   Chapter 15.9, Problem 22E

Chapter
Section
Textbook Problem

An important problem in thermodynamics is to find the work done by an ideal Carnot engine. A cycle consists of alternating expansion and compression of gas in a piston. The work done by the engine is equal to the area of the region R enclosed by two isothermal curves xy= a. xy = b and two adiabatic curves xy1.4 = c, xy1.4 = d, where 0 < a < b and 0 < c < d. Compute the work done by determining the area of R.

To determine

To compute: The work done by the engine.

Explanation

Given:

The region R enclosed by the isothermal curves are xy=a , xy=b .

The adiabatic curves are, xy1.4=c , xy1.4=d .

The region varies 0<a<b and 0<c<d .

Calculation:

The region R is defined as a<xy<b and c<xy1.4<d .

Choose the given curves as u=xy and v=xy1.4 .

Then, the region R becomes, a<u<b and c<v<d .

Let u=xy and v=xy1.4 .

Obtain the value of y by substituting x=uy in v.

v=xy1.4v=(uy)y1.4v=uy0.4y0.4=vu

Raise to the power (10.4) on both the sides.

y=(vu)10.4=(vu)2.5=v2.5u2.5

Substitute y=(vu)2.5 in u,

u=x(vu)2.5x=uu2.5v2.5x=u3.5v2.5

Obtain the Jacobian, (x,y)(u,v)=|xuxvyuyv|

Find the partial derivative of x and y with respect to u and v.

If x=u3.5v2.5 , then xu=3.5u2.5v2.5 and xv=2

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