Wall Energy Balance

The wall energy balance describes heat transfer between the packed bed wall and its surroundings.

C_{p,W} \rho_W \frac{\partial T_W}{\partial t} - \frac{2 r_i h_i}{r_o^2 - r_i^2} (T - T_W) + \frac{2 r_o h_o}{r_o^2 - r_i^2} (T_W - T_{amb}) - K_w \frac{\partial^2 T_W}{\partial z^2} = 0

Symbol	Description	Unit
$C_{p,W}$	Wall heat capacity	J kg⁻¹ K⁻¹
$\rho_W$	Wall material density	kg m⁻³
$T_W$	Wall temperature	K
$T$	Gas/Adsorbent temperature	K
$T_{amb}$	Ambient temperature	K
$h_i$	Internal heat transfer coefficient (gas–wall)	W m⁻² K⁻¹
$r_i$	Internal wall radius	m
$h_o$	External heat transfer coefficient (wall–ambient)	W m⁻² K⁻¹
$r_o$	External wall radius	m
$K_w$	Wall axial thermal conductivity	W m⁻¹ K⁻¹