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{ CHEMBURN.PDE

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

reaction.

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.

}

title

'Open Tube Chemical Reactor with Strip Heater'

select

painted { make color-filled contour plots }

variables Temp(threshold=0.1) C(threshold=0.1)
definitions Lz = 1 r1=1 heat=0 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 } yev=0.25
initial values Temp=T0 C=0 |

equations

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

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

boundaries

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 }

natural(Temp)=BI*(TW-Temp)

natural(C)=0

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

plots

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

grid(x,y)

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)

histories

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"

end