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MODELLING OF TWO PHASE ROCKET EXHAUST PLUMES AND OTHER PLUME PREDICTION DEVELOPMENTS A.G.SMITH and K.TAYLOR S & C Thermofluids Ltd

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**Overview Background to the PLUMES software**

Two phase rocket exhaust modelling Use of parabolic solver Assessment of parallel PHOENICS Transient plume modelling Conclusions

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Plumes modelling Combustion processes result in waste products - exhaust When the exhaust is released the resultant flow is known as the plume Although exhaust is waste - there are implications - impingement, infra-red, pollution - and a need to study

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**Based on PHOENICS CFD code**

PLUMES Developed for general plume flowfield prediction - Rocket exhausts - DERA Fort Halstead Air breathing engine exhausts - DERA Farnborough Land system exhausts - DERA Chertsey Ships - DERA Portsdown West Based on PHOENICS CFD code

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**Particles within exhaust plume**

Momentum (changes in bulk density and interphase friction) Temperature (Cp of particles, solidification, evaporation, further reaction) Increased radiative heat transfer (grey bodies as opposed to selective emissions) Further pollution issues

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**Particle modelling Most particles are small <10m**

Follow gas velocity (small lag) Follow gas temperature Extra set of momentum equations too much overhead - still only one diameter Use of particle tracking - cannot really study bulk effects

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**Two phase treatment - momentum**

Single set of momentum equations (accept velocity lag) Calculate a bulk density to modify overall momentum of exhaust mf = S (Mfi*smw/mmw) (1) mf is the overall mass fraction of any particulate species Mfi … mole fraction of any particulate species smw is the species molecular weight mmw is the overall mixture molecular weight.

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**Two phase momentum Particulate density - rp = mf / S (Mfi / ri) (2)**

Particulate volume fraction Vf = (mf/rp) / [(1-mf)/rg + mf/rp] (3) where rg is the gas mixture density Overall mean density r = Vf.rp + (1-Vf).rg (4)

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**Two phase temperature Small particles close to gas temperature**

Second energy equation not solved Cp calculated for particulates in the same way as for gaseous species - via ninth order polynomial

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**Results of initial 2 phase work**

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**Phase changes in plumes**

Chamber is high temperature and contains gaseous species as well as particulates Acceleration through convergent/divergent nozzle causes static temperature to fall Reactions slow and condensation/solidification Mixing of oxygen into plume Shock waves raise static temperature Secondary combustion Melting and evaporation

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**Phase change modelling**

Solid, liquid and gas species all solved within single phase Source terms added for heat and mass transfer to allow changes between each phase to take place

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**Phase change (liquid/solid)**

Q = Kh.As.(Tmp-T) (5) where Kh is a heat transfer coefficient and As is the surface area.T is temperature Kh = Nul/Dp (6) where l is the gas thermal conductivity and Dp the particle diameter. Nu is 2 for low Re - low slip velocity

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**Phase change (liquid/solid)**

If T < Tmp, the liquid-to-solid transfer (Sp) rate for each particle is then: Sp = Q/Hfs = Kh.As.(Tmp-T)/Hfs (7) where Hfs is the latent heat of fusion in J/kmol. Number of particles of a particular species and phase per unit volume is given by; np = rp /(pDp3/6) (8)

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**Phase change (liquid/solid)**

The liquid-to-solid transfer rate per unit volume (in kmol/s/m3) is then Svol = Sp * np = Kh.6/Dp.(Tmp-T) rp/Hfs (9) and rp = (Cl)*smw*r/rp (10) where Cl is the species concentration (in kmol/kg) of the liquid species, r is the bulk mean density and rp is the particle density.

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**Phase change (liquid/solid)**

The source term for each phase i, S = cell vol.Co.(Val - Ci) (11) Co = Kh.6/Dp/Hfs.|Tmp-T|*smw*r/rp (12) If T < Tmp, for the liquid phase Val = 0 for the solid phase Val = Cl +Cs This source term will also function as a melting rate if T>Tmp, but with Val = Cl+Cs for the liquid, and Val = 0 for the solid.

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**Phase change (gas/liquid)**

Sp = Km.As.(Csat-Cg).r (13) where Km is a mass transfer coefficient, As is the surface area. Cg is the gas species concentration in kmol/kg, r the bulk mean density and Cg > Csat if condensation is taking place. Csat is proportional to the saturation vapour pressure psat of the species: Csat*gmw = psat/p (14) Where p is the local static pressure and gmw the mean molecular weight of all the gaseous species.

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**Phase change (gas/liquid)**

The vapour pressure is a function of temperature and can be estimated as psat = e(a-b/T) (15) where a and b are constant for a particular species and can be determined if two points on the saturation line are known.

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**Phase change (gas/liquid)**

Km = Sh*D/Dp (16) where D is the diffusivity of the species in the mixture and Dp the particle diameter. The number of droplets of a particular species and phase per unit volume is given by equation 8. The gas-to-liquid transfer rate per unit volume (in kmol/s/m3) is therefore Svol = Sp * np = Km.6/Dp.(Csat-Cg).r. rp (17) where rp is defined in equation (10)

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**Phase change (gas/liquid)**

This transfer rate can be linearised for inclusion as a PHOENICS source term in the following way: The source term for each phase i, S = cell vol.Co.(Val - Ci) (11) where Co = Km.6/Dp.*smw*Cl.r2/rp (18) and for the gas phase Val = Csat for the liquid phase Val = Cg-Csat+Cl

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**Phase change results Plume reacting - no phase change**

Plume reacting + condensation and solidification

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Phase change results

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**Two phase - validation Particle velocities measured**

Full range of velocities observed Particle sizes measured

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**Application of Parabolic extensions**

IPARAB=5 for underexpanded free jets Significant increases in solution speed for 2D and 3D plumes Increased resolution of plume without large storage requirements Need to combine elliptic and parabolic solvers has become apparent

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**PARALLEL PHOENICS Domain decomposition is slabwise**

Plume flowfield predominantly slabwise PLUME software linked with PARALLEL PHOENICS (v3.1) on SGI Origin 200(MPI) Approximately 3x speed up for 4 processor Increase in performance good but hardware and software costs high

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**Transient plumes - the need**

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**Transient plumes - the model**

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**Transient plumes - method**

Lack of initial fields makes convergence difficult Use of small time steps (100microseconds) to resolve phenomena and stabilise the convergence of the solution

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**Conclusions PHOENICS based PLUME software development continued**

Limited two phase rocket exhaust prediction capability created Enhanced parabolic solver incorporated Parallel PHOENICS - potential speed increases Transient plumes now being modelled

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