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

2. Overview Background to the PLUMES software
Two phase rocket exhaust modelling
Use of parabolic solver
Assessment of parallel PHOENICS
Transient plume modelling
Conclusions

4. 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

5. 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

6. 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

7. 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

8. 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.

9. 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)

10. 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

11. Results of initial 2 phase work

12. 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

13. 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

14. 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

15. 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)

16. 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.

17. 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.

18. 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.

19. 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.

20. 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)

21. 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

22. Phase change results Plume reacting - no phase change
Plume reacting + condensation and solidification

23. Phase change results

24. Two phase - validation Particle velocities measured
Full range of velocities observed
Particle sizes measured

25. 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

26. 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

27. Transient plumes - the need

28. Transient plumes - the model

29. 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

30. 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