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Different types of losses are present during operation of Stirling engines. Due to their complex nature the analysis of these losses usually is not directly included in the simulation rather they are analysed and taken into account separately.

Conduction Losses

These losses are accounted for in large in the simulation process and are also the easiest to calculate. In a generalised analysis, it is often difficult to include every single conduction path. Besides in many components the temperature gradients are quite small and, hence, the conduction losses in these components can be neglected.

The conduction of the regenerator should be considered and calculated carefully and included in the overall heat balance due to its significant value. Conduction losses do not alter the temperature of the working fluid to a great extent and only affect the thermal efficiency of the engine. The losses due to conduction can be considered as an additional thermal load for the cooler and heater.

Gas Spring Hysteresis Losses

There are losses associated with the gas springs because the thermodynamic process that occurs in the gas spring is not totally reversible. A certain quantity of work is dissipated in the case of real gas springs, which can be represented as an area in the relevant Pressure – Volume diagram. This negative work in the diagram is the sum of the gas spring hysteresis

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loss which is a function of the temperature and viscous gradient effects in the cavity. The viscous effects are often neglected because they very small. The effect of the gas hysteresis loss can have very little effect on kinematical Stirling engines but may be considerable for free piston engines. The effect should be considered in the analysis of FPSEs due to the fact that the gas hysteresis loss increases damping effects which alter the amplitudes and phase angle in the displacer and piston displacements. The general differential energy equation for the gas in such calculations is (Hughes and Brighton, [106] ).

= ∇ + + ∅ (1.1)

where is the density of gas (kg/ ), is the gas specific heat at constant pressure (Jk/gK), T is temperature (K), t is time (s), k is thermal conductivity (W/ ), is the pressure (Pa), ∅ is the viscous dissipation (W/ )

Seal Leakage

The Stirling engines seals are either of close fitting clearance type or the conventional ring type. The seals are designed in a way that they must be run dry at all times that is without grease or oil lubrication. Therefore certain measures should be taken to reduce their wear. Local heating or gradual seizure of the moving surfaces may occur if the bearing pressure of the seal is not enough. The seals will experience leakages at some point to some level. The effects of the leakage on the thermodynamic cycle are an essential aspect which should be taken into account during engine designing. In cases where ring seals are employed, the leakage experienced is very minor and has no significant effect on the thermodynamic cycle. But the mechanical friction value is considerable. The leakage that occurs in a seal in terms of mass flow is given by

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The free piston Stirling engines are usually designed with sliding seals that act as springs and separates the working space from the internal gas volume. Also to prevent the pistons drifting away from its equilibrium position, centering ports are introduced, which causes gas leakage at some instances of the cycle. Therefore, free piston Stirling engines are more prone to leakage power losses than the kinematic machines.

Appendix Gap Losses

The appendix gap losses are caused by the clearance gap between the walls of the displacer and cylinder. There are three main sources of losses in the appendix gap:

(i) Gas enthalpy transfer (ii) Shuttle enthalpy transfer (iii) Hysteresis heat transfer

The shuttle heat transfer is a conduction loss caused by the oscillatory motion of the piston or displacer and it occurs down the walls of either of the moving elements. During operation when the displacer is at the top dead centre heat is accumulated by its walls and then dissipated in the cold zone when the piston is at the lowest position.

This type of losses were first calculated by Zimmerman and Longsworth [19].

3.1.13 Friction factor and heat transfer coefficient

The heat transfer coefficient and the flow friction factor can only be analysed for steady laminar flow conditions in plain geometric shape. The Stirling engines exhibit extreme unsteady flow conditions and empirical data is the only solution to calculate the above factors. This data is presented as correlations in terms of dimensionless groups.

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Dimensionless groups

Dimensional analysis and similarities are widely used to present data on heat transfer and flow [107]. The set of variables below are considered to analyse heat transfer process and flow-friction under forced convection, which are present in Stirling engines:

g, working gas mass flux ( − − ), d, hydraulic diameter (m), μ, dynamic viscosity of the working fluid ( − − ), h, coefficient of heat transfer ( − − − ), k, working fluid thermal conductivity ( − − − ) , _, specific heat capacity of working fluid at constant pressure ( − − .

The hydraulic diameter d is the single variable representing the heat exchanger geometry and its size. It is denoted by

≡

� (1.3)

where V is the void volume and � is the wall to gas or wetted area.

The hydraulic diameter will be equal to the internal diameter for flow in a circular pipe to occur.

The variables above are used to determine Reynolds number:

≡ | /�| (Reynolds number) (1.4)

This parameter represents the ratio of the inertial forcers to the viscous forces. The value of is used to determine if flow is laminar or turbulent. The heat transfer coefficient and the friction factor are defined by the type of flow, therefore is used for heat transfer and flow friction calculations.

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The Prandtl number is determined from the ratio of the kinematic viscosity to the thermal

diffusivity, and is related to the ratio of the thermal layer to the viscous boundary layers. ≡ �/ (Prandtl Number) (1.5)

Nusselt Number is a heat transfer measure by convection and is defined as product of Reynolds and Prandtl numbers.

≡ ℎ (Nusselt Number) (1.6)

≡ |ℎ/ | (Stanton Number) (1.7)

The Stanton number is often used as a substitute to the Nusselt number for analysing data of heat transfer. It can be determined as a function of the Nusselt, Prandtl and Reynolds number.

The Stanton number has a physical importance which can be referred to the ratio of the convective heat transfer to the thermal capacity of the working fluid.

≡ / (1.8)

The energy balance of a cooled or heated working fluid flowing through the heat exchanger can be given as:

ℎ� ₋ = � − (1.9)

where � is the wetted area, � is the free-flow area, _, are the respective wall and bulk fluid temperatures, _, are the respective input and output fluid temperatures.

57 Substituting equation 1.8 into equation 1.9

= � �

−

− (1.10)

The value of can be derived from the temperature measurements and heat exchanger dimensions without including the properties of the working fluid. The Number of transfer units can also be determined as

= ℎ� / � (1.11)

Therefore n = = � /� (1.12)

The influence of temperature on the fluid properties can be considered in simulations. The thermal conductivity k and the dynamic viscosity � changes with temperature as presented in [108].