Energy Balance

The energy balance assumes that the heat transfer between the gas and solid is very fast, and therefore the temperature of the gas and adsorbent are lumped together.

\left(\epsilon_b C_{p,g} \rho_g + C_{p,s} \rho_b\right) \frac{\partial T}{\partial t} - \lambda \frac{\partial^2 T}{\partial z^2} - \epsilon_b \frac{\partial P}{\partial t} + u C_{p,g} \rho_g \frac{\partial T}{\partial z} + \frac{4 h_i}{d_i} (T - T_w) - \frac{RT}{P}\rho_b\sum_{i=1}^{N}(-\Delta H_i)\frac{\partial{q_i}}{\partial{t}} = 0

Symbol	Description	Unit
$C_{p,g}$	Gas heat capacity	J kg⁻¹ K⁻¹
$C_{p,s}$	Adsorbent heat capacity	J kg⁻¹ K⁻¹
$T$	Gas/Adsorbent temperature	K
$T_w$	Wall temperature	K
$P$	Pressure	Pa
$u$	Superficial velocity	m s⁻¹
$\epsilon_b$	Bed porosity (gas phase only)	-
$\rho_g$	Gas density	kg m⁻³
$\rho_b$	Adsorbent bulk density	kg m⁻³
$\lambda$	Effective axial thermal conductivity	W m⁻¹ K⁻¹
$h_i$	Internal heat transfer coefficient (gas–wall)	W m⁻² K⁻¹
$d_i$	Wall internal diameter	m
$q_i$	Component i mass source	mol kg⁻¹ s⁻¹
$\Delta H_i$	Component i heat of adsorption	J mol⁻¹