Balance Laws

Conservation of Mass

$$ \iint_\Gamma\rho\bold v\cdot d\bold A+\large\partial_t \normalsize \iiint_\Omega\rho~dV=0 $$

Conservation of Momentum

$$ \iint_\Gamma\bold v\rho\bold v\cdot d\bold A+\large\partial_t \normalsize \iiint_\Omega\bold v\rho~dV=\Sigma\bold F $$

Conservation of Energy

$$ \dot Q -\dot W=\iint_\Gamma e\rho\bold v\cdot d\bold A+\large\partial_t \normalsize \iiint_\Omega e\rho~dV,~~~e=(h+\frac 12|\bold v|^2+gz) $$

Generation of Entropy

$$ -\Sigma\frac{d\dot Q}{T}+\iint_\Gamma s\rho\bold v\cdot d\bold A+\large\partial_t\normalsize\iiint_\Omega s\rho~dV=\dot s\ge0 $$

State and Process Relations

Ideal Gas Law

$$ pv=RT $$

Gibbs Second Relation

$$ Tds=du+pdv=dh-vdp $$

Non-Isentropic Process Relations for an Ideal Gas

$$ s_2-s_1=\int\frac{c_p}TdT+R\ln\frac{v_2}{v_1}=\int\frac{c_v}TdT-R\ln\frac{p_2}{p_1} $$

Isentropic Process Relations for Ideal Gases

$$ \Large\begin{pmatrix} \frac{T_2}{T_1} \end{pmatrix}^{\frac\gamma{\gamma-1}} \normalsize= \Large\begin{pmatrix} \frac{p_2}{p_1} \end{pmatrix} \normalsize= \Large\begin{pmatrix} \frac{\rho_2}{\rho_1} \end{pmatrix}^\gamma $$

Miscellaneous Identities

$$ R=c_p-c_v $$