First Law of Thermodynamics
Q = \overbrace{\Delta U + W }^{\text{Closed System}} = \Delta E_p + \Delta E_k + \overbrace{ \Delta U + W_\text{flow} }^{\Delta H} + W_\text{shaft}
where
W = \int_{v_1}^{v_2} p \text{d} v = \int_{p_1}^{p_2} v \text{d} p
Q = \int_{s_1}^{s_2} T \text{d} s
W_\text{max} = m \left[ \left( h_1 – h_2 \right) – T_L \left( s_1 – s_2 \right) \right]
Enthalpy
h = f(T,P,\text{phase})
\Delta h_f is Enthalpy of Formation
In general, h = u + p v , and for gasses h = u + zRT
\Delta h = \Delta u + W_\text{flow} = \Delta u \left( p_2 v_2 – p_1 v_1 \right)
For a constant pressure processes, \Delta h = m c_p \Delta T
For constant volumes processes, \Delta u = m c_v \Delta T
Entropy
\Delta s \ge \int_{T_1}^{T_2} \frac{\text{d}q}{T}, \quad \lim_{T \to 0 } s=0
For every real process, \sum \Delta s > 0
du = T ds – p dv
dh = T ds + v dp
For perfect gasses:
\Delta s = c_p \ln \left( \frac{T_2}{T_1} \right) – R \ln \left( \frac{p_2}{p_1} \right)
\Delta s = c_v \ln \left( \frac{T_2}{T_1} \right) + R \ln \left( \frac{v_2}{v_1} \right)
Polytropic Processes
p_1 v_1^2 = p_2 v_2^n
n = 0 constant pressure
n = 1 constant temperature
n = k isentripic
n = \infty constant volume
Isothermal Processes
0 = \Delta u = \Delta h
w = q = p_1 v_1 \ln \left( \frac{p_1}{p_2} \right)
\Delta s = \frac{q}{T} = R \ln \left( \frac{p_1}{p_2} \right)
Throttling Processes
\Delta T = q = w = \Delta h = \Delta u = 0
\Delta s = R \ln \left( \frac{p_1}{p_2} \right)