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Background Theory

Microkinetic modeling connects the elementary reaction kinetics to thermodynamic quantities measurable at reactor scale such as temperature, pressure, species mass, etc. Here, theoretical underpinnings of microkinetic modeling such as the governing equations of the reactor models are presented.

Reactor Models and Governing Equations

A reactor model is used to specify the conditions under which the chemical reactions take place. Mathematically, a reactor model imposes constraints to conserve fundamental physical quantities in a series of governing equations. The governing equations can be ordinary differential equations (ODE), algebraic relationships and in some cases, partial differential equations.
Different reactor models result in different governing equations. The governing equations of the reactor models implemented in OpenMKM are presented below.

Batch Reactor

A batch reactor is a fixed sized tank which is filled with the reactants and left to evolve under well mixed conditions. The reacting fluid composition, temperature, and pressure change as a function of time, but at any point of time, due to the well mixed condition, are uniform throughout the reactor.

Mass Balance

The mass of the reacting fluid in a batch reactor is fixed. As as result,

$$ \frac{dm}{dt} = \sum_k \frac{dm_k}{dt} = 0,$$

where $$m$$ and $$m_k$$ represent the total mass and individual mass of kth species respectively. Based on $$m$$ and $$m_k$$, the mass fraction of the kth species is defined as $$Y_k = m_k/m$$. The change in mass fraction of the kth species is given as

$$ \rho\frac{dY_k}{dt} = (\dot{\omega_k} + \dot{s_k} \frac{A_{cat}}{V}) W_k, $$

where $$\rho$$ is the density of the reacting fluid, $$V$$ is the volume of the reactor, $$Y_{k,0}$$ is the initial mass fraction of the kth species, $$\dot{\omega_k}$$ and $$\dot{s_k}$$ are the production rates of the kth species in the reacting fluid phase and on catalyst surface respectively, $$\frac{A_{cat}}{V}$$ is the ratio of catalyst surface area to reactor volume, W_k is the molar weight of the kth species.

Energy Balance

For modeling the kinetics of heterogeneous catalysis, the reactor is typically operated under isothermal conditions, where the temperature of the reactor is fixed. If the temperature of the reactor is allowed to change either due to heat released/absorbed due to chemical reactions or due to heat/cooling supplied from external sources, energy balance has to be satisfied.

$$\frac{dU}{dt} = \frac{dQ}{dt} + \frac{dW}{dt},$$

where $$U$$ is the internal energy of the reactor, $$Q$$ is the external heat supplied, and $$W$$ is the work done on the reactor. $$W$$ is 0 if the reactor volume is fixed, otherwise it is $$PdV$$.

The heat flux, $$\frac{dQ}{dt}$$, supplied to the reactor through an outer wall with an area, $$A_{wall}$$, and heat transfer coefficient, $$\hat{h}$$, from an external heat source at temperature, $$T_{ext}$$, is given as

$$\frac{dQ}{dt} = \hat{h}A_{wall}(T_{ext} - T), $$

where T is the reactor temperature. The internal energy of the reactor could be written as $$U = m \sum_k Y_k u_k$$, where $$u_k$$ is the specific internal energy of kth species w.r.t. unit mass given in terms of $$J/kg$$ or $$erg/g$$ for SI and CGS units respectively. Assuming fixed reactor size, a few more definitions to know are $$du_k = c_{v,k}dT$$ and $$c_v = \sum_k{Y_kc_{v,k}}$$, where $$c_v$$ is the mass specific heat at constant volume. Plugging the various definitions into the energy balance equation results in

$$\rho c_v \frac{dT}{dt} = -\sum_k{e_k W_k (\dot{\omega_k} + \dot{s_k} \frac{A_{cat}}{V})} + \hat{h}\frac{A_{wall}}{V}(T_{ext} - T).$$

CSTR

A continous stirred-tank reactor (CSTR) is a tank like batch reactor but with a continuous flow of reacting fluid into and out of the tank. The reacting fluid flows at a volumetric flow rate of $$r$$ and the reacting fluid is well mixed. This results in the reacting fluid spending an average residence time, $$\tau$$, inside the reactor, after which it get expelled from the reactor through the outlet. The volumetric flow rate, $$r$$ is also defined in terms of mass flow rate, $$\dot{m}_0$$, which is given as $$\dot{m}_0 = \rho_0 r$$. Similarly, flow rate, $$r$$ and residence time, $$\tau$$ are related as $$\tau = V/r$$. At the beginning, the reacting fluid contains only reactants, but as time progresses, it contains both reactants, products, and reaction intermediates. The reactor is typically operated at steady state conditions, where the composition of the reacting fluid inside the CSTR, which is typically different from the composition of the initial feed, does not change.

Mass Balance

$$ \rho\frac{dY_k}{dt} = \frac{\dot{m}0}{V} (Y{k,0}- Y_k) + (\dot{\omega_k} + \dot{s_k} \frac{A_{cat}}{V}) W_k. $$

Energy Balance

$$\rho c_p \frac{dT}{dt} = \frac{\dot{m}0}{V} \sum_k{Y{k,0}(h_k,0- h_k)} - \sum_k{h_k W_k (\dot{\omega_k} + \dot{s_k} \frac{A_{cat}}{V})} + \hat{h}\frac{A_{wall}}{V}(T_{ext} - T),$$

where $$c_p$$ is the specific heat for unit mass at constant pressure, $$W_k h_k$$ is the molar specific enthalpy of kth species, and $$h_k$$ is the specific enthalpy of kth species at unit mass. $$Y_{k,0}$$ and $$h_{k,0}$$ represent the initial mass fractions and initial specific enthalpies (w.r.t. unit mass) of kth species respectively.

PFR

A plug-flow reactor (PFR) is a continuous flow reactor with cross sectional area $$A_c$$. Reacting fluid enters from one side, flows along the axial direction, and exits from the other side. The state of the reacting fluid is uniform along the radial direction, but varies along the axial direction. Here the equations presented assume a fixed cross-sectional area for the PFR. Assuming steady state operating conditions, the governing equations are formulated for a small differential volume, $$dV$$, given as $$dV = A_c * dz,$$ where $$dz$$ is differential length along the axial direction. The axial flow rate of the fluid is represented with $$v$$ m/s. The volumetric flow rate, $$r$$, defined earlier is then given as $$r = A_c v$$.

Mass balance

$$\frac{d(\rho v)}{d z} = \sum_k{(\dot{w_k}+\dot{s_k}\frac{A_{cat}}{V})W_k}$$

The equation can be further simplified because there can be no net source/sink of mass in the reacting fluid phase, which implies $$\sum_k{\dot{w_k}W_k} = 0 $$. This results in

$$\frac{d(\rho v)}{d z} = \frac{A_{cat}}{V}\sum_k{\dot{s_k}W_k}.$$

At steady state, the amount of mass adsorbed onto a catalyst surface has to be equal to the amount of desorbed from the catalyst surface. This results in a further simplification of

$$\frac{d(\rho v)}{d z} = 0.$$

$$\rho\frac{dv}{d z} + v\frac{d\rho}{d z} = 0.$$

Individual Species Mass Balance

$$ \rho v \frac{dY_k}{dz} + \frac{A_{cat}}{V}Y_k\sum_k{\dot{s_k} W_k} = (\dot{\omega_k} + \dot{s_k} \frac{A_{cat}}{V}) W_k. $$

Energy Balance

$$\rho v c_p \frac{dT}{dz} = - \sum_k{h_k W_k (\dot{\omega_k} + \dot{s_k} \frac{A_{cat}}{V})} + \hat{h}\frac{A_{wall}}{V}(T_{ext} - T).$$

Surface Coverages on Catalyst Surface

The afore-mentioned conservation equations consider only the species in the reacting fluid. On a catalytic surface, the site density ($$\Gamma [=] mol/cm^2$$) and catalyst loading are considered as fixed. This results in an additional constraint involving the coverages of surface species, $$\theta_k$$.

$$\sum_k^{K_S}{\theta_k} = 1.$$

While the energy and mass balance equations are differential in nature, surface coverage conservation equation is algebraic.