cc8eee78-2f14-4d0c-ae59-bb9072ba4be7 176 6 9 decimal

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1.The Semenov Model

http://demonstrations.wolfram.com/TheSemenovModel/

Semenov developed a model that describes the thermal ignition phenomenon quantitatively

Denote the

x 1 el γ ≡

x 2 el T ≡

ε 0.15 :

where T is a dimensionless temperature, γ depends on the overall heat transfer coefficient, and ε depends on the activation energy

The dimensionles equation

f 1 el e 1 ε x 2 el * / -^x 1 el x 2 el 1 - (* - : 0 ≡

Kоординаты начальных точек

X0 1 el 0.1 :

X0 2 el e 1 ε T * / -^X0 1 el T 1 - (* - 0 ≡ T 015 solve :

X0 tmin tmax N D n f rows 1 + : f n el 0 : j 1 : j n ≤ j j 1 + : k 1 : k n ≤ k k 1 + : m j k el f j el x k el diff : for for Матрица Якоби x Jacobi m : Матрица  правых частей алгебраических уравнений b 1 - (x Jacobi 1 n 1 - n n submatrix * : Определители A x Jacobi 1 n 1 - 1 n 1 - submatrix : n 2 > Δ 1 el b A 1 n 1 - 2 n 1 - submatrix augment det - : Δ n 1 - el A 1 n 1 - 1 n 2 - submatrix b augment det - : k 2 : k n 2 - ≤ k k 1 + : Δ k el A 1 n 1 - 1 k 1 - submatrix b A 1 n 1 - k 1 + n 1 - submatrix augment det - : for Δ n el A det - : 41 line Δ 1 el b : (Δ n el A det : (& if i 1 : i n ≤ i i 1 + : D2 i el Δ i el Δ s el (2^(s 1 n sum sqrt / : for t x D1 D2 : result X0 tmin tmax N t x D1 rkfixed 1 N 2 X0 rows 1 + submatrix : 131 line :

tmin 0 :

tmax 48 - :

Δt 0.06 - :

N tmax Δt / :

N 800

B X0 tmin tmax N D :

t γ 0.021 0.0025 t * + :

t T root e 1 ε y * / -^t γ y 1 - (* - 0 ≡ y 050 solve : T t 18.8 > root 100100 stack root if : 21 line :

τ 028 range :

Dependence T of on the parameter γ The red dots show the stationary states

B t γ t T 1 el . 15 red 1 5 mat t γ 0 t γ 60 2 2 mat t γ t T 2 el . 15 red 1 5 mat t γ t T 3 el . 15 red 1 5 mat 5 1 sys

2.Continuously Stirred Tank Reactor

http://demonstrations.wolfram.com/ContinuouslyStirredTankReactorUsingArcLengthParameter/

x1-dimensionless concentration, x2-dimensionless temperature, Da-Damköhler number, γ-dimensionless activation energy, β-dimensionless heat transfer coefficient

B 10 :

Da 0.03084 :

γ 20 :

T0 5 - :

x 1 el x ≡

x 2 el T ≡

x 3 el Da ≡

Denote the

Kоординаты начальных точек

X0 1 el 0.004 :

X0 2 el 1.85 - :

X0 3 el 0.03084 :

tmin 0 :

tmax 9.8 - :

Δt 0.2 - :

N tmax Δt / :

The dimensionless equations

φ x 2 el - B x 3 el * 1 x 1 el - (* e x 2 el 1 x 2 el γ / + /^* + :

f 1 el x 1 el - x 3 el 1 x 1 el - (* e x 2 el 1 x 2 el γ / + /^* + : 0 ≡

tstart 6 :

tend 30 :

i tstart tend range β i el 0.2 0.04 i * + : f 2 el φ β i el x 2 el * - : K i el X0 tmin tmax N D : 31 line for

case insensitive ... very poor Greek ! ι tstart 2 tend * tstart - range :

Dependence T of on the parameter Da. By increasing the heat transfer coefficient the temperature drops

t tend ≥ t tstart : t t : if K t el 3 col K t el 2 col augment 2 1 sys

To replay: 1. delete 't' 2. retype 't'

replay step 1 t tend ≥ t 2 tend * t - : t # : if

replay step 2 t tend ≥ t 2 tend * t - : t t : if

tt 19 :

Export K tt el 3 col K tt el 2 col augment :

Select a Frame by 'tt' tt starts @ 6 ... ends 30

Export 0 2 1 sys