Introduction
The most of import unit operation in a chemical procedure is by and large a chemical reactor. Chemical reactions are either exothermal ( let go of energy ) or endothermal ( necessitate energy input ) and hence require that energy either be removed or added to the reactor for a changeless temperature to be maintained. Exothermic reactions are the most interesting systems to analyze because of possible safety jobs ( rapid additions in temperature, sometimes called “ ignition ” behaviour ) and the possibility of alien behaviour such as multiple steady-states ( for the same value of the input variable there may be several possible values of the end product variable ) .
In this faculty we consider a absolutely assorted, continuously stirred armored combat vehicle reactor ( CSTR ) , shown in Figure 1. The instance of a individual, first-order exothermal irreversible reaction, A — & gt ; B. We will demo that really interesting behaviour that can originate in such a simple system.
In Figure 1 we see that a fluid watercourse is continuously fed to the reactor and another fluid watercourse is continuously removed from the reactor. Since the reactor is absolutely assorted, the issue watercourse has the same concentration and temperature as the reactor fluid. Notice that a jacket environing the reactor besides has feed and exit watercourses. The jacket is assumed to be absolutely assorted and at a lower temperature than the reactor. Energy so passes through the reactor walls into the jacket, removed the heat generated by reaction.
There are many illustrations of reactors in industry similar to this 1. Examples include assorted types of polymerisation reactors, which produce polymers that are used in fictile merchandises such as polystyrene ice chests or plastic bottles. The industrial reactors typically have more complicated dynamicss than we study in this faculty, but the characteristic behaviour is similar.
The Modeling Equations
For simpleness we assume that the chilling jacket temperature can be straight manipulated, so that an energy balance around the jacket is non required. We besides make the undermentioned premises
- Perfect commixture ( merchandise watercourse values are the same as the majority reactor fluid )
- Changeless volume
- Changeless parametric quantity values
The changeless volume and parametric quantity value premises can easy be relaxed by the reader, for farther survey.
Parameters and Variables
A Area for heat exchange
CA Concentration of A in reactor
CAf Concentration of A in provender watercourse
cp Heat capacity ( energy/mass*temperature )
F Volumetric flowrate ( volume/time )
k0 Pre-exponential factor ( time-1 )
R Ideal gas invariable ( energy/mol*temperature )
R Rate of reaction per unit volume ( mol/volume*time )
T Time
T Reactor temperature
Tf Feed temperature
Tj Jacket temperature
Tref Reference temperature
U Overall heat transportation coefficient ( energy/ ( time*area*temperature ) )
V Reactor volume
DE Activation energy ( energy/mol )
( -DH ) Heat of reaction ( energy/mol )
R Density ( mass/volume )
The parametric quantities and variables that will look in the mold equations are listed
Overall stuff balance
The rate of accretion of stuff in the reactor is equal to the rate of stuff in by flow-the stuff out by flow.
Balance on Component A
The balance on constituent Angstrom is
where R is the rate of reaction per unit volume.
Energy Balance
The energy balance is
where Tref represents an arbitrary mention temperature for heat content.
State Variable signifier of Dynamic Equations
We can compose ( 1 ) and ( 2 ) in the undermentioned province variable signifier ( since dV/dt = 0 )
where we have assumed that the volume is changeless. The reaction rate per unit volume ( Arrhenius look ) is
where we have assumed that the reaction is first-order.
Steady-State Solution
The steady-state solution is obtained when dCA/dt = 0 and dT/dt = 0, that is
To work out these two equations, all parametric quantities and variables except for two ( CA and T ) must be specified. Given numerical values for all of the parametric quantities and variables we can utilize Newton ‘s method ( chapter 3 ) to work out for the steady-state values of CA and T. For convenience, we use an & A ; euml ; s & A ; iacute ; inferior to denote a steady-state value ( so we solve for CAs and Ts ) .
Dynamic Behavior
We noted in the old subdivision that were three different steady-state solutions to the instance 2 parametric quantity set. Here we wish to analyze the dynamic behaviour under this same parametric quantity set. Remember that numerical integrating techniques were presented in chapter 4.
The m-file to incorporate the mold equations iscstr_dyn.m, shown in Appendix 2. The bid to incorporate the equations is
[ T, x ] = ode45 ( ‘cstr_dyn ‘ , t0, tf, x0 ) ;
wheret0is the initial clip ( normally 0 ) , tfis the concluding clip, x0is the initial status vector.tis the clip vector andxis the province variable solution vector. Before executing the integrating it is necessary to specify the planetary parametric quantity vectorCSTR_PAR. To plot lone concentration or temperature as a map of clip, useplot ( T, x ( : ,1 ) ) andplot ( T, x ( : ,2 ) ) , severally.
Initial status 1
Here we use initial conditions that are close to the low temperature steady-state. The initial status vector is [ conc, temp ] = [ 9,300 ] . The curves plotted in Figure 2 show that the province variables converge to the low temperature steady-state.
Initial status 2
Here we use initial conditions that are close to the intermediate temperature steady-state. The initial status vector for the solid curve in Figure 3 is [ conc, temp ] = [ 5,350 ] , which converges to the high temperature steady-state. The initial status vector for the flecked curve in Figure 3 is [ conc, temp ] = [ 5,325 ] , which converges to the low temperature steady-state.
If we perform many simulations with initial conditions near to the intermediate temperature steady-state, we find that the temperature ever converges to either the low temperature or high temperature steady-states, but non the intermediate temperature steady-state. This indicates to us that the intermediate temperature steady-state isunstable. This will be shown clearly by the stableness analysis in subdivision 5.
Initial status 3
Here we use initial conditions that are close to the high temperature steady-state. The initial status vector is [ conc, temp ] = [ 1,400 ] . The curves plotted in Figure 4 show that the province variables converge to the high temperature steady-state.
In this subdivision we have performed several simulations and presented several secret plans. In subdivision 6 we will demo how these solutions can be compared on the same & A ; igrave ; stage plane & A ; icirc ; secret plan.
Linearization of Dynamic Equations
The stableness of the nonlinear equations can be determined by happening the undermentioned state-space signifier and finding the characteristic root of a square matrixs of Thea ( state-space ) matrix.
The nonlinear dynamic province equations ( 1a ) and ( 2a ) are
allow the province, and input variables be defined in divergence variable signifier
Stability Analysis
Performing the linearization, we obtain the undermentioned elements forA
where we define the undermentioned parametric quantities for more compact representation
From the analysis presented above, the state-space A matrix is
The stableness features are determined by the characteristic root of a square matrixs ofA, which are obtained by work outing det ( lI-A ) = 0.
det ( lI-A ) = ( l-A11 ) ( l-A22 ) -A12A21
=l2- ( A11+A22 ) l+A11A22-A12A21
=l2- ( trA ) l+det ( A )
the Eigen values are the solution to the second-order multinomial
l2- ( trA ) l+det ( A ) =0 ( 13 )
The stableness of a peculiar operating point is determined by happening theAmatrix for that peculiar operating point, and happening the Eigen values of the A matrix.
Here we show the Eigen values for each of the three instance 2 steady-state operating points.
Input / Output Transfer Function Analysis
The input-output transportation maps can be found from
G ( s ) =C ( sI-A ) -1B ( 14 )
where the elements of theBmatrix matching to the first input ( u1 = Tj-Tjs ) are
the reader should happen the elements of the B matrix that correspond to the 2nd and 3rd input variables ( see exercise 8 )
Here we show merely the transportation maps for the low temperature steady-state for instance 2. The input/output transportation map associating jacket temperature to reactor concentration ( province 1 ) is
and the input/output transportation map associating jacket temperature to reactor temperature ( province 2 ) is
Notice that the transportation map for concentration is a pure second-order system ( no numerator multinomial ) while the transportation map for temperature has a first-order numerator and second-order denominator. This indicates that there is a greater & A ; igrave ; slowdown & A ; icirc ; between jacket temperature and concentration than between jacket temperature and reactor temperature. This makes physical sense, because a alteration in jacket temperature must first impact the reactor temperature before impacting the reactor concentration.
Phase-plane Analysis
In subdivision 4 we provided the consequences of a few dynamic simulations, observing that different initial conditions caused the system to meet to different steady-state operating points. In this subdivision we construct a phase-plane secret plan by executing simulations for a big figure of initial conditions.
The phase-plane secret plan shown in Figure 6 was generated usingcstr_run.mandcstr.mfrom the appendix. Three steady-state values are clearly shown ; 2 are stable ( the high and low temperature steady-states, shown as & A ; euml ; o & A ; iacute ; ) , while one is unstable ( the intermediate temperature steady-state, shown as & A ; euml ; + & A ; iacute ; ) . Notice that initial conditions of low concentration ( 0.5 kgmol/m3 ) and comparatively low-to-intermediate temperatures ( 300 to 365 K ) all converge to the low temperature steady-state. When the initial temperature is increased above 365 K, convergence to the high temperature steady-state is achieved.
Now, see initial conditions with a high concentration ( 9.5 kgmol/m3 ) and low temperature ( 300 to 325 K ) ; these converge to the low temperature steady-state. Once the initial temperature is increased to above 325 K, convergence to the high temperature steady-state is achieved. Besides notice that, one time the initial temperature is increased to around 340 K, a really high wave-off to above 425 K occurs, before the system settles down to the high temperature steady-state. Although non shown on this phase-plane secret plan, higher initial temperatures can hold overshoot to over 500 K before settling to the high temperature steady-state. This could do possible safety jobs if, for illustration, secondary decomposition reactions occur at high temperatures. The stage plane analysis so, is able to & A ; igrave ; point-out & A ; icirc ; job initial conditions.
Besides notice that no initial conditions have converged to the intermediate temperature steady-state, since it is unstable. The reader should execute an eigenvalue/eigenvector analysis for theAmatrix at each steady-state ( low, intermediate and high temperature ) ( see exercise 3 ) . You will happen that the low, intermediate and high temperature steady-states have stable node, saddle point ( unstable ) and stable focal point behaviour ( see chapter 13 ) , severally.
It should be noted that feedback control can be used to run at the unstable intermediate temperature steady-state. The feedback accountant would mensurate the reactor temperature and pull strings the chilling jacket temperature ( or flowrate ) to keep the intermediate temperature steady-state. Besides, a feedback accountant could be used to do certain that the big wave-off to high temperatures does non happen from certain initial conditions.
Understanding Multiple Steady-state Behavior
In old subdivisions we found that there were three steady-state solutions for instance 2 parametric quantities. The aim of this subdivision is to find how multiple steady-states might originate. Besides, we show how to bring forth steady-state input-output curves that show, for illustration, how the steady-state reactor temperature varies as a map of the steady-state jacket temperature.
Heat coevals and heat remotion curves
In subdivision 3 we used numerical methods to work out for the steady-states, by work outing 2 equations with 2 terra incognitas. In this subdivision we show that it is easy to cut down the 2 equations in 2 terra incognitas to a individual equation with one terra incognita. This will give us physical penetration about the possible occurance of multiple steady-states.
Solving for Concentration of A as a map of Temperature
The steady-state concentration solution ( dCA/dt ) = 0 ) for concentration is
We can rearrange this equation to happen the steady-state concentration for any given steady-state reactor temperature, Ts
Solving for Temperature
The steady-state temperature solution ( dT/dt = 0 ) is
The footings in ( 17 ) are related to the energy removed and generated. If we multiply ( 17 ) by VrCp we find that
Qrem=Qgen
Energy Removed by flow and heat exchange Heat Generated by reaction
Note the signifier of Qrem
Notice that this is an equation for a line, where the independent variable is reactor temperature ( Ts ) . The incline of the line
is and the intercept is. Changes in jacket or provender temperature shift the intercept, but non the incline. Changes in UA or F consequence both the incline and intercept. is and the intercept is. Changes in jacket or provender temperature shift the intercept, but non the incline. Changes in UA or F consequence both the incline and intercept.
Now, see the Q gen term
Substituting ( 16 ) into ( 20 ) , we find that
Equation ( 21 ) has a characteristic S form for Q gen as a map of reactor temperature.
From equation ( 18 ) we see that a steady-state solution exists when there is an intersection of the Q paradoxical sleep and Q gen curves.
Consequence of Design Parameters
In Figure 6 we show different possible intersections of the heat remotion and heat coevals curves. If the incline of the heat remotion curve is greater than the maximal incline of the heat coevals curve, there is merely one possible intersection ( see Figure 6a ) . As the jacket or provender temperature is changed, the heat remotion lines displacements to the left or right, so the intersection can be at a high or low temperature depending on the value of jacket or provender temperature.
Notice that every bit long as the incline of the heat remotion curve is less than the maximal incline of the heat coevals curve, there will ever be the possibility of three intersections ( see Figure 6b ) with proper accommodation of the jacket or provender temperature ( intercept ) . If the jacket or provender temperature is changed, the remotion line displacements to the right or left, where merely one intersection occurs ( either low or high temperature ) . This instance is analyzed in more item in subdivision 7.3.
Multiple Steady-State Behavior
In Figure 7 we superimpose several possible additive heat remotion curves with the S-shaped heat coevals curve. Swerve A intersects the heat coevals curve at a low temperature ; curve B intersects at a low temperature and is tangent at a high temperature ; swerve C intersects at low, intermediate and high temperatures ; swerve D is tangent to a low temperature and intersects at a high temperature ; curve E has merely a high temperature intersection. Curves A, B, C, D and E are all based on the same system parametric quantities, except that the jacket temperature increases as we move from curve A to E ( from equation ( 7 ) we see that altering the jacket temperature changes the intercept but non the incline of the heat remotion curve ) . We can utilize Figure 7 to build the steady-state input-ouput diagram shown in Figure 8, where jacket temperature is the input and reactor temperature is the end product. Note that Figure 8 exhibits hysteresis behaviour, which was foremost discussed in chapter 15.
The term hysteresis is used to bespeak that the behaviour is different depending on the way that the inputs are moved. For illustration, if we start at a low jacket temperature the reactor operates at a low temperature ( indicate 1 ) . As the jacket temperature is increased, the reactor temperature additions ( points 2 and 3 ) until the low temperature bound point ( indicate 4 ) is reached. If the jacket temperature is somewhat increased farther, the reactor temperature leaps ( ignites ) to a high temperature ( indicate 8 ) ; farther jacket temperature increases consequence in little reactor temperature additions.
Contrast the input-output behaviour discussed in the old paragraph ( get downing at a low jacket temperature ) with that of the instance of get downing at a high jacket temperture. If one starts at a high jacket temperature ( indicate 9 ) there is a individual high reactor temperature, which decreases as the jacket temperature is decreased ( points 8 and 7 ) . As we move slighly lower than the high temperature bound point ( indicate 6 ) , the reactor temperature beads ( besides known asextinction ) to a low temperature ( indicate 2 ) . Further lessenings in jacket temperature lead to little lessenings in reactor temperature.
The hysteresis behaviour discussed above is besides known asignition-extinctionbehavior, for obvious grounds. Notice that part between points 4 and 6 appears to be unstable, because the reactor does non look to run in this part ( at least in a steady-state sense ) . Physical logical thinking for stableness is discussed in the undermentioned subdivision.
Decision and future work
Finally the decision is that a little survey on the uninterrupted stirred armored combat vehicle reactor and its theoretical account equation after traveling through we come to cognize its importance in the chemical technology field and besides its significance as a chemical reactor
The hereafter work is that we have to cipher and turn out the equation of the uninterrupted stirred armored combat vehicle reactor utilizing Laplace transmutation and look into it utilizing the MATLAB he equation of the uninterrupted stirred armored combat vehicle reactor utilizing Laplace transmutation and look into it utilizing the MATLAB
