1. Differential Form of Mass Transport Equation

From Lecture 4, we have derived Differential Form of Heat Transport Equation:

$$ \rho c_p \left[ \frac{\partial T}{\partial t}+ (\vec v \cdot \vec \nabla) T \right] = k\vec\nabla^2T + \dot{S_v} $$

By plugging in the equivalent term in Mass Transport:

“Heat Concentration” to Concentration

$$ \rho c_p T \to C $$

“Thermal Diffusivity” to Mass Diffusivity

$$ \frac{k}{\rho c_p} \equiv \alpha \to D $$

$$ \rho c_p \frac{\partial T}{\partial t} \to \frac{\partial C}{\partial t} $$

We get the Differential Form of Mass Transport Equation:

$$ \frac{\partial C}{\partial t} + (\vec v \cdot \vec \nabla)C

D\vec \nabla^2 C + \dot S_v $$

Meaning, respectively:

$$ \text{Rate of change of Concentration over time and by advection} = \text{Rate of Solute Accumulation by diffusion} + \text{Rate of Solute Generation} $$

<aside> 💡 Valid for Constant and Uniform D and Dilute solute

</aside>

Equations describing Heat, Mass and Momentum

Differential Form of Heat Transport Equation

$$ \rho c_p \left[ \frac{\partial T}{\partial t}+ (\vec v \cdot \vec \nabla) T \right] = k\vec\nabla^2T + \dot{S_v} \\ \frac{\partial T}{\partial t}+ (\vec v \cdot \vec \nabla) T = \alpha\vec\nabla^2T + \dot{S_T} $$

• “Thermal Diffusivity”

$$ \alpha = \frac{k}{\rho c_p} $$

Differential Form of Mass Transport Equation

• Source Term

$$ S_T = \frac{\dot S_v}{\rho c_p} $$

$$ \frac{\partial C}{\partial t} + (\vec v \cdot \vec \nabla)C

D\vec \nabla^2 C + \dot S_v $$

• Diffusivity

$$ D $$

• Source Term

$$ \dot S_v $$

Navier-Stokes Equation - “Transport” of Momentum

$$ \frac{\partial \vec v}{\partial t} + (\vec v \cdot \vec \nabla)\vec v = \gamma \vec \nabla^2 \vec v + \vec S_v $$

• “Momentum Diffusivity”

$$ \gamma = \frac{\mu}{\rho} $$

• Source Term

$$ \vec S_v = \frac{1}{\rho}(-\vec \nabla p + \vec g) $$

Boundary Conditions for MT