2. Work and Heat Engines Thermodynamics allows

us to study ideal models for a heat engine that con-

verts heat energy to mechanical energy. For an ide-

alised engine, we can assume that gases obey ideal

gas laws such as PV = nRT, or for adiabatic expan-

sions, PV

= constant (

1:4 for air). Note that

P = pressure, V = volume and T = temperature

of the gas, and n and R can be taken to be con-

stants. The term adiabatic simply implies that in

the process of compressing (or expanding) a gas by

changing its pressure and volume, the temperature

of the gas also changes.

The work done by the system in one complete cycle

in an ideal heat engine is given by the area enclosed

in the cycle when illustrated on a P ô€€€ V (pressure

versus volume) diagram. By denition, as the gas

changes from volume Va to Vb the work done by the

gas is

Work =

Z Vb

Va

P dV :

An example of a heat engine is the Rankine cycle,

which is an idealised version of a steam engine. De-

tails of such a cycle are shown in the diagram.

V

P

0:05 0:15 0:25

10

20

c = (0:01; 20:0)

d = (0:03; 20:0)

b = (0:01; 1:03)

a = (0:25; 1:03)

In this cycle, liquid water at low temperature and

pressure is heated at constant volume within a

boiler. This occurs along path bc. The water is con-

verted to steam and expands along path cd. The ex-

pansion continues adiabatically along path da. The

water is then cooled and condensed back to liquid

along path ab. There is no work done along path bc

because the volume remains constant.

(a) Using the ideal gas law PV = nRT, show that

the appropriate expression for work along an

isothermal (constant temperature) path from

to is

Work = nRT ln(V ô€€€ V) :

(b) Show that the appropriate expression for work

along an adiabatic path from to is

Work = (PV ô€€€ PV)=(1 ô€€€

) :

(c) Determine the total work done in single cycle of

the Rankine cycle, using the information pro-

vided in the gure above.

us to study ideal models for a heat engine that con-

verts heat energy to mechanical energy. For an ide-

alised engine, we can assume that gases obey ideal

gas laws such as PV = nRT, or for adiabatic expan-

sions, PV

= constant (

1:4 for air). Note that

P = pressure, V = volume and T = temperature

of the gas, and n and R can be taken to be con-

stants. The term adiabatic simply implies that in

the process of compressing (or expanding) a gas by

changing its pressure and volume, the temperature

of the gas also changes.

The work done by the system in one complete cycle

in an ideal heat engine is given by the area enclosed

in the cycle when illustrated on a P ô€€€ V (pressure

versus volume) diagram. By denition, as the gas

changes from volume Va to Vb the work done by the

gas is

Work =

Z Vb

Va

P dV :

An example of a heat engine is the Rankine cycle,

which is an idealised version of a steam engine. De-

tails of such a cycle are shown in the diagram.

V

P

0:05 0:15 0:25

10

20

c = (0:01; 20:0)

d = (0:03; 20:0)

b = (0:01; 1:03)

a = (0:25; 1:03)

In this cycle, liquid water at low temperature and

pressure is heated at constant volume within a

boiler. This occurs along path bc. The water is con-

verted to steam and expands along path cd. The ex-

pansion continues adiabatically along path da. The

water is then cooled and condensed back to liquid

along path ab. There is no work done along path bc

because the volume remains constant.

(a) Using the ideal gas law PV = nRT, show that

the appropriate expression for work along an

isothermal (constant temperature) path from

to is

Work = nRT ln(V ô€€€ V) :

(b) Show that the appropriate expression for work

along an adiabatic path from to is

Work = (PV ô€€€ PV)=(1 ô€€€

) :

(c) Determine the total work done in single cycle of

the Rankine cycle, using the information pro-

vided in the gure above.

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