0. Biot Number for H.T. and M.T.

Heat Transport

$$ \text{Biot} \equiv \frac{hw}{k} $$

• $h$ - Convective Heat Transport Coefficient [$Wm^{-2}K^{-1}$]
• $k$ - Thermal Conductivity [$Wm^{-1}K^{-1}$]

Mass Transport

$$ \text{Biot} \equiv \frac{k_mw}{D} $$

• $k_m$ - Convective Mass Transport Coefficient [$ms^{-1}$]
• $D$ - Diffusion Coefficient [$m^2s^{-1}$]

1. Integral vs. Differential Form

Macroscopic CV

• Good for analysing global behaviour of a large system
  • e.g. Inflows and Outflows of pipe network
• Does not typically provide internal details

Microscopic CV

• Provides internal details
• In general, requires solving PDEs
  • allow calculating changes in 3D space and time

Divergence Theorem

The Divergence Theorem is a mathematical statement that the net quantity of any vector leaving a volume must be equal to net flow of that vector across the surface bounding that volume

$$ \oint_\text{CS}(\vec{f}\cdot\hat{n})dA = \int_\text{CV}(\vec{\nabla}\cdot\vec{f})dV $$

From Mathematics Alpha: Sum of net flow through the bounding surface is equivalent to Sum of divergence inside a body

Divergence Theorem transforms a Surface Integral into a Volume Integral

Therefore the 2 Surface Integrals in Integral Form of RTT for H.T.: 1) Rate of Heat Loss due to Heat Flux, 2) Rate of Heat Loss by Fluid Flow across CS can be transferred to Volume Integrals.

$$ \frac{\partial}{\partial t}\int_\text{CV}\rho c_p T dV = \int_\text{CV} \dot{S}V dV -\underbrace{\oint\text{CS}(\vec{q} \cdot \hat{n})dA} - \underbrace{\oint_\text{CS}\rho c_p T (\vec{v}\cdot\hat{n}) dA} $$

$$ \oint_\text{CS}(\vec{q} \cdot \hat{n})dA \equiv \int_\text{CV}(\vec{\nabla}\cdot\vec{q})dV $$

$$ \oint_\text{CS}\rho c_p T (\vec{v}\cdot\hat{n}) dA \equiv \int_\text{CV}[\vec{\nabla}\cdot(\rho c_pT\vec{v})]dV $$

For a small CV:

Divergence indicates the net flow of $\vec{f}$ out of the CV