Unit Operations

A chromatography simulation is built from interconnected unit operations. Each unit type contributes its own governing equations to the system. Units are connected via a directed graph, with flow routed from upstream to downstream.

Source (Inlet)

Source (or inlet) unit operations can be thought of as buffer flasks that feed the system with feed and buffer solutions. They are a parametric boundary condition that provides time-dependent concentration and flow to downstream units. Sources have no internal states.

Outlet profiles:

c_{\text{out}}(t) = c_0 + c_1 \cdot t

Q(t) = f_0 + f_1 \cdot t

where $c_0, c_1$ are the concentration intercept and slope, and $f_0, f_1$ are the flow rate intercept and slope. Parameters are re-specified per phase, allowing piecewise-linear inlet profiles.

Continuous Stirred Tank Reactor (CSTR)

Continuous stirred tank reactors are typically used to model mixers, pumps, large column adaptors, and other chambers with relatively large volumes. A CSTR is modeled as a perfectly mixed reactor tank with volume $V$ . The concentration of component $i$ in the tank evolves according to:

\frac{dc_i}{dt} = \frac{Q}{V} \left( c_{i,\text{in}} - c_i \right)

where $Q$ is the volumetric flow rate through the tank, and $c_{i,\text{in}}$ is the inlet concentration of component $i$ .

Plug Flow Reactor (PFR) / Delay

Plug flow reactors are typically used to model tubings. PFRs are 1D axial transport units with optional axial dispersion. The Delay unit is a special case with zero dispersion.

\frac{\partial c}{\partial t} + u \frac{\partial c}{\partial z} = D \frac{\partial^2 c}{\partial z^2}

where $u = Q / A$ is the interstitial velocity and $A = \pi (d/2)^2$ is the cross-sectional area.

The plug flow reactor uses a finite volume scheme similar to the column models but simplified:

Velocity: $u = Q / A$ (no porosity factor)
Optional axial dispersion via central difference

Column Models

Columns are the core separation units where solutes interact with stationary phase particles. There are different types of column models that trade off complexity and accuracy, but all are based on the same general convection-dispersion-binding structure.

\frac{\partial c}{\partial t} = \underbrace{-u \frac{\partial c}{\partial z}}_{\text{convection}} + \underbrace{D \frac{\partial^2 c}{\partial z^2}}_{\text{dispersion}} - \underbrace{\beta \frac{dq}{dt}}_{\text{binding}}

where $\beta$ is the phase ratio converting between solid-phase and liquid-phase concentration bases (model-dependent, e.g. $(1 - \varepsilon_t) / \varepsilon_t$ for the LRM).

The different column models differ in how they describe mass transfer within and around particles. For example, some models include a pore diffusion term, whilst others assume instantaneous equilibrium between the mobile and stationary phases. You can learn more about the different column model types in the Column Models page.

Column models are accompanied with binding models, which describe the adsorption and desorption kinetics of molecules onto the adsorber. See the Binding Models page for more information.

Detectors

Detectors are purely algebraic pass-through units that propagate inlet concentrations and compute signals during post-processing.

Detector	Signal
UV	Beer-Lambert: $A_i = \varepsilon_i \cdot l \cdot c_i$
Conductivity	Based on ion properties, charge, and temperature
pH	Based on hydrogen concentration

Unit Connections and Flow Routing

Units are connected via a directed graph. The solver topologically sorts units so that upstream units are evaluated before downstream ones.

Flow splitting: When a unit feeds multiple downstream units, flow is split equally.
Flow mixing: When a unit has multiple upstream connections, concentrations are mixed by flow-weighted average:

c_{\text{in}} = \frac{\sum_k c_k \cdot Q_k}{\sum_k Q_k}

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