4.7. Stirred Reactor Theory

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4.7. Stirred Reactor Theory

In the discussion of premixed turbulent flames, the case of infinitely fast mixing of reactants and products was introduced. Generally this concept is referred to as a stirred reactor. Many investigators have employed stirred reactor theory not only to describe turbulent flame phenomena, but also to determine overall reaction kinetic rates [23] and to understand stabilization in high-velocity streams [65]. Stirred reactor theory is also important from a practical point of view because it predicts the maximum energy release rate possible in a fixed volume at a particular pressure.

Consider a fixed volume V into which fuel and air are injected at a fixed total mass flow rate

\dot{m}

$\dot{m}$

and temperature T₀. The fuel and air react in the volume, and the injection of reactants and outflow of products (also equal to

\dot{m}

$\dot{m}$

) are so oriented that within the volume there is instantaneous mixing of the unburned gases and the reaction products (burned gases). The reactor volume attains some steady temperature T_R and pressure P. The temperature of the gases leaving the reactor is, thus, T_R as well. The pressure differential between the reactor and the exit is generally considered to be small. The mass leaving the reactor contains the same concentrations as those within the reactor and thus contains products as well as fuel and air. Within the reactor there exists a certain concentration of fuel (F) and air (A) and also a fixed unburned mass fraction, ψ. Throughout the reactor volume, T_R, P, (F), (A), and ψ are constant and fixed; that is, the reactor is so completely stirred that all elements are uniform everywhere. Figure 4.49 depicts the stirred reactor concept in a generalized manner.

The stirred reactor may be compared to a plug flow reactor in which premixed fuel–air mixtures flow through the reaction tube. In this case, the unburned gases enter at temperature T₀ and leave the reactor at the flame temperature T_f. The system is assumed to be adiabatic. Only completely burned products leave the reactor. This reactor is depicted in Figure 4.50.

The volume required to convert all the reactants to products for the plug flow reactor is greater than that for the stirred reactor. The final temperature is, of course, higher than the stirred reactor temperature.

Figure 4.49 Variables of a stirred reactor system of fixed volume.

Figure 4.50 Variables in plug flow reactor.

It is relatively straightforward to develop the controlling parameters of a stirred reactor process. If ψ is defined as the unburned mass fraction, it must follow that the fuel–air mass rate of burning

{\dot{R}}_{B}

${\dot{R}}_{B}$

${\dot{R}}_{B} = \dot{m} (1 - ψ)$ ${\dot{R}}_{B} = \dot{m} (1 - ψ)$

and the rate of heat evolution

\dot{q}

$\dot{q}$

$\dot{q} = q \dot{m} (1 - ψ)$ $\dot{q} = q \dot{m} (1 - ψ)$

where q is the heat of reaction per unit mass of reactants for the given fuel-air ratio. Assuming that the specific heat of the gases in the stirred reactor can be represented by some average quantity

{\bar{c}}_{p}

${\bar{c}}_{p}$

, one can write an energy balance as

$\dot{m} q (1 - ψ) = \dot{m} {\bar{c}}_{p} (T_{R} - T_{0})$ $\dot{m} q (1 - ψ) = \dot{m} {\bar{c}}_{p} (T_{R} - T_{0})$

For the plug flow reactor or any similar adiabatic system, it is also possible to define an average specific heat that takes its explicit definition from

${\bar{c}}_{p} \equiv q / (T_{f} - T_{0})$ ${\bar{c}}_{p} \equiv q / (T_{f} - T_{0})$

To a very good approximation these two average specific heats can be assumed equal. Thus, it follows that

$(1 - ψ) = (T_{R} - T_{0}) / (T_{f} - T_{0}), ψ = (T_{f} - T_{R}) / (T_{f} - T_{0})$ $(1 - ψ) = (T_{R} - T_{0}) / (T_{f} - T_{0}), ψ = (T_{f} - T_{R}) / (T_{f} - T_{0})$

The mass burning rate is determined from the ordinary expression for chemical kinetic rates; that is, the fuel consumption rate is given by

$d (F) / d t = - (F) (A) Z^{'} e^{- E / R T_{R}} = - {(F)}^{2} (A / F) Z^{'} e^{- E / R T_{R}}$ $d (F) / d t = - (F) (A) Z^{'} e^{- E / R T_{R}} = - {(F)}^{2} (A / F) Z^{'} e^{- E / R T_{R}}$

where (A/F) represents the air–fuel ratio. The concentration of the fuel can be written in terms of the total density and unburned mass fraction

$(F) = \frac{(F)}{(A) + (F)} ρ ψ = \frac{1}{(A / F) + 1} ρ ψ$ $(F) = \frac{(F)}{(A) + (F)} ρ ψ = \frac{1}{(A / F) + 1} ρ ψ$

which permits the rate expression to be written as

$\frac{d (F)}{d t} = - \frac{1}{{[(A / F) + 1]}^{2}} ρ^{2} ψ^{2} (\frac{A}{F}) Z^{'} e^{- E / R T_{R}}$ $\frac{d (F)}{d t} = - \frac{1}{{[(A / F) + 1]}^{2}} ρ^{2} ψ^{2} (\frac{A}{F}) Z^{'} e^{- E / R T_{R}}$

Now the great simplicity in stirred reactor theory is realizable. Since (F), (A), and T_R are constant in the reactor, the rate of conversion is constant. It is now possible to represent the mass rate of burning in terms of the preceding chemical kinetic expression:

$\dot{m} (1 - ψ) = - V \frac{(A) + (F)}{(F)} \frac{d (F)}{d t}$ $\dot{m} (1 - ψ) = - V \frac{(A) + (F)}{(F)} \frac{d (F)}{d t}$

$\dot{m} (1 - ψ) = + V [(\frac{A}{F}) + 1] \frac{1}{{[(A / F) + 1]}^{2}} (\frac{A}{F}) ρ^{2} ψ^{2} Z^{'} e^{- E / R T_{R}}$ $\dot{m} (1 - ψ) = + V [(\frac{A}{F}) + 1] \frac{1}{{[(A / F) + 1]}^{2}} (\frac{A}{F}) ρ^{2} ψ^{2} Z^{'} e^{- E / R T_{R}}$

From the equation of state, by defining

$B = \frac{Z^{'}}{[(A / F) + 1]}$ $B = \frac{Z^{'}}{[(A / F) + 1]}$

and substituting for (1 − ψ) in the last rate expression, one obtains

$(\frac{\dot{m}}{V}) = (\frac{A}{F}) {(\frac{P}{R T_{R}})}^{2} [\frac{{(T_{f} - T_{R})}^{2}}{T_{f} - T_{0}}] \frac{B e^{- E / R T_{R}}}{T_{R} - T_{0}}$ $(\frac{\dot{m}}{V}) = (\frac{A}{F}) {(\frac{P}{R T_{R}})}^{2} [\frac{{(T_{f} - T_{R})}^{2}}{T_{f} - T_{0}}] \frac{B e^{- E / R T_{R}}}{T_{R} - T_{0}}$

By dividing through by P², one observes that

$(\dot{m} / V P^{2}) = f (T_{R}) = f (A / F)$ $(\dot{m} / V P^{2}) = f (T_{R}) = f (A / F)$

$(\dot{m} / V P^{2}) (T_{R} - T_{0}) = f (T_{R}) = f (A / F)$ $(\dot{m} / V P^{2}) (T_{R} - T_{0}) = f (T_{R}) = f (A / F)$

which states that the heat release is also a function of T_R.

This derivation was made as if the overall order of the air–fuel reaction were 2. In reality, this order is found to be closer to 1.8. The development could have been carried out for arbitrary overall order n, which would give the result

$(\dot{m} / V P^{n}) (T_{R} - T_{0}) = f (T_{R}) = f (A / F)$ $(\dot{m} / V P^{n}) (T_{R} - T_{0}) = f (T_{R}) = f (A / F)$

Figure 4.51 Stirred reactor parameter $(\dot{m} / V P^{2})$ $(\dot{m} / V P^{2})$ (T_R − T₀) as a function of reactor temperature T_R.

A plot of

(\dot{m} / V P^{2}) (T_{R} - T_{0})

$(\dot{m} / V P^{2}) (T_{R} - T_{0})$

versus T_R reveals a multivalued graph that exhibits a maximum, as shown in Figure 4.51. The part of the curve in Figure 4.51 that approaches the value T_x asymptotically cannot exist physically since the mixture could not be ignited at temperatures this low. In fact, the major part of the curve, which is to the left of T_opt, has no physical meaning. At fixed volume and pressure it is not possible for both the mass flow rate and temperature of the reactor to rise. The only stable region exists between T_opt to T_f. Since it is not possible to mix some unburned gases with the product mixture and still obtain the adiabatic flame temperature, the reactor parameter must go to zero when T_R = T_f.

The value of T_R, which gives the maximum value of the heat release, is obtained by maximizing the last equation. The result is

$T_{R,opt} = \frac{T_{f}}{1 + (2 R T_{f} / E)}$ $T_{R,opt} = \frac{T_{f}}{1 + (2 R T_{f} / E)}$

For hydrocarbons, the activation energy falls within a range of 120–160 kJ/mol and the flame temperature in a range of 2000–3000 K. Thus,

$(T_{R,opt} / T_{f}) \sim 0.75$ $(T_{R,opt} / T_{f}) \sim 0.75$

Stirred reactor theory reveals a fixed maximum mass loading rate for a fixed reactor volume and pressure. Any attempts to overload the system will quench the reaction. It is also worth noting that stirred reactor analysis for both non-dilute and dilute systems gives the maximum overall energy release rate possible for a given fuel–oxidizer mixture in a fixed volume at a given pressure. Attempts have been made to determine chemical kinetic parameters from stirred reactor measurements. The usefulness of such measurements at high temperatures and for non-dilute fuel–oxidizer mixtures is limited. Such analyses are based on the assumption of complete instantaneous mixing, which is impossible to achieve experimentally. However, for dilute mixtures at low and intermediate temperatures where the Damkohler number (Da = τ_m/τ_c) is small, studies have been performed to investigate the behaviors of reaction mechanisms [66].

Using the notation of Chapter 2, Section 2.8.3, the stirred reactor species and energy equations may be written in terms of the individual species equations and the energy equation. The species equations are given by

$\frac{d m_{j}}{d t} = V {\dot{ω}}_{j} M W_{j} + {\dot{m}}_{j}^{*} - {\dot{m}}_{j} j = 1, \dots, n$ $\frac{d m_{j}}{d t} = V {\dot{ω}}_{j} M W_{j} + {\dot{m}}_{j}^{*} - {\dot{m}}_{j} j = 1, \dots, n$

(4.78)

where m_j is the mass of the jth species in the reactor (and reactor outlet), V is the volume of the reactor,

{\dot{ω}}_{j}

${\dot{ω}}_{j}$

is the chemical production rate of species j, MW_j is the molecular weight of the jth species,

{\dot{m}}_{j}^{∗}

${\dot{m}}_{j}^{∗}$

is the mass flow rate of species j in the reactor inlet, and

{\dot{m}}_{j}

${\dot{m}}_{j}$

is the mass flow rate of species j in the reactor outlet.

For a constant mass flow rate and mass in the reactor, Eqn (4.78) may be written in terms of species mass fractions, Y_j, as

$\frac{d Y_{j}}{d t} = \frac{{\dot{ω}}_{j} M W_{j}}{ρ} + \frac{\dot{m}}{ρ V} (Y_{j}^{*} - Y_{j}) j = 1, \dots n$ $\frac{d Y_{j}}{d t} = \frac{{\dot{ω}}_{j} M W_{j}}{ρ} + \frac{\dot{m}}{ρ V} (Y_{j}^{*} - Y_{j}) j = 1, \dots n$

(4.79)

At steady-state (dY_j/dt = 0), and therefore

$Y_{j} = \frac{V {\dot{ω}}_{j} M W_{j}}{\dot{m}} + Y_{j}^{*} j = 1, \dots, n$ $Y_{j} = \frac{V {\dot{ω}}_{j} M W_{j}}{\dot{m}} + Y_{j}^{*} j = 1, \dots, n$

(4.80)

The energy equation for an adiabatic system at steady-state is simply

$\dot{m} \sum_{j = 1}^{n} Y_{j}^{*} h_{j}^{*} (T^{*}) - \dot{m} \sum_{j = 1}^{n} Y_{j} h_{j} (T) = 0$ $\dot{m} \sum_{j = 1}^{n} Y_{j}^{*} h_{j}^{*} (T^{*}) - \dot{m} \sum_{j = 1}^{n} Y_{j} h_{j} (T) = 0$

where

$h_{j} (T) = h_{f, j}^{0} + \int_{T_{r e f}}^{T} c_{p, j} d T$ $h_{j} (T) = h_{f, j}^{0} + \int_{T_{r e f}}^{T} c_{p, j} d T$

Equation (4.80) is a set of nonlinear algebraic equations and may be solved using various techniques [67]. Often the nonlinear differential Eqn (4.79) are solved to the steady-state condition in place of the algebraic equations using the stiff ordinary differential equation solvers described in Chapter 2 [68]. See Appendix I for more information on available numerical codes.

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Table of Contents for 4.7. Stirred Reactor Theory

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4.7. Stirred Reactor Theory

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4.7. Stirred Reactor Theory