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Mass and energy equations for non-flow; cyclic process

Short answer

For a closed system, mass is constant and Q - W = ΔU + ΔKE + ΔPE. In a cyclic process, ΔE =0, hence ∮ δQ = ∮ δW, i.e. net heat added equals net work done.

State the equations of conservation of mass and conservation of energy for non flow process and deduce it for cyclic process undergoing in a closed system.

Non-flow process (closed system, no mass crosses the boundary)

Conservation of mass: the mass of the system stays constant:
m = constant, i.e. dm = 0 (m₁ = m₂)

Conservation of energy (first law for a closed system): the net energy supplied as heat and work equals the change in the energy of the system:

Q₁₂ − W₁₂ = E₂ − E₁ = ΔU + ΔKE + ΔPE

Differential form: δQ − δW = dE; for a stationary system, δQ − δW = dU. Sign convention: heat into the system +ve, work by the system +ve.

Reduction for a cyclic process in a closed system

In a cycle the system returns to its initial state: m, E and all other properties regain initial values, so
m₂ = m₁ and E₂ − E₁ = 0 (∮ dE = 0).

Substituting in the first law:

∮ δQ − ∮ δW = 0

∮ δQ = ∮ δW i.e. Q_net = W_net

So in a cycle, the net heat transfer equals the net work transfer. For a power cycle: W_net = Q_H − Q_L; for a refrigeration cycle: W_in = Q_H − Q_L.

✓ Answer checked by Claude Opus 5.5 (high effort)

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