Thermodynamics Concept Page - 15

Diagram

PV diagram of a Carnot cycle

Diagram

TS diagram of a Carnot Cycle

Definition

Carnot's cycle as a process of maximum efficiency

The Carnot cycle is reversible and thus represents the upper limit on efficiency of an engine cycle. Practical engine cycles are irreversible and thus have inherently lower efficiency than the Carnot efficiency when operated between the same temperatures TH${\displaystyle T_H}$  and TC${\displaystyle T_C}$ . One of the factors determining efficiency is how heat is added to the working fluid in the cycle, and how it is removed. The Carnot cycle achieves maximum efficiency because all the heat is added to the working fluid at the maximum temperature TH${\displaystyle T_H}$ , and removed at the minimum temperature TC${\displaystyle T_C}$ .

Definition

Efficiency of a carnot cycle

η=WQ1=1−Q2Q1=1−T2T1$\eta ={\displaystyle \frac{W}{Q_1}}=1-{\displaystyle \frac{Q_2}{Q_1}}=1-{\displaystyle \frac{T_2}{T_1}}$
where Q1$Q_1$ is the energy supplied to the system and  Q2$Q_2$ is the energy released by the system.

Definition

Refrigerator

A heat pump or refrigerator is the reverse of heat engine.

Definition

Coefficient of performance of a heat pump

The coefficient of performance (CP) for a heat pump is the ratio of the energy transferred for heating to the input electric energy used in the process.
CP=QHW$CP={\displaystyle \frac{Q_H}{W}}$
where QH$Q_H$ is the heat delivered to the hot reservoir

Definition

Performance of a heat pump

In a refrigerator or a heat pump, the system extracts heat Q2$Q_2$ from the cold reservoir and releases Q1$Q_1$ amount of heat to the hot reservoir, with work W$W$ done on the system. The co-efficient of performance of a refrigerator is given by α=Q2W=Q2Q1−Q2$\alpha ={\displaystyle \frac{Q_2}{W}}={\displaystyle \frac{Q_2}{Q_1-Q_2}}$ 

Definition

Work done and heat involved in different steps of a Carnot cycle

Step 1: Isothermal expansion:
ΔU=q+w$\mathrm{\Delta }U=q+w$
0=q+w$0=q+w$
q=−w=PΔV=nRT∫V2V11VdV=nRT(V2V1)$q=-w=P\mathrm{\Delta }V=nRT{\int }_{V_1}^{V_2}{\displaystyle \frac{1}{V}}dV=nRT\left({\displaystyle \frac{V_2}{V_1}}\right)$
Step 2: Adiabatic expansion
ΔU=q+w=0+w=−PΔV=∫V3V2PΔV=C∫V3V2dVVγ=CV1−γ3V1−γ21−γ$\mathrm{\Delta }U=q+w=0+w=-P\mathrm{\Delta }V={\int }_{V_2}^{V_3}P\mathrm{\Delta }V=C{\int }_{V_2}^{V_3}{\displaystyle \frac{dV}{V^{\gamma }}}=C{\displaystyle \frac{V_3^{1-\gamma }{V}_2^{1-\gamma }}{1-\gamma }}$
Step 3:Isothermal compression
−q=w=∫V4V3PdV=−nRTln(V4V3)$-q=w={\int }_{V_3}^{V_4}PdV=-nRT\ln \left({\displaystyle \frac{V_4}{V_3}}\right)$
Step 4: Adiabatic compression
q=0$q=0$

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