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First law for cyclic process; power and refrigeration cycles

Short answer

A cycle is a sequence of processes returning a system to its initial state, with zero net property change. First law for a cycle: net heat transfer equals net work (∮δQ = ∮δW). In a power cycle, W_net = Q_H - Q_L, efficiency η = W_net/Q_H. In a refrigeration cycle, work input W_in = Q_H - Q_L, COP_ref = Q_L/W_in.

Define a cycle. Explain first law of thermodynamics for control mass undergoing cyclic process. Write down statements of first law for power cycle and refrigeration cycle.

Cycle

A cycle is a series of processes in which a system starts from an initial state, passes through other states and returns to the initial state. All properties return to their initial values: ΔE = 0, ∮dP = 0 and so on. On a P-V diagram, a cycle is a closed curve.

First law for a control mass undergoing a cycle

The first law for a process is Q − W = ΔE. For a cycle, ΔE = 0 (E is a property), so

∮ δQ = ∮ δW i.e. Q_net = W_net

The net heat transferred to the system in a cycle equals the net work done by it.

Power cycle

A cycle that produces net work from heat input (clockwise on P-V; e.g. a steam power plant). Heat Q_H is received from a high-temperature source and Q_L is rejected to a sink:

W_net = Q_H − Q_L (W_net > 0), efficiency η = W_net/Q_H = 1 − Q_L/Q_H

Refrigeration (heat pump) cycle

A cycle that needs net work input to remove heat Q_L from a cold space and reject Q_H to a hot space (anticlockwise on P-V):

W_in = Q_H − Q_L (W_net < 0)

COP of refrigerator = Q_L/W_in; COP of heat pump = Q_H/W_in.

  Power cycle                 Refrigeration cycle
  Hot reservoir T_H           Hot reservoir T_H
       | Q_H                       ^ Q_H
       v                           |
     [ E ] --> W_net             [ R ] <-- W_in
       | Q_L                       ^ Q_L
       v                           |
  Cold reservoir T_L          Cold reservoir T_L

✓ Answer checked by Claude Opus 5.5 (high effort)

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