Consider a
piston cylinder arrangement as given in the Figure 2.4. If the pressure of the
fluid is greater than that of the surroundings, there will be an unbalanced
force on the face of the piston. Hence, the piston will move towards right.
Force acting on
the piston = Pressure ´ Area
= pA
\ Work done = Force ´ distance
= pA ´ dx
= pdV
where dV -
change in volume.
This work is known as displacement work or pdV work corresponding
to the elemental displacement dx. To
obtain the total work done in a process, this elemental work must be added from
the initial state to the final state.
Mathematically,
Evaluation of Displacement Work
Constant Pressure Process
Figure 2.5 shows a piston cylinder arrangement containing a fluid.
Let the fluid expands such that the pressure of the fluid remains constant
throughout the process. Figure 2.6 shows the process in a p-V diagram.The
mathematical expression for displacement work can be obtained as follows:
= p(V2 – V1)
...(2.1)
This expression shows that the area under a curve in a p-V diagram
gives work done in the process.
2.6.2. Constant volume process
Consider a gas contained in a rigid vessel being heated. Since
there is no change in volume, the displacement work .
Hyperbolic process
Let the product of pressure and volume remains constant at all the
intermediate states of a process. In the
p-V diagram it will be a hyperbola as given in Figure 2.7.
1W2 =
= where C=pV
= C
= C ln (V2/V1)
1w2 = p1V1ln(V2/V1) (or) p2V2ln (V2/V1) ...(2.2)
For Ideal gases when temperature remains constant, pV will be
constant i.e., isothermal process are hyperbolic processes for an ideal gas.
Polytropic Process
Any process can
be represented by the general form pVn =
constant. Based on the valve of n,
the process differs as given below;For other values of n, the process is known
as polytropic process. Figure 2.8 shows the polytropic process of various
possible polytropic index ‘n’ on p-V coordinates. Expression for displacements
work for a polytropic process .
