Compressor in Figure 1

\[T_2 \,>\, T_1\]

\[S_2 \,=\, S_1\]

\[W \,=\, H_2 \,-\, H_1\]

Cooler in Figure 1

\(H\) = enthalpy per kg

\(S\) = entropy per kg

\(T\) = temperature

\(W\) = work per kg

\(\dot{Q}\) = rate of heat transfer (kJ/s)

\(\dot{m}_1\) = mass flow rate entering the first compressor (kg/s)

Compressors in Figure 2

\[T_{1a} > T_1\]

\[S_{1a} = S_1\]

\[W_1 = H_{1a}\, -\, H_1\]

\[T_{1c} > T_{1b}\]

\[S_{1c} = S_{1b}\]

\[W_2 = H_{1c} \,-\, H_{1b}\]

\[T_2 > T_{1d}\]

\[S_2 = S_{1d}\]

\[W_3 = H_2 \,-\, H_{1d}\]

Coolers in Figure 2

\[\dot{m}_1(H_{1b} \,-\, H_{1a}) = \dot{Q}_1\]

\[\dot{m}_1(H_{1d} \,-\, H_{1c}) = \dot{Q}_2\]

\[\dot{m}_1(H_3 \,-\, H_2) = \dot{Q}_3\]

Heat exchanger (Figures 1 and 2)

\[\dot{m}_3(H_4 \,-\, H_3) \,+\, \dot{m}_8(H_3 \,-\, H_2) = 0\]

\[(H_4 \,-\, H_3) \,+\, (1\,-\, y)(H_8 \,-\, H_7) = 0\]

where \(y\) = fraction of flow exiting the Joule-Thomson expansion that is liquid

Joule-Thomson expansion (Figures 1 and 2)

\[H_4 = H_3\]

Flash Drum (Figures 1 and 2)

\[H_5 = y\,H_6 \,+\, (1\,-\,y)H_7\]