EXAM MIE516

1

Question 2
Approach: Integrate Equations (6.10) & (6.15) of the text book using Euler method to find Fuel concentration and reactor temperature as a function of time. Euler method is an explicit
Numerical scheme, meaning that the unknown values in each time‐step (temperature and species) are related to the known parameters in the previous time‐step. First let’s find the conservation equations. Stoichiometric methane combustion reaction is:
CH4+2(O2+3.76N2) → CO2+2H2O+7.52N2
&
& ω O 2 = 2ω F
&
ωN 2 = 0
&
& ω CO 2 = −ω F
&
& ω H 2O = −2ω F
∴

&
∑ω

=0

Re ac tan ts & Pr oducts

We will have one species conservation equation (IC.6.15) for each of CH4, O2, N2, CO2 and H2O concentrations and one equation for temperature. All of these equations use fuel consumption rate which is in the following form (from table 5.1):
⎛ mole ⎞
&
{ω F }n = { .3E8 × exp(−24358 / T ) × [CH 4]−0.3 [O 2]1.3 }n ⎜
1
⎟
⎝ cc. sec ⎠

Where “n” indicates the time‐step number. Assuming zero heat transfer to the reactor, the temperature equation is:

&
⎧ − ∑ (h f ,i + c p ,i (T − Tref ))ω i ⎫
⎪
⎪ Re ac.& Pr od .
⎧ dT ⎫
⎨ ⎬ =⎨
⎬
X
⎩ dt ⎭ n ⎪
∑Pr[od . i ]c p,i
⎪
Re ac .&
⎭n
⎩
{T }n+1 = {T }n + ⎧ dT ⎫ × dt
⎨ ⎬
⎩ dt ⎭ n

Where Tref=298K. Species equations are in the following form if we utilize Euler’s method:
⎧
⎧ d[ X i ] ⎫
⎛ 1 dT ⎞⎫
&
⎟⎬
⎨
⎬ = ⎨ωi − [ X i ]⎜
⎝ T dt ⎠⎭ n
⎩ dt ⎭ n ⎩ ⎧ d[ X i ] ⎫
{[ X i ]}n+1 = {[ X i ]}n + ⎨
⎬ × dt
⎩ dt ⎭ n

Initial condition:
T=1000K,

X_H2O=X_CO2=0,

X_FUEL=0.095, X_O2=0.19, X_N2=0.715

Table of constants: parameter UNITS
CH4
Cp
J/mole/K 71.6 at 1000K

O2
34.936

N2
32.762

CO2
54.36

H2O
41.315

hf
J/mole ‐74831 0
0
‐393546 ‐241845
MW
kg/mole 0.016
0.032
0.028
0.044
0.018
Here, 1ms was chosen as the time‐step, but any value that shows the same trend is OK.
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