Mass Balance

Overall Mass Balance

The overall mass balance ensures mass conservation throughout the packed bed.

\[ \epsilon_t(\frac{1}{P}\frac{\partial P}{\partial t} - \frac{1}{T}\frac{\partial T}{\partial t}) + \frac{T}{P}\frac{\partial}{\partial z}(\frac{u P}{T}) + \frac{RT}{P}\rho_b\sum_{i=1}^{N}\frac{\partial{q_i}}{\partial{t}} = 0 \]

Symbol	Description	Unit
\(P\)	Pressure	Pa
\(T\)	Temperature	K
\(\epsilon_t\)	Total void fraction	-
\(u\)	Superficial velocity	m s⁻¹
\(q_i\)	Component i mass source	mol kg⁻¹ s⁻¹
\(t\)	Time	s
\(z\)	Axial coordinate	m
\(R\)	Ideal gas constant	J mol⁻¹ K⁻¹

Component Mass Balance

The component mass balance accounts for axial dispersion, convection, and adsorption/desorption accordnig to the rate model selected. Axial dispersion can be constant or calculated from different empirical equations. How \(\rho_b\), \(\epsilon_t\) and \(\epsilon_b\) follow from the deck's parameters, and which settings to check when comparing with another code, is on Matching Another Model.

\[ \epsilon_t (\frac{\partial y_i}{\partial t} + \frac{y_i}{P}\frac{\partial P}{\partial t} - \frac{y_i}{T}\frac{\partial T}{\partial t}) + \frac{T}{P} \frac{\partial}{\partial z}(\frac{u y_i P}{T}) - \epsilon_b \frac{T}{P}\frac{\partial}{\partial z}(\frac{D_{ax} P}{T} \frac{\partial y_i}{\partial z}) + \frac{RT}{P} \rho_b \frac{\partial q_i}{\partial t} = 0 \]

Symbol	Description	Unit
\(y_i\)	Mole fraction of component \(i\) in gas phase	-
\(P\)	Pressure	Pa
\(T\)	Temperature	K
\(\epsilon_b\)	Bed void fraction	-
\(u\)	Superficial velocity	m s⁻¹
\(D_{ax}\)	Axial disperion coefficient	m² s⁻¹
\(q_i\)	Component i mass source	mol kg⁻¹ s⁻¹
\(t\)	Time	s
\(z\)	Axial coordinate	m
\(R\)	Ideal gas constant	J mol⁻¹ K⁻¹

Inlet and outlet conditions

At an inlet the boundary condition is of the Danckwerts type: the total flux of each component entering the bed (convection plus dispersion) equals the feed's,

\[ u\thinspace C\thinspace y_{i,in} = u\thinspace C\thinspace y_i\rvert_{0^+} - \epsilon_b D_{ax} C \frac{\partial y_i}{\partial z}\bigg\rvert_{0^+} \]

so the gas in the first cell can differ from the feed, and the moles of each component the bed takes in are exactly the moles the inlet stream supplies (the amounts the stream logs, the performance metrics and the cyclic-steady-state mass balance count). At an outlet or a closed end the composition has zero gradient. The energy balance treats heat conduction at the inlet the same way. Numerics gives the discrete form.