Canonical Reactors I¶

• Zero-dimensional reactors.
  • No spatial dependence.
  • Evolve in time (or space for the PFR).
• Batch reactors
  • Constant TP
  • Constant TV
  • Constant HP
  • Constant UV
• Plug flow reactors (PFR)
• Perfectly stirred reactor (PSR), also called a continuous stirred tank reactor (CSTR)
  • These are flow reactors.

Approach¶

• Couple kinetics with conservation laws for heat and mass balances.
• These reactors provide convenient, well-characterized baselines for more complicated flow processes.
• All equations below are on a mass basis: $h$ is J/kg, etc.

Batch reactor: TP¶

• Batch reactor is a closed system $\rightarrow$ constant mass.
• Specify an initial state: $T$, $P$, $y_i$.
• Mass balance: $$\dot{m}_{accum} = \dot{m}_{in} - \dot{m}_{out} + \dot{m}_{generation}$$
• $\dot{m}_{in} = \dot{m}_{out} = 0$, so we have $$\dot{m}_{accum} = \dot{m}_{generation}$$
• Now, write in the details, where $\dot{m}_i^{\prime\prime\prime}$ is the net mass generation rate of species $i$ with units of kg/m$^3$*s, and $V$ is volume. $$\frac{dm_i}{dt} = \dot{m}^{\prime\prime\prime}_i V,$$

$$\frac{dmy_i}{dt} = \dot{m}^{\prime\prime\prime}_i V,$$

• $m=\rho V$ is constant: $$\frac{dy_i}{dt} = \frac{\dot{m}^{\prime\prime\prime}_i V}{\rho V},$$

• Also, we have $$\rho = \frac{MP}{RT},$$

• That is, we have $N_{sp}$ ODEs for each species mass fraction. All other terms are written in terms of the $y_i$ and the known $T$, $P$.
• The last blue equation indicates the functional dependence of the species rates, which are given in terms of kinetic expressions for the given combustion mechanism.
• At constant pressure, the volume will vary. The volume per unit mass is simply $1/\rho$, where $\rho$ at any time is given by the ideal gas law in terms of the composition at any time.

Batch reactor: TV¶

• If volume (or more likely, volume per unit mass, so that the 0-D system is intensive) is given instead of pressure, then the system density is constant (and is known from the initial state.)
• The previous equations hold, but $\rho$ is known.
• The ideal gas law can be used to solve for $P$, which now varies.

Batch reactor: HP¶

• For an adiabatic, constant pressure reactor, the previous equations all hold, but now temperature will vary.
• The energy conservation equation becomes

• The system enthalpy is known from the initial state.
• At any given time, the temperature may be found by solving $h=h(T,y_i)$ for $T$ given the known $h$, and the current $y_i$.
• In Cantera, if h0 is our enthalpy, this calculation is done internally, and we would simply write:

gas.HPY = h0, P, y
T = gas.T

Temperature equation¶

• Alternatively, we can derive a temperature equation as follows.

$$\frac{dh}{dt} = 0,$$

$$h = h(T, y_i),$$

$$dh = \underbrace{\frac{\partial h}{\partial T}}_{c_p}dT + \sum_i\underbrace{\frac{\partial h}{\partial y_i}}_{h_i}dy_i,$$

Now divide through by $dt$:

$$\frac{dh}{dt} = c_p\frac{dT}{dt} + \sum_ih_i\frac{dy_i}{dt} = 0,$$

Use $dy_i/dt = \dot{m}_i^{\prime\prime\prime}/\rho$, and solve for $dT/dt$:

Also, $$c_p = \sum_iy_ic_{p,i},$$

Batch reactor: UV¶

• If we have an adiabatic, constant volume batch reactor, then then energy equation is written in terms of internal energy $u$ instead of enthalpy, and $c_v$ instead of $c_p$.
• Also, $\rho$ is constant as for the TV system above.

Also, $$c_v = \sum_iy_ic_{v,i},$$

Plug flow reactors (PFR)¶

• Instead of a batch reactor evolving in time, we have a flow reactor evolving in space (flow in a tube, say).
• Radial mixing is perfect, and the "plug" of fluid moves down the tube without mixing axially.
• Hence, the plug behaves as a batch reactor, reacting in time. But the plug is at a different spatial location at each time.
• Convert time $t$ to spatial location $z$:

$$\frac{d}{dt} = \frac{d}{dz}\frac{dz}{dt} = v\frac{d}{dz}.$$

• That is, we transform from time to space using the local velocity: $v = dz/dt\rightarrow dt = dz/v$.
• Now, for a PFR, the mass flux is constant (even though the local velocity may vary):

$$\dot{m}^{\prime\prime} = \rho v\rightarrow v = \dot{m}^{\prime\prime}/\rho$$

Hence,

• So, in all the previous equations, wherever you see $d/dt$, replace it with the relation above.
• You will have to specify the mass flux.

Perfectly stirred reactor (PSR)¶

• This is a flow reactor with an inlet and an outlet.
• The outlet composition is the same as the reactor composition.
• This can be constant temperature or adiabatic.
• Characterized by the reactor size (volume $V$) and flow rate $\dot{m}$.
  • These can be combined as $\tau = \rho V/\dot{m}$, where $\tau$ is the PSR residence time.
  • The PSR has one key parameter, $\tau$, which is the mixing timescale of the reactor.
• We can write steady or unsteady versions of the reactor.

Unsteady PSR¶

• Assumptions:
  • Mass $m$ in the PSR is constant.
  • $\dot{m}_{in}=\dot{m}_{out}$.
• Species balance equation: $$\dot{m}_{accum} = \dot{m}_{in} - \dot{m}_{out} + \dot{m}_{generation},$$

$$\frac{dmy_i}{dt} = \dot{m}y_{i,in} - \dot{m}y_{i} + \dot{m}_i^{\prime\prime\prime}V,$$

$$\frac{dy_i}{dt} = \frac{\dot{m}}{m}(y_{i,in} - y_{i}) + \frac{\dot{m}_i^{\prime\prime\prime}V}{m},$$

• Energy balance equation.
  • If the reactor is isothermal, we just specify a constant temperature.
  • For an adiabatic reactor (with constant $m$ and $\dot{m}_{in}=\dot{m}_{out}$, we have

$$\frac{dh}{dt} = h_{in} - h.$$

• If the initial $h$ is equal to $h_{in}$ (so make the initial composition equal to the inlet composition for consistency), then we have $$\frac{dh}{dt} = 0,$$ $$h = h_0 = \text{constant}.$$

• Temperature can be found implicitly by solving $h=h(T,y_i)$, as noted above.

• Alternatively, we can formulate and solve a temperature equation:

Temperature equation¶

• The temperature equation parallels the treatment for a batch reactor, where we also had the $dh/dt=0$ equation:

$$\frac{dh}{dt} = c_p\frac{dT}{dt} + \sum_ih_i\frac{dy_i}{dt} = 0,$$

• Solve for $dT/dt$ and use our PSR equation above for $dy_i/dt$:

$$\frac{dT}{dt} = -\frac{1}{c_p}\sum_ih_i\frac{dy_i}{dt}.$$

Solution approach¶

• Note that the equations for an unsteady PSR are very similiar to those for a batch reactor, and we can use the same code to solve both systems.
• Often, an unsteady PSR is solved to steady state by solving the ODE system for long enough.
• Alternatively, we can directly solve the steady problem by setting the $d/dt$ terms to zero. We then have a coupled system of nonlinear algebraic equations that we can solve using, e.g., Newton's method.

PSR Note¶

• Consider again the unsteady PSR equations for species:

$$\frac{dy_i}{dt} = \frac{(y_{i,in} - y_{i})}{\tau} + \frac{\dot{m}_i^{\prime\prime\prime}}{\rho}.$$

• At steady state we have $$\frac{(y_{i} - y_{i,in})}{\tau} = \frac{\dot{m}_i^{\prime\prime\prime}}{\rho},$$

$$(\text{mixing term}) = (\text{reaction term}).$$

• That is, mixing balaces reaction.
• $\tau$ is the mixing timescale and $1/\tau$ is a mixing rate. Hence, we have (mixing rate)=(reaction rate).
• As $\tau$ decreases, the mixing rate increases, and eventually the chemical reaction rate cannot match it.
  • At that point, the PSR blows out.
  • The balance still holds, but we get the trivial solution 0 = 0.
  • We can decrease $\tau$ by increasing $\dot{m}$, or by decreasing volume $V$.

Chemical timescale¶

• Combustion chemistry is complex and there is no one reaction rate (or no one reaction timescale).
  • However, at the point just before blowout, we can take $\tau_0$ as the characteristic chemical timescale.
  • We can then compare $\tau_0$ to some other mixing timescale to define a Damkohler number: $Da = \tau/\tau_0$.

Analogy with diffusive mixing¶

• Consider the unsteady diffusion equation:

$$\frac{\partial y_i}{\partial t} = D\frac{\partial^2y_i}{\partial x^2}.$$

• If we nondimensionalize this equation, then we need a timescale $\tau$, a lengthscale $L$, and a scale $y_{i,ref}$ (which cancels).

• The nondimensional form is $$\frac{1}{\tau}\frac{\partial y_i^*}{\partial t^*} = \frac{D}{L^2}\frac{\partial^2y_i^*}{\partial x^{*2}}.$$

• Here, starred variables are nondimensional.

• If we chose appropriate reference values, then the magnitude of terms $\partial y_i^*/\partial t^*$ and $\partial^2y_i^*/\partial x^{*2}$ are both $\mathcal{O}(1)$.

• This gives $$\tau = \frac{L^2}{D}$$ as the characteristic diffusion timescale.

• Hence, we can write:

$$D\frac{\partial^2 y_i}{\partial x^2}\sim D\frac{\Delta y_i}{L^2} \sim \frac{\Delta y_i}{\tau}$$

But this is just our PSR mixing term. The point is, that the PSR mixing term can be used as an analogy to more complex diffusive mixing in terms of a mixing timescale, where mixing and reaction processes are in a kind of balance.

HP Batch Reactor code¶