Errors of the method

utilization of the variable step size and order features of the ordinary dierential equation

solver. The result of the seond hemistry operator is a predited solution at

t + 1

^{, whih}

the transportoperator then uses as initialonditionsto omplete the step.

It seems natural, whenperforming any kind ofimplementation ina CFD ode or any other

typeofsoftware, tostartalulatingsimpleproblems beforeattemptingmoreomplexones.

The time and eort required for a alulation is shorter and the results an be more easily

veried with available 1D tools. This way, the rst step in order to ahieve the oupling of

the CFD-solver and the CHEMKIN-pakage has been the simulation of 1D ongurations

assumingonstant properties ofthe gas (onstantdensityand visosity),and negleting the

speies diusion and heat transfer inorder to reprodue the perfetlystirred reator model

in aone-dimensional simulation.

By means of the program SENKIN, we have obtained the solution for

Y k

^and

T

^from

the following system of dierentialequations:

ρ dY k

dt = ˙ ω k

ρc p k

dT

dt = − X

k

˙ ω k h k

(3.1)

appliedtotheombustionmehanismsof hydrogenandmethane. Theseequationsrepresent

a partiular ase of the speies (2.13) and energy (2.4) transport equation in whih all

omponents of the veloity vetor are equalto zero in the whole domain. They orrespond

to the denition of a perfetly stirred reator, whose desription has been given in setion

1.4.2.3. The solutions given by the 1D tool SENKIN will be used to verify the results

obtained by means of the operator-splittingsheme implemented inFASTEST-3Dfor a 1D

onguration.

A seond step in order to verify the validity of the implementation is the alulation

of hydrogen and methane ombustion in a plug-ow reator (PFR). This model onsiders

following assumptions:

•

^{Axial ow}

•

^{Perfet mixingin radialdiretionbut nomixinginaxial diretion}

•

^{Constant density}

•

^{Steady state}

Symmetry

Inlet

Outlet

Symmetry

Figure3.1.: Linear reator used for

H 2

^and

CH 4

^{1D ombustion}

The speies and energy transport equations for the plug-ow reator(with

D _k , λ = 0

⁾

an be simpliedto:

ρu ∂Y _k

∂x = ˙ ω k

ρuc p k

∂T

∂x = −

N

X

k =1

h k ω ˙ k

(3.2)

By foringthe veloitytobe

u = 1

^{inthe wholedomain,equations 3.2are equivalentto}

equations 3.1where the variable

t

^{has been} substituted by the spatialvariable

x

^.

Theomputationalgeometryhasbeenrepresentedingure3.1. Itonsistsofaartesian

grid withalltheells havingequaldimensionsandwithamuhlargernumberofthemalong

the owdiretion (50 for hydrogen and 500 for methane), while the ross setionis divided

in

3 × 3

^{ells. The boundary onditions used inthe alulation have been presented in the}

same gure.

It is well known that mesh size and time step have a very big impat on errors

asso-iated to the operator-splitting sheme ([37℄, [93℄ and [48℄). Several simulations have been

performed for dierent values of these parameters, as well as for dierent temporal

dis-retization shemes,inorder toahievethe requiredknowledge tobeapplyed latterinmore

omplex 2D ongurations. Sine the operator-splitting sheme is applied usually to

un-steady alulations (even inthe ase of steadyproesses), the temporaldisretizationof the

problem issupposedto inuene the aurayof the method,even if not asritiallyas the

spatial disretization [40℄.

Bysalingthegridweanobservethe inueneofthespaingonthe nalsolution. The

time step of the splitting orresponds to twie the time step of the CFD-solver and, thus,

the sheme providesa Strang-type yle:

∆t _{cf d} + 2 × ∆t _chem + ∆t _{cf d}

This expression is equivalent to equations 2.79. All 1D and 3D simulations shown in this

workmakeuseofthesameapproahand,therefore,nodesriptionofhemialormehanial

time step willbe madein followingsetions.

To determine the inuene of the grid spaing and the time step inthe results,the two

parametrial analysis presented in the next setions have been performed. They show the

solution of the PFR-problem for stoihiometri mixtures of fuel and air for dierent values

of grid spaing and time step.

Theresultsaregivenasafuntionofthetemporaldisretization(

∆t

^),spatial

disretiza-tion (

∆x

^{) onstant allalong the reator, and of the}

Courant − F riedrich − Lewis

⁽

CF L

⁾

number, whihis the ratio of a time step to the ellresidene time:

CF L = u∆t

∆x

^(3.3)

The onvergene riterion for the simulations performed with the ode FASTEST-3D

has been seleted to be equal to

10 ^{− 3}

^{. This value is kept as well for the 2D} simulations presented in the next hapter.

3.1. Stoihiometri

H 2 − air

^ombustion

Hydrogenwilllikelybeometheprimaryenergyarrierinthefuture. Ononehand,hydrogen

has the potential for utting greenhouse gas emissions as well as reduing rural, urban

and regional air pollution. Besides, there is presently onsiderable interest worldwide in

promoting hydrogen as a fuel for transport. As oil beomes more expensive, hydrogen

may eventually replae it as a transport fuel and in other appliations. This development

beomesmore likelyasfuel ells are developed, with hydrogen asthe preferred fuel,though

storage at vehile sale is a major hallenge. Meanwhile hydrogen an be used in internal

ombustion engines [35℄. The simpliityof some ombustion mehanisms, whih reprodue

withreasonableauraythereationstaking plaeinhydrogenombustion,makesitavery

attrativetopi for omputationalanalysis.

For the ase of hydrogen ombustion, a stoihiometri mixture of fuel and air evolves

aording tothe speies andtemperatureprolesshown ingure3.2and 3.3. These are the

results provided by the SENKIN ode that will be used as referene for the veriation of

the 1D resultsgiven by FASTEST-3D.

The initial temperature is set to

1400

^{K. The fuel-air mixture is} stoihiometri:

Y H ² =

0.028

^,

Y O ² = 0.226

^and

Y N ² = 0.745

^{and the timestep amounts to}

1 × 10 ^{− 6} s

^{. Ignition starts}

after

2 × 10 ^{− 5} s

^{. The} temperaturereahes

2800

^{K after}

2 × 10 ^{− 4} s

^.

1e-05 0.0001 0.001 0.01 0.1 1

0 2e-05 4e-05 6e-05 8e-05 0.0001

MASS FRACTION

TIME (s)

O2 H2 H2O H O OH

Figure3.2.: Speies evolutionin stoihiometri

H 2

^{ombustiongiven by SENKIN}

1200 1400 1600 1800 2000 2200 2400 2600

0 2e-05 4e-05 6e-05 8e-05 0.0001

TEMPERATURE (K)

TIME (s)

Figure 3.3.: Temperatureevolution in stoihiometri

H 2

^{ombustiongiven by SENKIN}

Theresultingspeiesmassfrations(

Y k

^)andtemperature(

T

^{)timeevolutionshavebeen}

ompared with the ones resultingfrom the CFD simulations intwo ases:

•

^{Closed system where time is the only}independent variable (PSR)

•

^{Reating owwith onstant properties and nodiusion of speies (PFR)}

In bothases, theequationssolved by FASTEST-3Dare equivalenttothe onessolved inthe

SENKIN ode (equations 3.1). The equivalene between the perfetly stirred reator and

the plug-ow reator with onstant properties is ahieved through the hange of variable

x = u · t

^and eliminatingthe spatial diusion of speies and temperature. The density and the veloity are onstant along the reator. As

u = 1

^{inthe whole burner, the evolutionsin}

time forthePSRandinspaeforthe PFRhavetobeoinident. Arst-orderfullyimpliit

algorithmhas been hosen for the time integration.

Im Dokument Numerical Simulation of Reacting Flows under Laminar Conditions with Detailed Chemistry (Seite 42-47)