4.1 Background
4.1.3 Differential Pressure Flow Measurement Devices
There are a variety of differential pressure flow measurement devices used to measure the flow rate of liquids and gases. For accuracies sufficient for the nuclear core power calorimetry, a calibrated ASME PTC 6 flow nozzle such as the one pictured in Figure 4.3, or custom flow tube would typically be used. In recent years, other methods, including sonic measurement using sensors mounted externally on the pipe, have become available and approved by ASME.
ASME Performance Test Codes (PTC) 19.5 and PTC 6 together with ASME MFC‐3M
“Measurement of Fluid Flow in Pipes Using Orifice, Nozzle, and Venturi” are excellent refer- ences for the manufacture, installation, and use of all types of flow‐measurement devices.
It is not the intent of this text to duplicate the information contained in the ASME documents.
Independent reading and research into these and other industry standards is encouraged. There
are a few practices, and equations from the references, which are pertinent to this case study. In 1984, ASME PTC 6 allowed use of a primary flow element in the final feedwater, downstream of a deaerating feedwater heater, for acceptance testing as long as an inspection port was included in the flow section. To maintain the highest accuracy, the nozzle was to be inspected prior to, and after the performance test, to ensure the nozzle was not damaged, and did not require cleaning. Any deposits on the flow nozzle prior to the test were to be cleaned with a high‐pressure water‐jet device. Inspections showing an iron oxide deposit should ensure the thickness was less than specified by PTC 6 and that the surface of the nozzle remained smooth throughout performance testing. A rough surface would decrease the nozzle discharge coeffi- cient, and result in an indication of flow that was higher than the true flow through the nozzle.
For continuous real‐time measurement of feedwater flow required for steam calorimetry in a nuclear power generating facility, regular inspections of the feedwater nozzle are impractical.
In this case study, the inspection ports were not installed.
Equation (4.1) shows the general equation for flow through a differential flow meter (from ASME 19.5):
m n d
CY Pgc
2
4 4
2
1 (4.1)
Values and units for the parameters in equation (4.1) are given in Table 4.1.
For the measurement of two‐phase flow, as in the case of the flow from an SGU in a nuclear plant, ASME PTC 6 includes a modified flow equation (equation 4.2):
m n d CY P
x g f f
2
4 1 5
4
1 . (4.2)
Figure 4.3 ASME flow nozzle. Source: Reproduced by permission of Triad Measurement and Equipment.
When measuring gas flow with throat tap nozzles, and venture tubes, the adiabatic expansion factor (Y) is found from equation (4.3) per ASME MFC‐3M. For liquids, Y is equal to one.
Y
2 4
4 2
1
1 1 1
1 1
/
/
/ 0 5.
(4.3)
The uncertainty of the Y, also from ASME MFC‐3M is shown by equation (4.4).
relativeuncertainty of Y P
% 4 100 8 P (4.4)
Restrictions for the use of equations (4.3) and (4.4) are set forth in ASME MFC‐3M.
The isentropic exponent, κ, in equation (4.3) for steam can be approximated by taking a finite differential from the equation (4.5). Some steam‐table functions available for use in spreadsheets can provide the isentropic exponent given for specified thermodynamic states:
P P
s
(4.5) Between 4 MPa and 7.6 MPA and steam qualities (x) between 0.9 and 0.9994, κ can be approx- imated as a function of pressure in MPa with equations (4.6) and (4.7)
a P2 2 a P a1 0 (4.6)
where
a x
a x
a
0 1
2
2 4
0 1194 1 1533
10 1 6133 1 114
10 7 7
. ln .
. ln .
. 553 ln x 2 3415.
(4.7)
Pipe and flow element diameters are measured at ambient temperatures. These measure- ments must be corrected for operating temperatures. The lineal thermal expansion factor for
Table 4.1 Flow equation units of measure.
Symbol SI Customary
m kg/s lbm/h
n 1 (kg/(m s2 Pa) 300 (ft2/s2) [(in2 s2)/(ft2 h2)]0.5
D m in
C Dimensionless Dimensionless
ρ kg/m3 lbm/ft3
P Pa psi
gc: proportionality constant 1 dimensionless 32.174 [(lbm ft)/(lbf s2)]
Source: From ASME 19.5.
various materials is given by the equation (4.8 with constants provided in Table 4.2 (ASME19.5).
106 a bT cT2 dT3 (4.8)
Length at the operating temperature is calculated with equation (4.9) from ASME PTC 19.5:
LT LB 1 T B (4.9)
where:
LT = length at operating temperature;
LB = length at base temperature;
T = operating temperature (°C);
B = base temperature, generally 20 °C.
When a flow section containing a differential flow element is calibrated, the laboratory usually establishes the flow element discharge coefficient at a throat Reynolds number of about one million. The operating Reynolds number is most often many times higher than during calibration; therefore, the calibrated value for C must be adjusted to the operating con- ditions. For a calibrated ASME PTC 6 flow nozzle or custom flow tube, the constant Cx is calculated to satisfy equation (4.10) for the calibrated values of C. Each value of C from an individual calibration, determines a value for Cx at the corresponding Reynolds numbers.
There may be several different calibration tests required for a single nozzle and each set of upstream and downstream pressure taps. The average value for Cx for a specific set of pressure taps is then used to find the value of the discharge coefficient, C, at any Reynolds number greater than one million using equation (4.10).
C Cx 0 185. Rd 0 2. 1 361 239, /Rd 0 8. (4.10) (ASME PTC 6)
where:
C: discharge coefficient at specific throat Reynolds numbers;
Cx: a constant.
Equation (4.10) results in an iterative calculation of flow. The iteration procedure begins with the calibrated value of C, yielding a calculated flow rate. At the Reynolds number Table 4.2 Constants for linear thermal expansion factor for temperature in (°C).
Material a b c d
Austenitic stainless steel 16.224 0.0063076 −5.9575E‐06 3.6098E‐9 Carbon: carbon‐moly steels 10.728 0.0081725 −1.6951E‐6 −2.0374E‐9 Source: ASME PTC 19.5.
determined at the flow rate, C is re‐established, and a new flow calculated. The process repeats until the flow rate converges to a satisfactory tolerance. Convergence is generally very rapid.
The permanent pressure loss from an ASME flow nozzle can be approximated per ASME 19.5 according to equation (4.11). If the flow section includes a diffuser behind the nozzle, approximately 70% of the loss may be recovered. If a venture or custom flow tube with a diffuser is used, ASME standards recommend that between 5% and 20% of the measured differential pressure will be a permanent pressure loss.
pressure loss
differential pressure 1 0 014. 2 06. 2 1 18. 3 (4.11) ASME MFC‐3M suggests that the permanent pressure loss through a nozzle can be approxi- mated by equation (4.12). Equations (4.11) and (4.12) yield similar results:
pressure loss 1 1
1 1
4 2 2
4 2 2
( (
) )
C C
C C P (4.12)