric equation, where known mass flow, specific heat capacity, input and output temperature of cooling medium are flowing in the heat exchanger.

$$Q = \dot{m}.c.\Delta t\tag{15}$$

$$
\Delta t = t\_2 - t\_1 \tag{16}
$$

where Δt [�C]—temperature difference, t1 [ �C]—input temperature, t2 [ �C]—output temperature, ṁ[J kg�<sup>1</sup> K�<sup>1</sup> ]—mass flow of liquid, and c [J kg s�<sup>1</sup> ]—special thermal capacities of liquid.

In Figure 25, results of the experimental determination of influence of porous wick structure and working fluid on the heat pipe heat transfer ability at horizontal position and heat source 80�C are shown. It is seen that the heat pipe with working fluid water is able to transfer highest thermal performance in range 150–200 W. The best working wick structure in the water heat pipe is porous wick structure made from Cu powder with grain size 63 μm. On the other side, the porous wick structure made from Cu powder with grain size 35 μm is better for the heat pipes with working fluids such as acetone and ethanol that are able to transfer thermal performance

Figure 24. Scheme of measuring unit.

areas. Positive gravity action zone is represented by angle of inclination from vertical position 0–75, zero gravity action zone (horizontal position) is represented by angle of inclination from vertical position 90, and negative gravity action zone is represented by angle of inclination from vertical position 105–180. There is seen that all wick heat pipe has good ability heat transfer in all zones. The best working wick heat pipe in positive and zero gravity action zone is heat pipe with wick structure made form Cu powder 63 μm. The best working heat pipe in zone with negative gravity action is wick heat pipe with wick structure made from Cu powder 100 μm.

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The hat flux transferred through the heat pipe depends mainly on the temperature difference and the corresponding thermal resistances. The real transferred heat is affected by the hydrodynamic and thermal processes that take place in the heat pipe at the various operating conditions. The heat flux transferred by the heat pipe can reach limit values that depend on these processes. There are five known limitations that limit the overall heat transfer in different parts of the heat pipe depending on the working temperature. In Figure 27, an ideal model of all heat transfer limitations that define area of maximum heat flux transferred by heat pipe

The mathematical model consists of calculation of the heat pipe heat transfer limitations. Heat pipe heat transfer limitations depend on the working fluid, the wick structure, the dimensions of the heat pipe, and the heat pipe operation temperature. Each heat transfer limitation expresses part of total heat flux heat pipe, which is influenced by hydrodynamic and thermal processes occurring in the heat pipe. Each of limitations exists alone and they are oneself non-influence together. To design mathematical model for the calculation of heat flux transferred by heat pipe,

Figure 27. Heat transfer limitations of water wick heat pipe with sintered wick structure (heat pipe inner diameter 20 mm, total length 2 m, axial orientation 90, sphere diameter of copper powder 0.85 mm, porosity 0.55, and width of

5.4. Calculation of heat transfer limitation of the heat pipe

depending on operating temperature is shown [4].

wick structure 6 mm).

Figure 25. Influence of wick structure on the heat pipe heat transfer ability at heat source 80C.

around 120 W. The experiment did not show the best one porous wick structure for selected working fluids, because each porous structure has various porosity and pore size which depend on the manufacturing process and each working fluid has different physical properties. There did not exist only one the best heat pipe with the best wick structure or the best working fluid because each heat pipe with various combination of the porous structure and working fluid is unique due its different properties.

In Figure 26, influence working position on the heat transfer ability of wick heat pipe with various porous wick structures is shown. Working position of heat pipe can divided into three

Figure 26. Dependence thermal performance on working position of wick heat pipes with various wick structures.

areas. Positive gravity action zone is represented by angle of inclination from vertical position 0–75, zero gravity action zone (horizontal position) is represented by angle of inclination from vertical position 90, and negative gravity action zone is represented by angle of inclination from vertical position 105–180. There is seen that all wick heat pipe has good ability heat transfer in all zones. The best working wick heat pipe in positive and zero gravity action zone is heat pipe with wick structure made form Cu powder 63 μm. The best working heat pipe in zone with negative gravity action is wick heat pipe with wick structure made from Cu powder 100 μm.

### 5.4. Calculation of heat transfer limitation of the heat pipe

around 120 W. The experiment did not show the best one porous wick structure for selected working fluids, because each porous structure has various porosity and pore size which depend on the manufacturing process and each working fluid has different physical properties. There did not exist only one the best heat pipe with the best wick structure or the best working fluid because each heat pipe with various combination of the porous structure and working fluid is

Figure 25. Influence of wick structure on the heat pipe heat transfer ability at heat source 80C.

In Figure 26, influence working position on the heat transfer ability of wick heat pipe with various porous wick structures is shown. Working position of heat pipe can divided into three

Figure 26. Dependence thermal performance on working position of wick heat pipes with various wick structures.

unique due its different properties.

166 Porosity - Process, Technologies and Applications

The hat flux transferred through the heat pipe depends mainly on the temperature difference and the corresponding thermal resistances. The real transferred heat is affected by the hydrodynamic and thermal processes that take place in the heat pipe at the various operating conditions. The heat flux transferred by the heat pipe can reach limit values that depend on these processes. There are five known limitations that limit the overall heat transfer in different parts of the heat pipe depending on the working temperature. In Figure 27, an ideal model of all heat transfer limitations that define area of maximum heat flux transferred by heat pipe depending on operating temperature is shown [4].

The mathematical model consists of calculation of the heat pipe heat transfer limitations. Heat pipe heat transfer limitations depend on the working fluid, the wick structure, the dimensions of the heat pipe, and the heat pipe operation temperature. Each heat transfer limitation expresses part of total heat flux heat pipe, which is influenced by hydrodynamic and thermal processes occurring in the heat pipe. Each of limitations exists alone and they are oneself non-influence together. To design mathematical model for the calculation of heat flux transferred by heat pipe,

Figure 27. Heat transfer limitations of water wick heat pipe with sintered wick structure (heat pipe inner diameter 20 mm, total length 2 m, axial orientation 90, sphere diameter of copper powder 0.85 mm, porosity 0.55, and width of wick structure 6 mm).

it is necessary to know basic and derived parameters of the heat pipe and its wick structure and physical properties of the working fluid liquid and vapor phase.

#### 5.4.1. Capillary limitation

Capillary limitation involves a limitation that affects the wick heat pipe operation, which results from the capillary pressure acting on the condensed working fluid in the capillary structure. At the contact of liquid and wick structure surface, the capillary pressure is formed. This causes the liquid phase of the working fluid to flow from the condenser to the evaporator. Decreasing the pores of the capillary structure increases the capillary pressure as well as hydraulic resistance. The capillary limit occurs when the capillary forces at the interface of the liquid and vapor phases in the evaporator and condenser section of the heat pipe are not large enough to overcome the pressure losses generated by the friction. If the capillary pressure in the heat pipe during the operation is insufficient to provide the necessary condensate flow from the condenser to the evaporator, the capillary structure in the evaporator is dried and thus the further evaporation of the working substance is stopped. In general, the capillary limit is the primary limit that influences the heat pipe performance and is expressed by the relationship [39].

$$\dot{Q}\_{\text{c}} = \frac{\sigma\_{l\cdot} \rho\_l l\_{\text{v}}}{\mu\_i} \cdot \frac{K.A\_{\text{v}}}{l\_{\text{eff}}} \cdot \left(\frac{2}{r\_{\text{eff}}} - \frac{\rho\_{l\cdot} \text{g} \, l\_{\text{t}} \cos \Psi}{\sigma\_l}\right) \tag{17}$$

where Aw is the wick cross-sectional area (m2 ), K is the wick permeability (m2 ), μ<sup>l</sup> is the liquid viscosity (N s/m2 ), r<sup>l</sup> is the liquid density (kg/m3 ), g is the acceleration due to gravity (9.8 m/s2 ), reff is the wick capillary radius in the evaporator (m), and lt is the total length of the pipe (m) [7].

Furthermore, if the heat pipe has properly operated, the maximum capillary pressure has to be greater than the total pressure loss in the heat pipe and it is expressed by the relationship

$$\left(\left(\Delta P\_{\rm c}\right)\_{\rm max} \ge \Delta P\_{\rm tot}\right) \tag{18}$$

capillary structure in the low-velocity vapor flow. The most frequent cases of exceeding the boundary of the viscous limit occur when the heat pipe operates at temperature close to the solidification of the working fluid. In this case, working fluid evaporation in the evaporator and heat transfer in the form of vapor flow through the adiabatic section into the condenser of the heat pipe did not occur. It is assumed that the vapor is isothermal ideal gas, the water vapor pressure on the end of the condenser is equal to zero, which provides the absolute limit for the pressure in the condenser. The viscous limit is referred as the condition of the vapor

where lv is the latent heat of vaporization (J/kg), rv is the cross-sectional radius of the vapor core (m), leff is the effective length of the heat pipe (m), μ<sup>v</sup> is the vapor viscosity in the

In cases when the viscous limit is reached for many conditions, the condenser pressure could

The sonic limit characterizes the state in which the velocity of the evaporated vapor flow at the outlet of the evaporator reaches the sound velocity. Generally, this phenomenon occurs on the start of heat pipe operation at a low vapor pressure of the working fluid. Assuming that the vapor of the working fluid is the ideal gas and the vapor flow at the sound velocity throughout the heat pipe cross section is uniform, the sonic limit is determined by the relationship (22). The sonic limit does not depend on the heat pipe orientation and type of the heat pipe, and the same formula is applied for the gravity and wick heat pipe. The most difficult in the sonic limit determination is

The sonic limit is mainly associated with liquid metal heat pipe startup or low-temperature heat pipe operation due to very low vapor densities that occur in these cases. For the low temperature or cryogenic temperatures, the sonic limit is not a typically factor, except for heat

) is the vapor density, Pv (Pa) is pressure at the end of heat pipe evaporator,

).

determining quantities of vapor density and pressure on inlet to the condenser [41].

), Pv (Pa) is the vapor pressure, and r<sup>v</sup> (kg/m3

ð20Þ

169

ð21Þ

ð22Þ

) is the density at the end of

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phase flow at low velocity and is expressed by the relationship

not be a zero. Then the following expression is applied:

where Pv,c is the vapor pressure in the condenser [40].

and Av is the cross-sectional area of the vapor core (m2

evaporator (N s/m<sup>2</sup>

5.4.3. Sonic limitation

where r<sup>v</sup> (kg/m<sup>3</sup>

the heat pipe evaporator [4].

The maximum capillary pressure ΔPc developed in wick structure of the heat pipe is defined by the Laplace-Young equation.

$$
\Delta \lambda\_{\varepsilon}^{\nu} = \frac{2\sigma}{r\_{\text{eff}}} . \cos \theta \tag{19}
$$

where reff is the effective pores radius of the wick structure and θ is contact angle liquid phase of the working fluid in wick structure, where θ = 0� is the best wetting contact angle [4].

#### 5.4.2. Viscous limitation

When the heat pipe operates at low operating temperatures, the saturated vapor pressure may be very small and has the same range as the required pressure drop necessary for the vapor to flow from the evaporator to the condenser of the heat pipe. This results in a condition expressed by the viscous limit about balance of the vapor pressure and viscous forces in the capillary structure in the low-velocity vapor flow. The most frequent cases of exceeding the boundary of the viscous limit occur when the heat pipe operates at temperature close to the solidification of the working fluid. In this case, working fluid evaporation in the evaporator and heat transfer in the form of vapor flow through the adiabatic section into the condenser of the heat pipe did not occur. It is assumed that the vapor is isothermal ideal gas, the water vapor pressure on the end of the condenser is equal to zero, which provides the absolute limit for the pressure in the condenser. The viscous limit is referred as the condition of the vapor phase flow at low velocity and is expressed by the relationship

$$\mathcal{Q}\_{\mathfrak{a}} = \mathcal{A}\_{\mathfrak{v}} J\_{\mathfrak{v}} \left( \frac{\rho\_{\mathfrak{v}} \sigma\_{\mathfrak{l}}}{2 \mathcal{F}\_{\mathfrak{c}, \mathfrak{b} \times \mathfrak{c}}} \right)^{0.5} \tag{20}$$

where lv is the latent heat of vaporization (J/kg), rv is the cross-sectional radius of the vapor core (m), leff is the effective length of the heat pipe (m), μ<sup>v</sup> is the vapor viscosity in the evaporator (N s/m<sup>2</sup> ), Pv (Pa) is the vapor pressure, and r<sup>v</sup> (kg/m3 ) is the density at the end of the heat pipe evaporator [4].

In cases when the viscous limit is reached for many conditions, the condenser pressure could not be a zero. Then the following expression is applied:

$$\underline{Q}\_{lr} = \frac{4\pi J\_{\text{eff}} \cdot \mathcal{A}\_{\text{eff}} \cdot \mathcal{T}\_{\text{v}} \cdot \mathcal{O}\_{\text{f}}}{J\_{\text{v}} \cdot \mathcal{P}\_{\text{v}} \cdot \ln \frac{r\_i}{r\_o}} \left(\frac{1}{r\_n} - \frac{1}{r\_{\text{eff}}}\right) \tag{21}$$

where Pv,c is the vapor pressure in the condenser [40].

#### 5.4.3. Sonic limitation

ð17Þ

),

ð19Þ

), μ<sup>l</sup> is the liquid

it is necessary to know basic and derived parameters of the heat pipe and its wick structure and

Capillary limitation involves a limitation that affects the wick heat pipe operation, which results from the capillary pressure acting on the condensed working fluid in the capillary structure. At the contact of liquid and wick structure surface, the capillary pressure is formed. This causes the liquid phase of the working fluid to flow from the condenser to the evaporator. Decreasing the pores of the capillary structure increases the capillary pressure as well as hydraulic resistance. The capillary limit occurs when the capillary forces at the interface of the liquid and vapor phases in the evaporator and condenser section of the heat pipe are not large enough to overcome the pressure losses generated by the friction. If the capillary pressure in the heat pipe during the operation is insufficient to provide the necessary condensate flow from the condenser to the evaporator, the capillary structure in the evaporator is dried and thus the further evaporation of the working substance is stopped. In general, the capillary limit is the primary limit that influences the heat pipe performance and is expressed by the relation-

reff is the wick capillary radius in the evaporator (m), and lt is the total length of the pipe (m) [7]. Furthermore, if the heat pipe has properly operated, the maximum capillary pressure has to be greater than the total pressure loss in the heat pipe and it is expressed by the relationship

The maximum capillary pressure ΔPc developed in wick structure of the heat pipe is defined

where reff is the effective pores radius of the wick structure and θ is contact angle liquid phase of the working fluid in wick structure, where θ = 0� is the best wetting contact angle [4].

When the heat pipe operates at low operating temperatures, the saturated vapor pressure may be very small and has the same range as the required pressure drop necessary for the vapor to flow from the evaporator to the condenser of the heat pipe. This results in a condition expressed by the viscous limit about balance of the vapor pressure and viscous forces in the

), K is the wick permeability (m2

), g is the acceleration due to gravity (9.8 m/s2

ð Þ ΔP<sup>c</sup> max ≥ ΔPtot (18)

physical properties of the working fluid liquid and vapor phase.

5.4.1. Capillary limitation

168 Porosity - Process, Technologies and Applications

ship [39].

viscosity (N s/m2

where Aw is the wick cross-sectional area (m2

by the Laplace-Young equation.

5.4.2. Viscous limitation

), r<sup>l</sup> is the liquid density (kg/m3

The sonic limit characterizes the state in which the velocity of the evaporated vapor flow at the outlet of the evaporator reaches the sound velocity. Generally, this phenomenon occurs on the start of heat pipe operation at a low vapor pressure of the working fluid. Assuming that the vapor of the working fluid is the ideal gas and the vapor flow at the sound velocity throughout the heat pipe cross section is uniform, the sonic limit is determined by the relationship (22). The sonic limit does not depend on the heat pipe orientation and type of the heat pipe, and the same formula is applied for the gravity and wick heat pipe. The most difficult in the sonic limit determination is determining quantities of vapor density and pressure on inlet to the condenser [41].

$$Q\_s = 0.474...A\_v J\_v \left(\rho\_v P\_v\right)^{0.5} \tag{22}$$

where r<sup>v</sup> (kg/m<sup>3</sup> ) is the vapor density, Pv (Pa) is pressure at the end of heat pipe evaporator, and Av is the cross-sectional area of the vapor core (m2 ).

The sonic limit is mainly associated with liquid metal heat pipe startup or low-temperature heat pipe operation due to very low vapor densities that occur in these cases. For the low temperature or cryogenic temperatures, the sonic limit is not a typically factor, except for heat pipes with very small vapor channel diameters. The sonic limitation is referred as an upper limit of the axial heat transport capacity and does not necessarily result in dry out of the wick structure in heat pipe evaporator or total heat pipe failure [4].

#### 5.4.4. Entrainment limitation

Increasing the heat flux transferred by heat pipe increases the vapor flow velocity of the working fluid too and this results in a more pronounced interaction of the vapor and liquid phase inside the heat pipe. The interfacial surface becomes unstable and the viscous force on the surface of the liquid overcomes the forces of the surface tension. The waves are created on the liquid phase surface at first from which the droplets gradually tears off. At a certain vapor flow velocity, the liquid flow interruption into the evaporator section occurs. The condenser section of heat pipe is overfilled by vapor and liquid phase and the evaporator is overheated due to lack of the working fluid. The limit value of the heat flux, when the heat pipe condenser is overfilled by vapor and liquid, corresponds to interaction limit [42]. Entrainment limitation of the wick heat pipe is related to the condition when the vapor flows against the liquid flow in the wick structure, which may result in insufficient liquid flow in the wick structure [43]. Entrainment limitation of the wick heat pipe is expressed by relationship:

$$Q\_v = A\_v I\_v \left(\frac{\rho\_v \sigma\_l}{2\rho\_{c,\text{ave}}}\right)^{0.5} \tag{23}$$

Approximate determination of the boiling limitation for the wick heat pipe is expressed by the

where λeff is the effective thermal conductivity of the wick structure which is composed of the wick thermal conductivity and working fluid thermal conductivity (W/m K), Tv is temperature of vapor saturation (K), rv is the vapor core radius, ri is the inner container radius (m), and rn is the bubble nucleation radius in range from 0.1 to 25.0 μm for conventional metallic heat pipe

To calculate heat pipe heat transport limitations, it is necessary to know thermophysical properties of working fluid in heat pipe, basic heat pipe parameters, thermal conductivity of heat pipe material, working temperature of heat pipe, axial orientation of heat pipe, and other

> Av ¼ πr 2

> > 2

where lt is total length of heat pipe [m], le is evaporation length of heat pipe [m], lad adiabatic length of heat pipe [m], lc is condensation length of heat pipe [m], leff is effective length of heat

rv is cross-sectional radius of vapor core [m], ri is inner container radius [m], and h is wick

The other parameters needed to calculate heat pipe heat transport limitations are basic parameters of sintered wick structure and other parameters calculated from basic parameters of wick

<sup>K</sup> <sup>¼</sup> <sup>d</sup><sup>2</sup>

:ε3

2:λ<sup>l</sup> þ λ<sup>m</sup> � 2:ð Þ 1 � ε :ð Þ λ<sup>l</sup> � λ<sup>m</sup> 2:λ<sup>l</sup> þ λ<sup>m</sup> þ ð Þ 1 � ε :ð Þ λ<sup>l</sup> � λ<sup>m</sup>

of wick structure [m], λeff is effective thermal conductivity, λ<sup>l</sup> is thermal conductivity of

Aw ¼ π r

lt ¼ le þ lad þ lc (26)

<sup>v</sup> (28)

], Aw is wick cross-sectional area [m2

<sup>i</sup> � ð Þ ri � <sup>h</sup> <sup>2</sup> (29)

reff ¼ 0:21 � ds (30)

], d is sphere diameter [m], ε is porosity [�], reff is effective radius

<sup>150</sup>:ð Þ <sup>1</sup> � <sup>ε</sup> <sup>2</sup> (31)

leff ¼ 0:5 le þ l ð Þþ <sup>c</sup> lad (27)

heat pipe parameters calculated from basic heat pipe parameters.

pipe [m], Av is cross-sectional area of the vapor core [m<sup>2</sup>

λeff ¼ λ<sup>l</sup>

working fluid liquid, and λ<sup>m</sup> is thermal conductivity of wick material [47].

ð25Þ

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],

(32)

relationship [46]

container materials [4].

5.4.6. Heat pipe parameters

structure width [m].

where K is permeability [m<sup>2</sup>

structure.

where rc,ave is the average capillary radius of the wick structure and, in many cases, it is approximated to reff, and σ<sup>l</sup> is the liquid surface tension (N/m) [4].

#### 5.4.5. Boiling limitation

When heating the surface of the heat pipe wall with a layer of liquid in the saturation boundary, a three basic heat transfer regimes can occur. At low temperature difference of the heated surface and interfacial surface of the liquid, a natural convection and evaporation from the liquid surface occur. When increasing the temperature difference, a bubble boiling and gradual transformation to the film boiling occur. In heat pipe, a surface evaporation at low heat flux densities and bubble boiling at higher densities occur. Although the heat transfer intensity is greatest in the bubble boiling, for most types of wick heat pipes, the bubble boiling is not desired because it interferes with the liquid wicking into the wick structure. On the other hand, in a heat pipe with a grooved capillary structure, a gravity heat pipe with bubble boiling is favorable [44]. The heat flux in which the bubble boiling occurs in the wick heat pipes and the film boiling occurs in the gravity heat pipe is referred as the boiling limit. The gravity heat pipe is expressed by the relationship [45]:

$$Q\_b = 0, 16.A\_v.l\_v \sqrt[4]{\sigma\_l \mathcal{g}\_v \rho\_v^2 (\rho\_l - \rho\_v)}\tag{24}$$

Determination of the boiling limit of the wick heat pipe is problematic, because it depends on a number of technological and operating conditions. The most reliable determination of the boiling limit is experimental determination for the particular wick structure and working fluid.

Approximate determination of the boiling limitation for the wick heat pipe is expressed by the relationship [46]

$$\underline{Q}\_{\flat} = 0.16. A\_{\flat} J\_{\vee} \sqrt[4]{\sigma\_l \, \lg \rho\_{\vee}^2 (\rho\_l - \rho\_{\vee})} \tag{25}$$

where λeff is the effective thermal conductivity of the wick structure which is composed of the wick thermal conductivity and working fluid thermal conductivity (W/m K), Tv is temperature of vapor saturation (K), rv is the vapor core radius, ri is the inner container radius (m), and rn is the bubble nucleation radius in range from 0.1 to 25.0 μm for conventional metallic heat pipe container materials [4].

#### 5.4.6. Heat pipe parameters

ð23Þ

(24)

pipes with very small vapor channel diameters. The sonic limitation is referred as an upper limit of the axial heat transport capacity and does not necessarily result in dry out of the wick

Increasing the heat flux transferred by heat pipe increases the vapor flow velocity of the working fluid too and this results in a more pronounced interaction of the vapor and liquid phase inside the heat pipe. The interfacial surface becomes unstable and the viscous force on the surface of the liquid overcomes the forces of the surface tension. The waves are created on the liquid phase surface at first from which the droplets gradually tears off. At a certain vapor flow velocity, the liquid flow interruption into the evaporator section occurs. The condenser section of heat pipe is overfilled by vapor and liquid phase and the evaporator is overheated due to lack of the working fluid. The limit value of the heat flux, when the heat pipe condenser is overfilled by vapor and liquid, corresponds to interaction limit [42]. Entrainment limitation of the wick heat pipe is related to the condition when the vapor flows against the liquid flow in the wick structure, which may result in insufficient liquid flow in the wick structure [43].

where rc,ave is the average capillary radius of the wick structure and, in many cases, it is

When heating the surface of the heat pipe wall with a layer of liquid in the saturation boundary, a three basic heat transfer regimes can occur. At low temperature difference of the heated surface and interfacial surface of the liquid, a natural convection and evaporation from the liquid surface occur. When increasing the temperature difference, a bubble boiling and gradual transformation to the film boiling occur. In heat pipe, a surface evaporation at low heat flux densities and bubble boiling at higher densities occur. Although the heat transfer intensity is greatest in the bubble boiling, for most types of wick heat pipes, the bubble boiling is not desired because it interferes with the liquid wicking into the wick structure. On the other hand, in a heat pipe with a grooved capillary structure, a gravity heat pipe with bubble boiling is favorable [44]. The heat flux in which the bubble boiling occurs in the wick heat pipes and the film boiling occurs in the gravity heat pipe is referred

q

Determination of the boiling limit of the wick heat pipe is problematic, because it depends on a number of technological and operating conditions. The most reliable determination of the boiling limit is experimental determination for the particular wick structure and working fluid.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σl:g:r<sup>2</sup> <sup>v</sup> <sup>r</sup><sup>l</sup> � <sup>r</sup><sup>v</sup> ð Þ <sup>4</sup>

as the boiling limit. The gravity heat pipe is expressed by the relationship [45]:

Qb ¼ 0, 16:Av:lv

structure in heat pipe evaporator or total heat pipe failure [4].

Entrainment limitation of the wick heat pipe is expressed by relationship:

approximated to reff, and σ<sup>l</sup> is the liquid surface tension (N/m) [4].

5.4.4. Entrainment limitation

170 Porosity - Process, Technologies and Applications

5.4.5. Boiling limitation

To calculate heat pipe heat transport limitations, it is necessary to know thermophysical properties of working fluid in heat pipe, basic heat pipe parameters, thermal conductivity of heat pipe material, working temperature of heat pipe, axial orientation of heat pipe, and other heat pipe parameters calculated from basic heat pipe parameters.

$$l\_t = l\_e + l\_{ad} + l\_c \tag{26}$$

$$l\_{\rm eff} = 0.5(l\_{\rm c} + l\_{\rm c}) + l\_{\rm ad} \tag{27}$$

$$A\_v = \pi r\_v^2 \tag{28}$$

$$A\_w = \pi \left( r\_i^2 - \left( r\_i - h \right)^2 \right) \tag{29}$$

where lt is total length of heat pipe [m], le is evaporation length of heat pipe [m], lad adiabatic length of heat pipe [m], lc is condensation length of heat pipe [m], leff is effective length of heat pipe [m], Av is cross-sectional area of the vapor core [m<sup>2</sup> ], Aw is wick cross-sectional area [m2 ], rv is cross-sectional radius of vapor core [m], ri is inner container radius [m], and h is wick structure width [m].

The other parameters needed to calculate heat pipe heat transport limitations are basic parameters of sintered wick structure and other parameters calculated from basic parameters of wick structure.

$$\mathbf{r\_{eff}} = 0.21 \cdot \mathbf{d\_s} \tag{30}$$

$$K = \frac{d^2 \, . \varepsilon^3}{150.(1 - \varepsilon)^2} \tag{31}$$

$$
\lambda\_{\rm eff} = \lambda\_l \frac{2.\lambda\_l + \lambda\_m - 2.(1 - \varepsilon).(\lambda\_l - \lambda\_m)}{2.\lambda\_l + \lambda\_m + (1 - \varepsilon).(\lambda\_l - \lambda\_m)} \tag{32}
$$

where K is permeability [m<sup>2</sup> ], d is sphere diameter [m], ε is porosity [�], reff is effective radius of wick structure [m], λeff is effective thermal conductivity, λ<sup>l</sup> is thermal conductivity of working fluid liquid, and λ<sup>m</sup> is thermal conductivity of wick material [47].

#### 5.5. Verification of mathematical model

The mathematical model was created according to above equations of limitations and input heat pipe parameters. Result of mathematical model is graphic dependencies of heat transport limitations on heat pipe working temperature. Mathematical model results of heat transport limitations of specific types of heat pipes were compare with results from measurement of heat pipe performance at temperatures 50�C and 70�C. In Figure 28, graphic comparison results of heat transport limitations determining total performance of heat pipe from mathematical model with measured performance of ethanol wick heat pipe with sintered wick structure and sphere diameter of copper powder 0.1 mm are shown. Dotted line creates boundary of heat pipe performance by capillary limitation and dashed line is boiling limitation. The full line is measured results of heat pipe thermal performance at temperature 50�C and 70�C. Figure 29 confirms the verification of mathematical model, where it is seen that the measured values of the transferred heat flux by heat pipe with sintered wick structure at temperatures 50�C and 70�C, are in approximately the same area as a calculated values of capillary limitation by mathematical model. In Figures 28 and 29, it is seen that the dotted line and full line are

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Results of the heat pipe calculation are some interesting graphs of the maximal heat flux transferred by heat pipe depending on the wick structure parameters. It could be used in design optimization of the heat pipe wick structure. The curves present an area of maximal

Next graphic dependencies of heat pipe performance are created from mathematical model for ethanol wick heat pipe with sintered wick structure and various porosity, sphere diameter of copper powder, and wick structure width. In Figure 30, the influence of porosity on heat pipe performance is shown. Porosity of wick structure can change by adding some additives to sintered technology. There is clearly seen a rise in heat pipe performance with increasing porosity of wick structure. Heat pipe with the higher permeability of the wick structure can transfer higher heat flux. But with increasing permeability of wick structure, entrainment of liquid flow to evaporator by vapor flow can occur. This may cause dry out of heat pipe

In Figure 31, the influence of sphere diameter copper powder on sintered wick structure is shown. Using the bigger sphere diameter of copper powder to sintered technology, higher porosity wick structure is created. It can be said that increasing porosity is directly proportional to sphere dimension of copper powder, and to make more porosity wick structures, adding additives to sintered technology is not needed. In this case, an increase of heat pipe

approximately in the same region at temperatures 50�C and 70�C.

heat flux transferred by heat pipe depending on operating temperature.

evaporation section and decrease total heat pipe performance.

performance with used bigger sphere dimension of copper powder is seen.

Figure 30. Dependence of heat pipe performance from wick structure porosity of the sintered wick heat pipe.

5.6. Results of the mathematical model

Figure 28. Verification of mathematical model by measuring of heat pipe performance (ethanol wick heat pipe with sintered wick structure and sphere diameter of copper powder 0.1 mm and axial orientation of heat pipe ψ 180�).

Figure 29. Verification of mathematical model by measuring of heat pipe performance (water wick heat pipe with sintered wick structure and sphere diameter of copper powder 0.63 mm axial orientation of heat pipe ψ 180�).

confirms the verification of mathematical model, where it is seen that the measured values of the transferred heat flux by heat pipe with sintered wick structure at temperatures 50�C and 70�C, are in approximately the same area as a calculated values of capillary limitation by mathematical model. In Figures 28 and 29, it is seen that the dotted line and full line are approximately in the same region at temperatures 50�C and 70�C.
