Williams–Boltzmann equation explained

The Williams–Boltzmann equation, also known as the Williams spray equation is a kinetic equation modeling the statistical evolution of evaporating or burning droplets or solid particles in a fluid medium. It was derived by Forman A. Williams in 1958.^{[1] [2]} The Williams–Boltzmann equation must be solved concurrently with the hydrodynamic equations such as the Navier–Stokes equations with forcing terms accoutning for the presence of sprays.

Mathematical description

Consider a spray of liquid droplets or solid particles with

M chemical species, all of which are assumed spherical in shape with radius r ; the spherical assumption can be relaxed if needed. For liquid droplets to be nearly spherical, the spray has to be dilute (total volume occupied by the droplets is much less than the volume of the ambient fluid) and the Weber number We = 2 r\rho ^{2 /\sigma} _{g| v - u |} , where \rho _{g} is the gas density, v is the spray droplet velocity, u is the gas velocity and \sigma is the surface tension of the liquid spray, should be We \ll 10 .

The droplet/particle number density function for a

j -th chemical species is denoted by f _{j=f} _{j(r,x,v,T,t)} such thatf _{j(r,x,v,T,t)} dr d x d v dT

represents the probable number of droplets/particles of chemical species

j (of M total species), that one can find with radii between r and r + dr , located in the spatial range between x and x + d x , traveling with a velocity in between v and v + d v and having the temperature in between T and T + dT at time t . Then the spray equation for the evolution of this density function is given by^[3] (\partial f _{j})/(\partial t) + \nabla _{x ⋅ (vf} _{j)} + \nabla _{v} ⋅ (F _{jf} _{j)} =- (\partial)/(\partial r) (R _{jf} _{j)} - (\partial)/(\partial T) (E _{jf} _{j)} + Q _{j} + \Gamma _{j,} j = 1 , 2 ,\ldots,M.

where

F _{j} = \left( (d v)/(dt) \right) _{j} is the force per unit mass acting on the j ^{th} species spray (acceleration applied to the sprays),R _{j = \left( (dr)/(dt)} \right) _{j} is the rate of change of the size of the j ^{th} species spray,E _{j = \left( (dT)/(dt)} \right) _{j} is the rate of change of the temperature of the j ^{th} species spray due to heat transfer,^[4] Q _{j} is the rate of change of number density function of j ^{th} species spray due to nucleation, liquid breakup etc.,\Gamma _{j} is the rate of change of number density function of j ^{th} species spray due to collision with other spray particles.

A simplified model for liquid propellant rocket

This model for the rocket motor was developed by Probert,^[5] Williams^[6] and Tanasawa.^{[7] [8]} It is reasonable to neglect

Q _{j,} \Gamma _{j} , for distances not very close to the spray atomizer, where major portion of combustion occurs. Consider a one-dimensional liquid-propellent rocket motor situated at x =0 , where fuel is sprayed. Neglecting E _{j} (density function is defined without the temperature so accordingly dimensions of f _{j} changes) and due to the fact that the mean flow is parallel to x axis, the steady spray equation reduces to(\partial)/(\partial r) (R _{jf} _{j)} + (\partial)/(\partial x) (u _{j} f _{j)} + (\partial)/(\partial u _{j}) (F _{jf} _{j)} = 0

where

u _{j} is the velocity in x direction. Integrating with respect to the velocity results(\partial)/(\partial r) \left(\int R _{j} f _{j} d u _{j} \right) + (\partial)/(\partial x) \left(\int u _{j} f _{j} d u _{j} \right) + [F _{jf} _{j]} ^{inf ty} ^{0} =0

The contribution from the last term (spray acceleration term) becomes zero (using Divergence theorem) since

f _{j →} 0 when u is very large, which is typically the case in rocket motors. The drop size rate R _{j} is well modeled using vaporization mechanisms asR _{j} =- (\chi _{j})/(r ^{k _{j}}) , \chi _{j} \geq 0 , 0 \leq k _{j} \leq 1

where

\chi _{j} is independent of r , but can depend on the surrounding gas. Defining the number of droplets per unit volume per unit radius and average quantities averaged over velocities,G _{j} = \int f _{j} du _{j,} \bar{R} _{j} = (\int R _{j} f _{j} d u _{j})/(G _{j}) , \bar{u} _{j} = (\int u _{j} f _{j} d u _{j})/(G _{j})

the equation becomes

(\partial)/(\partial r) (\bar{R} _{jG} _{j)} + (\partial)/(\partial x) (\bar{u} _{j} G _{j)} =0 .

If further assumed that

\bar{u} _{j} is independent of r , and with a transformed coordinateη _{j} = \left[r ^{k _{j+1}} + (k _{j+1)} \int ^{x} _{0} (\chi _{j})/(\bar{u) _{j}} dx \right] ^{1 /(k _{j+1)}}

If the combustion chamber has varying cross-section area

A(x) , a known function for x> 0 and with area A _{o} at the spraying location, then the solution is given byG _{j(η} _{j)} = G _{j,o} ( η _{j) (A _{o} \bar{u)/(_{j,o})} } \left(\frac\right)^.

where

G _{j,0} = G _{j(r,0),} \bar{u} _{j,0} = \bar{u} _{j(x=0)} are the number distribution and mean velocity at x =0 respectively.

See also

• Boltzmann equation
• Spray (liquid drop)
• Liquid-propellant rocket
• Smoluchowski coagulation equation

Notes and References

1. Williams . F. A. . Spray Combustion and Atomization . Physics of Fluids . AIP Publishing . 1 . 6 . 1958 . 0031-9171 . 10.1063/1.1724379 . 541. 1958PhFl....1..541W .
2. Williams . F.A. . Progress in spray-combustion analysis . Symposium (International) on Combustion . Elsevier BV . 8 . 1 . 1961 . 0082-0784 . 10.1016/s0082-0784(06)80487-x . 50–69.
3. Book: Williams, F. A. . Combustion theory : the fundamental theory of chemically reacting flow systems . Addison/Wesley Pub. Co . Redwood City, Calif . 1985 . 978-0-201-40777-8 . 26785266 .
4. Emre . O. . Kah . D. . Jay . Stephane . Tran . Q.-H. . Velghe . A. . de Chaisemartin . S. . Fox . R. O. . Laurent . F. . Massot . M. . Eulerian Moment Methods for Automotive Sprays . Atomization and Sprays . Begell House . 25 . 3 . 2015 . 1044-5110 . 10.1615/atomizspr.2015011204 . 189–254.
5. Probert . R.P. . XV. The influence of spray particle size and distribution in the combustion of oil droplets . The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science . Informa UK Limited . 37 . 265 . 1946 . 1941-5982 . 10.1080/14786444608561330 . 94–105.
6. Williams, F. A. "Introduction to Analytical Models of High Frequency Combustion Instability,”." Eighth Symposium (International) on Combustion. Williams and Wilkins. 1962.
7. Tanasawa, Y. "On the Combustion Rate of a Group of Fuel Particles Injected Through a Swirl Nozzle." Technology Reports of Tohoku University 18 (1954): 195–208.
8. TANASAWA . Yasusi . TESIMA . Tuneo . On the Theory of Combustion Rate of Liquid Fuel Spray . Bulletin of JSME . 1 . 1 . 1958 . 1881-1426 . 10.1299/jsme1958.1.36 . 36–41. free.