(sec:reactors)= # Reactors The governing equations of the reactor models implemented in Omnisoot are briefly presented in the following sections. The control volume encompasses both the gas mixture and soot particles, as illustrated in {numref}`fig:reactors`. The equations ensure conservation of the total mass and energy of the gas–particle system, which may also gain or lose heat through the reactor walls. ```{figure} ../../assets/images/reactors.png :width: 100% :align: center :name: fig:reactors Schematics of (a) the constant-volume reactor, (b) the constant-pressure reactor, (c) the perfectly stirred reactor, and (d) the plug-flow reactor. ``` (sec:cvr)= ## Constant-Volume Reactor (CVR) As shown in panel (a) of {numref}`fig:reactors`, the CVR represents a closed, constant-volume system in which chemical reactions convert part of the gas mixture into soot particles. The mass-balance equation is written as ```{math} :label: eqn:contconstuv \frac{\mathrm{d}m}{\mathrm{d}t} = (1-\varphi)V \sum_i \dot{s}_i W_i. ``` Here, $m$ is the mass of the gas mixture. The rate of change of $m$ is determined by the net mass transfer between the gas mixture and soot particles. Similarly, the transport equation for gaseous species $k$ is expressed as ```{math} :label: eqn:speciesconstuv \frac{\mathrm{d}Y_k}{\mathrm{d}t} = \frac{1}{\rho} \left( \dot{\omega}_k + \dot{s}_k \right)W_k - \frac{Y_k}{\rho} \sum_i \dot{s}_i W_i. ``` The transport equation for a generic soot variable, $\psi$, can be written as ```{math} :label: eqn:sootconstuv \frac{\mathrm{d}\psi}{\mathrm{d}t} = S_{\psi} - \frac{\psi}{\rho} \sum_i \dot{s}_i W_i. ``` The second term on the right-hand side of Equations {eq}`eqn:speciesconstuv` and {eq}`eqn:sootconstuv` represents the changes in $Y_k$ and $\psi$, respectively, caused by the removal or addition of gas mass through soot-related processes. Energy conservation for the gas mixture is written in terms of the rate of change of temperature. An external heat-transfer rate, $\dot{Q}$, is included to account for possible heat loss or gain through the reactor walls. ```{math} :label: eqn:energyconstuv \begin{aligned} \frac{\mathrm{d}T}{\mathrm{d}t} &= \frac{1} {\rho c_v+\rho_{\mathrm{soot}}f_v c_{\mathrm{soot}}} \left[ -\sum_k e_k \left( \dot{\omega}_k+\dot{s}_k \right)W_k \right. \\ &\qquad\left. + e_{\mathrm{soot}} \sum_k \dot{s}_k W_k + \frac{\dot{Q}}{V(1-\varphi)} \right]. \end{aligned} ``` Here, $\rho_{\mathrm{soot}}f_v c_{\mathrm{soot}}$ and $e_{\mathrm{soot}}\sum_k \dot{s}_k W_k$ account for the soot contribution to the system heat capacity and the energy associated with soot-related mass transfer, respectively. The influence of these terms on the gas and soot properties was investigated by simulating the pyrolysis of 30% $\mathrm{CH_4}$ in Ar with and without accounting for soot sensible energy. As shown in panel (a) of {numref}`fig:sseeffect`, neglecting soot sensible energy results in an overprediction of the temperature by nearly 150 K and of the mobility diameter by a factor of three over the 80 ms simulation. The overpredicted temperature alters the gas-phase chemistry, leading to a noticeable decrease in the residual methane and benzene mole fractions, as shown in panel (b) of {numref}`fig:sseeffect`. ```{subfigure} AB :layout-sm: A|B :gap: 1rem :width: 100% :align: center :name: fig:sseeffect ![Temperature and soot mobility diameter.](../../assets/images/sse_temp_dm.png) ![Methane and benzene mole fractions.](../../assets/images/sse_gasresid.png) Comparison of (a) the temperature and soot mobility diameter, $d_m$, and (b) the mole fractions of methane, $\mathrm{CH_4}$, and benzene, A1, during the simulated pyrolysis of 30% $\mathrm{CH_4}$ in Ar when soot sensible energy is included (labeled “WSSE”) or neglected (labeled “W/OSSE”). The CVR is used with the Caltech mechanism, the Reactive Dimerization inception model, and the monodisperse particle dynamics model (MPBM). ``` (sec:ipr)= ## Pressure Reactor (IPR) As shown in panel (b) of {numref}`fig:reactors`, the pressure reactor is a closed system similar to the CVR. The main difference is that its boundary can move, allowing the system volume to change in response to an externally imposed pressure that may remain constant or vary with time. In either case, the pressure is prescribed a priori. Heat transfer may also occur through the reactor walls, thereby changing the energy of the gas–particle system. The mass, species, and soot-variable equations for the IPR are the same as those for the CVR. However, the energy equation is written in terms of enthalpy, $h$, rather than internal energy, $e$: ```{math} :label: eqn:energypressure \begin{aligned} \frac{\mathrm{d}T}{\mathrm{d}t} &= \frac{1} {\rho c_p+\rho_{\mathrm{soot}}f_v c_{\mathrm{soot}}} \left[ -\sum_k h_k \left( \dot{\omega}_k+\dot{s}_k \right)W_k \right. \\ &\qquad\left. + h_{\mathrm{soot}} \sum_k \dot{s}_k W_k + \frac{\mathrm{d}P}{\mathrm{d}t} + \frac{\dot{Q}}{V(1-\varphi)} \right]. \end{aligned} ``` Here, $\mathrm{d}P/\mathrm{d}t$ represents the effect of a time-varying pressure on the gas temperature. This term is zero when the pressure remains constant throughout the process. (sec:psr)= ## Perfectly Stirred Reactor (PSR) Gas enters the reactor with an inlet mass flow rate, composition, and temperature of $\dot{m}_{\mathrm{in}}$, $Y_{\mathrm{in}}$, and $T_{\mathrm{in}}$, respectively, and reacts homogeneously with the mixture inside the reactor. The reacting mixture is assumed to have spatially uniform temperature and composition, denoted by $T$ and $Y$, respectively. The temperature, composition, and soot properties of the outlet stream are assumed to be the same as those of the mixture inside the reactor. In the absence of soot formation, the inlet and outlet mass flow rates are equal, that is, $\dot{m}_{\mathrm{in}}=\dot{m}_{\mathrm{out}}$. When soot is formed, $\dot{m}_{\mathrm{out}}$ is slightly smaller than $\dot{m}_{\mathrm{in}}$. The nominal residence time of the PSR is calculated as ```{math} :label: eqn:taupsr \tau_{\mathrm{psr}} = \frac{\rho V}{\dot{m}_{\mathrm{in}}}. ``` Conservation of mass in the PSR can be expressed by accounting for the inlet and outlet mass fluxes and the transfer of mass between the gas mixture and soot particles: ```{math} :label: eqn:contpsr \frac{\mathrm{d}m}{\mathrm{d}t} = \dot{m}_{\mathrm{in}} - \dot{m}_{\mathrm{out}} + V(1-\varphi) \sum_i \dot{s}_i W_i. ``` The gas composition is obtained by solving the species transport equations: ```{math} :label: eqn:speciespsr \frac{\mathrm{d}Y_k}{\mathrm{d}t} = \frac{\dot{m}_{\mathrm{in}}} {\rho V(1-\varphi)} \left( Y_{k,\mathrm{in}}-Y_k \right) + \frac{1}{\rho} \left[ \left( \dot{\omega}_k+\dot{s}_k \right)W_k - Y_k \sum_i \dot{s}_i W_i \right]. ``` The transport equation for a generic soot variable, $\psi$, is written as ```{math} :label: eqn:sootpsr \frac{\mathrm{d}\psi}{\mathrm{d}t} = \frac{\dot{m}_{\mathrm{in}}} {\rho V(1-\varphi)} \left( \psi_{\mathrm{in}}-\psi \right) + S_{\psi} - \frac{\psi}{\rho} \sum_i \dot{s}_i W_i. ``` The energy equation for the PSR is ```{math} :label: eqn:energypsr \begin{aligned} \frac{\mathrm{d}T}{\mathrm{d}t} &= \frac{1} {\rho c_p+\rho_{\mathrm{soot}}c_{p,\mathrm{soot}}f_v} \left[ \frac{\dot{m}_{\mathrm{in}}} {V(1-\varphi)} \left( h_{\mathrm{in}}-h \right) \right. \\ &\qquad - \frac{\dot{m}_{\mathrm{in}}} {V(1-\varphi)} \sum_k \left( Y_{k,\mathrm{in}}-Y_k \right)h_k \\ &\qquad - \sum_k \left( \dot{\omega}_k+\dot{s}_k \right)W_k h_k \\ &\qquad\left. + h_{\mathrm{soot}} \sum_i \dot{s}_i W_i + \frac{\dot{Q}}{V(1-\varphi)} \right]. \end{aligned} ``` (sec:pfr)= ## Plug-Flow Reactor (PFR) The PFR is an idealized representation of a channel or duct in which the temperature, composition, and soot properties of a steady-state, one-dimensional flow evolve along the reactor axis. The flow is assumed to be well mixed across each cross-section, so there are no radial gradients in these properties. Axial diffusion is neglected. The continuity equation for the PFR is written as ```{math} :label: eqn:contpfr \frac{\mathrm{d}\dot{m}}{\mathrm{d}z} = (1-\varphi)A \sum_i \dot{s}_i W_i. ``` The momentum equation is expressed as ```{math} :label: eqn:momenpfr u(1-f_v) \sum_i \dot{s}_i W_i + \rho u(1-\varphi) \frac{\mathrm{d}u}{\mathrm{d}z} = - \frac{\mathrm{d}}{\mathrm{d}z} \left[ p(1-\varphi) \right] - \frac{\tau_w}{R_H}. ``` Here, $\tau_w$ and $R_H$ are the wall shear stress and hydraulic radius of the reactor, respectively. The wall shear stress is determined from the friction factor, $f$, as ```{math} :label: eqn:wallshearpfr \tau_w = \frac{1}{2}\rho u^2 f. ``` The friction factor can be calculated over the full range of Reynolds numbers, from laminar to turbulent flow, using the explicit formula proposed by {cite:t}`haaland1983simple`: ```{math} :label: eqn:fpfr \frac{1}{f^{1/2}} = -1.8\log \left[ \frac{6.9}{Re} + \left( \frac{\epsilon/D_H}{3.7} \right)^{1.11} \right]. ``` Here, $\epsilon$ and $D_H$ are the wall roughness and hydraulic diameter of the reactor, respectively. The species transport equation is ```{math} :label: eqn:speciespfr \frac{\mathrm{d}Y_k}{\mathrm{d}z} = \frac{1}{\rho u} \left[ \left( \dot{\omega}_k+\dot{s}_k \right)W_k - Y_k \sum_i \dot{s}_i W_i \right]. ``` The transport equation for a generic soot variable, $\psi$, is written as ```{math} :label: eqn:sootpfr \frac{\mathrm{d}\psi}{\mathrm{d}z} = \frac{S_{\psi}}{u} - \frac{\psi}{\rho u} \sum_i \dot{s}_i W_i - \frac{4}{D_H} \frac{k_{\mathrm{dep}}^i\psi}{u}. ``` Here, $k_{\mathrm{dep}}^i$ is the deposition velocity of soot particles in section $i$, calculated as ```{math} :label: eqn:kdep k_{\mathrm{dep}}^i = \frac{Sh\,D^i}{D_H}. ``` For laminar flow, $Sh=3.66$. For turbulent flow, the Sherwood number is calculated using the Berger and Hau correlation {cite:p}`berger1977mass`: ```{math} :label: eqn:shdep Sh = 0.0165Re^{0.86}Sc^{1/3}. ``` The energy equation is expressed as ```{math} :label: eqn:energypfr \begin{aligned} \frac{\mathrm{d}T}{\mathrm{d}z} &= \frac{1} {\rho u c_p+\rho_{\mathrm{soot}}u f_v c_{p,\mathrm{soot}}} \left[ -\sum_k h_k \left( \dot{\omega}_k+\dot{s}_k \right)W_k \right. \\ &\qquad\left. + h_{\mathrm{soot}} \sum_k \dot{s}_k W_k + q^{\prime\prime}\frac{P_c}{A} \right]. \end{aligned} ``` Here, $q^{\prime\prime}$ is the reactor-wall heat flux.