A Review of Critical Flow Venturis

Sonic Nozzles

Aamth

RTcc

av

p

0

*R0

th RTCAPm M

zRTPM

y = -4.5881x + 1.0002

0.976

0.978

0.98

0.982

0.984

0.986

0.988

0.99

0.992

0.002 0.0025 0.003 0.0035 0.004 0.0045 0.005

Re-1/2

Cd

0.976

0.978

0.98

0.982

0.984

0.986

0.988

0.99

0.992

0 50000 100000 150000 200000

Re

Cd

Discharge Coefficient Cd

0

*R0

th RTCAPm M

dth

PS Cmm

PSm

Isentropic, 1-Dimensional Flow, Perfect Gas

Velocity at each cross section of a convergent-divergent critical venturi (Reynolds 1886, Rayleigh 1916)

121

2

121

*

211

211

,1,

MaMaMaf

fAA

Isentropic, 1-Dimensional Flow, Perfect Gas, Fully Expanded (no shocks)

0TT

0PP

)75.0(aw RT

Inviscid Core FlowSmith and Matz 1962, “A non-one-dimensional flow exists because of the centrifugal forces created by the turning of the flow in the contraction section.”

1-D:

2-D:

Hall 1962, Kliegel and Levine 1969 0.12 %

44

33

22

cored, 1

C where 2, 3, and 4 are gas species dependent components and

Λ is the expansion parameter (R or 1 + R).

Velocity Boundary Layer: Cd scales with Re-1/2

x

x.Re72081

Blasius: boundary layer on a flat plate

Laminar: Tang 1969, Geropp 1971, similarity transformations

Turbulent: Stratford 1964, integral boundary layer technique

Mickan 2006

Transition at Re 1 x 106

nmnm aaC 2221bld, ReRe1

where Ω = r*/R is the throat curvature ratio (nominally 0.25 for an ASME / ISO venturi), a1 and a2 are coefficients, and m and n are exponents whose values depend on whether the flow is laminar or turbulent.

0.955

0.960

0.965

0.970

0.975

0.980

0.985

0.990

0.995

1.000

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014

Re -1/2

mex

pt /

m2 =

Cd

0.955

0.960

0.965

0.970

0.975

0.980

0.985

0.990

0.995

1.000

0.01 0.1 1 10 100 1000

m expt (g/s)

mex

pt /

m2

0.955

0.960

0.965

0.970

0.975

0.980

0.985

0.990

0.995

1.000

0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06

Re

mex

pt /

m2

21Re /dC

Reynolds Number Scaling

Boundary Layer Transition

Ishibashi and Arnberg, The Effect of Inlet Geometry on the Critical Flowrate of Toroidal Throat Venturi Nozzle, CFVN Workshop, Quedlinburg, Germany, June, 2005

Analytical Cd predictions agree well with experiments

Johnson and Wright 2008

Mickan 2006

Gas Species Effects

• Solid lines from Nakao and Takamoto, Discharge Coefficients of Critical Flow Venturi Nozzles for CO2 and SF6, Transactions of the ASME, December, 2000, d = 0.295 to 2.36 mm.

• Points from NIST experiments, d = 0.387 mm.

Cd can be treated as numerous uncoupled physical phenomena

Cd = CR*Cinv Cvbl CTbl C Cvib + higher order terms

Vibrational relaxation (CO2 , SF

6 )

Thermal expansion of CFV material

Thermal boundary layer

Velocity boundary layer

Real gas effectsInviscid core flow