I need Derivation on Carnot Cycle …

For this questions you  will need some mathematics and knowledge of engineering thermodynamics. Appropriate links are provided at the bottom of this section. As we saw previously all Carnot heat engine have the same efficiency irrespective of the working gas we are using. Hence, we might as well use an ideal gas with temperature-independent specific heats because its properties are precisely known and can be represented in terms of fairly easy equations. All noble gases ( helium, neon, argon, krypton, xenon, and the radioactive radon ) come extremely close to being an ideal gas at temperatures and pressure relevant to Stirling engine.

(6)       p V = m R_s T         Ideal gas law

(7)       Δu = c_v ΔT         Change of internal energy is proportional to change in temperature

<th>Figure 5 : Carnot Cycle</th>

1 → 2 : isothermal compression. Because temperature is constant the internal energy of the gas does not change according to Eq.(2). According to Eq.(1) we also have <i>p V = m R Tc= constant</i>. This makes it possible to evaluate the work integral Wc =&int; p dV This results in the equation : (8)       Qc=Wc = m Rs Tc ln ( V1 / V2 ) 3 → 4 : isothermal expansion. This follows exactly the logic as the process 1 → 2. Hence, (9)       Qh=Wh = m Rs Th ln ( V4 / V3 ) 2 → 3 : adiabatic compression. With κ = cp/cv for such process : (10)       V2/V3 = (Th/Tc)1/(κ-1) 4 → 1 : adiabatic expansion. Follows the same logic as process 2 → 3 with the result: (11)       V1/V4 = (Th/Tc)1/(κ-1)

We use Eq.(8) and (9) to find a first expression for the efficiency of the Carnot cycle (see also Eq.(3) ) :

(12)       η = 1 - Q_c/Q_h = 1 - (T_c ln ( V1 / V2 )) / (T_h ln ( V4 / V3 ))

From Eq. (10) and (11) we get :

(13)       V₁/V₂ = V₄/V₃

and therefore :

(14)       η = 1 - T_c/T_h

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