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  This problem models an extremely nonlinear chemical reaction in an open tube

  reactor with a gas flowing through it. The problem illustrates the use of

  FlexPDE to solve mixed boundary value - initial value problems and involves

  the calculation of an extremely nonlinear chemical reaction.


  While the solutions sought are the 3D steady state solutions, the problems

  are mixed boundary value / initial value problems with vastly different

  phenomena dominating in the radial and axial direction.


  The equations model a cross-section of the reactor which flows with the

  gas down the tube.  There is therefore a one to one relation between the

  time variable used in the equations and distance down the tube given by

  z = v*t.


  The chemical reaction has a reaction rate which is exponential in

  temperature, and shows an explosive reaction completion, once an

  'ignition' temperature is reached.  The problem variable 'C' represents

  the fractional conversion (with 1 representing reaction completion).

  The reaction rate 'RC' is given by


       RC(C,Temp) = (1-C)*exp[gamma*(1-1/Temp)]


  where the parameter GAMMA is related to the activation energy of the



  The gas is initially at a temperature of 1, in our normalized units, with

  convective cooling at the tube surface coupled to a cooling bath at a

  temperature of 0.92.


  The problem is cylindrically symmetric about the tube axis. Because of

  the reaction the axis of the tube will remain hotter than the periphery,

  and eventually the reaction will ignite on the tube axis, sending

  completion and temperature fronts propagating out toward the wall. For

  small GAMMA, these fronts are gentle, but for GAMMA greater than about

  twelve the fronts becomes very steep and completion is reached rapidly

  and sharply creating very rapid transition from a very high reaction rate

  reaction rate to a zero reaction rate.  The adaptive gridding and

  adaptive evolution 'time' stepping capabilities of FlexPDE come into

  play in this extreme nonlinear and process nonisotropic problem,

  allowing a wave of dense gridding in time to accompany the completion

  and temperature fronts across the tube.


  In this problem we introduce a heating strip on the two vertical

  faces of the tube, for a width of ten degrees of arc.  These strips are

  held at a temperature of 1.2, not much above the initial gas temperature.

  The initial timesteps are held small while the abrupt temperature gradient

  at the heating strips diffuses into the gas.


  As the cross-section under study moves down the reactor, the heat generated

  by the reaction combines with the heat diffusing in from the strip heater to cause

  ignition at a point on the x-axis and cause the completion front and temperature

  front to progate from this point across the cross-section.


  We model only a quarter of the tube, with mirror planes on the X- and Y-axes.

  The calculation models a cross-section of the tube, and this cross-section

  flows with the gas down the tube.


  The "cycle=10" plots allow us to see the flame-front propagating across

  the volume, which happens very quickly, and would not be seen in a

  time-interval sampling.


  While the magnitudes of the numerical values used for the various

  constants including gamma are representative of those found with real

  reactions and real open tube reactors they are not meant to represent

  a particular reaction or reactor.





'Open Tube Chemical Reactor with Strip Heater'



  painted     { make color-filled contour plots }  







 Lz = 1  



 gamma = 16  

 beta = 0.2  

 betap = 0.3  

 BI = 1  

 T0 = 1  

 TW = 0.92  

{ the very nasty reaction rate: }

 RC = (1-C)*exp(gamma-gamma/Temp)  

 xev=0.96     { some plot points }  



initial values  





 Temp:     div(grad(Temp)) + heat + betap*RC = dt(Temp)  

 C:        div(grad(C)) + beta*RC = dt(C)  



region 1

  start (0,0)


  { a mirror plane on X-axis }

  natural(Temp) = 0

  natural(C) = 0

  line to (r1,0)


  { "Strip Heater" at fixed temperature }

  { ramp the boundary temp in time, because  discontinuity is costly to diffuse }

  value(Temp)=T0 + 0.2*uramp(t,t-0.05)


  natural(C)=0               { no mass flow on strip heater }

  arc(center=0,0) angle 5


  { convective cooling and no mass flow on outer arc }



  arc(center=0,0) angle 85


  { a mirror plane on Y-axis }

  natural(Temp) = 0

  natural(C) = 0

  line to (0,0) to close


time 0 to 1



for cycle=10                 { watch the fast events by cycle }


  contour(Temp) fixed range (0.9,2.5)

  contour(C) as "Completion" fixed range(0,1.1)


for t= 0.1 by 0.05 to 0.2 by 0.01 to 0.3 0.5 endtime   { show some surfaces during burn }

  contour(Temp) fixed range (0.9,2.5)

  surface(Temp) fixed range (0.9,2.5)

  contour(C) as "Completion" fixed range(0,1.1)

  surface(C) as "Completion" fixed range(0,1.1)



history(Temp) at (0,0) (xev/2,yev/2) (xev,yev) (yev/2,xev/2) (yev,xev)  

history(C) at (0,0)  (xev/2,yev/2) (xev,yev) (yev/2,xev/2) (yev,xev) as "Completion"