Potassium Sulfate Crystallization Model
Overview
Model describes K$_2$SO$_4$ crystallization using method of moments, tracking crystal size distribution moments (μ$_0$-μ$_3$) and solute concentration (c).
Core Equations
\begin{align} \frac{d\mu_0}{dt} &= B_0 \ \frac{d\mu_1}{dt} &= G_{\infty} (a\mu_0 + b\mu_1 \times 10^{-4}) \times 10^4 \ \frac{d\mu_2}{dt} &= 2G_{\infty} (a\mu_1 \times 10^{-4} + b\mu_2 \times 10^{-8}) \times 10^8 \ \frac{d\mu_3}{dt} &= 3G_{\infty} (a\mu_2 \times 10^{-8} + b\mu_3 \times 10^{-12}) \times 10^{12} \ \frac{dc}{dt} &= -0.5\rho\alpha G_{\infty} (a\mu_2 \times 10^{-8} + b\mu_3 \times 10^{-12}) \end{align}
Rate Dependencies
\begin{align} C_{eq} &= -686.2686 + 3.579165(T+273.15) - 0.00292874(T+273.15)^2 \ S &= c \times 10^3 - C_{eq} \ B_0 &= k_a \exp\left(\frac{k_b}{T+273.15}\right) (S^2)^{k_c/2} (\mu_3^2)^{k_d/2} \ G_{\infty} &= k_g \exp\left(\frac{k_1}{T+273.15}\right) (S^2)^{k_2/2} \end{align}
Key Parameters
- Nucleation: $k_a$=0.92, $k_b$=-6800, $k_c$=0.92, $k_d$=1.3
- Growth: $k_g$=48, $k_1$=-4900, $k_2$=1.9
- Size-dependent: a=0.51, b=7.3
- Physical: α=7.5, ρ=2.7 g/cm$^3$
Temperature (T) serves as control variable to achieve desired crystal size distribution.
Observation
The observation of the crystallisation environment provides information on the state variables and their associated setpoints (if they exist) at the current timestep. The observation is an array of shape (1, 7 + N_SP) where N_SP is the number of setpoints. Therefore, the observation when there a setpoint exists for $CV$ and $Ln$ is
[mu0, mu1, mu2, mu3, conc, CV, Ln, CV_SP, Ln_SP].
The observation space is defined by the following bounds corresponding to the ordered state variables:
An example, tested set of initial conditions are as follows:Action
The action space is a ContinuousBox of [[-1],[1]] which corresponds to a change in cooling temperature.
Reward
The reward is a continuous value corresponding to square error of the state and its setpoint. For multiple states, these are scaled with a factor (r_scale)and summed to give a single value.
Reference
The original model was created by de Moraes et. al. (2023).