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# Lumped rate model without pores (LRM)¶

The lumped rate model without pores [3, 4] deviates from the lumped rate model with pores (see Section Lumped rate model with pores (LRMP)) by neglecting pores completely. The particle phase $$c^p$$ is removed and the porosity $$\varepsilon_t$$ is taken as total porosity

(10)\begin{aligned} \varepsilon_t = \varepsilon_c + \left( 1 - \varepsilon_c \right) \varepsilon_p. \end{aligned}

The phase ratio is denoted by $$\beta_t = \varepsilon_t / (1 - \varepsilon_t)$$ accordingly. The model equations are given by

\begin{aligned} \frac{\partial c^l_i}{\partial t} + \frac{1}{\beta_t} \frac{\partial}{\partial t} \sum_{m_i} c^s_{i,m_i} &= -u \frac{\partial c^l_i}{\partial z} + D_{\text{ax},i} \frac{\partial^2 c^l_i}{\partial z^2} + f_{\text{react},i}^l\left( c^l, c^s \right) + \frac{1}{\beta_t} f_{\text{react},i}^s\left( c^l, c^s \right), \end{aligned}

where $$\beta_t = \varepsilon_t / (1 - \varepsilon_t)$$ denotes the (total) phase ratio. The equations are complemented by Danckwerts boundary conditions [8]

\begin{split}\begin{aligned} u c_{\text{in},i}(t) &= u c^l_i(t,0) - D_{\text{ax},i} \frac{\partial c^l_i}{\partial z}(t, 0) & \forall t > 0,\\ \frac{\partial c^l_i}{\partial z}(t, L) &= 0 & \forall t > 0. \end{aligned}\end{split}

Both quasi-stationary and dynamic binding models are supported:

\begin{split}\begin{aligned} \text{quasi-stationary: }& & 0 &= f_{\text{ads}}\left( c^l, c^s\right), \\ \text{dynamic: }& & \frac{\partial q}{\partial t} &= f_{\text{ads}}\left( c^l, c^s\right) + f_{\text{react}}^s\left( c^l, c^s \right). \end{aligned}\end{split}

By default, the following initial conditions are applied for all $$z \in [0,L]$$:

\begin{aligned} c^l_i(0, z) &= 0, & c^s_{i,m_i}(0,z) &= 0. \end{aligned}

Note that by setting $$\varepsilon_t = 1$$, removing all bound states by setting $$N_{\text{bnd},i} = 0$$ for all components $$i$$, and applying no binding model, a dispersive plug flow reactor (DPFR) is obtained. For the specification of flow rate and direction, the same holds as for the general rate model (see Section Specification of flow rate / velocity and direction).

For information on model parameters see Lumped Rate Model Without Pores.