Heat Engines:

The heat engine is a device utilized to convert thermal energy (i.e. heat) into mechanical work and then exhausts heat that can't be utilized to do work.

Basic Operation of Heat Engine:

Working body absorbed heat from hot reservoir at relatively high temperature. Part of absorbed heat is utilized by working body to do mechanical work. Unused energy is then ejected as heat at lower temperature. Process of converting thermal energy to mechanical work by heat engine is shown in figure given below.

Heat engine has of two heat reservoirs, one hot at T_H and other cold at T_C. Interaction between these two reservoirs and working body leads to conversion of heat energy to mechanical work. Another explanation is that working body absorbs heat Q_H at temperature TH, uses part of it do mechanical work, and then ejected unused heat energy (Q_C ) at temperature T_C through cold reservoir.

Thermodynamic Efficiency E of Real Engines:

Efficiency of a heat engine is

E = W/QH = work output/heat in (in one cycle)

The efficiency E estimates fraction of heat pumped in working body which is converted to mechanical work by working body. Efficiency of real heat engines is always less than unity. Changes in energy of working body are related to changes in thermodynamic properties. Using combined first and second laws of thermodynamics,

dU = TdS - PdV = dQ - dW

Where dQ is heat in working body and dW is mechanical work (that is work output). But working body operates in the cycle (i.e. cyclic process), returning system back to initial state.

0 =

_cycledU =_cycleTdS -_cyclePdV

Where_cycledU = U_final - U_initial = 0 as system returns to initial state.

∫PdV = W= is work done in a cycle.

_cycledQ = Q_H - Q_C = Q total heat transfer to working body in the cycle.

Therefore, E = 1 - Q_C/Q_H = 1 - heat released/heat absorbed

This is generalized form of efficiency for heat engine.

Otto Cycle/Engine:

Otto cycle has two adiabatic processes and two constant volume (isochoric) processes or strokes. PV diagram of Otto cycle is shown in figure 8.2 and as indicated in PV diagram, heat is absorbed during one of isochoric processes and heat is rejected during other isochoric process.

Description of Processes:

Process de is the adiabatic compression i.e no heat is added as volume of working substance decreases from volume V_d to V_e. Temperature rises from T_d to Te according to equation

T_dV_d^γ-1 = T_eV_e^γ-1

Process ef is the isochoric process during which heat QH is added as temperature of working substance changes from T_e to T_f and pressure also increases from P_e to P_f.

Process f_g is adiabatic expansion i.e. no heat is added as the volume of the working substance increases from volume V_e to V_d. The temperature decreases from T_f to T_g according to equation T_fV_e^γ-1 = T_gV_d^γ-1

Processes g_d is an isochoric process during which heat Q_C is ejected as temperature changes from T_g to T_d.

Efficiency of Otto Engine:

Usually, efficiency of heat engine is given in equation as

E = 1 - Q_C/Q_H

Heat is added during isochoric process e_f and temperature increases from T_e to T_f and pressure increase also from P_e to P_f. Heat added Q_H is

Q_H = ∫_Te ^Tf C_VdT = C_V(T_f - T_e)

Two adiabatic processes where involved in cycle and these provide:

T_gV_d^γ-1 = T_fV_e^γ-1

And T_dV_d^γ-1 = T_eV_e^γ-1

Therefore the efficiency of Otto cycle is

E = 1-(V_e/V_d)^{(CP - CV)/CV}

Where C_V and C_P are specific heat at constant volume and pressure respectively.

Stirling Engine:

The PV diagram for an ideal Stirling heat engine is shown in figure given below. The cycle consists of two isochoric processes and two isothermal processes. Three important components of Stirling engine are:

Heat Exchangers: As the name implies, these transfer heat between the working gas and the outside of the system.

Displacer Mechanism: The purpose of this is to move the working gas between the hot and cold ends of the machine through the regenerator.

Regenerator: This is a device normally placed between hot and cold portions of the machine that is in contact with the hot and cold reservoirs respectively. It consists of packing of steel wool or a series of metal baffles of low thermal conductivity. The purpose of this device is to act as thermal barrier and also as thermal store for the cycle.

Description of the Processes:

Process fg is the isothermal (constant temperature) expansion during which heat Q_H is absorbed at temperature T_H. Due to expansion, work is done during process (i.e. high pressure working gas absorbs heat from heat absorbing heat-exchanger and expands isothermally, therefore work is done).

Process gd is the isochoric (constant volume) process. Displacer transfers all working gas isochorically through the regenerator to cold end of machine. Heat is absorbed from gas as it passes through regenerator; therefore temperature decreases from T_H to T_C and pressure also decrease from P_g to P_d

Process de is the isothermal compression. During this process, work is done on gas and this compresses gas isothermally at temperature T_C, then heat Q_C is ejected to cold reservoir through heat rejecting heat exchanger.

Process ef is the isochoric process. During this process, displacer transfers all working gas isochorically through regenerator to hot end of machine. Heat is added to gas as it passes through regenerator, therefore increasing temperature of gas from T_C to T_H and pressure also increases from P_e to P_f.

Efficiency of Stirling Engine:

Usually, the efficiency E is

E = W/Q_H = Work output/heat in

Total work done in Stirling-cycle engine is

W = -PdV

Integral in equation is over closed loop. From PV diagram, two isochoric processes takes place during cycle at T_H and T_C (i.e. work is done only during isothermal expansion and compression processes). No work is done during isochoric processes in cycle.

W = -nRlnV_g/V_e(T_H - T_C)

Work done represents energy out of system, and so has negative value according to sign convention we have been using.

Heat Flow into the Ideal Stirling Engine:

Heat flowing into and out of Stirling-cycle engine can be estimated by considering integral of temperature with respect to entropy:

Q = ∫TdS

For this cycle, heat is transfer into and out of the system only during two isothermal processes. In closed cycle isothermal expansion process f_g we have

QH = ∫_Sf^SgT_HdS

This integral can be most easily estimated by considering first law of thermodynamics

The ratio V_g/V_e = V_d/V_f is known as expansion ratio of working gas. Inverse of this is known as compression ratio. Efficiency of the ideal Stirling engine is thus

E = (nRlnV_g/V_e(T_H - T_C))/(nRT_HlnV_g/V_e)

And it gives E = (T_H - T_C)/T_H

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