Flow Meter Element

Description and Quick Guide

The Flow Meter element uses a change in area to produce a change in velocity and thus a change in static pressure. The static pressure can be measured by pressure taps and used to determine the flow rate in the tube. The Flow Meter element in Flow Simulator can account for the pressure drop through the device. The flow measurement is not the main purpose of this element since all Flow Simulator elements return a flow rate.

There are four types of flow meters that can be modeled with this element. You can choose different loss correlations (Cd) for each type.

Thin Orifice Plate
Venturi
Nozzle
Venturi-Nozzle

The orifice plate has the highest overall total pressure drop but it is popular due to its simplicity and low cost. The other meter types have much less total pressure drop but have a longer length and higher cost.

There are very specific flow and geometry guidelines for the flow meter (ref 1). The loss correlations are only valid if these guidelines are followed.

This element can be used for compressible gas or incompressible liquid. The compressible gas element does check for flow choking (Mach 1 in the vena-contracta) and limits the flow rate if choking occurs. This element does not consider rotation or elevation change.

Flow Meter Element Inputs

Table 1. Element Specific Input Variables
Index	UI Name (.flo label)	Description
1	Subtype (SUBTYPE)	The type of flow meter. Thin Orifice Plate Venturi Nozzle Venturi-Nozzle
2	Cd Equation (LOSS_OPTION)	The discharge coefficient equation. The options available depend on the Subtype. 0: Specified Cd 11: ISO 12: Reader-Harris 21: Cast 22: Machined 23: Welded 31: ISA 1932 32: Long Radius 40: ISO
3	Cross-Section Shape (CS_SHAPE)	The type of geometry inputs. 1: Circular with Area Specified 2: Circular with Diameter Specified
4	Pipe Geometry Option (PIPE_GEOM_OPT)	Options for pipe geometry information. 0: User Input 1: Automatic The automatic option uses the diameter of the elements attached to the flow meter element.
5	Throat Area (THROAT_AREA)	Only used if CS_SHAPE = 1. The minimum physical area of the flow meter. This is the orifice diameter for an orifice plate subtype (in^2).
6	Pipe Area (PIPE_AREA)	Only used if CS_SHAPE = 1 and PIPE_GEOM_OPT = 0. The pipe area (in^2).
7	Throat Diameter (THROAT_DIA)	Only used if CS_SHAPE = 2. The minimum physical diameter of the flow meter. This is the orifice diameter for an orifice plate subtype (in).
8	Pipe Diameter (PIPE_DIA)	Only used if CS_SHAPE = 2 and PIPE_GEOM_OPT=0. The pipe diameter (in).
9	Forward Flow Cd (CD_FWD)	Only used if LOSS_OPTION = 0. The user-specified discharge coefficient for forward flow.
10	Reverse Flow Cd (CD_REV)	The user-specified discharge coefficient for reverse flow. The flow is not typically reversed through a flow meter.
11	Portion of Ustrm Cham. Dyn. Head Lost (DQ_IN)	Inlet dynamic head loss. Refer to the General solver theory sections for more details about this input.
12	Element Inlet Orientation: Tangential Angle (THETA)	Angle (deg) between the element center line at the entrance of the element and the reference direction. If the element is rotating or directly connected to one or more rotating elements, the reference direction is defined as parallel to the engine center line and the angle is the projected angle in the tangential direction. Otherwise, the reference direction is arbitrary but assumed to be the same as the reference direction for all other elements attached to the upstream chamber. Theta for an element downstream of a plenum chamber has no impact on the solution except to set the default value of THETA_EX. See also THETA_EX.
13	Element Inlet Orientation: Radial Angle (PHI)	Angle (deg) between the element center line at the entrance of the element and the THETA direction (spherical coordinate system). Phi for an element downstream of a plenum chamber has no impact on the solution except to set the default value of PHI_EX. See also PHI_EX.
14	Element Exit Orientation: Tangential Angle (THETA_EXIT)	Angle (deg) between the element exit center line and the reference direction. THETA_EX is an optional variable to be used if the orientation of the element exit differs from that of the element inlet. The default value (THETA_EX = -999) results in the assumption that THETA_EX = THETA. Other values are interpreted in the manner presented in the description of THETA.
15	Element Exit Orientation: Radial Angle (PHI_EXIT)	Angle (deg) between the element exit center line and the THETA_EX direction. PHI_EX is an optional variable to be used if the orientation of the element exit differs from that of the element inlet. The default value (PHI_EX = -999) results in the assumption that PHI_EX = PHI. Other values are interpreted in the manner presented in the description of PHI.
16 17 18	Exit K Loss: Axial (K_EXIT_Z) Tangential (K_EXIT_U) Radial (K_EXIT_R)	Head loss factors in the Z, U, and R directions based on the spherical coordinate system of theta and phi. Z = the axial direction. (theta=0 and phi=0) U = the tangential direction. (theta=90 and phi=0) R = the radial direction. (theta=0 and phi=90) Valid values of K_EXIT_i (i = Z, U, R) range from zero (default) to one. The three loss factors reduce the corresponding three components of velocity exiting the element. $V_{a c t u a l e x i t i d i r} = V_{n o l o s s e x i t i d i r} * \sqrt{1 - K_E X I T_i d i r}$ Default value provides no loss, K_EXIT_i=0.
19	Fluid Compressibility Mode (FLUID_MODE)

Flow Meter Element Theory

The flow rate and Cd equations can be found in the references.

Flow Rate

\dot{m} = C d * E * A r e a_{t h r o a t} * \sqrt{2 * ρ_{u p} * Δ P}

for incompressible liquid

\dot{m} = C d * Y * E * A r e a_{t h r o a t} * \sqrt{2 * ρ_{u p} * Δ P}

for compressible gas

Where:

C d = \frac{a c t u a l \dot{m}}{i d e a l \dot{m}} = d i s c h a r g e c o e f f i c i e n t

E = \frac{1}{\sqrt{1 - β^{4}}}, β = \frac{D i a_{t h r o a t}}{D i a_{p i p e}}

Y = e x p a n s i o n f a c t o r f o r c o m p r e s s i b l e g a s

ρ_{u p} = f l u i d d e n s i t y u p s t r e a m o f p i p e

Δ P = P_{s t a t i c, u p} - P_{s t a t i c, v e n a - c o n t r a c t a}

Static Pressure at the Vena-Contracta

Flow Simulator uses the upstream total pressure and downstream static pressure for flow calculations. The flow meter calculation uses the upstream static pressure and a vena-contracta static pressure. There will be a static pressure increase from the vena-contracta to the known downstream static pressure. The relationship between these pressures can be found from equation 2-9 in reference 1:

\frac{P_{s t a t i c, u p} - P_{s t a t i c, d o w n}}{P_{s t a t i c, u p} - P_{s t a t i c, v e n a - c o n t r a c t a}} = \frac{P_{s t a t i c, u p} - P_{s t a t i c, d o w n}}{Δ P} = \frac{\sqrt{1 - β^{4} * (1 - C d^{2})} - C d * β^{2}}{\sqrt{1 - β^{4} * (1 - C d^{2})} + C d * β^{2}}

The Discharge Coefficient

Each type of flow meter has a unique empirically derived discharge coefficient. The equations used for each type are listed here.

Orifice Plate, ISO: This equation is in references 2 and 3 and is specifically for taps 1 diameter upstream and ½ diameter downstream of the orifice plate. Cd will be around 0.6
$C d = 0.595 + 0.0312 * β^{2.1} - 0.184 * β^{8} + 0.09 * \frac{β^{4}}{1 - β^{4}} - 0.01685 * β^{3} + 91.71 * \frac{β^{2.5}}{R e_{u p}^{0.75}}$
Orifice Plate, Reader-Harris: This equation is in references 1 and 4 and is specifically for taps 1 diameter upstream and ½ diameter downstream of the orifice plate. Cd will be around 0.6.
$C d = 0.5961 + 0.0261 * β^{2} - 0.216 * β^{8} + 0.000521 * {(\frac{1 x 10^{6}}{R e_{u p}} β)}^{0.7} + (.0188 + .0063 * A A) * β^{3.5} * {(\frac{1 x 10^{6}}{R e_{u p}})}^{0.3} + (.0043 + .08 * e^{- 10} - 0.123 * e^{- 7}) * (1 - 0.11 * A A) * \frac{β^{4}}{1 - β^{4}} - .0031 * (M 2 - 0.8 * M 2) * β^{1.3} + T e r m 1$; Where:
$A A = {(\frac{19000 * β}{R e_{u p}})}^{0.8}$
$M 2 = \frac{2 * 0.47}{1 - β}$; $i f D i a_{p i p e} > 2.8, T e r m 1 = 0.11 * (0.75 - β) * (2.8 - D i a_{p i p e}) e l s e T e r m 1 = 0$
Venturi, Cast: This constant Cd is in references 1, 2, and 4 and is for a venturi meter that has a surface finish from casting.
$C d = 0.984$
Venturi, Machined: This constant Cd is in references 1, 2, and 4 and is for a venturi meter that has a surface finish from machining.
$C d = 0.995$
Venturi, Welded: This constant Cd is in references 1, 2, and 4 and is for a venturi meter that has a surface finish from welding.
$C d = 0.985$
Nozzle, ISA 1932: This equation is in references 1 and 4 and is for a nozzle that has a circular shape.
$C d = 0.99 - 0.2262 * β^{4.1} - (0.00175 * β^{2} - 0.0033 * β^{4.15}) * {(\frac{1 x 10^{6}}{R e_{u p}})}^{1.15}$
Nozzle, Long Radius: This equation is in references 1, 2, 3, and 4 and is for a nozzle that has an elliptical shape.
$C d = 0.9965 - 0.00653 * β^{0.5} * {(\frac{1 x 10^{6}}{R e_{u p}})}^{0.5}$
Venturi-Nozzle, ISO: This equation is in references 1, 3, and 4 and is for an ISA 1932 nozzle followed by a straight section and conical expansion.
$C d = 0.9858 - 0.196 * β^{4.5}$

The Expansion factor, Y

The flow meter flow rate equations must use an expansion factor (also know as the expansibility factor) to account for the compressibility of a gas. The expansion factor is based on empirical correlations and is different for different types of flow meters.

Orifice Plates: This equation is in references 1 and 4.
$Y = 1 - (0.351 + 0.256 * β^{4} + 0.93 * β^{8}) * (1 - τ^{\frac{1}{γ}})$
$τ = \frac{P_{s t a t i c, v e n a - c o n t r a c t a}}{P_{s t a t i c, u p}}$
Venturi, Nozzle, Venturi-Nozzle: This equation is in references 1, 2, and 4.
$Y = \sqrt{(\frac{γ τ^{\frac{2}{γ}}}{γ - 1}) (\frac{1 - β^{4}}{1 - β^{4} * τ^{\frac{2}{γ}}}) (\frac{1 - τ^{\frac{(γ - 1)}{γ}}}{1 - τ})}$

Flow Meter Element Outputs


Name	Description	Units
PIPE_AREA	Physical area of a pipe upstream and downstream of the restriction.	in^2, m^2
THROAT_AREA	Physical area of the throat of the restriction (orifice, nozzle, or venturi).	in^2, m^2
PIPE_DIA	Physical diameter of a pipe upstream and downstream of the restriction.	in, m
THROAT_DIA	Physical diameter of the throat of the restriction (orifice, nozzle, or venturi).	in, m
FLOWMETER_CD	Discharge coefficient.	(unitless)
NRPD	Non-Recoverable Pressure Drop.	psia, MPa
K_LOSS_INCOMPR_RSLT	$K = \frac{P_{t o t a l, u p} - P_{t o t a l, d o w n}}{.5 * ρ_{u p} * V e l_{u p}^{2}}$	(unitless)
K_LOSS_COMPR_RSLT		(unitless)
EXP_FACTOR	Expansion factor, Y, for compressible gas only.	(unitless)
PipeIn, AREA	Physical area of a pipe upstream of the restriction.	in^2, m^2
PipeIn, PT	Total pressure upstream, based on the chamber total pressure and angle between the velocity and the element.	psia, MPa
PipeIn, PS	Static pressure upstream.	psia, MPa
PipeIn, TT	Total temperature upstream, based on the chamber's total temperature.	Deg F, Deg K
PipeIn, TS	Static temperature upstream.	Deg F, Deg K
PipeIn, VEL	Fluid velocity upstream.	ft/sec, m/sec
PipeIn, MACH	Fluid Mach number upstream, compressible gas only.	(unitless)
PipeIn, VOLFLOW	Volumetric flow rate upstream, incompressible liquid only.	GPM, m^3/sec
PipeIn, REYN	Reynolds number upstream.	(unitless)
PipeIn, THETA	Tangential flow angle upstream.	degrees
PipeIn, PHI	Radial flow angle upstream.	degrees
PipeIn, RHO	Fluid density upstream.	lbm/ft^3, kg/m^3
Throat, AREA	Physical area of the restriction.	in^2, m^2
Throat, PT	Total pressure at the vena-contracta, assumed to be the same as upstream.	psia, MPa
Throat, PS	Static pressure at the vena-contracta.	psia, MPa
Throat, TT	Total temperature at the vena-contracta, assumed to be the same as upstream.	Deg F, Deg K
Throat, TS	Static temperature at the vena-contracta.	Deg F, Deg K
Throat, VEL	Fluid velocity at the vena-contracta.	ft/sec, m/sec
Throat, MACH	Fluid Mach number at the vena-contracta, compressible gas only.	(unitless)
Throat, VOLFLOW	Volumetric flow rate at the vena-contracta, incompressible liquid only.	GPM, m^3/sec
Throat, REYN	Reynolds number at the vena-contracta.	(unitless)
Throat, THETA	Tangential flow angle at the vena-contracta, assumed to be the same as upstream.	degrees
Throat, PHI	Radial flow angle at the vena-contracta, assumed to be the same as upstream.	degrees
Throat, RHO	Fluid density at the vena-contracta.	lbm/ft^3, kg/m^3
PipeEx, AREA	Physical area of the pipe downstream of the restriction. Always the same as the upstream area.	in^2, m^2
PipeEx, PT	Total pressure downstream.	psia, MPa
PipeEx, PS	Static pressure downstream, same as the downstream chamber static pressure.	psia, MPa
PipeEx, TT	Total temperature downstream, same as upstream.	Deg F, Deg K
PipeEx, TS	Static temperature downstream.	Deg F, Deg K
PipeEx, VEL	Fluid velocity downstream.	ft/sec, m/sec
PipeEx, MACH	Fluid Mach number downstream, compressible gas only.	(unitless)
PipeEx, VOLFLOW	Volumetric flow rate downstream, incompressible liquid only.	GPM, m^3/sec
PipeEx, REYN	Reynolds number downstream.	(unitless)
PipeEx, THETA	Tangential flow angle downstream.	degrees
PipeEx, PHI	Radial flow angle downstream.	degrees
PipeEx, RHO	Fluid density downstream.	lbm/ft^3, kg/m^3

References

ASME MFC-3M-2004, Measurement of Fluid Flow in Pipes Using Orifice, Nozzle, and Venturi; 2017.
Blevins, R. D., Applied Fluid Dynamics Handbook, Krieger Publications, 2003.
White, Frank M., Fluid Mechanics, 8th Ed., McGraw - Hill, 2015.
Crane, Flow of Fluids Through Valves, Fittings, and Pipe; Technical Paper 410.