1. Reynolds Transport Theorem

Recall RTT:

$$ \underbrace{\frac{dB_\text{system}}{dt}}\text{Term A} = \underbrace{\frac{\partial}{\partial t}\int\text{CV}\rho\beta dV}\text{Term B} + \underbrace{\oint\text{CS}\rho \beta (\vec{v}\cdot\hat{n}) dA}_\text{Term C} $$

• Term A - Rate of change of B within System (Lagrangian)
• Term B - Rate of change of B within CV (Eulerian)
• Term C - Rate of change of B within CV due to fluid flow (Eulerian)

<aside> 💡 RTT applies to Fluid Mechanics, Heat Transport and Mass Transport!!

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<aside> 💡 RTT balances changes of an extensive property within a Lagrangian System with intensive property in Eulerian Control Volume

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2. Integral Form of Mass Transport Equation

RTT of Solute Mass

$$ B_\text{system} \to N \qquad \beta \to \frac{N}{m} $$

• $B_\text{system}$ - moles of solute [$mol$]
• $\beta$ - moles of solute per unit mass of mixture [$molkg^{-1}$]
• $\rho\beta=C$ - concentration of solute [$kgm^{-3}\cdot molkg^{-1} = molm^{-3}$]

<aside> 💡 Dilute Assumption: Volume of CV contain solution $\approx$ Volume of Solvent

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$$ \text{RTT} \quad \longmapsto \quad \frac{dN}{dt} = \frac{\partial}{\partial t}\int_\text{CV}C dV + \oint_\text{CS}C (\vec{v}\cdot\hat{n}) dA $$

Similar to Heat Transport, we know, for mass transport:

$$ \frac{dN}{dt} = \dot{J_D}+\dot{S_V} $$

Therefore: