Rosemount 485 Annubar Annubar Primary Element Flow Calculations ANNUBAR PRIMARY ELEMENT FLOW EQUATIONS The Annubar primary element flow equations are all derived from the hydraulic equations which are shown in equations 2.9 and 2.0 of the Annubar (document number 00807-000-9, Rev BA). For a detailed example of a derivation of an Annubar primary element equation, see the Rosemount 485 Annubar Flow Test Data Book (document number 00809-000-93, Rev CA). Equation : Volume rate of flow - Liquids (Actual Conditions) Q a C I Q h w OR h a w ------ 2 C I C I F na K D 2 F aa ----- G f NOTE: For description of standard volumetric flow equations, see page 2-5 of the Annubar (document number 00809-000-9, Rev BA). Equation 2: Mass rate of flow - Liquids W C I W h w OR h w ----- 2 C I C I F na K D 2 F aa ρ f Equation 3: Mass rate of flow - Gas and Steam W C I h w h W w ----- OR 2 C I C I F na K D 2 Y a F aa ρ f Equation 4: Volume rate of flow - Gas (Standard Conditions) Q s C I Q h w P f OR h w ---- s P ------ f C I 2 C I F na K D 2 Y a F pb F tb F tf F g F pv F aa Equation 5: Volume rate of flow - Gas (Actual Conditions) Q a C I Q h w h a w ------ OR 2 C I F na K D 2 Y a F aa ---- ρ f C I www.rosemount.com
For a detailed description of each term in the above equations, see Nomenclature on page 6. Please note that each of the above equations has a C' constant. It is not intended that the C' constant of one equation is equal to the C' constant of another equation. The numerical value of any C' constant is the product of the appropriate factors for that equation only. The following tabulations of the flow equations will serve as handy work pads. Also, the table numbers where the necessary information can be found are given in the headings of these tabulations. Several completed examples of flow calculations are given beginning on page 6. NOTE The 485 Annubar primary needs no correction for the Reynolds Number. 2
Table. Equation for Liquid Volume Rate of Flow Rate of Flow Unit Conversion Factor Annubar Flow Coefficient Internal Pipe Diameter Thermal Expansion Factor (Table 9) Flowing Specific Gravity Differential Pressure Annubar Flow Constant C I Q a F K D 2 F ----- na x x x aa x h G x w f GPM 5.6664 (in) 2 lbm/ft 3 inch H 2 O at 68 F GPH 339.99 (in) 2 lbm/ft 3 inch H 2 O at 68 F GPD 859.7 (in) 2 lbm/ft 3 inch H 2 O at 68 F BPH (42 gal) 8.0949 (in) 2 lbm/ft 3 inch H 2 O at 68 F BPD (42 gal) 94.28 (in) 2 lbm/ft 3 inch H 2 O at 68 F ft 3 /min 0.75749 (in) 2 lbm/ft 3 inch H 2 O at 68 F CFH 45.4494 (in) 2 lbm/ft 3 inch H 2 O at 68 F CFM 0.7575 (in) 2 lbm/ft 3 inch H 2 O at 68 F lmp. GPM 4.783 (in) 2 lbm/ft 3 inch H 2 O at 68 F LPH 0.3958 (mm) 2 kg/m 3 mm H 2 O at 20 C LPM 0.0065966 (mm) 2 kg/m 3 mm H 2 O at 20 C LPS 0.0000994 (mm) 2 kg/m 3 mm H 2 O at 20 C m 3 /D 0.0094993 (mm) 2 kg/m 3 mm H 2 O at 20 C m 3 /H 0.000396 (mm) 2 kg/m 3 mm H 2 O at 20 C m 3 /M 6.5967E-06 (mm) 2 kg/m 3 mm H 2 O at 20 C m 3 /s.0995e-07 (mm) 2 kg/m 3 mm H 2 O at 20 C 3
Table 2. Equation for Liquid Mass Rate of Flow Rate of Flow Unit Conversion Factor Annubar Flow Coefficient Internal Pipe Diameter Thermal Expansion Factor (Table 9) Flowing Density Differential Pressure Annubar Flow Constant C I W F K D 2 na x x x F aa x P f x h w PPD 864.56 (in) 2 lbm/ft 3 inch H 2 O at 68 F PPH 358.94 (in) 2 lbm/ft 3 inch H 2 O at 68 F PPM 5.9823 (in) 2 lbm/ft 3 inch H 2 O at 68 F PPS 0.0997 (in) 2 lbm/ft 3 inch H 2 O at 68 F T(met)/hr.25E-05 (mm) 2 kg/m 3 mm H 2 O at 20 C kg/d 0.30025 (mm) 2 kg/m 3 mm H 2 O at 20 C Kg/H 0.025 (mm) 2 kg/m 3 mm H 2 O at 20 C kg/m 0.0002085 (mm) 2 kg/m 3 mm H 2 O at 20 C kg/s 3.475E-06 (mm) 2 kg/m 3 mm H 2 O at 20 C Table 3. Equation for Gas and Steam Mass Rate of Flow Rate of Flow Unit Conversion Factor Annubar Flow Coefficient Internal Pipe Diameter Annubar Expansion Factor Thermal Expansion Factor (Table 9) Flowing Density Differential Pressure Annubar Flow Constant C I W F K D 2 na x x x Y a x F aa x P f x h w PD 864.56 (in) 2 lbm/ft 3 inch H 2 O at 68 F PPH 358.94 (in) 2 lbm/ft 3 inch H 2 O at 68 F PPM 5.9823 (in) 2 lbm/ft 3 inch H 2 O at 68 F PPS 0.0997 (in) 2 lbm/ft 3 inch H 2 O at 68 F T(met)/hr.25E-05 (mm) 2 kg/m 3 mm H 2 O at 20 C kg/d 0.30025 (mm) 2 kg/m 3 mm H 2 O at 20 C Kg/H 0.025 (mm) 2 kg/m 3 mm H 2 O at 20 C kg/m 0.0002085 (mm) 2 kg/m 3 mm H 2 O at 20 C kg/s 3.475E-06 (mm) 2 kg/m 3 mm H 2 O at 20 C 4
Table 4. Equation for Gas and Steam Mass Rate of Flow (Standard Conditions) Rate of Flow Unit Conversion Factor Annubar Flow Coefficient Internal Pipe Diameter Annubar Expansion Factor Pressure Base Factor Temperature Base Factor Flowing Temperature Factor Specific Gravity Factor Supercomp Factor (Table 8) Thermal Expansion Factor (Table 2) Flowing Pressure Differential Pressure Annubar Flow Constant C I Q s F K D 2 Y na a F pb F tb F P tf F g F pv F aa f h w x x x x x x x x x x x SCFD 8.6. (in) 2 psia inch H 2 O at 68 F SCFH 338.7 (in) 2 psia inch H 2 O at 68 F SCFM 5.6362 (in) 2 psia inch H 2 O at 68 F NL/H.227 (mm) 2 kpa mm H 2 O at 20 C NL/M 0.087 mm) 2 kpa mm H 2 O at 20 C NM 3 /D 0.026945 (mm) 2 kpa mm H 2 O at 20 C NM 3 /H 0.00227 mm) 2 kpa mm H 2 O at 20 C NM 3 /M.872E-05 (mm) 2 kpa mm H 2 O at 20 C NM 3 /S 3.86E-07 mm) 2 kpa mm H 2 O at 20 C Table 5. Equation for Gas and Steam Rate of Flow (Actual Conditions) Rate of Flow Unit Conversion Factor Annubar Flow Coefficient Internal Pipe Diameter Annubar Expansion Factor Thermal Expansion Factor (Table 9) Flowing Density Differential Pressure Annubar Flow Constant C I Q a K D 2 F na x x x Y a x F aa x ---- h P x w f ACFD 864.56 (in) 2 lbm/ft 3 inch H 2 O at 68 F ACFH 358.94 (in) 2 lbm/ft 3 inch H 2 O at 68 F ADFM 5.9823 (in) 2 lbm/ft 3 inch H 2 O at 68 F AL/H 2.500 (mm) 2 kg/m 3 mm H 2 O at 20 C AL/M 0.2085 (mm) 2 kg/m 3 mm H 2 O at 20 C Am 3 /D 0.30025 (mm) 2 kg/m 3 mm H 2 O at 20 C Am 3 /H 0.025 (mm) 2 kg/m 3 mm H 2 O at 20 C Am 3 /M 0.0002085 (mm) 2 kg/m 3 mm H 2 O at 20 C Am 3 /S 3.475E-06 (mm) 2 kg/m 3 mm H 2 O at 20 C 5
NOMENCLATURE D F aa F g Internal diameter of pipe, inches (mm) Thermal Expansion Factor. This factor corrects for the flowing area change of the pipe at the Annubar location due to temperature effects. For 36 stainless steel Annubar primary elements mounted in carbon steel pipe, F aa.0000 for temperatures between 3 and 06 F. See Table 2 on page 5. Specific Gravity Factor. This factor corrects the flow equation whenever the gas is not air. The factor can be calculated as: F g --- G F na F pb where, G specific gravity of flowing gas, air.000. For a more complete description of specific gravity, see page 2-4 and Appendix C of the Annubar (document number 00807-000-9, Rev BA). Units Conversion Factor. This factor is used to convert the flow rate to the desired or wanted set of units. Appendix C Annubar (document number 00807-000-9, Rev BA) shows an example of how the numerical value of F na is derived from the hydraulic equation for a given set of input units. Pressure Base Factor. The Pressure Base Factors are calculated to give gas volumes at a pressure base of 4.73 psia (0.56 kpa abs). The pressure base factor can be calculated as: 4.73 0.56 F pb -- base pressure, psia OR F pb -- base pressure, kpa abs F pv Supercompressibility Factor. The Supercompressibility Factor accounts for the deviation from the ideal gas laws. In the flow equations, gas volumes are assumed to vary with pressure and temperature in accordance with Boyle s and Charles' laws (the ideal gas laws). Actually, the volume occupied by individual gases deviate, by a slight degree, from the volumes which the ideal gas laws indicate. The amount of deviation is a function of the composition of the gas and varies primarily with static pressure and temperature. The actual deviation may be obtained by a laboratory test conducted on a sample of the gas, carefully taken at line conditions of pressure and temperature. The National Bureau of Standards, Circular 564, gives the compressibility factor (Z) of air and other pure gases. The relationship between supercompressibility factor and compressibility factor is as follows: F pv -- Z Table 0 on page 3 gives an abbreviated listing of the supercompressibility factors for air. Practical relationships have been established by which this deviation can be calculated and tabulated for natural gases containing normal mixtures of hydrocarbon components, considering the presence of small quantities of carbon dioxide and nitrogen and also relating the deviation to the heating value of gas. The A.G.A. manual (NX-9), Determination of Supercompressibility Factors for Natural Gas, should be used for determination of F pv. Table on page 4 gives an abbreviated listing of the supercompressibility factors for natural gas. 6
F tb Temperature Base Factor. The Temperature Base Factors are calculated to give gas volumes at a base temperature of 60 F (520 R) or 6 C (289 K). The factor can be calculated as: temperature base F tb -- ( F) + 460 520 OR temperature base F tb -- ( C) + 273 289 F tf Flowing Temperature Factor. The units conversion factor (F na ) for volumetric flow of gases at standard conditions has been calculated assuming that the gas temperature flowing around the Annubar primary element is 60 F (520 R) or 6 C (289 K). If measurement is made at any other flowing temperature, then the flowing temperature factor must be applied. The factor can be calculated as: F 520 520 tf --------- OR flowing temperature ( F) + 460 520 289 F tf --------- flowing temperature ( C) + 273 G Specific Gravity of Flowing Liquid. Ratio of the density of the flowing fluid to the density of water at 60 F which is 63.3707 lbm/ft 3. See Appendix A of the Annubar (document number 00807-000-9, Rev BA) for specific gravities of various liquids. h w Differential pressure as measured by the Annubar primary element. For this handbook, the differential pressure is expressed as the height, in inches, of a water column at 68 F at standard gravity (g c 32.74 ft/sec 2 9.807 m/sec 2 ). K P f Q a Q s h w inches H 2 O at 68 F (mm at 20 C) Flow Coefficient. Equation 8 on page 0 defines the flow coefficient of an Annubar primary element. It is related to the diameter of the pipe and is generally expressed as a function of Reynolds Number. See page 2-7 of the Annubar (document number 00807-000-9, Rev BA) for an explanation of Reynolds Number. Flowing Pressure. This is the static pressure, in absolute units, existing in the pipe. For this handbook, the pressures are expressed in psia and kpa abs. Actual Volumetric Flow Rate. This term is the flow rate of the fluid passing the Annubar primary element in actual volume units per units of time. Examples are actual cubic feet per hour (ACFH), GPM, etc. Standard Volumetric Flow Rate. This term is the flow rate of the fluid passing the Annubar primary element in standard volume units per unit of time. For some gases, especially fuel gases, the cubic foot is the unit of measurement. A cubic foot of gas has no absolute or comparative value unless the pressure and temperature of the gas are specified when it fills a cubic foot. A common unit used for evaluating rates of flow is standard cubic foot per hour, (SCFH). This unit states how many cubic feet of gas per hour would be flowing around the Annubar primary element if the flowing pressure and temperature were equal to the base pressure and temperature. For this handbook, the base pressure is 4.73 psia (0.56 kpa abs) and the base temperature is 60 F (520 R) or 6 C (289 K). 7
W Y A Mass Rate of Flow. This term is the flow rate of the fluid passing the Annubar primary element in mass units per unit time. Expansion Factor. When a gas flows around an Annubar primary element, the change in velocity is accompanied by a change in density. The expansion factor must be applied to correct for this change. The expansion factor also accounts for small changes in the internal energy of the molecules due to the temperature difference between the upstream and downstream pressure ports of the Annubar primary element. The variation of the expansion factor is small and the ratio of specific heats for commercial gases is sufficiently constant to warrant using a constant ratio of specific heat. Use the following algorithm to calculate the value of the gas expansion factor. This equation adjusts for density and internal energy effects of the gas as it flows around the Annubar primary element. Equation 6: Gas Expansion Factor Y a ( 0.0332( B) 2 0.00342) h w -------- P f ϒ Equation 7: Blockage Equation ρ f 4d ------- B πd Blockage D Internal Pipe Diameter in inches (cm) d See Table Table 8 on page 2 h w Differential pressure in inches (mm) of water column P F Flowing line pressure in psia (kpa abs) γ Ratio of specific heats Examples of gases with a specific heat ratio of.4 are: air, CO, H 2, NO, N 2 and O 2. Examples of gases with a specific heat ratio of.3 are: natural gas, ammonia, CO 2, Cl 2, H 2 S, N 2 O, SO 2, and steam. Y a is needed in all gas flow equations and requires the differential pressure can be calculated first. If the differential pressure is not known, Y a is assumed to be.000 and the differential pressure is calculated. Iteration is then necessary to determine its final value. Flowing Density. For this handbook, the densities are expressed in lbm/ft (kg/m 3 ). Appendix A of the Annubar (document number 00807-000-9, Rev BA gives densities of various fluids. 8
Flow Coefficient Reynolds Number Dependency Flow Coefficient Theory Signal How Sensor Shape Affects the Differential Pressure How Pipe Blockage Affects Differential Pressure When the Rosemount 485 Annubar primary element is used within the acceptable Reynolds Number range defined by Rosemount, the Annubar Primary element's flow coefficient will be independent of changing Reynolds Number. Any variations in the K-value with changing Reynolds Number are due to scatter and fall within ±.75% of the published K-value. A 485 Annubar primary element's K-factor independence of Reynolds number allows the user to measure a large range of Reynolds Numbers without need of a correction factor. The 485 Annubar's K-factor independence can be attributed to a constant separation point along the edges of its T-shaped sensor and the probe's ability to take a proper average of its sensing slots. Rosemount is the first company to identify and utilize the theoretical equations linking self-averaging pitot tube flow coefficients to pipe blockage. This K-to-Blockage theoretical link establishes a higher degree of confidence in 485 Annubar K-factors than in flow meters that use only an empirical data base for determining their flow coefficients. The signal generated by an Annubar primary element can be divided into two major parts: the differential pressure contribution due to the Annubar primary element's shape (H S ) the differential pressure contribution due to the Annubar primary element's blockage in the pipe (H b ). An Annubar primary element placed in an infinitely large pipe (with no confining walls) will still produce a differential pressure. This differential pressure is nearly twice that of a standard pitot tube, and is the result of a reduced low pressure on the downstream side. The upstream, or high pressure is caused by the fluid impacting the front of the Annubar primary element and is known as the stagnation pressure. The downstream, or low pressure is caused by the fluid traveling past the Annubar primary element, creating suction on the rear side. This suction phenomenon can be attributed to a boundary layer flow separation. An Annubar primary element is an obstruction in the pipe and therefore, reduces the cross-sectional area in which the fluid can pass. This reduced area causes the fluid to accelerate and hence, reduces its pressure. Therefore, the downstream pressure measurement of an Annubar primary element will be affected by the Annubar primary element's blockage in the pipe. Since an Annubar primary element uses the internal diameter of the pipe it is being inserted into as a throat diameter in its calculation of a flow rate, the Annubar primary element K-factor must compensate for the amount of obstructed area the sensor itself causes in the pipe. This is analogous to the velocity of approach factor for an orifice plate or a venturi meter. By writing a mass balance and an energy balance around the Annubar primary element, and by dividing the differential pressure produced by the Annubar primary element into H s and H b, one can derive the relationship between the Annubar primary element K-factor and the Annubar primary element's blockage in the pipe. The derivation involves partial differential pressure components, and the integration of a k-blockage equation. The result is the following K vs. Blockage equation: 9
Equation 8: K vs. Blockage K A ( C 2 B) ------------------------------------------------- C ( C 2 B) The constants C and C 2 must be determined experimentally. Once C and C 2 are determined, the equations above becomes the theoretical link between the Annubar primary element K-factor (K) and the Annubar primary element's blockage in the pipe (B). The values for constants C and C 2 are shown in the table below: Table 6. 485 Sensor Constants The Importance of the Flow Coefficient, or K vs. Blockage Theory Coefficient Sensor Size Sensor Size 2 Sensor Size 3 C.55.492.5856 C 2.4229.479.338 As with any other meter, the 485 Annubar primary element's accuracy is only as good as its flow coefficient (K-factor). Rosemount has tested a representative sample of Flowmeters and empirically determined flow coefficients. For Annubars, these flow coefficients are plotted against the meter's blockage. Curve fitting techniques are applied to the base line data to generate equations that predict flow coefficients in untested line sizes and untested Reynolds Number ranges. Please see the 485 Annubar Flow Test Data Book (document number 00809-000-93, Rev CA) for a more detailed discussion of this topic. Provided the theory is based on the proper physics, these relationships are immune to minor variation in test data. Using a theoretical basis (in addition to empirical testing) for the prediction of untested flow coefficients provides a much higher degree of confidence in the untested values. The following graphs show that empirical data agree with a plot of the K vs. Blockage Equation. 0
Figure 7. K vs. BLOCKAGE
Operating Limitations Table 8. Reynolds Number and Probe Width For an Annubar primary element to operate accurately, the flowing fluid must be separate from the probe at the same location (along the edges of the T-shape sensor). Drag coefficients, lift coefficients, separation points, and pressure distributions around bluff bodies are best compared by calculating the rod Reynolds Number. There is a minimum rod Reynolds Number at which the flowing fluid will not properly separate from the edges of a T-shape sensor. The minimum rod Reynolds Numbers for the Rosemount 485 are: Sensor Size Probe Width (d) Minimum Reynolds Number 0.590-in. (.4986 cm) 6500 2.060-in (2.6924 cm) 2500 3.920-in (4.8768 cm) 25000 Above these rod Reynolds Numbers, 485 Annubar primary elements will operate accurately. To determine the rod Reynolds Number at any given flowrate, use the following relationship: dvρ Re rod ---------- dvρ 2µ OR Re rod ------------- 00µ where, ρ fluid density in lbm/ft 3 (kg/m 3 ) d probe width in inches (cm) V velocity of fluid in feet per second (m/s) µ fluid viscosity in lbm/ft-sec (kg/m-s) When determining the minimum operating flow rate for an Annubar primary element, one should also consider the capability of the secondary instrumentation (differential pressure transmitters, manometers, etc.). The upper operating limit for Rosemount 485 Annubar primary elements is reached when any one of the following criteria is met:. The fluid velocity reaches the structural limit of the Annubar primary element. 2. The fluid velocity reaches a choked flow condition at the Annubar primary element (for gases). 3. Cavitation occurs on the downstream side of the Annubar primary element. 2
Table 9. Supercompressibility Factor, F pv F pv k ρ G 05 k 2 ( + G ) + ----------------------------------------- 3.825 T f Range of Specific Constants Gravity, G k k2 0.600 < G < 2.480 2.020 0.60 < G < 0.650 3.320.80 0.65 < G < 0.750 4.660.600 0.75 < G < 0.900 7.90.260 0.90 < G <.00.630.070.0 < G <.500 7.480 0.900 Table 0. Supercompressibility Factor: Air, Specific Gravity.0 Pressure psia Temperature 40 F 20 F 0 F 20 F 40 F 60 F 80 F 00 F 20 F 40 F 60 F 80 F 200 F 220 F 4.7.0092.0077.0065.0056.0048.004.0035.003.0027.0024.002.009.006.005 00.063.056.0437.0372.039.0275.0239.0208.082.060.04.025.02.000 200.93.007.0856.0732.0629.0544.0472.042.036.038.028.0249.0222.098 300.744.477.259.079.0930.0805.0700.062.0537.0473.049.0372.033.0296 400.2270.929.649.46.223.060.0924.0808.070.0626.0554.0493.0439.0392 500.2774.2365.2026.744.508.30.43.00.0880.0777.0689.062.0546.0488 600.3260.2785.239.2062.786.554.358.9.048.0926.082.0730.0652.0583 700.3728.392.2746.2373.2058.793.568.377.23.073.0952.0847.0756.0677 800.48.3587.309.2675.2324.2028.775.560.376.28.08.0963.0860.077 900.4620.397.3428.297.2585.2258.979.74.537.36.209.077.0963.0863 000.5046.4345.3756.3260.2840.2483.279.98.695.502.335.9.065.0955 00.5460.4709.4077.3543.3090.2705.2376.2094.85.642.460.303.66.046 200.5863.5064.4390.3820.3336.2923.2569.2266.2005.779.584.44.266.36 300.6257.54.4697.4092.3577.337.2760.2436.257.96.706.524.365.226 400.664.575.4998.4358.384.3348.2948.2604.2308.205.827.633.463.34 470.6905.5984.5204.4542.3977.3493.3078.272.242.244.9.709.53.376 3
Table. Supercompressibility Factor: Hydrocarbon Gas, Specific Gravity 0.6 Pressure psia Temperature 40 F 20 F 0 F 20 F 40 F 60 F 80 F 00 F 20 F 40 F 60 F 80 F 200 F 220 F 4.7.0027.0023.009.006.004.002.00.0009.008.0007.0006.005.0005.0004 00.085.055.03.02.0096.0082.007.0062.0054.0048.0042.0037.0033.0030 200.0368.0309.026.0222.090.064.042.024.008.0095.0084.0074.0066.0059 300.0547.0459.0389.033.0284.0245.022.085.062.042.026.0.0099.0089 400.0722.0608.055.0440.0377.0325.0282.0246.025.090.067.048.032.08 500.0896.0755.0640.0547.0469.0405.0352.0307.0269.0236.0209.085.065.047 600.066.0899.0764.0652.056.0484.042.0367.0322.0283.0250.0222.097.076 700.234.042.0886.0757.065.0563.0489.0427.0374.0329.029.0258.0230.0205 800.399.83.007.086.074.064.0557.0486.0427.0376.0332.0295.0262.0234 900.562.322.26.0964.0830.078.0625.0546.0479.0422.0373.033.0295.0263 000.723.460.244.066.098.0795.0692.0604.0530.0467.043.0367.0327.0292 00.882.596.36.67.006.087.0758.0663.0582.053.0454.0403.0359.032 200.2038.730.477.267.093.0947.0825.072.0633.0558.0494.0439.039.0349 300.293.863.59.366.79.023.089.0779.0684.0604.0534.0474.0423.0378 400.2345.994.705.465.265.097.0956.0837.0735.0649.0574.050.0455.0407 500.2496.224.87.562.350.72.02.0894.0786.0693.064.0546.0486.0435 600.2645.2253.928.659.434.245.086.095.0836.0738.0654.058.058.0463 700.2792.2380.2039.755.58.39.50.008.0886.0782.0693.066.0550.0492 800.2937.2506.248.850.60.39.24.064.0936.0827.0733.065.058.0520 900.308.2630.2256.945.683.464.278.20.0986.087.0772.0686.062.0548 2000.3223.2754.2364.2038.765.536.34.76.035.095.08.072.0644.0576 200.3364.2876.2470.23.847.607.404.232.085.0958.0850.0756.0675.0604 2200.3503.2997.2576.2223.928.678.467.287.34.002.0889.079.0706.0632 2300.364.37.2680.235.2008.749.529.342.82.045.0927.0825.0737.0660 2400.3778.3236.2784.2406.2088.89.59.397.23.089.0966.0860.0768.0688 2500.393.3354.2887.2496.267.889.653.45.279.32.004.0894.0799.075 2600.4047.3470.2989.2585.2245.958.74.506.328.75.043.0928.0829.0743 2700.480.3586.3090.2674.2324.2027.775.560.376.27.08.0963.0860.0770 2800.43.370.39.2763.240.2095.835.63.423.260.9.0997.0890.0798 2900.444.385.329.2850.2479.264.896.667.47.302.57.03.092.0825 3000.4570.3928.3390.2937.2555.223.956.720.58.344.94.064.095.0853 4
Table 2. F aa Thermal Expansion Factor Temperature ( F) of Piping Material Aluminum Copper Type 430 2% CRMO 5% CRMO Bronze Carbon Steel Type 36 Type 304 Correcton Factor, F aa 264 37 0.993 204 322 245 0.994 55 230 90 276 0.995 08 63 37 89 0.996 63 02 86 9 0.997 9 44 34 55 0.998 25 9 44 3 4 7 6 7 0.999 68 68 68 68 68 68 68 68.000 3 27 57 46 5 22 44 30.00 246 222 232 75 28 86.002 332 296 32 225 289 240.003 45 366 389 273 358 292.004 494 434 460 32 425 343.005 568 50 527 369 489 39.006 64 566 594 47 55 439.007 73 629 662 63 488.008 783 690 730 675 536.009 85 750 795 735 584.00 98 8 858 794 63.0 956 87 98 85 674.02 054 928 979 907 727.03 2 984 040 96 777.04 89 038 02 05 799.05 5
Flow Calculation Examples: Problem: Oil with a specific gravity of 0.825 is flowing at a rate of 6000 GPM. The 20-in. standard wall (ID - 9.26-in.) carbon steel pipeline has a pressure of 75 psig and a temperature of 00 F. What is the differential pressure (h w ) that a Sensor Size 2 Rosemount 485 Annubar primary element would measure? Solution: h w Q a 2 ------ C (from Equation on page ) Q a 600 GPM C I F na K D 2 F aa ----- G f F na 5.6664 (from Table on page 3) K ( C 2 B) ------------------------------------------------- (from Equation 8 on page 0) C ( C 2 B) 4d 4(.060) B ------- ---------------------- πd 9.25π 0.070 (from Equation 7 on page 8) C.492 C 2.479 (from Table 6 on page 0) K (.479 0.070) ------------------------------------------------------------------------------------------------------ 0.6058 (.492) (.479 0.070) D 2 9.26 2 370.9476 F aa.000 ----- G f --------------.0 0.825 (from Table 6 on page 0) C l 5.6664 0.6058 370.9476.000.0 40.9625 and: 6000 h w --------------------------- 2 40.9625 8.36 inchh 2 O 6
Problem: Oil with a specific gravity of 0.825 is flowing at a rate of 22,700 LPM. The 50 cm inside diameter carbon steel pipeline has a pressure of 57 kpa and a temperature of 38 C. What is the differential pressure (h w ) that a Sensor Size 2 Rosemount 485 Annubar primary element would measure? Solution: h w Q a 2 ------ C (from Equation on page ) Q a 22700 LPM C I F na K D 2 F aa ----- G f F na 0.0065966 (from Table on page 3) K ( C 2 B) ------------------------------------------------- (from Equation 8 on page 0) C ( C 2 B) 4d 4( 2.6924) B ------- -------------------------- 0.0686 (from Equation 7 on page 8) πd 50π C.492 C 2.479 (from Table 6 on page 0) K (.479 0.0686) ------------------------------------------------------------------------------------------------------ 0.6065 (.492) (.479 0.0686) D 2 500 2 250000 F aa.000 ----- G f --------------.0 0.825 (from Table 6 on page 0) C l 0.0065966 0.6065 250000.000.0 0.23 and: h w -------------------- 22700 2 0.23 424.9 mmh 2 O 7
Problem: Steam at 500 psia and 620 F is flowing in a 24-in. ID carbon steel pipe. The measured differential pressure on a Sensor Size 3 Rosemount 485 Annubar primary element is 5-in H 2 O. What is the flowrate in PPH? Solution: W C I h w C I F na K D 2 Y a F aa ρ f (from Equation 3 on page ) F na 358.94 (from Table on page 3) K ( C 2 B) ------------------------------------------------- (from Equation 8 on page 0) C ( C 2 B) B ------- 4d 4 ---------------------- (.920) 0.09 πd 24π (from Equation 7 on page 8) C.5856 C 2.338 (from Table 6 on page 0) K (.338 0.09) --------------------------------------------------------------------------------------------------------- 0.5848 (.5856) (.338 0.09) D 2 24 2 576 Y a ( 0.0332( B) 2 0.00342) h w -------- (from Equation 6 on page 8) P f ϒ B ------- 4d 4 ---------------------- (.920) 0.09 πd 24π H w 5inH 2 O P f 500psia ϒ.3 Y a 0.0332( 0.09) 5 ( 0.00342) ----------------------- 500.3 0.9999 F aa.008 ρ f 0.843 0.972 so C l 358.94 0.5848 576 0.9999.008 0.972 77.96 W 77.96 5 432890.93 PPH 8
Problem: Steam at 3500 kpa abs and 350 C is flowing in a 60.96 cm ID carbon steel pipe. The measured differential pressure on a Sensor Size 3 Rosemount 485 Annubar primary element is 75.04 mm H 2 O. What is the flowrate in kg/hr? Solution:) W C I h w C I F na K D 2 Y a F aa ρ f (from Equation 3 on page ) F na 0.025 (from Table 3 on page 4) K ( C 2 B) ------------------------------------------------- (from Equation 8 on page 0) C ( C 2 B) B ------- 4d -------------------------- 4( 4.949) 0.027 πd 60.96π (from Equation 7 on page 8) C.5856 C 2.338 (from Table 6 on page 0) K (.338 0.027) --------------------------------------------------------------------------------------------------------- 0.5848 (.5856) (.338 0.027) D 2 609.6 2 3762.6 Y a ( 0.0332( B) 2 0.00342) h w -------- (from Equation 6 on page 8) P f ϒ B ------- 4d 4 -------------------------- ( 4.949) 0.027 πd 60.96π H w 75.04mmH 2 O P f 3500psia ϒ.3 Y a 0.0332( 0.027) 75.04 ( 0.00342) --------------------------- 3500.3 0.999 F aa.000 ρ f 3.0249 3.609 (P f 3.0249 kg/m 3 per ASME Steam Tables) so C l 0.025 0.5848 3762.6 0.999.000 3.609 9803.59 W 9803.59 75.04 26250.27 ( kg) h 9
Problem: Natural gas with a specific gravity of 0.63 is flowing in a 2-in. schedule 80 carbon steel pipe. the operating pressure is 264 psia and he operating temperature is 20 F. For a Sensor Size 2 Rosemount 485 Annubar primary element, determine the differential pressure (h w ) for a flowrate of 6 MM SCFH at a base temperature of 60 F and a pressure of 4.73 psia. Solution: Q h w ---- s P ------ f C I 2 (from Equation 4 on page ) Q s 6000000SCFH P f 264psia C I F na K D 2 Y a F pb F tb F tf F g F pv F aa F na 338. (from Table 4 on page 5) K ( C 2 B) ------------------------------------------------- (from Equation 8 on page 0) C ( C 2 B) B ------- 4d ---------------------- 4(.060) 0.86 πd.37π (from Equation 7 on page 8) C.492 C 2.479 (from Table 6 on page 0) K (.479 0.86) ------------------------------------------------------------------------------------------------------ 0.5835 (.492) (.479 0.86) D 2.376 2 29.4 The differential pressure h w is required to calculate Y a. Since h w is not known, assume Y a and verify the results. 4.73 F pb -- base pressure, psia 4.73 -------------- 4.73 temperature base ( F) + 460 F pb -- 520 F tf -- 520 flowing temperature ( F) + 460 F g --- ----------.2599 G 0.63 F pv -- -----------------.0637 Z 0.8838 F aa.00 60 ---------------------- + 460 520 520 ---------------------------------------- 0.9469 20( 520) + 460 Compressibility factor for natural gas from A.G.A Report Number 8 (from Table 2 on page 5) 20
C l 338.7.5835 29.4 0.9469.2599.0637.00 32436.74 h w ---- P Q ------ s f C I 6000000 ------------ ------------------------ 2 264 32436.74 27.07inH 2 O Now the value of Y a, assumed above, can be checked: Y a ( 0.0332( B) 2 0.00342) h w --------- (from Equation 6 on page 8) P f ϒ where B ------- 4d ---------------------- 4(.060) 0.86 πd.37π H w 27.07inchH 2 O P f 264psia ϒ.3 Y a 0.0332( 86) 2 27.07 ( 0.00342) --------------------------- 264.3 The assumed and calculated value are the same. Therefore, the value of h w 27.07 inch H 2 O is the correct answer. 2
Problem: Natural gas with a specific gravity of 0.63 is flowing in a 330 mm ID carbon steel pipe. The operating pressure is 8700 kpa abs and he operating temperature is 50 C. For a Sensor Size 2 Rosemount 485 Annubar primary element, determine the differential pressure (h w ) for a flowrate of 700 Nm 3 /m at a base temperature of 6 C and a pressure of 03 kpa. Solution: Q h w ---- s P ------ f C I 2 Q s 700( ( Nm 3 ) m) (from Equation 4 on page ) P f 8700kPa C I F na K D 2 Y a F pb F tb F tf F g F pv F aa F na.872 x 0-5 K ( C 2 B) ------------------------------------------------- (from Equation 8 on page 0) C ( C 2 B) 4d 4( 26.924) B ------- -------------------------- 0.43 (from Equation 7 on page 8) πd 300π C.492 C 2.479 (from Table 6 on page 0) K.479 0.43 ------------------------------------------------------------------------------------------------------ 0.5856 (.492) (.479 0.43) (from Table 4 on page 5) D 2.376 2 29.4 The differential pressure h w is required to calculate Y a. Since h w is not known, assume Y a and verify the results. 0.56 F pb -- base pressure, kpa abs 0.56 --------------------- 0.9860 03 temperature base ( C) + 273 F pb -- 289 F tf -- 289 flowing temperature ( C) + 273 6 + 273 ---------------------- 289 ------------------------- 289 0.9459 50 + 273 F g --- ----------.2599 G 0.63 F pv -- --------------.0684 Z 0.876 F aa.00 (from Table 2 on page 5) 22
C l.872x0 5 0.5856 90000 0.9860 0.9459.2599.0684.00.239 h w ---- P Q ------ s f C I 700 ------------ -------------- 2 8700.239 26.39mmH 2 O Now the value of Y a, assumed above, can be checked: Y a ( 0.0332( B) 2 0.00342) h w --------- (from Equation 6 on page 8) P f ϒ 4d 4( 26.924) B ------- -------------------------- 0.43 πd 300π H w 26.39mmH 2 O P f 8700kPa ϒ.3 Y a 0.0332( 0.43) 2 26.39 ( 0.00342) --------------------------- 8700.3 The assumed and calculated value are the same. Therefore, the value of h w 26.39 mm H 2 O is the correct answer. 23
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