Flow Measurements BMEGETMW03 laser-optical flow measurements - - PDF document

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Flow Measurements BMEGEÁTMW03 “laser-optical flow measurements”

Handout by Jenő Miklós SUDA, PhD suda@ara.bme.hu

• Dept. Fluid Mechanics BME

DOWNLOAD from: http://www.ara.bme.hu/oktatas/tantargy/NEPTUN/BMEGEATMW03

Contents, main topics of the lecture:

• 1. Introduction, basic questions: what to measure, why, how
• 2. Lasers
• 3. Advantage / disadvantage of the technique
• 4. Characterisation of particle-laden mixtures
• 5. Particle dynamics, equation of motion
• 6. Seeding / tracer problematic
• 7. LDV, PDA, PIV, PTV(S)
• 4. Characterisation of particle-laden mixtures

Aerosols

Definition of aerosols: Aerosols are defined as being gas-particle mixtures in quasi-stable state. The mixture contains partly gas as primary (or carrier) phase and partly solid or liquid particulate matter (as secondary phase). The quasi-stable state means that the characteristics of the mixture in a given volume (e.g. particle number, mass concentration) do not change significantly, i.e. “nearly stable” in time. Changing of the characteristics of the mixture can occur due to the:

• settling out of larger particles from the given volume of the mixture, or
• diffusion and agglomeration of the small particles.

Both may cause increase or decrease of mass of the particles in the fixed volume, hence may cause changing of the characteristics of the mixture. Diameter (x [µm]) range of the particles in aerosols: 0,01µm ≤ x ≤ 50 µm Note, that the lower & upper limiting values are not strict limiting values: “0,01µm” & “50µm” means that approx. a few hundreds & few times ten microns can be considered as the limits of the diameter range of aerosols.

Note: 1µm = 10-3 mm = 10-6 m The resolution of sensitivity of a human finger tip is about 40 microns. The human hair’s diameter is 40÷100 microns. The average height of surface roughness of a bearing ball is aprrox. 0,01 micron. The diameter of seeding particles for Laser Doppler Velocimetry and flow visualization (spherical oil smoke droplets) is approx. 1÷3 microns.

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2 Types and sizes of particles Dust: size range: x ≥ 0,2 [µm] description: solid particles, produced by breaking or attrition, abrasion, wearing of solid substances, perceptible to the eye, the diameter is larger than the wave length of light. Smoke (fume): size range: x ≤ 1 [µm] description: solid or liquid particles or droplets, originated from condensation or chemical reaction, in most cases chain-like structures. Produced at combustion, chemical processes etc. Mist (fog): size range: 0,1 ≤ x ≤ 200 [µm] description: liquid droplets originated from steam condensation or by atomization, spraying. The mist droplets and the saturated steam of that liquid are in equilibrium state.

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4 Size of particles: In case of spherical particles the diameter is denoted by x. Elsewhere usually it is denoted by dp. How to define the size of non-spherical particles? It is needed to introduce the equivalent diameter. Various types of equivalent diameter can be defined based on – geometrical, – aerodynamic and – optical equivalence. For particle dynamics the most relevant is to know the xae aerodynamic equivalent diameter, that is defined to be the diameter of a spherical particle from the same material (ρp=same) as the real particle, settling with the same ws settling velocity in the same gas (ρg=same).

Aerodynamic equivalent diameter, xae

Average relative distance (a/x) between neighboring particles in gas: Let’s calculate the c [kg/m3] mass concentration of n particles evenly distributed in a particle-gas mixture having a volume of Vg+p. Let’s assume that each particle is sitting in the center of a cube. (see Figure below). The concentration can be calculated:

3 p 3 3 p 3 p g p p p g p p g p

a 6 x a n 6 x n V V V m V m c ρ ⋅ π ⋅ = ⋅ ρ ⋅ π ⋅ ⋅ = ρ ⋅ = ≅ =

+ + +

∑ ∑ ∑

where c [kg/m3] mass concentration, a [m] average distance between particles, ρp [kg/m3] density of particles, n is number of particles. For the average relative distance (a/x) between neighboring particles in gas we get:

3 p

c 6 x a ⋅ π ⋅ ρ = , a Vcube x a a

ws ρp xae g ρg

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5 Example: If we consider monodisperse particle distribution where all particles have x=3µm diameter with ρp=2000kg/m3, and c = 10 g/m3 (that is relatively very high concentration of particles), the a/x = 47. a) It means that aerosols are very dilute mixtures: the neighboring particles are far from each other (approx. a=5ms when x=10cm would be the particle diameter). Hence, the possibility of collision, momentum exchange between two particles is relatively small. b) In 1cm3 volume there are 350000 particles for c=10 g/m3. Even in case of small 0,1 g/m3 concentration we get 3500 particle in 1cm3 volume. Notwithstanding that it is a dilute mixture the number of particles is very high in given volume even for low concentration. That is important to know when very strict demand is defined on air quality (e.g. at surgery rooms, when dealing with toxic or infective particles is concerned) Conclusion:

• in case of usual particle concentration values the particle-laden flows are very dilute mixtures.

(the distance between neighboring particles is very large).

• particles are present with very high number even in particle-gas mixtures having very low

concentration. Characterization of particle assembly: Particle size distribution curves: Considering polydisperse particle distribution with size range of xmin < x < xmax Cumulative or undersize distribution related to number of particles: Q0=N/Ntot =f(x). Q1, Q2, Q3. Subscript denotes: 0: related to number of… 1: related to length of... (1D – one dimensional quantity) 2: related to surface of…(2D – two dimensional quantity) 3: related to volume or mass of…(3D – three dimensional quantity) If Q0=N/Ntot=f(x) and the overall number of particles Ntot are known, the number of particles in the size range between [x] and [x+∆x] can be calculated: x dx dQ N N

tot

∆ ⋅ ⋅ = ∆ . Cumulative or undersize distribution related to number of particles: Q0. The value of Q0(x) for given x gives information how many percentage of Ntot particles have smaller diameter that x. Taking the tangent of the Q0 curve (see Figure below) is denoted by: q0= dx dQ0 , we get for the number of particles in the range between [x] and [x+∆x]: x q N x dx dQ N N

tot tot

∆ ⋅ ⋅ = ∆ ⋅ ⋅ = ∆ c[g/m3] x a N [db/cm3] 10 47 350.000 1 101 35.000 0.1 218 3.500

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6 Relation/conversion between distributions related to various quantities: For example conversion of Q0 to Q3:

( )

∫ ∫ ∫ ∫

= π π =

max min min max min min

x x 3 x x 3 x x tot 3 x x tot 3 3

dx q x dx q x dx dx dQ N 6 x dx dx dQ N 6 x x Q . When q0 cumulative distribution related to the particle number is given (see Figure below), we can

• btain the average diameter of the particle distribution (

x ) related to particle number:

∫ ∫

⋅ = ⋅ ⋅ =

max min max min

x x x x tot tot

dx q x dx q N x N 1 x .

N Ntot xmin xmax x[µm] 1 0.5 x50,0 Q0=N/Ntot N(x) Q0(x)

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• 5. Particles in gas flow: particle dynamics

Effect of particles on the gas flow Navier-Stokes equation extended with considering the influence of the particles’ forces acting on the carrier gas phase: t v gradp 1 g v rot x v 2 v grad t v

g 2

+ ∆ ν + ρ − = − + ∂ ∂ where [

]

gas

kg N t / is the force acting to the gas from particles in 1 kg of gas:

g

F n t ρ − = n[piece/m3]: particle number concentration

[ ]

piece / N F : aerodynamic force acting on one particle

[ ]

3

m / kg ρ : gas density The effect of particle phase on the flow field can be neglected, if 1 c

g

〈〈 ρ and the particle acceleration dt u d

p is in the same order of magnitude as the carrier gas-phase acceleration dt

v d . Therefore in case of dt v d dt u d c

p g

〈〈 ρ , the effect of particle phase on the flow field can be neglected. From other viewpoint we can say that in this dilute mixture one single particle moving in the gas cannot change the gas’ momentum, but one particle’s movement is influenced by the carrier gas, see in next chapter: defining the aerodynamic force acting on the particles. Aerodynamic (drag) force acting on a single particle: Particle Reynolds number: ν ⋅ = x w Rep is small, viscosity is dominant in a fluid flow around the particle (using the relative co-ordinate system fixed to the particle) Stokes: If Rep < 0.1, the well known Stokes relation for Fd drag force acting on a particle (having diameter x) moving with w relative velocity in the gas (µ is dynamic viscosity) is: w x 3 F F

Stokes d

⋅ ⋅ µ ⋅ π ⋅ = = where w u v + = v absolute (gas) velocity u particle velocity w relative velocity Drag coefficient of the particle?

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8 4 x w 2 F A w 2 F c

2 2 d ref 2 d d

π ρ = ⋅ ρ = Substituting the Stokes relation for Fd into equation of cd, we get a very simple form for the drag coefficient of a sphere:

p d

Re 24 c = . The above form can be used only in cases when the particle Reynolds-number is lower than 0.01. Researchers in this field obtained various forms for particle Reynolds-number corrected Stokes’ drag coefficient based on further experiments or nowadays morely based on numerical simulations. For example: a) the Oseen’s relation: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + =

p p e

Re 16 3 1 Re 24 c , that is valid when Rep < 5. b) the Michaelides’s relation:

( )

687 , p p d

Re 15 , 1 Re 24 c ⋅ + ⋅ = is valid when 0.1 < Rep < 1000. Momentum equation for particles moving with w relative velocity in gas Due to Newton’s 2nd law the particle’s momentum equals to the sum of the acting forces. Forces: gravity force and drag force (Usually we may neglect the bouyancy force).

d g p

F F dt u d m + = w x 3 g 6 x dt u d 6 x

p 3 p 3

µ π + ρ π = ρ π Dimensionless equation of motion of the particle: As a usual non-dimensionalizing formulation procedure let’s multiply the equation with

2

v l , where l0 is a characteristic length (e.g. l0=x), and v0 is a characteristic velocity (e.g. v0= average gas flow velocity)

2 p 3 p 3

v l w x 3 g 6 x dt u d 6 x ⋅ µ π + ρ π = ρ π . Then we get the form of equation of motion:

p 2 2

v w v l x 18 v l g v / l t d v u d ρ µ + = . Dimensionless momentum equation for particles ( ' denotes dimensionless quantities, e.g. v u u = ′ ).

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9 w v l x 18 v l g t d u d

p 2 2

′ ρ µ + = ′ ′ Settling velocity of the particle (ws): Settling of particle of

p

ρ density in a gas of ρg density:

s g 3 p 3

w x 3 g 6 x g 6 x µ π + ρ π = ρ π Settling velocity: µ ρ − ρ = 18 g ) ( x w

p 2 s

. 18 g x w if

p 2 s p

µ ρ = ⇒ ρ 〉〉 ρ Correction of settling velocity due to the diffusion effect in submicron size-range: ws,corr= Cu · ws where x A 2 1 Cu λ + = is the Cunningham coefficient (or Cunningham correction factor), where A≈1.4, and λ is the mean free path of molecules, at room-temperature λ = 6.5 ∗10-2 µm).

0,00000001 0,0000001 0,000001 0,00001 0,0001 0,001 0,01 0,1 1 0,01 0,1 1 10 100 Szemcseátmérő, d p [µ m ] w s [m/s ] 1 10 100 Cu [-] ρ p= 3000 kg/m3 2500 2000 1500 1000 Cunningham-tényező Cu (d p) Cu -tényezővel korrigált w s görbék

ws settling velocity (cont. lines) ad ws,corr corrected settling velocity (dashed lines) Settling velocity as function of particle diameter and density

By neglecting the effect of the gravity field strength the dimensionless equation of motion of particles will turn to another form using the ws settling velocity: ′ ⋅ = ′ ρ µ ≅ ′ ρ µ + = ′ ′ w v w l g w v l x 18 w v l x 18 g v l t d u d

s p 2 p 2 2

w v w l g t d u d

s

′ ⋅ = ′ ′ Introducing ψ inertia parameter will help us to evaluate the particle motion in gas flow:

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10

s

l g v w ⋅ = ψ Dimensionless momentum equation for particles with inertia parameter: ) u v ( 1 w 1 t d u d ′ − ′ ψ = ′ ψ = ′ ′ case A) dashed line in the upper figure When ψ→0, for small (x) and/or light (ρp) particles, which settling velocity is small, or ws → , and if ∞ → ′ ⇒ ≠ − ′ t d ' u d ) ' u v ( , hence particle move along the gas streamline, particle follow the carrier gas flow. case B) dash-dot line in the upper figure When ψ→∞, for large and/or heavy particles, which settling velocity is large, . , 1 → ′ ′ → t d u d ly consequent ψ hence particle move along its initial path, leaving the gas streamline.

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11 APPENDIX Relative velocity of the particle: u w − = v Particle REYNOLDS-number (

p

Re ) µ ρ ν

g p p p

d w d w Re ⋅ ⋅ = ⋅ =

0,000001 0,00001 0,0001 0,001 0,01 0,1 1 10 100 0,01 0,1 1 10 100

Porszemcse átmérő, d p [µ m ] Reynolds-szám, Re p [-]

w= 0,0001 m/s w= 0,001 m/s w= 0,01 m/s w= 0,1 m/s w= 0,2 m/s w= 0,5 m/s w= 0,8 m/s w= 1 m/s w= 2 m/s w= 5 m/s w= 8 m/s w=10 m/s dp=1.54 mikron

+ sign: data for particle generated by a professional SAFEX smoke generator for LDA measurements (dp=1.5µm) STOKES-formula for spherical particle ( 25 , <

p

Re ) so-called STOKES-regime: w F

p e

d µ π 3 =

p e

Re c 24 = OSEEN ‘s formula to extend the validity above Stokes-regime: ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⋅ =

p p e

Re Re c 16 3 1 24 5 <

p

Re ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⋅ = 6 1 24

3 2

p p e

Re Re c 400 3 < <

p

Re same by MICHAELIDES (1997):

( )

687 ,

15 , 1 24

p p e

Re Re c ⋅ + ⋅ = 1000 1 , < <

p

Re

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12 Charatceristic parameters of the primary & secondary phase:

• carrier fluid (primary phase)
• seeding /tracer particles (secondary phase)

Volume ratio:

p p p g p p

c a d V V ρ π α = = =

3 3

6 Mass loading ratio:

g p p g p

c M ρ ρ α ρ = =

g p p

M ρ ρ α = , or

p g p

M ρ ρ α = where: cp: particle mass concentration ρg: density of gas (carrier phase) ρp: density of particle (material)

ρp [kg/m

3]

ρg[kg/m

3]

αp

800 1500 2500

M

0,8 1,0 1,2 0,0001 1,3·10

• 10

6,7·10

• 11

4,0·10

• 11

0,0001 1,3·10

• 7

1,0·10

• 7

8,3·10

• 8

0,001 1,3·10

• 9

6,7·10

• 10

4,0·10

• 10

0,001 1,3·10

• 6

1,0·10

• 6

8,3·10

• 7

0,01 1,3·10

• 8

6,7·10

• 9

4,0·10

• 9

0,01 1,3·10

• 5

1,0·10

• 5

8,3·10

• 6

0,1 1,3·10

• 7

6,7·10

• 8

4,0·10

• 8

0,1 1,3·10

• 4

1,0·10

• 4

8,3·10

• 5

1 1,3·10

• 6

6,7·10

• 7

4,0·10

• 7

1 1,3·10

• 3

1,0·10

• 3

8,3·10

• 4

10 1,3·10

• 5

6,7·10

• 6

4,0·10

• 6

10 1,3·10

• 2

1,0·10

• 2

8,3·10

• 3

cp

[g/m

3]

100 1,3·10

• 4

6,7·10

• 5

4,0·10

• 5

cp

[g/m

3]

100 1,3·10

• 1

1,0·10

• 1

8,3·10

• 2

c=0,0001 g/m3 c=0,001 g/m3 c=0,01 g/m3 c=0,1 g/m3 c=1 g/m3 c=10 g/m3 c=100 g/m3 1E-08 1E-07 1E-06 1E-05 1E-04 1E-03 1E-02 1E-01 1E+00 1E-11 1E-10 1E-09 1E-08 1E-07 1E-06 1E-05 1E-04 1E-03

Térfogati arány, α p [-] Tömegarány, M [-]

800kg/m3 1500kg/m3 2500kg/m3

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1 10 100 1000 10000 1 10 100 1000 10000 Sűrűség, ρp [kg/m 3 ] 0,0001 0,001 0,01 0,1 1 1,2 kg/m3 1,0 kg/m3 0,8 kg/m3

Table ρp [kg/m

3]

ρp [kg/m

3]

a / dp

800 1500 2500

N [db/mm

3]

800 1500 2500 0,0001 1612 1988 2357 0,0001 0,07 0,04 0,02 0,001 748 923 1094 0,001 0,7 0,4 0,2 0,01 347 428 508 0,01 7 4 2 0,1 161 199 236 0,1 71 38 23 1 75 92 109 1 707 377 226 10 35 43 51 10 7074 3773 2264

cp

[g/m

3]

100 16 20 24

cp

[g/m

3]

100 70736 37726 22635

ELGHOBASHI (1994): „Turbulence modulation map”: particle STOKES-number (Stp=τp/τe) in function of the αp τp: characteristic (response) time of the particle τe: characteristic time of the carrier fluid dilute mixtures:

3

10− <

p

α dense mixtures:

3

10− >

p

α

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0,000001 0,00001 0,0001 0,001 0,01 0,1 1 10 100 0,01 0,1 1 10 100

Porszemcse átmérő, d p [µ m ] Stokes-szám, St p [-]

800 kg/m3, v0= 1,0 m/s 1000 kg/m3, v0= 1,0 m/s 1500 kg/m3, v0= 1,0 m/s 2000 kg/m3, v0= 1,0 m/s 2500 kg/m3, v0= 1,0 m/s 3000 kg/m3, v0= 1,0 m/s dp=1.54 mikron

Irodalom: ELGHOBASHI, S.E. (1994) On predicting particle-laden turbulent flows. Appl. Sci. Res. Vol. 52, pp.309-329. MICHAELIDES, E.E. (1997) Review – The transient equation of motion for particles, bubbles and

• dropets. Transactions of the Americal Society of Mechanical Engineers, J. Fluids Eng., Vo..

119, pp.233-247.

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15

0,000001 0,00001 0,0001 0,001 0,01 0,1 1 10 100 0,01 0,1 1 10 100

Porszemcse átmérő, d p [µ m ] Reynolds-szám, Re p [-]

w= 0,0001 m/s w= 0,001 m/s w= 0,01 m/s w= 0,1 m/s w= 0,2 m/s w= 0,5 m/s w= 0,8 m/s w= 1 m/s w= 2 m/s w= 5 m/s w= 8 m/s w=10 m/s dp=1.54 mikron

w F

p Stokes

d µ π 3 = , µ ρ ν

g p p p

d w d w Re ⋅ ⋅ = ⋅ = , 4 2

2 2

π ρ

p g Stokes d

d w F c = ,

p d

Re c 24 = OSEEN ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⋅ =

p p d

Re Re c 16 3 1 24 5 <

p

Re (1) OSEEN ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + ⋅ = 6 1 24

3 2

p p d

Re Re c 400 3 < <

p

Re (2) MICHAELIDES

( )

687 ,

15 , 1 24

p p d

Re Re c ⋅ + ⋅ = 1000 1 , < <

p

Re (3) Settling velocity of the particle (and from when effect of buoyancy force is neglected): µ ρ µ ρ ρ 18 18 ) (

2 2

g d g d w

p p p g p s

≅ − =

0,00000001 0,0000001 0,000001 0,00001 0,0001 0,001 0,01 0,1 1 0,01 0,1 1 10 100 Szemcseátmérő, d p [µ m ] w s [m/s ] 1 10 100 Cu [-] ρ p= 3000 kg/m3 2500 2000 1500 1000 Cunningham-tényező Cu (d p) Cu -tényezővel korrigált w s görbék