Thermodynamic Relationships

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

$$ \Delta T = q = w = \Delta h = \Delta u = 0 $$

$$ \Delta s = R \ln \left( \frac{p_1}{p_2} \right) $$