Cone Meter

Introduction

This application calculates the values associated with the flow of gas through a cone-style meter. A cone meter uses a cone-shaped element suspended in the meter tube; gas passes through the annular space around the cone, creating a measurable pressure drop that is used to determine the volumetric flow rate. The calculator can solve for flow rate, cone diameter, or differential pressure (whichever is designated as the “unknown”), and it reports the meter tube velocity and throat velocity as supporting results.

There are currently no industry standards covering the installation or flow calculation of cone meters; each manufacturer publishes its own method. Values such as the Meter Coefficient and Cone Diameter are unique to each meter and are taken from the manufacturer’s data sheet. The supported calculation methods are derived from manufacturer (McCrometer) literature, and the User must confirm that the selected method is appropriate for the device being analyzed.

Background

Various methods and devices are used to measure the flow of gas at a specific location; these devices are generally referred to as “meters.” A cone meter relies on the known characteristics of gas flowing around a cone-shaped object inserted in a meter tube. The geometry forces the gas to pass through the annular area between the tube wall and the cone, producing a pressure differential that can be measured and related to a volumetric flow rate. Because no specific industry standard governs these devices, the installation details and calculation procedures are supplied by each manufacturer.

Diameter Temperature Compensation

The cone and tube diameters are physical dimensions established at a known reference temperature. When the gas flows at a temperature different from that reference, the metal expands or contracts, changing the effective dimensions. GASCalc adjusts both the cone diameter and the tube diameter for this thermal effect before computing the flow.

Throat and Tube Velocity

The meter tube/pipe velocity is the velocity of the flow through the specified tube diameter, evaluated at the static pressure and unadjusted for flowing temperature. The throat velocity is the velocity of the flow stream through the annular space between the tube and the cone, also evaluated at the static pressure and unadjusted for flowing temperature; this value is approximate. If the throat velocity exceeds the Normal Velocity Limit set in the Check Limits preferences, the value is displayed in red text to flag a possible issue.

Equations

The calculator provides several methods for calculating the parameters associated with a cone meter. Each method is unique and may have specific limitations, so the User must determine the appropriate equation for the application being analyzed. The supported equations are described below.

Diameter Temperature Compensation

The tube and cone diameter values are adjusted to compensate for expansion and contraction caused by differences between the reference temperature and the flowing temperature, using the following equations (derived from Reference 1):

\begin{aligned} d &= d_R \times (1 + \alpha \times (T_F - T_R)) \\ D &= D_R \times (1 + \alpha \times (T_F - T_R)) \end{aligned}

\begin{aligned} d &= d_R \times (1 + \alpha \times (T_F – T_R)) \\ D &= D_R \times (1 + \alpha \times (T_F – T_R)) \end{aligned}

Where:
d − Meter Cone Outside Diameter (feet)
d_R − Meter Cone Diameter at the Reference Temperature (feet)
D − Meter Tube Inside Diameter (feet)
D_R − Meter Tube Diameter at the Reference Temperature (feet)
α − Coefficient of Thermal Expansion (in-in/°F)
T_F − Temperature at Flowing (Metered) Conditions (°R)
T_R − Reference Temperature for Meter Tube and Cone Dimensions (°R)

Standard Volumetric Flow Rate

The flow computed at flowing conditions is converted to the specified base (standard) conditions using the following equation (derived from Reference 2):

Q_B = Q_F \times \left[\frac{P_F}{P_B}\right] \times \left[\frac{T_B}{T_F}\right] \times \left[\frac{Z_B}{Z_F}\right] \times MF

Q_B = Q_F \times \left[\frac{P_F}{P_B}\right] \times \left[\frac{T_B}{T_F}\right] \times \left[\frac{Z_B}{Z_F}\right] \times MF

Where:
Q_B − Metered Volume at the Specified Base Pressure and Temperature (ft³)
Q_F − Metered Volume at Flowing (Metered) Conditions (ft³)
P_F − Pressure at Flowing (Metered) Conditions (psia)
P_B − Base Pressure (psia)
T_B − Base Temperature (°R)
T_F − Temperature at Flowing (Metered) Conditions (°R)
Z_B − Compressibility Factor at Specified Base Conditions
Z_F − Compressibility Factor at Flowing (Metered) Conditions
MF − Meter Adjustment Factor

McCrometer Methods

This method calculates the volumetric flow through the meter at flowing conditions using the following equations (derived from References 1 and 3). The flow rate is governed by the equation constant, the gas expansion factor, the differential pressure, and the flowing density:

Q_F = K_1 \times Y \times \left[\frac{DP}{\rho_F}\right]^{0.5}

Q_F = K_1 \times Y \times \left[\frac{DP}{\rho_F}\right]^{0.5}

Where:
Q_F − Metered Volume at Flowing (Metered) Conditions (ft³)
K₁ − Equation Constant
Y − Expansion Factor
DP − Pressure Differential at Flowing (Metered) Conditions (psia)
ρ_F − Gas Density at Flowing Conditions as Defined by Reference 1 (lbm/ft³)

The equation constant combines the meter geometry and the meter coefficient:

K_1 = \frac{\pi}{4} \times (2g)^{0.5} \times \left[\frac{D^2 \times \beta^2}{(1 - \beta^4)^{0.5}}\right] \times C_D

K_1 = \frac{\pi}{4} \times (2g)^{0.5} \times \left[\frac{D^2 \times \beta^2}{(1 – \beta^4)^{0.5}}\right] \times C_D

Where:
K₁ − Equation Constant
g − Gravitational Constant (ft/sec²)
D − Meter Tube Inside Diameter (feet)
β − Beta Ratio (dimensionless)
C_D − Meter Coefficient (dimensionless)

The beta ratio relates the cone and tube diameters:

\beta = \left[1 - \frac{d^2}{D^2}\right]^{0.5}

\beta = \left[1 – \frac{d^2}{D^2}\right]^{0.5}

Where:
β − Beta Ratio (dimensionless)
d − Meter Cone Outside Diameter (feet)
D − Meter Tube Inside Diameter (feet)

The gas density at flowing conditions is determined from the specific gravity, flowing pressure, flowing temperature, and compressibility:

\rho_F = 2.6988 \times SG \times \frac{P_F}{Z_F \times T_F}

\rho_F = 2.6988 \times SG \times \frac{P_F}{Z_F \times T_F}

Where:
ρ_F − Gas Density at Flowing Conditions as Defined by Reference 1 (lbm/ft³)
SG − Ideal Gas Specific Gravity
P_F − Pressure at Flowing (Metered) Conditions (psia)
Z_F − Compressibility Factor at Flowing (Metered) Conditions
T_F − Temperature at Flowing (Metered) Conditions (°R)

Expansion Factor — Adiabatic Expansion Method

For the Adiabatic Expansion Method, the expansion factor is calculated as follows:

Y = \left[\frac{(1 - \beta^4) \times \dfrac{k}{k-1} \times R^{\frac{2}{k}} \times \left(1 - R^{\frac{k-1}{k}}\right)}{\left[1 - \beta^4 \times R^{\frac{2}{k}}\right] \times (1 - R)}\right]^{0.5} \qquad R = 1 - \frac{DP}{P_F}

Y = \left[\frac{(1 – \beta^4) \times \dfrac{k}{k-1} \times R^{\frac{2}{k}} \times \left(1 – R^{\frac{k-1}{k}}\right)}{\left[1 – \beta^4 \times R^{\frac{2}{k}}\right] \times (1 – R)}\right]^{0.5} \qquad R = 1 – \frac{DP}{P_F}

Where:
Y − Expansion Factor
β − Beta Ratio
k − Ratio of Specific Heats
R − Pressure Ratio
DP − Pressure Differential at Flowing (Metered) Conditions (psia)
P_F − Pressure at Flowing (Metered) Conditions (psia)

Expansion Factor — V-Cone Method

For the V-Cone Method, the expansion factor is calculated as follows:

Y = 1 - \left[(0.649 + 0.696\beta^4) \times \frac{DP}{k \times P_F}\right]

Y = 1 – \left[(0.649 + 0.696\beta^4) \times \frac{DP}{k \times P_F}\right]

Where:
Y − Expansion Factor
β − Beta Ratio
DP − Pressure Differential at Flowing (Metered) Conditions (psia)
k − Ratio of Specific Heats
P_F − Pressure at Flowing (Metered) Conditions (psia)

Expansion Factor — Wafer-Cone Method

For the Wafer-Cone Method, the expansion factor is calculated as follows:

Y = 1 - \left[(0.755 + 6.787\beta^8) \times \frac{DP}{k \times P_F}\right]

Y = 1 – \left[(0.755 + 6.787\beta^8) \times \frac{DP}{k \times P_F}\right]

Where:
Y − Expansion Factor
β − Beta Ratio
DP − Pressure Differential at Flowing (Metered) Conditions (psia)
k − Ratio of Specific Heats
P_F − Pressure at Flowing (Metered) Conditions (psia)

The flowing pressure is the sum of the gauge pressure and the atmospheric pressure: P_F = P_FG + P_ATM, where P_FG is the gauge pressure at flowing conditions (psig) and P_ATM is the atmospheric pressure at the metered location (psia). Unless otherwise noted, all values are in consistent dimensional units.

Case Guide

Part 1: Create Case

Input Parameters

Part 2: Outputs/Reports

Results

Note: Depending on the number of decimal places displayed in your software, calculated values may not appear exactly as in published examples; results should match when rounded to the least common decimal place. The number of decimal places shown for any calculated item can be set under File > Preferences > Decimals.

References

• McCrometer, Flow Calculations For The V-Cone and Wafer-Cone Flowmeters, Lit #24509-54, Rev 3.2, 2008.
• American Gas Association, Measurement, GEOP Series Book M-1, 1993.
• McCrometer, Advanced Differential Pressure Flow Technology, Article ARTC02031710, 2002.

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