Heat transfer and fluid flow in the heat pipe system result in thermodynamic irreversibility generating entropy. The minimum entropy generation principle can be used for optimum design of flat heat pipe. The objective of the present work is to minimise the total entropy generation rate as the objective function with different parameters of the flat heat pipe subjected to some constraints. These constraints constitute the limitations on the heat transport capacity of the heat pipe. This physical nonlinear programming problem with nonlinear constraints is solved using LINGO 15.0 software, which enables finding optimum values for the independent design variables for which entropy generation is minimum. The effect of heat load, length, and sink temperature on design variables and corresponding entropy generation is studied. The second law analysis using minimum entropy generation principle is found to be effective in designing performance enhanced heat pipe.

Over the last few decades, entropy generation minimisation principle [

The heat pipe consists of a sealed container with a lining of porous wick structure adjacent to the wall containing liquid working fluid and a hollow space inside it having vapour medium at the operating conditions [

The heat pipe is axially divided into three parts, namely, the evaporator, the condenser, and a transport (adiabatic) section as shown in Figure

Schematic of flat heat pipe.

The investigations made in the area of heat pipe and entropy generation minimisation are presented in this section. A detailed overview regarding the heat pipe characteristics, its performance, and challenges has been reported by Faghri [_{2}O_{3} and TiO_{2} nanofluids at different concentrations as working substance. A reduction in entropy generation was observed for a good range of nanofluid concentrations. The recent works on entropy generation studies performed on porous media and viscous filled fluid in buoyancy induced flows in channels and enclosures were summarised by Oztop and Al-Salem [

Although these published research papers over the past few decades show very encouraging results, efforts are not put in to draw a strong conclusion on entropy generation minimisation in heat pipe.

This paper proposes the scope of carrying out parametric investigations to evaluate the effect of various parameters of the heat pipe for optimising the entropy generation rate. This work is motivated by the fact that demand for high precision heat pipes has been increasing rapidly as they are more needed in cooling electronic equipment and spacecraft application. Thus the objective of the present study is to determine the active sites which cause entropy generation in flat heat pipe, estimation of entropy, and its minimisation for enhancing the performance of the system.

The major factor which results in generation of entropy in a flat heat pipe is heat transfer through a finite temperature difference. The increase in temperature difference will increase the irreversibility associated with the system, thereby increasing the entropy generation rate.

The thermal resistance circuit of a flat heat pipe is shown in Figure

Thermal circuit of flat heat pipe.

From (

The constraints constitute the limitations on the heat transport capability.

The limits of heat transfer play a significant role in the design of the heat pipe since the lowest limit defines the maximum heat transport limitation at a given temperature. The parameters limiting heat transport in the conventional heat pipe are capillary limit, sonic limit, viscous limit, entrainment limit, and boiling limit.

The ability of the capillary wick structure to provide continuous circulation of working fluid is limited with a constraint or limit called capillary limit or hydrodynamic limitation. When the heat transfer is increased above this limit, pumping rate of the working fluid will not be sufficient to provide enough liquid to the evaporator section. This occurs due to the fact that maximum capillary pressure that the wick can sustain is lower than the sum of liquid and vapour pressure drops in the heat pipe. The expression for the capillary limitation is given as

At higher temperatures, when the vapour flow velocity becomes equal to or higher than the sonic velocity, it results in a choked flow condition. This limit is called sonic limit. There is a maximum axial heat transfer rate at this limit and subsequently it does not increase by decreasing the condenser temperature under choked flow condition. The expression for the limit can be written as

There exists shear forces at the liquid-vapour interface, since both the vapour and the liquid inside the heat pipe move in the opposite direction. When the corresponding velocities of vapour and liquid are relatively high, a limit is reached where liquid will be torn off from the wick surface and entrained to the vapour region flowing towards the condenser section. When the entrainment begins and becomes predominant, it results in dry out of the wick at the evaporator section. The entrainment heat load for flat heat pipe is obtained as

At low operating temperature the vapour pressure difference between the evaporator and condenser end will be small. Viscous forces may be dominant and limit the operation of the heat pipe. This flow condition where total vapour pressure in the vapour region becomes insufficient to sustain an increased flow is called viscous limit. The expression for viscous limit is obtained as

Heat transfer across the liquid saturated wick is accompanied by a transverse temperature gradient in the liquid. When this heat flux in the evaporator becomes high, the wall temperature becomes excessively high resulting in boiling of liquid in the evaporator. The vapour bubbles that form in the wick structure cause hot spots and obstruct the circulation of fluid which leads to dry out of the wick in the evaporator. This limitation of heat transfer due to transverse heat flux density is termed as boiling limitation. Unlike the other heat pipe limits discussed above which are due to axial heat flux limitation, the boiling limitation is a transverse heat flux limit. The expression for the boiling heat transfer limit is obtained as

Thus the present optimization problem can be formulated to find

In the present work a flat heat pipe is analysed with following assumptions.

The operation of the heat pipe is carried out at steady state.

The fluid is laminar and incompressible.

The heat transfer through the liquid wick is modelled as pure conduction with an effective thermal conductivity.

The temperature difference which exists within the liquid-vapour interface between the vapour core and wick structure is small and neglected.

The axial heat conduction through the container wall and wick is neglected.

^{2}K, assuming evaporator and condenser section being maintained at desired temperature by circulating water in a constant temperature bath.

Thermophysical properties taken as input parameters for design.

Thermophysical properties | Design value |
---|---|

Density of water | 985 kg/m^{3} |

Density of water vapour | 0.13 kg/m^{3} |

Viscosity of water | 0.000797 N⋅s/m^{2} |

Viscosity of water vapour | 0.0000134 N⋅s/m^{2} |

Surface tension of water | 0.070 N/m |

Latent heat of vapourisation | 2300000 J/kg |

Thermal conductivity of water | 0.608 W/mK |

A flat heat pipe of one-meter length (combining length of evaporator, adiabatic section, and condenser) is considered for the analysis. The length of adiabatic section is varied from 0 to 0.5 m to study the effect of variation in entropy generation rate. The wick thickness is taken as a design variable and its value is assumed to be varying from 0.0005 to 0.0015 m. The lower and upper limit values are specified in the program as an additional constraint. Meanwhile, the length of evaporator and condenser is influenced by the length of adiabatic section, since the latter is a design variable. The linear dimension in the transverse direction (i.e., perpendicular to the flow direction) cannot be defined since it is determined by the sum of the container wall, wick, and vapour core thicknesses, which varies according to design requirement in each case. However upper bound limit to the transverse length of the heat pipe is fixed to 0.05 m as a design constraint.

The present problem is a nonlinear programming problem with nonlinear inequality constraint. The optimisation or rather minimisation is to obtain a set of design variables which gives minimum entropy generation. The problem is modelled using optimisation modelling software “LINGO 15.0.” It provides completely integrated built-in solver for solving the nonlinear optimisation model to get global optimum solution. The entropy generation rate is calculated with respect to different heat pipe parameters.

The variation of entropy generation rate is studied for different heat pipe parameters such as heat load, sink temperature, and various lengths of adiabatic section using the software. From the present analysis, it is inferred that entropy generated in a heat pipe is mainly due to heat transfer, and the entropy generated due to fluid friction of the vapour and liquid flow is negligibly small. Hence the dimensionless parameter called Bejan number (ratio of entropy generated due to heat transfer to total entropy generated or sum of entropy generated by heat transfer and fluid friction) tends to 1 in this problem, which clearly indicates that heat transfer irreversibility contributes almost 100% to the total entropy generation. Figure ^{−3} to ^{3}K for the heat input range of 100 to 600 W without the transport section. The entropy rate is found to be almost doubled when the length of adiabatic section is varied from 0 to 0.5 m, for the corresponding heat flux value.

Variation in entropy generation rate against heat load for different lengths of adiabatic section.

The decrease in entropy generation rate with the increase in condenser temperature without the transport region is depicted in Figure ^{3}K for a heat load of 100 W over the entire sink temperature range. But at higher heat load, the variation is found to be comparable. At heat load of 500 W the variation is found to be ^{3}K for the entire sink temperature range. Entropy generation rate is found to increase by 25 times when the heat load is increased from 100 to 500 W. The graph shows similar variation pattern for the entire range of sink temperature. For a constant heat load the variation in entropy generation rate against the length of adiabatic section for different sink temperatures is depicted in Figure

Variation in entropy generation rate against sink temperature for different heat loads.

Variation in entropy generation rate against length of adiabatic section for different sink temperatures.

Figure

Variation in entropy generation rate against mesh number for different heat loads.

Variation in permeability against mesh number for different heat loads.

The variation of entropy generation rate with respect to different heat loads for various wick thickness is shown in Figure ^{3}K. But the difference increases to ^{3}K for the heat load of 500 W. The comparison of entropy generation rate at different wick thicknesses shows that the variation is more prominent for higher values of heat load.

Variation in entropy generation rate against heat load for different wick thicknesses.

Figure

Variation in entropy generation rate against sink temperature for different wick thicknesses.

Entropy generation rate associated with the flat heat pipe has been estimated and minimised for reducing the irreversibility. The model has been formulated and solved as a nonlinear programming problem with nonlinear functional constraints using commercial software (LINGO 15.0). The analysis shows that fluid friction and heat transfer contribute to entropy generation and has to be considered for the improvement in the performance of the device. Entropy generation is found to increase with heat load and length of the adiabatic section, while a decrease in the former is observed with increase in sink temperature. The following design modification can be recommended to improve the system performance by reducing entropy generation rate:

Reducing the length of the adiabatic section as far as possible.

Increasing the condenser temperature.

Decreasing the wick thickness.

Increasing the mesh number.

Area (m^{2})

Heat transfer coefficient (W/m^{2}K)

Latent heat of vaporisation (J/kg)

Thermal conductivity (W/mK)

Permeability (m^{2})

Length (m)

Mass flow rate (kg/s)

Mesh number (1/m)

Pressure (Pa)

Capillary pressure (Pa)

Heat transfer rate (W)

Radius (m)

Resistance (K/W)

Entropy (J/kgK)

Thickness (m)

Temperature (K)

Velocity (m/s)

Width (m).

Viscosity (Ns/m^{2})

Density (kg/m^{3})

Surface tension (N/m)

Porosity

Contact angle (degree)

Inclination angle (degree).

Adiabatic section

Condenser section

Evaporator section

Effective

Generated

Hydraulic radius

Source

Liquid

Sink

Nucleation

Outside

Vapour

Wick

Boiling

Capillary

Container

Entrainment

Sonic

Viscous.

The authors declare that there is no conflict of interests regarding the publication of this paper.