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Bernoulli Equation (valid for incompressible fluids, no flid friction, no change in internal energy) $$ E_t = \frac{p}{\rho} + \frac{v^2}{2} + zg $$ Ergun Equation $$ \frac{\Delta p}{L} = \frac{1 – \varepsilon}{\varepsilon^3} \frac{G^2}{\phi D \rho} \left( \frac{150 (1 – \varepsilon) \mu }{\phi D G} + 1.75 \right)$$

$$ c = – \frac{1}{v} \left. \frac{\partial v}{\partial p} \right|_{T} $$ For water, \( c \approx 1 \times 10^{-5} \text{/psi} \) For an isothermal ideal gas $$ c \propto \frac{1}{p} $$ For an adiabatic ideal gas, where \( k \) is the taio of specific heats (1.4 for air) $$ c \propto \frac{1}{k p }…

For Phase Equilibria \( T_\text{liq} = T_\text{vap} \) \( p_\text{liq} = p_\text{vap} \) \( f^\text{liq} = f^\text{vap} \) Relative Volatility \( K_i = \frac{y_i}{x_i} \) \( \alpha_{ij} = \frac{K_i}{K_j} \) Eutectic \( L \rightarrow \alpha(s) + \beta(s) \) Peritectic \( \alpha(s) + L \rightarrow \beta(s) \) Azeotrope \( y_i = x_i\) At an azeotrope the…

Partial Pressure \( p_i = y_i p\) Henry’s Law \( x_i = C p_i \), at low values of \( x_i \). Also \( C = \frac{1}{k_i} \) Raoult’s Law \( x_i = \frac{p_i}{p_{i,\text{sat}}} \), at high values of \( x_i\) Amagat’s Law \( V = V_A + V_B + V_C + \ldots \) Dalton’s…

van der Waal’s $$ \left( p + \frac{a}{V^2} \right) \left( V – b \right) = nRT $$ Virial $$ pv = RT \left( 1 + \frac{B}{v} + \frac{C}{v^2} + \frac{D}{v^3} + \ldots \right) $$ $$ pv = RT \left( 1 + B^\prime p + C^\prime p^2 + D^\prime p^3 + \ldots \right) $$ According to…

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} =…

Kalman filters are a sequential algorithm that can be used to generate updated estimates using a the last state of a series of measures. A classic example is to estimate the new position of an object using a new set of observations combined with the previous estimates of position, velocity and acceleration. Kalman filters are…

By definition: $$ r = \sqrt{\frac{k t}{\phi \mu c_t}} $$ or in oil field units $$ r = \sqrt{\frac{k t}{948 \phi \mu c_t}} $$ If \( k = 0.02 \text{mD}\), \( \mu_g = 0.022\text{cP}\), \( c_t = 2 \times 10^{-4} \text{psi}^{-1} \), \( \phi = 0.04 \), then Setting \( r = 300 \text{m (984.25ft)}\)…

Dimensionless Pressure for Time Intervals (partially compressible): $$ p_{wD} = \frac{k h \left[ P_w (t_e) – P_w (t_e + \Delta t) \right] }{141.2 q B \mu} $$ $$ \Delta t_D = \frac{0.0002637 k \Delta t}{\phi \mu c_t L_f^2} $$ Dimensionless Pressure for Compressible Flow: $$ p_D = \frac{k h}{1422 q_g (T + 460) } \Delta…