- Model for the Loophole
- Parameters
- Overview
- Filter speed control
- Absorption of SAH
- Evaluation

Parameters

Parameter | Definition | Value |

$n_m$ | Amount of 1,3,7-trimethyluric acid | Variable |

$n_t$ | Amount of theacrine | Variable |

$n_s$ | Amount of SAH | Variable |

$v_w$ | Volume of water | Variable |

$p_0$ | Air pressure in container | Variable |

$p_1$ | Controlled pressure above the tunnels | Variable |

$Q$ | Flow rate of the loophole complex | Variable |

$v$ | Reaction speed | Variable |

$c_\text{in}$ | Original caffeine concentration | Variable |

$c_\text{out}$ | Target caffeine concentration | Variable |

$n$ | Amount of tunnels | Variable |

$r_t$ | Radius of tunnels | Variable |

$T_o$ | Operating temperature | $343$K |

$\eta_0$ | Adhesion coefficient of coffee | $0.407\times10^{-3}$Pa$\cdot$s |

$\rho_0$ | Density of coffe | $0.978\times10^3$kg/m^{3} |

$g$ | Gravitational acceleration | $9.80$m/s^{2} (in Beijing) |

$\sigma$ | Equivalent concentration of Cdh | $3$mM |

$c_0$ | Equivalent concentration of CkTcS | $0.05$mM |

$N_\text{med}$ | Total amount of adsorption medium^{(*)} |
$1.50$mmol |

$l_t$ | Length of tunnels | $0.1$m |

(*) We choose Tulsion® ADS-600 as the adsorption medium now.

Overview

Our hardware works as follows:

*Loophole. Step 1*Coffee was firstly stored at the loophole, which can be considered as the combination of a cylinder and a truncated cone. A temperature is used to control the switch. Initially the switch is turned**OFF**, and when the temperature is below our operating temperature To, the switch will be turned**ON**.*Loophole. Step 2 Coffee*Coffee solution passes through the filter. Meanwhile,*Reaction. Step 1*operates. The flow rate of coffee can be controlled by the vertical movement of its top.*Loophole. Step 3*Another filter screen was set on the cup, where*Reaction. Step 2*occurs and SAH will be adsorbed by some median

In this model we neglect the factor of temperature because our hardware is thermostatic

Filter speed control

Inspired by intravenous infusion devices, we construct a speed control system based on pressure. We assume that the liquid is adhesive and the coefficient is $\eta_0$. We assume that the density of water, gravitational acceleration, and air pressure are $\rho_0$, $g$ and $p_0$, respectively, then if the pressure above the water in tunnels are controlled to $p_1$, from Poiseuille's Law we have \begin{equation} Q=\dfrac{n\pi r_t^4(p_1-p_0+\rho_0gl_t)}{8\eta_0l_t} \end{equation} If we consider the tunnel as a system with input $c_{in}Q$ and output $c_{out}Q$. If we assume the diffusion rate is much faster than reaction rate as in part 1, then we can assume the concentration in the tunnel is $c_{out}$ everywhere. In addition, as $r_t$ is very small in our device, although the enzymes are only distributed on the boundary of the tunnels, we can still assume that they are distributed evenly inside the tunnels. Therefore, we can apply Michaelis-Menten kinetics to the reaction, and assume that the enzymes are distributed in the whole space instead of on the surface only, that is, \begin{equation} v=\dfrac{n\pi r_t^2l_t\sigma K_{cat,1}c_{out}}{K_{m,1}+c_{out}} \end{equation} Therefore, when the system is stable, we have $c_{in}Q=v+c_{out}Q$, so if we set a particular $c_{out}$ target, then we have \begin{equation} p_0-p_1=\rho_0gl_t-\dfrac{8\eta_0\sigma l_t^2K_{cat,1}c_{out}}{r_t^2(K_{m,1}+c_{out})(c_{in}-c_{out})} \end{equation} Our hardware can control $p_0-p_1$ using a feedback control system.

Absorption of SAH

For *Loophole. Step 3*, the solution flows down to the cup for \texttt{Reaction.Step 2}. For the
adsorption, we consider that it is a two-order reaction.
\begin{align}
\dfrac{\mathrm{d}n_m}{\mathrm{d}t}&=(c_{in}-c_{out})Q-\dfrac{K_{cat,2}c_{2}n_m}{K_{m,2}+n_m/v_w}\\
\dfrac{\mathrm{d}n_t}{\mathrm{d}t}&=\dfrac{K_{cat,2}c_{2}n_m}{K_{m,2}+n_m/v_w}\\
\dfrac{\mathrm{d}n_s}{\mathrm{d}t}&=\dfrac{K_{cat,2}c_{2}n_m}{K_{m,2}+n_m/v_w}-k_{ad}n_s(N_{med}-n_t+n_s)/v_w\\
\dfrac{\mathrm{d}v_w}{\mathrm{d}t}&=Q
\end{align}

Evaluation

To evaluate our design, we still take a common example, where our goal is to eliminate all the caffeine in a cup of Starbucks Grande Caffe Americano, that is, $c_{in}=2.45$mM. We set our goal that $c_{out}=0.100$mM. Note that $c_{out}$ should never set to zero even if the user wants to degrade all of the caffeine, otherwise, from Eq.(2) we can find that the reaction speed drops to zero, which is far from real situations. Therefore, we should set $c_{out}$ to a small value so that the mistake will not be amplified much. As the IC50 value of SAH is 0.900$\mu$M, therefore, we can set our goal that the amount of SAH should not exceed 0.100$\mu$mol. In addition, we might want to drink coffee in half an hour, so the operating time of adsorption should not exceed 30.0min. If we control the flow rate $Q=0.5$mL$\cdot$s$^{-1}$, then the concentration of 1,3,7-trimethyluric acid and SAH is shown below.