Isotherms | Skarstrom Docs

Isotherm models

q_i^* = \frac { m_1 b_1 \exp( \frac {q_1}{R T} ) P_i }{ 1 + b_1 \exp( \frac {q_1}{R T} ) P_i }

Symbol	Description	Unit
$q_i^*$	Adsorbed amount of component $i$ at equilibrium	mol kg⁻¹
$m_1$	Isotherm parameter 1	mol kg⁻¹
$b_1$	Isotherm parameter 2	bar⁻¹
$q_1$	Isotherm parameter 3	J mol⁻¹
$P_i$	Partial pressure of component $i$	bar
$T$	Temperature	K
$R$	Ideal gas constant	J mol⁻¹ K⁻¹

q_i^* = \frac { m_1 b_1 \exp( \frac {q_1}{R T} ) C_i }{ 1 + b_1 \exp( \frac {q_1}{R T} ) C_i }

Symbol	Description	Unit
$q_i^*$	Adsorbed amount of component $i$ at equilibrium	mol kg⁻¹
$m_1$	Isotherm parameter 1	mol kg⁻¹
$b_1$	Isotherm parameter 2	mol mol⁻¹
$q_1$	Isotherm parameter 3	J mol⁻¹
$C_i$	Molar concentration of component $i$	mol m⁻³
$T$	Temperature	K
$R$	Ideal gas constant	J mol⁻¹ K⁻¹

q_i^* = \frac { m_1 b_1 \exp( \frac {q_1}{R T} ) P_i }{ 1 + b_1 \exp( \frac {q_1}{R T} ) P_i } + \frac { m_2 b_2 \exp( \frac {q_2}{R T} ) P_i }{ 1 + b_2 \exp( \frac {q_2}{R T} ) P_i }

Symbol	Description	Unit
$q_i^*$	Adsorbed amount of component $i$ at equilibrium	mol kg⁻¹
$m_1$	Isotherm parameter 1	mol kg⁻¹
$b_1$	Isotherm parameter 2	bar⁻¹
$q_1$	Isotherm parameter 3	J mol⁻¹
$m_2$	Isotherm parameter 4	mol kg⁻¹
$b_2$	Isotherm parameter 5	bar⁻¹
$q_2$	Isotherm parameter 6	J mol⁻¹
$P_i$	Partial pressure of component $i$	bar
$T$	Temperature	K
$R$	Ideal gas constant	J mol⁻¹ K⁻¹

q_i^* = \frac { m_1 b_1 \exp( \frac {q_1}{R T} ) C_i }{ 1 + b_1 \exp( \frac {q_1}{R T} ) C_i } + \frac { m_2 b_2 \exp( \frac {q_2}{R T} ) C_i }{ 1 + b_2 \exp( \frac {q_2}{R T} ) C_i }

Symbol	Description	Unit
$q_i^*$	Adsorbed amount of component $i$ at equilibrium	mol kg⁻¹
$m_1$	Isotherm parameter 1	mol kg⁻¹
$b_1$	Isotherm parameter 2	mol mol⁻¹
$q_1$	Isotherm parameter 3	J mol⁻¹
$m_2$	Isotherm parameter 4	mol kg⁻¹
$b_2$	Isotherm parameter 5	mol mol⁻¹
$q_2$	Isotherm parameter 6	J mol⁻¹
$C_i$	Molar concentration of component $i$	mol m⁻³
$T$	Temperature	K
$R$	Ideal gas constant	J mol⁻¹ K⁻¹

q_i^* = k \exp( \frac {Q}{R T} ) P_i

Symbol	Description	Unit
$q_i^*$	Adsorbed amount of component $i$ at equilibrium	mol kg⁻¹
$k$	Isotherm parameter 1	mol kg⁻¹
$Q$	Isotherm parameter 2	J mol⁻¹
$P_i$	Partial pressure of component $i$	bar
$T$	Temperature	K
$R$	Ideal gas constant	J mol⁻¹ K⁻¹

\frac{\partial q_i}{\partial t} = k (q_i^* - q_i)

Symbol	Description	Unit
$k$	Mass transfer coefficient	s⁻¹
$q_i^*$	Equilibrium adsorbed concentration	mol kg⁻¹

The macropore mass transfer coefficient is calculated using the following equation:

k = 60 \frac{D_{\text{eff}}}{d_p^2}

where the effective diffusivity $D_{\text{eff}}$ combines Knudsen diffusivity $D_k$ and molecular diffusivity $D_m$ through:

\frac{1}{D_{\text{eff}}} = \frac{1}{D_{k,\text{eff}}} + \frac{1}{D_{m,\text{eff}}}

The effective Knudsen and molecular diffusivities account for the porosity and tortuosity of the medium:

D_{k,\text{eff}} = D_k \frac{\epsilon}{\tau}

D_{m,\text{eff}} = D_m \frac{\epsilon}{\tau}

The Knudsen diffusivity $D_k$ is estimated as:

D_k = 97 \, r_p \sqrt{\frac{T}{M}}

Symbol	Description	Unit
$k$	Mass transfer coefficient	s⁻¹
$D_{\text{eff}}$	Effective diffusivity	m²/s
$D_{k,\text{eff}}$	Effective Knudsen diffusivity	m²/s
$D_{m,\text{eff}}$	Effective molecular diffusivity	m²/s
$D_k$	Knudsen diffusivity	m²/s
$D_m$	Molecular diffusivity (input)	m²/s
$\epsilon$	Porosity of the medium	–
$\tau$	Tortuosity of the medium	–
$d_p$	Particle diameter	m
$r_p$	Pore radius	m
$T$	Temperature	K
$M$	Molecular weight	g/mol