barcode reader integration with asp.net FIGURE 31-7 Equilibrium positions for pendulum. in Software

Drawer Code128 in Software FIGURE 31-7 Equilibrium positions for pendulum.

FIGURE 31-7 Equilibrium positions for pendulum.
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OF NONLINEAR SYSTEMS
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A CHEMICAL REACTOR
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Consider the stirred-tank chemical reactor* of Fig. 3 1.8. The contents of the reactor are assumed to be perfectly mixed, and the reaction taking place is A+B (31.12) which occurs at a rate RA = kCAe-E RT (31.13) where RA = moles of A decomposing per hour per cubic foot of reacting mixture k = reaction velocity constant, hr- CA = concentration of A in reacting mixture, moles/ft3 E = activation energy, a constant, Btu/mole R = universal gas law constant T = absolute temperature of reacting mixture The reaction is exothermic; AH Btu of heat are generated for each mole of A that reacts. Hence, in order to control the reactor, cooling water is supplied to a cooling coil. The actual reactor temperature is compared with a set point, and the rate of cooling-water flow adjusted accordingly. To indicate this control mathematically, we write that Q(T) Btu/hr of heat are removed through the cooling coil. In Chap. 32 we shall make a more detailed analysis of the dynamic behavior of the reactor. For the present preliminary analysis, it is not necessary to look carefully at Q(T), and hence it is merely assumed that, as T rises, more heat is removed in the coil. Let xb = mole fraction of A in feed stream xBo = mole fraction of B in feed stream
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*This example is based on the work of R. Aris and N. R. Amundson (1958).
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Feed: reactant
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Reaction
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FIGURE 31-8 Schematic of exothermic chemical reactor.
NONLINEAR
CONTROL
Then (1 - ~4 - x~,) is the fraction of inerts in the feed stream. A mass balance on A, (A in feed) - (A in product) - (A reacting) = (A accumulating in reactor) takes the form
FpxA, - F~XA - kpVe -EIRTXA = pv$h
where F = feed rate, ft3/hr XA = mole fraction of A in reactor p = density of reacting mixture, moles/ft3 V = volume of reacting mixture, ft3
(31.14)
To arrive at Eq. (31.14) we have used Eq. (31.13) and made the following assumptions: 1. The density of the reacting mixture is constant, unaffected by the conversion of A to B. 2. The feed and product rates F are equal and constant. 3. Together, 1 and 2 imply that V, the volume of reacting mixture, is constant. 4. Perfect mixing occurs, so that XA is the same in the reactor and product stream. A similar mass balance may be derived for substance B. However, Eq. (31.12) shows that one mole of B appears for every mole of A destroyed. Hence XB - XB,, = XA,, - XA (31.15) Equation (3 1.15) permits us to circumvent the mass balance for x B, since knowing XA we can calculate xs directly. The energy balance on the reactor (Sensible heat in feed) - (sensible heat in product) + (heat generated by reaction) - (heat removed in cooling coil) = (energy accumulating in reactor) can be written as FpC,(To - T) + kpV(AH)erEfRTxA - Q(T) = PVC,: where TO = temperature of feed stream T = temperature in reactor C, = specific heat of reacting mixture In writing Eq. (31.16), it is assumed that 1. The specific heat of the reacting mixture is constant, unaffected by the conversion of A to B. (31.16)
EXAMPLES
OF NONLINEAR SYSTEMS
2. he perfect mixing means that the temperatures of the reacting mixture and product stream are the same. 3. The heat of reaction AH is constant, independent of temperature and composition. We remark here that these assumptions, as well as those made in Eq. (31.14), may be relaxed without affecting the conceptual aspects of the phase analysis. They are made only to keep the example as uncluttered as possible, without being trivial. Equations (3 1.14) and (3 1.16) may be rearranged to the system dXA xh - XA) - ke-E RTXA dt= dT ! (To _ T) + Fe-EfRTxA - f$ dt= P P
(31.17)
As a typical application of this system of equations, we might consider starting up the reactor, initially filled with a mixture at composition x~(0) and temperature T(O). Suppose the feed rate, feed composition, feed temperature, and flow rate of cooling water are held constant and the reactor is operated in this manner until steady state is reached. To describe the transient behavior of the chemical reactor, one can solve Eqs . (3 1.17) by integrating them numerically, using a typical stepwise procedure such as the Euler or Runge-Kutta method. This will result in functions XA(t) and T(t) for values of t from zero to some value (if one exists) at which, for practical purposes, XA(t) and T(t) cease to change with t. Alternatively, we may consider a phase-plane analysis of Eqs. (31.17) and seek solutions in the form of XA versus T curves. Note that division of the first of Eqs. (3 I. 17) by the second gives
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