Annexv

Annex V

AV- Theory of Vented Deflagration, Grigorash 2002 How to obtain the derivation equations of vented deflagration used in this model is explained below. This theory was developed by Grigorash in 2002.

First of all, let us refer the real distorted flame front area Ff(t) at any moment t of time to the surface Fs(t), see Eq. (AV-1), where rb is the radius of the sphere occupied by burnt gas. The ratio of both flame front areas is called turbulence factor (see Eq. (AV-2)).

Fs (t) = 4 πo rb2 F (t) χ(t) = f Fs (t)

(AV-1) (AV-2)

The dependence of burning velocity is on the temperature and pressure of the gaseous mixture, Eq. (AV-3). m

T  p Su = Sui  u     Tui   pi 

n

(AV-3)

From the adiabatic equation pVγ = cte together with the ideal gas state equation, one can easily derive the following relationship, Eq. (AV-4).

T p =  Ti  pi 

γ−1

γ

1− 1 γ

=π

(AV-4)

This proportion allows one to express the burning velocity in the form Su = Sui πε

(AV-5)

Where the values of thermokinetic factor ε = m + n – m/γu along with the burning velocity can be found in literature.

Defining σu as Eq. (AV-6), where ρi and ρu are the initial and the current unburnt gas densities and taking the adiabatic equation written now in the form p(m/ρ)γ = cte, for both the initial and the current densities, one can see that σu = π1/ γ u

(AV-6)

For defining σb, we denote nu = mu/mi and nb = mb/mi the relative masses; wu = Vu/V and wb = Vu/V the relative volumes of unburnt and burnt gases. Volume conservation within the gas gives wu + wb = 1. Therefore, for the average relative density of burnt gases throughout the enclosure we can obtain Eq. (AV7).

A-49

Annex V

σb =

ρb n nb σu = b = ρi w b σu − n u

(AV-7)

For further simplicity of derivation it is useful to imagine, in mathematical sense only, that we deal with spherical flame propagation in a spherical vessel of radius a, in m.

1/ 3

 3V  a=   4 πo 

(AV-8)

Then the dimensionless radius of our imaginable spherical flame can be calculated as 1/ 3

1/ 3  r V V n ( τ)  −1/ γ u 1/ 3 r ( τ ) = b =  b  =  1 − u  =  1 − u ) (AV-9)  = (1 − n u ( τ ) π a V  V σ  u  Using notation for χ, Eq. (AV-5) and Eq. (AV-6) and according to the assumed flamelet model of turbulent combustion, the burning rate of fresh gases can be rewritten as Eq. (AV-10).

dm u = −Ff ρuSu dt

dm u = −4 πo rb2ρi σu χSui πε dt

(AV-10)

The latter formula expresses the instantaneous change in the unburnt gases mass due to burning. In the following paragraphs, we take into account for the other change to this mass: due to venting.

For doing that in each vent, the orifice equations for the calculation of the mass flow rate for subsonic and sonic regimes give a mass rate of discharge Gju of unburnt and Gjb of burnt gases. With the notation for the relative pressure and density, the mass rate of discharge of any gas is given in Eq. (AV-11).

2 G = µFpiρ1/ i R

(AV-11)

Where R is the outflow parameter given by Eq. (AV-12).

1/ 2

 2 γ  p 2 / γ  p ( γ+1) / γ    R= πσ  a  −  a  γ − 1 p π p π     i   i   

γ+1   γ−1   2  and R =  γ  πσ   γ + 1    

1/ 2

(AV-12)

for subsonic regime and sonic regime, respectively. The transition from the subsonic to the sonic regime of discharge happens when the pressure exceeds the critical value of:

p π≥ a  pi

  1 + γ  γ /( γ−1)    2 

(AV-13)

A-50

Annex V The unburnt versions of R and the critical pressure are obtained by Eq. (AV12) and Eq. (AV-13) with substituting the unburnt and burnt versions of γ and σ in these equations.

Considering the existence of N vents in the enclosure and the simultaneous discharge of unburnt and burnt gases throughout the vents, the rate of change becomes: N dm u = −4 πo rb2ρi σu χSui πε − ∑ (1 − A j ) Guj dt j=1

(AV-14)

Since the limits of summation will be the same everywhere, we omit them from the summation sign.

Using Eq. (AV-6), Eq. (AV-8) and Eq. (AV-9) and going over to relative mass and dimensionless time in Eq. (AV-14), the rate of change of the unburnt gases mass is obtained, Eq. (AV-15).

 a (1 − A j ) Gui  2/ 3 dn u = −3 (1 − n u π−1/ γ u ) χπε+1/ γu + ∑  dτ 3m iSui  

(AV-15)

With Eq. (AV-11) and the formula of Newton-Laplace of the speed of sound at initial condition, it is possible to rewrite the discharge term in Eq. (AV-15) as following:

∑

a (1 − A j ) Gui 3m iSui

= WR u

∑ (1 − A )µ F ∑µ F j

j

j

j

(AV-16)

j

Where W is the transient venting parameter, dependent on the varying current venting areas, Eq. (AV-17).

W=

cui ∑ µ jFj

( 36πo )

1/ 3

V 2 / 3Sui γ u

(AV-17)

Now, with Eq. (AV-15) and Eq. (AV-16), Eq. (AV-17) becomes 2/ 3 dn u = 3 χπε+1/ γu (1 − n u π−1/ γ u ) + R u W∑ (1 − A j ) µ j Fj / ∑ µ j Fj    dτ

(AV-18)

Following the same reasoning, the equation for the rate of change of mass of burnt gases is given by Eq. (AV-19).

2/ 3 dn b = 3 χπε+1/ γu (1 − n u π−1/ γu ) − R b W∑ A jµ jFj / ∑ µ jFj    dτ

(AV-19)

A-51

Annex V

Eq. (AV-18) and Eq. (AV-19) are two of the three governing equations of this model. The third equation that describes the change in the relative pressure is obtained from the laws of conservation, as follows.

According to the conservation of mass.

nb + nu + ∫

t

0

∑

A jGbj + (1 − A j ) Gbj mi

dt = 1

(AV-20)

According to the conservation of energy per mass unit.

The mass flux of the unburnt gases though every vent reduces the energy of the system by the sum of the amount of the internal energy of the escaped gas and the work required to move this mass through the vent cross-section. ui =

t  p  A jGbj  p  (1 − A j ) Guj  u dn + u dn + u + + u +  dt   ∑  ∑ b b u u b u ∫nb ∫nu ∫0  ρb  mi ρb  mi    (AV-21)

The internal energy of mass unit of the unburnt and burnt gases changes in consequence of adiabatic compression or expansion: u u = u i + ( Tu − Tui ) cvu Mui−1

(AV-22)

u b = u bi + ( Tb − Tbi ) c vb Mbi−1

(AV-23)

Since the combustion takes place at constant initial pressure, the enthalpy is conserved:

u i + R gas Tui M−ui1 = u bi + R gas Tbi Mbi−1

(AV-24)

This leads to

u b = u i + R gas Tui M−ui1 − R gas Tbi M−bi1 + ( Tb − Tbi ) c vb Mbi−1

(AV-25)

Substitution of Eq. (AV-22) to Eq. (AV-25) to the energy equation (AV-21) followed by simplications using the mass conservation, the gas state equation in the form p/ρ = RT/M and the specific heats equation cp-cv = Rgas gives the following: Tui T T Tb Tu = cpb bi n b + cpu ui n u − cvb ∫ dn b − c vu ∫ dn u + n n b M u M Mui Mbi Mui bi ui tc tc ( T − Tb ) ( T − Tu ) A jGbjdt + ∫ pu ui + ∫ pb bi ∑ ∑ (1 − A j ) Gujdt 0 0 m i Mbi m i Mui

R gas

(AV-26)

A-52

Annex V Assuming an uniform distribution of temperature in the combustion products, and using the ideal gas state equation and Eq. (AV-3) and Eq. (AV-4), we obtain

∫

nb

Tb T dn b = ui σuγ u −1 ( σu − n u ) Mbi Mui

(AV-27)

Multiplying both parts of Eq. (AV-26) by Mui, dividing them by cvbTui and simplify the result using Eq. (AV-27), Eq. (AV-1) and Eq. (AV-3) and the following relations (γ=cp/cv, γ-1=Rgas/cv): 1 − A j )Guj  t γ u − γ b 1−1/ γ u γ u ( γ u − 1)  1−1/ γ u ∑ ( π + dt +  n u + ∫0 (1 − π ) γu − 1 γ u − 1  mi   t  π1/ γu − n u   ∑ A jGbj  +γ b E i n b + ∫ E i − π1−1/ γu  dt (AV-28)  0 n m  b  i    π + γ b − 1 = nu

Differentiation of this equation with respect to the relative time τ, and simplification using equations (AV-15), (AV-18) and (AV-19) leads to the equation for dimensionless pressure, Eq. (AV-29).

χ ( τ ) Zπ dπ = 3π dτ

ε+1/ γ u

(1 − n π

)

−1/ γ u 2 / 3

u

π1/ γ u −

γu − γb nu γu

− γ u WR Σ

(AV-29)

where W is defined in the Eq. (AV-16), R∑ is the outflow contribution RΣ = Ru

∑ (1 − A ( τ ) ) µ F ( τ ) + R ∑ µ F ( τ) j

j

j

j

j

 π1/ γ u − n u  ∑ A j ( τ ) µ jFj ( τ )  b  nb ∑ µ jFj ( τ )  

(AV-30)

and Z is the auxiliary quantity  γ ( γ − 1)  Z = γ b E i − u b π γ b ( γ u − 1)  

1−γ u γu

+

γb − γu γu − 1

(AV-31)

Thus, equation (AV-18), (AV-19) and (AV-29) constitute the system of governing equations

AV.2 Simple Correlation used for Comparison -

A-53

No 1

Equation

pred = ( CAs ) A 2

pred

  1 =   f  dV exp ( gpstat ) 

1/ h v

A

pred = 1.804·10  D Sfl ( E o − 1)  A −4

2

2

−2 v

Remarks

Restrictions

pred is expressed in bar g; C (in bar ) – coefficient

Valid for pstat ≤ 0.1m

depending on explosive mixture

bar g

Simpson (equation

pstat and pred are expressed in bar g V and Av are

Restrictions:

transporting the

expressed in m3 and m2

0.1≤ pstat ≤ 0.5

Bartknecht nomograms)

d, f, g and h are constants depending on the

pstat + 0.1 ≤ pred ≤ 2

nature of explosive mixture

1 ≤ V ≤ 5000

NFPA 68 1/ h

2

3

References

−2 v

Runes

1/2

3

pred in bar g; D and Av in m , flame speed Sfl in m/s

4

  pred = pstat = 2.43  A   So  pred ≥ 1b ar g

−0.6993

Bradley, formula 1 for

A=

  pred = pstat = 12.46  A  for  So  pred ≤ 1b ar g

6

pred = 4.82P

0.375 stat

pred

A   S  o 

  = 0.365  A   So 

−1

−1.25

Cd A v dimensionless vent area As

 S S ρ So = uo  uo − 1  = uo ( E o − 1) co  ρbo  co

−2

5

pstat and pred in bar g

dimensionless venting parameter Brandley, formula 2

Both pstat and pred are expressed in bar g

Cubbage and Simmonds,

pred in bar g

formula cited by Brandley

Table AV-1 Relationship used for reduced pressure calculation.

The maximum pressure does not exceed the vent opening pressure

No 7

Equation

  pred = 0.150pstat + 0.365  A   So  2

8

9

References −1

Rasbash, formula cited

Remarks

Restrictions

pred is expressed in bar g

by Bradley −1

pred

 0.375χ 0.675 E 7o / 6   A  =    Eo − 1    So 

πred

 Br ( E o − 1) µ  p − po = red = 9.8   1/ 3 po γ u χ   ( 36πo )

−2.4

Yao, formula cited by

pred is expressed in bar g. Bradley recommends to

Bradley

use a turbulence factor χ = 4.

Molkov, formula 1

pred and pstat is expressed in bar g

Br =

Av V2 / 3

(dimensionless reduced overpressure)

co Bradley  1 − 1/ γ b  Suo  E o −  1 − 1/ γ u  

number

 (1 + 10V1/ 3 ) (1 + 0.5Br )  χ = 0.9   µ 1 + πv   10

πred = Brt−2.4 for πred < 1 πred = 7 − 6Brt−0.5 for πred > 1

0.37

Molkov, formula 2

Table AV-2 (continued): Relationship used for reduced pressure calculation.

Annex V

A-56