Example 7E - 2: PVT Relationships for Isentropic, IG Processes

7E-2 :	PVT Relationships for Isentropic, IG Processes	8 pts

Consider the Carnot Power Cycle shown in the PV Diagram, below. The working fluid is air and the specific heat ratio, g, is constant.

Show that…
a.) V₂/V₁ = V₃/V₄
b.) (T₂ / T₃)^g = (P₂ / P₃)⁽^g^-1)
c.) T₂ / T₃ = (V₃ / V₂)⁽^g^-1)

Read :

(a) Note that h = W_cycle/Q_in = (W₁₂ + W₃₄)/Q₁₂ because Q₂₃ and Q₄₁ are equal and have opposite signs. Determine W₁₂ in terms of V₁, V₂, and T_H (temperature of high temp reservoir) and also W₃₄ in terms of V₃, V₄, and T_C. Note the relationship between Q₁₂ and W₁₂ determined from an energy balance during step 1-2. Compare h obtained this way with the Carnot cycle efficicency and you will arrive at the desired conclusion.

(c) Easier to do part (c) before (b). Apply the 1st Law to step 2-3. Note that dU = m C_V dT, C_V = (R/MW)/(g-1), PV = nRT and dW = P dV. You will arrive at the form (1/T) dT and (1/V) dV on both sides. Integrate to obtain the desired result.

(b) Just use the result from part (c) along with the ideal gas EOS to convert V to P.

Given:

Q₂₃

Carnot Cycle

T₁ = T₂ = T_H

Q₄₁

T₃ = T₄ = T_C

Step:

1-2

Isothermal Expansion

2-3

Adiabatic Exapansion

3-4

Isothermal Compression

4-1

Adiabatic Compression

Find:

Show that:

(a) V₄V₂ = V₁V₃

(b) T₂/T₃ = (P₂/P₃)⁽⁽^g^-1)/^g

Diagram:

Given in the problem statement.

Assumptions:

1 -

The system consists of an ideal gas.

2 -

The specific heat ratio is constant (required in part (b) only).

3 -

The cycle is executed in a closed system ( not required, but it makes the solution simpler).

4 -

Changes in kinetic and potential energies are negligible.

5 -

Boundary work is the only type of work that crosses the system boundary.

6 -

The system undergoes a Carnot cycle (reversible).

Equations / Data / Solve:

Part a.)

It may be hard to determine where to start with the proof but following the provided hints will help you.

Starting with the thermal effiicency:

Eqn 1

Now we need to determine W₁₂ in terms of V₁, V₂, and T_H and also W₃₄ in terms of V₃, V₄, and T_C.

PV work done by the system during the isothermal expansion and compression processes can be evaluated as follows:

Eqn 2

Assuming the system consists of an ideal gas substitute P = nRT/V:

Eqn 3

Eqn 5

Integrating:

Eqn 4

Eqn 6

Apply the 1st Law for a closed system with negligible changes in kinetic and potential energies to get Q₁₂:

Eqn 7

Since the internal energy of an ideal gas depends on temperature only and the temperature is constant along Process 1-2, U₂ = U₁ and the energy balance reduces to: