BY : Dr S V Zhubrin

DATE : March 2002

FOR : Multi-physics demonstration case

The case presents the implementation in PHOENICS of an extension for SCRS model to handle the turbulent flow, mixing and evaporation of a finely dispersed water droplets in a co-flowing combusting streams resulting from the admission of two separate coaxial turbulent jets of gaseous fuel and oxidant into a duct.

The flow, heat and mass transfer are considered as homogeneous phenomena, with evaporating droplets being treated as non-gas species which have the same velocity components as the gas and the constant temperature of the saturation state.

The specific problem considered is the injection of a two-phase mixture of the methane and water into a stream of of pre-heated air/water mixture flowing in a rectangular duct.

The streams are ignited at the entry and steady state combustion takes place with the water droplets evaporated completely as they pass along the duct.

The present case consists of a rectangular duct with unrestricted exhaust in which the gaseous methane-fuel burn in a stream of heated air. The methane premixed with the certain amount of finely dispersed water droplets is introduced through a slot on the symmetry plane of the duct, and the secondary air, also premixed with water fines, is introduced through a slot inlet surrounding the fuel entry.

The present treatment of two-phase flow is based on the homogeneous approach considering no slip velocity between the liquid and gas phases. It is assumed that there is no influence of droplets on the turbulence structure represented via standard K-e model of turbulence and the gaseous combustion is supposed to proceed in accordance with the "mixed-is-burned" rule.

The temperature of droplets is supposed to be constant and equal to the water saturation under the given pressure. The local rate of evaporation is assumed to be determined by the rate of exchange of heat between hot gas eddies and eddies containing liquid phase, and thus related to the local rate of turbulent dissipation.

The independent variables of the problem are the two components of cartesian coordinate system, X and Y.

The main dependent (solved for) variables are:

- Two velocity components, U1 and V1
- Pressure, P1
- Kinetic energy of turbulence,KE, and its dissipation rate, EP
- Specific gas enthalpy, H1.
- Mixture fraction, MIXF
- The sum of liquid water and evaporated water mass fractions, YWPV
- Mass fraction of liquid water, YWAT.

The model employed postulates a physically controlled, one-step
instant reaction, with fuel, CH_{4}, and oxygen,
O_{2}, unable to coexist at the same location:

CH_{4} + 2O_{2} + Diluent = CO_{2} + 2H_{2}O + Diluent,

The stoichiometric ratio,* s*, of oxygen to methane
is 4.0, i.e. 4 kg of oxygen are required to complete combustion
of 1 kg of methane since the molecular masses of CH

The combustion products are water vapour, H_{2}O, and
and carbon dioxide,CO_{2}. The mixture of N2, liquid water fines, WAT, and a water vapour associated with evaporating liquid water, VAP,
is regarded as a single substance, i.e. simple total diluent, entering no chemical reaction at all.

The mixture fraction, MIXF, is represented as

MIXF = (* s*YCH4 - YO2 + YO2

from which the stoichiometric value, F_{stoic},
can easily be deduced by substituting YCH4=YO2=0:

F_{stoic} = YO2_{in}(* s*YCH4

In above, YCH4_{in} and YO2_{in} are the mass fractions
of the methane and oxygen in the fuel- and air-supply streams accordingly.
They are defined by the inlet stream compositions.

Instead of solving two conservation equations for mass fractions of liquid water, YWAT, and the vapour emerging from it, YVAP, the composite variable, YWPV = YVAP + YWAT, is used. The advantage is that the balance of YWPV contains no source term.

The evaporation rate, R_{ev} in kg m^{-3}s^{-1}, featuring in the conservartion equation for YWAT is modelled as follows:

- In the regions where the gas temperature is significantly higher than the water-vapour saturation temperature the rate of evaporation is limited by the rate of dissipation of liquid containing eddies, i.e
R

_{ev}= C_{a}RHO1(EP/KE)YWAT - In the regions where the mass fraction of water in a liquid state is high and the superheating of the gas mixture is low the rate of evaporation is limited by the dissipation of excess superheated eddies, i.e.
R

_{ev}= C_{a}C_{b}RHO1C_{p,mix}(T_{g,mix}- T_{s})H^{-1}_{fg}(EP/KE)wherein C

_{p,mix}is the mixture specific heat, Jkg^{-1}K^{-1}, T_{g,mix}stands for mixture temperature, K, and H_{fg}is the latent heat of evaporation, Jkg^{-1}. - The equation that gives the lowest evaporation rate is the one that determines the local rate of evaporation.

C_{a} = 4.0 and C_{b} = 0.5

- if MIXF is less or equal F
_{stoic}, then

YCH4=0.0 and

YO2 = YO2_{in}(1-MIXF/F_{stoic}) - if MIXF is greater than F
_{stoic}, then

YO2=0.0 and

YCH4 = YO2_{in}(MIXF/F_{stoic}-1)**s**^{-1} - for any MIXF,

YN2 = YN2_{in}(1 - MIXF)

The mass fractions of carbon dioxide is calculated from the continuity of mixture composition and the stoichiometry of combustion products as follows:

YCO2 = (1 - YCH4 - YO2 - YN2 - YWPV)*44/80

The total mass fraction of water vapour is calculated similarly as:

YH2O = (1 - YCH4 - YO2 - YN2 - YWPV)*36/80 + YVAP

wherein YVAP = YWPV - YWAT.

The water density, RWAT, is taken as constant and gas density, RGAS, is computed from the local pressures, temperatures and local mixture molecular masses, so that the mixture density will then be:

RHO1^{-1} = RGAS^{-1}(1-YWAT) + RWAT^{-1}YWAT

where

RGAS = M_{mix}R^{-1}_{gas}T_{g,mix}^{-1}P1

Here R_{gas} stands for universal gas constant.

The molecular mass of the mixture is composition-dependent:

M^{-1}_{mix} =
YCH4/16+YO2/32+YN2/28+YCO2/44+YH2O/18

The water specific heat, CPWAT, is taken as constant, and the gas-phase specific heat is assumed to be equal for all gas components and is a linear function of gas temperature. The mixture specific heat will then be:

C_{P,mix} = (1-YWAT)(1059+0.25(T_{g,mix}-300))+
CPWAT*YWAT

The combustion heat of a methane,H°, and the latent heat of evaporation, H_{fg}, are taken as constants.

The specific mixture enthalpies are related to gas temperatures as follows:

H1 = C_{P,mix}T_{g,mix} +
H°YCH4 + H_{fg}YVAP

At all inlets, values are given of all dependent variables together with the prescribed flow rates.

The gas composition continuity requires that the followings should always be held:

- at the fuel-supply inlet: YCH4+YWAT+YVAP=YCH4+YWPV=1
- at the air-supply inlet: YN2+YO2+YWAT+YVAP = YN2+YO2+YWPV = 1

Fixed exit pressure. As the fluid is assumed incompressible, this pressure is set equal to zero and the computed pressures are relative to this pressure.

The smooth-wall 'wall functions' are used to provide the non-slip conditions for momentum equations.

It is assumed that there is no heat exchange to the wall, ie. an adiabatic boundary conditions are employed.

The plots show the flow distribution, temperature and mixture composition as represented by the model.

Pictures are as follows :

- Velocity vectors
- Temperature contours
- Density contours
- Liquid water mass fraction
- Mass fraction of evaporated vapour
- Total mass fraction of water vapour
- Methane mass fraction
- Oxygen mass fraction
- Nitrogen mass fraction
- Carbon dioxide mass fraction

All model settings have been made by InForm of PHOENICS 3.5.