The unsteady-state mole balance is given by Equation (4-46)

where we use shorthand notation:

A typical plot of C_A vs. T for steady-state operation (i.e., M_B = 0) is shown in Figure R9.4-1. For example, if the reaction is first-order and irreversible, the equation for the "M_B = 0 curve" would be

This curve (i.e., M_B = 0) divides the phase plane into two regions:

where concentration increases with time, and
where concentration decreases with time.

We now consider the relative magnitude of two positive terms, When is positive and concentration increases with time. When(i.e. ), the derivative is negative and the concentration decreases with time.

To continue our analysis of the approach to steady state we need to consider the energy balance next. For operation of a CSTR with, the unsteady energy balance, Equation (9-11), can be rearranged in the form

Unsteady
mole
balance

where

Figure R9.4-2 shows a typical plot of C_A vs.T for E_B = 0. For an irreversible first-order reaction, the equation for the "E_B = 0 curve" would be

This curve divides the phase plane into two regions:

1. where temperature increases with time, and
2. where temperature decreases with time.

Figure R9.4-2
Regions of the Phase Plane

Temperature can either decrease or increase with time depending on the relative magnitudes of and.

Next we combine Figures R9.4-1 and R9.4-2 to yield Figure R9.4-3, which shows the curve for M_B = 0 and E_B = 0 for a particular set of parameter values (e.g.,).

(Note that other parameter values will give a different set of intersections of M_B = 0 and E_B = 0). The steady-state value is, of course, the intersection of the M_Band E_Bcurves. We can divide the phase plane into four quadrants:

The dashed square around the steady state in Figure R9.4-3 is expanded and shown in Figure R9.4-4 identifying the various quadrants. Temperature is increasing in quadrants I and II and decreasing in quadrants III and IV, while the concentration of species A is increasing in quadrants I and IV and decreasing in II and III. With this information we may make qualitative sketches of the temperature-concentration pathways or trajectories.

Figure R9.4-4 shows a sketch of two different approaches to the steady state that result from starting the reactor up at the same initial temperature, T_i but at two different initial concentrations. Starting at T_i and C_A = 0 in quadrant IV, we know that C_A, will increase and T will decrease until we cross the solid line of the E_B = 0 curve into quadrant I, where both C_A and T increase until we cross the M_B = 0 curve into quadrant II, where C_A decreases and T increases. The C_A vs.T phase-plot trajectory continues its spiral approach to the steady state(C_As,T_s) as it crosses into quadrant III, where both C_A and T decreases and then it reenters quadrant IV. This trajectory is shown in Figure R9.4-5. A similar type of reasoning can be used to trace out the trajectory starting at T_i and C_Ai.