2. The Transport Model

CONTENTS

2.1 The model
2.2 Implementation
2.3 Activation

2.1 The model

The default transport model built into PHOENICS may be used with the thermodynamic and transport properties calculated in the CHEMKIN and TRANLIB routines. However, for laminar reaction- diffusion processes it is frequently the case that a more accurate model is required.

As an alternative to the default model, the "Curtiss-Hirschfelder" model of mass transport, including thermal diffusion (the Soret effect) has been supplied as an option.

However, the present implementation of the "Curtiss-Hirschfelder" model is under investigation at CHAM, and so it is recommended that Ficks law be employed, as done for example in CHEMKIN Library cases C204 and C205.

The default model calculates the diffusive mass-flux, J(k), from the formula

J(k) = -Rho.Dm(k).div(Y(k))

where Dm(k) is the mixture-averaged diffusivity for species k, and where div represents the divergence operator. The Curtiss-Hirschfelder model calculates the diffusive mass flux from

J(k) = -Rho.Dm(k).(W(k)/Wm).(div(X(k))-kT(k).div(T)/T)

where kT(k) is the thermal diffusion ratio for species k, and Wm is the mean molecular-weight.

There is a difficulty with the thermal diffusion term which is that the term is proportional, through the dependence of kT on Y(k), to the mass-fraction itself rather than to the spatial derivative.

The thermal diffusion term behaves more as a convective flux than as a diffusive flux, and so the same difficulties that are associated with the convective terms can be expected to be encountered. The implication is that the thermal-diffusion flux should be upwinded.

For the enthalpy equation there is a further potential complication. In addition to the conductive heat flux, there is also an enthalpy flux associated with the mass-diffusion. The additional term is

K
Sum(div(J(k).h'(k))
k=1

Again this term is more a convection-like term than a diffusion term and, again, upwinding of each term is required to ensure stability and accuracy. We do not solve for the last species in the set, and so we must make a modification to the term. It is known that

K
Sum(J(k)) = 0
k=1

so
K-1
J(K) = - Sum(J(k))
k=1

which leads to the expression

K
Sum(div(J(k).(h'(k)-h'(K)))
k=1

for the enthalpy flux associated with mass-diffusion.

The user is advised that the present implementation of the transport model

2.2 Implementation

The enhanced transport algorithm involves coding sequences in group 8 of EARTH. The finite difference form for the species gradient term is;

J'(k) = -Rho.<Dm(k)/Wm>.Wm.(Y(k)-Y(k;+).Wm(+)/Wm)

where here the <...> indicates a mean value calculated on the cell-face, and (+) indicates the neighbour-cell values.

The finite difference form for the thermal diffusion term is

J''(k) = Rho.<kT(k).D(k)/Y(k)>.log(T(+)/T)

which is then upwinded so that

• if J'' > 0 ;

  Su(k;+) = Su(k;+) + J''(k).Y(k)

  Ap(k;+) = Ap(k;+) + J''(k)

• if J'' >0 ;

  Su(k) = Su(k) - J''(k).Y(k;+)

  Ap(k) = Ap(k) - J''(k)

• In section 9 (for diffusion coefficients): the thermal diffusion flux is calculated from the diffusion coefficient, which has already been modified within EARTH to take account of blockages and solids, and the diffusion coefficient is multiplied by the mean molecular weight
• In section 11 (for diffusion neighbours): the diffusion neighbour is multiplied by the neighbour value of the mean molecular weight and divided by the in-cell value of the mean molecular weight. Also, the upwinded contribution from the current species to the enthalpy flux due to mass diffusion is calculated and added to a store.

In addition, the modifications to the diffusion coefficient, ie. the division by mean molecular weight, necessitate an entry to GXCHKI to multiply the laminar wall function coefficient for a mass-fraction by the mean molecular weight.

2.3 Activation

The enhanced transport model is activated by setting 2 variables in the Q1 file:

• ENULA=GRND9 ; enhanced mixture-averaged transport model
• ENULB=GRND9 ; thermal diffusion (Soret effect)

  The default settings (ENULA=0.0;ENULB=0.0) imply that the standard PHOENICS model of diffusive transport is activated.

If the modified diffusion law is in use, any laminar wall- functions for mass-fractions should be setup using a PATCH-name starting with CKWAL and GRND9 in the COefficient slot of the COVAL. For example,

PATCH(CKWALIN,SWALL,1,NX,1,NY,1,NZ,1,LSTEP) COVAL(CKWALIN,H2,GRND9,0.025)

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