Method and system for evaluation of fire suppression systems performance
An initial (near-field) spray generated by a sprinkler under study is fully characterized using a laser-supported Shadowgraphy/PTV system. Near-field spray characteristics are established from local measurements, which are mapped in a spherical coordinate system consistent with the kinematics of the spray. A novel data compression scheme is introduced to generate analytical functions describing the sprinkler spray based on the measurements. These analytical functions are useful for initiating the sprinkler spray in computational fluid dynamics (CFD) based spray dispersion and fire suppression modeling. The near-field spray measurements and associated data compression approach are validated by comparing volume density measurements 1 meter below the sprinkler with volume density predictions generated from spray dispersion calculations initiated with the analytical spray functions.
1. A method for evaluating a fire suppression system performance, comprising the steps of:
(a) measuring near-field characteristics of an initial spray generated by a sprinkler under study equipped with a sprinkler deflector, said characteristics being measured in vicinity of the sprinkler discharge at various operating conditions and different geometries of the sprinkler deflector, said near-field characteristics including 3-dimensional characterization of volume flux distributions, drops' size distributions, and drops' velocity distributions in said initial spray azimuthally correlated with geometric features of said sprinkler deflector;
(b) entering said measured characteristics of the initial spray in a computer system configured to process said measured characteristics in accordance to a measurement analysis algorithm,
specifying said initial spray by a number of individual drops determined based on the measured characteristics attained in said step (a), wherein said number of drops is generated on a surface of a sphere originated from a center of said sprinkler deflector with a radius equal to an initiation distance for complete spray formation,
assigning azimuthal angle ψ, elevation angle θ, dimensionless drop size d, and dimensionless drop velocity V to each of said number of individual drops, and
describing the initial spray in terms of the volume probability density as
∫ θ ∫ ψ ∫ u ∫ d f V (θ,ψ, u,d ) dθ·dψ·du·dd= 1,
where the azimuthal angle ψ is determined by randomly choosing an outcome space ranging from 0 to 1, and selected according to the cumulative distribution function:
F V (ψ′)=∫ 0 ψ′ f V (ψ) dψ,
where f V (ψ) represents the volume probability density for ψ integrated over elevations angles θ, drops sizes, and drops velocities in:
f V (ψ)=∫ θ ∫ u ∫ d f V (θ,ψ, u,d ) dθ·du·dd ; and
(c) transforming said 3-dimensional characterization of said initial spray into a compact representation thereof through the steps of:
generating analytical functions describing spatial variation of said drops' density, size, and velocity in correspondence to an elevation angle, wherein said analytical functions include Legendre polynomials, Gaussian functions, and Fourier series, each defined through respective coefficients determined by processing said measured near-field characteristics, wherein said respective coefficients provide average values and profile shapes for said measured spray characteristics, for said Legendre polynomials and Gaussian functions, and wherein said respective coefficients are determined from the sprinkler's nozzle geometry for said Fourier series.
2. The method of claim 1 , further comprising the step of:
in said step (c), generating a product database for said sprinkler under study.
3. The method of claim 1 , further comprising the step of:
in said step (a), applying Shadowgraphy/PTV (Particle Tracking Velocimetry) measurements.
4. The method of claim 1 , further comprising the step of:
in said step (a), directing a pulsed laser beam onto the initial spray,
focusing a digital camera on the initial spray,
synchronizing the pulsed laser and the digital camera actuation to acquire double images of drops in said initial spray separated by a predetermined image separation time interval,
applying spatial calibration and image processing to said double images of drops to result in drops' sizes in each said double image, and
acquiring drops' velocities through comparison of drops' trajectories obtained from said double images and said image separation time interval.
5. The method of claim 1 , further comprising the step of:
in said step (b), combining individual images and said multiple imaging areas.
6. The method of claim 1 , further comprising the steps of:
in said step (a), measuring radial volume density distributions; and
in said step (b), obtaining, in said computer system, a maximum density radial location R of said initial spray as:
R=U (2 h/g ) 1/2 ,
where h is the measurement elevation below a nozzle of the sprinkler, g is a gravitational constant, and U is the maximum initial spray velocity, and
calculating, in said computer system, a linear density of a dispersed volume flow q′ as:
q
′
=
q
″
Q
/
π
R
2
(
2
r
′
)
,
where q″ is the area volume density, and Q is the nozzle flow rate.
7. The method of claim 1 , further comprising the steps of:
specifying the elevation angle θ through random selection from the outcome space ranging between 0 and 1 of the conditional probability cumulative distribution function:
F V (θ|ψ′)=∫ 0 θ f V ({dot over (θ)}|ψ′) d {dot over (θ)}
where (θ/ψ′) represents the conditional volume probability density of θ at a specific azimuthal area ψ′ given by:
f
V
(
θ
|
ψ
′
)
=
∫
u
∫
d
f
V
(
θ
,
ψ
′
,
u
,
d
)
ⅆ
u
·
ⅆ
d
∫
θ
∫
u
∫
d
f
V
(
θ
,
ψ
′
,
u
,
d
)
ⅆ
θ
·
ⅆ
u
·
ⅆ
d
.
8. The method of claim 2 , further comprising the step of:
incorporating said product database into a fire suppression system design system.
9. The method of claim 4 , further comprising the steps of:
in said step (a), traversing and rotating said sprinkler to form an extended spherical interrogation region covering multiple imaging areas, each azimuthally aligned with the geometry features of said sprinkler deflector.
10. The method of claim 7 , further comprising the step of:
in said step (c), calculating, in said computer system, a local drop size distribution as:
F V ( d |θ′,ψ′)=CVF( d )=∫ 0 d f V ({dot over ( d )}|θ′,ψ′) d{dot over (d)}
where CVF(d) is a Cumulative Volume Fraction.
11. The method of claim 10 , further comprising the step of:
in said step (c) calculating, in said computer system, a local drop velocity as:
F V ( u|θ′,ψ′,d ′)=∫ 0 u f V ( {dot over (u)}|θ′,ψ′,d ′) d{dot over (u)}.
12. The method of claim 11 , further comprising the steps of:
in said step (a), measuring azimuthally variable characteristics of said initial spray and
in said step (c), where said sprinkler deflector has a plurality of tines and spaces, applying Fourier series to said measured azimuthally variable characteristics to calculate a continuous interpolation function between said characteristics measured for adjacent spaces and tines, as:
F
V
(
ψ
)
=
A
(
ψ
)
F
V
(
ψ
Tine
)
+
(
1
-
A
(
ψ
)
)
F
V
(
ψ
Space
)
A
(
ψ
)
=
a
0
2
+
∑
n
=
1
∞
a
n
cos
n
π
T
/
2
ψ
where T is the angle sum of an individual tine and individual space,
α o and α n are Fourier coefficients for a square wave determined from said deflector geometry calculated for a first tine of the deflector as:
a
n
=
2
T
∫
-
T
tine
/
2
T
tine
/
2
cos
n
π
T
/
2
ψ
ⅆ
ψ
.
13. The method of claim 12 , further comprising the steps of:
in said step (c), determining the elevation angle locations of random drops in said initial spray as:
f
V
(
θ
|
ψ
′
)
=
f
0
2
π
σ
exp
(
(
θ
-
θ
0
)
2
2
σ
2
)
+
∑
n
=
0
∞
C
n
(
θ
)
P
n
(
cos
(
θ
)
)
,
where f o is a magnitude of the local volume flux peak,
θ 0 is an elevation angle location of the local volume flux peak,
σ characterizes a width of the local volume flux peak,
P n are the Legendre polynomial coefficients determined from the measurements performed in said step (a).
14. The method of claim 13 , further comprising the steps of:
wherein in said step (c), said compact basis functions describe the local characteristic drop size, d v50 , and distribution width parameter, γ, as:
f
(
d
v
50
|
θ
′
,
ψ
′
)
=
∑
n
=
0
∞
C
n
(
d
v
50
)
P
n
(
cos
(
θ
′
)
)
,
and
f
(
γ
|
θ
′
,
ψ
′
)
=
∑
n
=
0
∞
C
n
(
γ
)
P
n
(
cos
(
θ
′
)
)
wherein local drop size distributions are generated from said d v50 and γ by applying a combined Log-Norman-Rosin-Ramnler function:
CVF
(
d
)
=
{
1
2
π
∫
0
d
γ
/
1.15
d
.
exp
(
-
(
ln
(
d
.
/
d
v
50
)
)
2
2
(
1.15
/
γ
)
2
)
ⅆ
d
.
(
d
<
d
v
50
)
1
-
exp
(
-
0.693
(
d
/
d
v
50
)
γ
)
(
d
>
d
v
50
)
.