#### Heat Engines, Physics tutorial

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 TH and other cold at TC. 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 QH at temperature TH, uses part of it do mechanical work, and then ejected unused heat energy (QC ) at temperature TC 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

WherecycledU = Ufinal - Uinitial = 0 as system returns to initial state.

∫PdV = W= is work done in a cycle.

cycledQ = QH - QC = Q total heat transfer to working body in the cycle.

Therefore, E = 1 - QC/QH = 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 Vd to Ve. Temperature rises from Td to Te according to equation

TdVdγ-1 = TeVeγ-1

Process ef is the isochoric process during which heat QH is added as temperature of working substance changes from Te to Tf and pressure also increases from Pe to Pf.

Process fg is adiabatic expansion i.e. no heat is added as the volume of the working substance increases from volume Ve to Vd. The temperature decreases from Tf to Tg according to equation TfVeγ-1 = TgVdγ-1

Processes gd is an isochoric process during which heat QC is ejected as temperature changes from Tg to Td.

Efficiency of Otto Engine:

Usually, efficiency of heat engine is given in equation as

E = 1 - QC/QH

Heat is added during isochoric process ef and temperature increases from Te to Tf and pressure increase also from Pe to Pf. Heat added QH is

QH = ∫Te Tf CVdT = CV(Tf - Te)

Two adiabatic processes where involved in cycle and these provide:

TgVdγ-1 = TfVeγ-1

And TdVdγ-1 = TeVeγ-1

Therefore the efficiency of Otto cycle is

E = 1-(Ve/Vd)(CP - CV)/CV

Where CV and CP 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 QH is absorbed at temperature TH. 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 TH to TC and pressure also decrease from Pg to Pd

Process de is the isothermal compression. During this process, work is done on gas and this compresses gas isothermally at temperature TC, then heat QC 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 TC to TH and pressure also increases from Pe to Pf.

Efficiency of Stirling Engine:

Usually, the efficiency E is

E = W/QH = 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 TH and TC (i.e. work is done only during isothermal expansion and compression processes). No work is done during isochoric processes in cycle.

W = -nRlnVg/Ve(TH - TC)

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 fg we have

QH = ∫SfSgTHdS

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

The ratio Vg/Ve = Vd/Vf 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 = (nRlnVg/Ve(TH - TC))/(nRTHlnVg/Ve)

And it gives E = (TH - TC)/TH

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