Diffusion flames and fire plumes Dimensionless heat release rate:

ρ∗

∞ ∞

= 2c

cp

QQc T D gD

In empirical formulas: use in kW! cQ

4.1 Laminar jet flames Flame height l ~ 0,5V

4.2 Turbulent jet flames Flame height

⎡ ⎤⎛ ⎞= + −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

1/ 25,3 (1 ) airT F

f fi f i f

Ml T C Cd C mT M

With di = diameter of jet burner TF = flame temperature Ti = initial temperature m = molar stoichiometric ratio reactants to products (nitrogen included) r = stoichiometric molar air to fuel ratio ri = initial air to fuel ratio

+=

+11

if

rCr

4.3 Flames from natural fires General: c cQ m H A′′= Δ f

Summary of formulae from ‘Diffusion flames and fire plumes’ 1

4.3.1 The buoyant plume Necessary HRR for plume to reach the ceiling

3 5 / 2 3 /1,06 10conv aQ H−= ⋅ ⋅ Δ 2T

Temperature increase on axis, at height z, with ambient temperature 293 K. Valid in rising plume, above the flame (z > l)

2 / 3

0 5 / 326 convQTz

Δ = or 2 / 3

0 5 / 322 cQTz

Δ =

Virtual origin

2 / 50 1,02 0,083 cz Q

D D= − +

4.3.2 The fire plume Frequency of oscillating flames

[ ](0,50 0,04) gf HD

= ± z

Flame height as function of (dimensionless) HRR

2 / 50,23 cl Q= ⋅

If diameter is known: 2 / 5

2 / 5

0,23 1,02

3,7 1,02

cl Ql QD

∗

⎧ D= − ⋅⎪⎨

= −⎪⎩

Estimate flame volume from

31200c

f

QQ kV

′′′ = = W m

Summary of formulae from ‘Diffusion flames and fire plumes’ 2

Data on the axis (experiment: 0,3 m2)

Velocity 01/ 5 2 / 5

u zkQ Q

η⎛ ⎞= ⎜ ⎟⎝ ⎠

Temperature 2 12

02 / 5

0

2g T k zT C Q

η −Δ ⎛ ⎞⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Region 2 / 5z Q k η C

Flame < 0,08 6,8 1/2 0,9 Intermittent 0,08 – 0,2 1,9 0 0,9 Plume > 0,2 1,1 -1/3 0,9

4.3.3 Upward flows Upwards mass flow rate in plume

[ ]1/ 3 5 / 3 2 / 3 5 / 30,071 1 0,026conv convm Q z Q z kg−⎡ ⎤= +⎣ ⎦ s

4.3.4 Interaction of the fire plume with compartment boundaries Wall

Ceiling

Gas temperature below ceiling at distance r from the plume axis

( )2 / 3

max

2 / 3

max 5 / 3

5,380,18

16,90,18

c

c

Q rr H T T

HQr H T T

H

∞

∞

> − =

⋅≤ − =

4.3.5 The effect of wind on the fire plume

Summary of formulae from ‘Diffusion flames and fire plumes’ 3

4.4 Some practical applications

4.4.1 Radiation from flames Heat flux at distance R from a point source:

2

22 2

cos4

2

fr c

Aq m HR

lR d

χ θπ

′′ ′′= Δ

⎛ ⎞= +⎜ ⎟⎝ ⎠

With Af = fuel area Empirical formula by Shokri en Beyler (with d = distance to target)

1.592

, 15,4r Tdq kD

−⎛ ⎞′′ W m⎡ ⎤= ⎜ ⎟ ⎣ ⎦⎝ ⎠

4.4.2 The response of ceiling-mounted fire detectors

1/ 20 0RTI uτ=

τ ∞⎛ ⎞−= ⋅ ⎜ ⎟−⎝ ⎠

max

max

lnL

T TtT T

max max

max,0 0

L LT T TT T TT T T

∞

∞

∞

Δ = −Δ = −Δ = −

[ ]

[ ]

1/ 3 1/ 2

max 5 / 6

1/ 3

max

0,1970,18

0,18 0.946

Q Hr H u mr

Qr H u m sH

> =

⎛ ⎞≤ = ⎜ ⎟

⎝ ⎠

s

Summary of formulae from ‘Diffusion flames and fire plumes’ 4