Calculation of flow rate from differential pressure devices orifice plates Peter Lau EMATEM - Sommerschule Kloster Seeon August -, 008 Contents Introduction - Why using dp-devices when other principles provide higher accuracy? Situation of very high temperatures. How is flow and pressure linked? Why do we need a discharge coefficient? What must be known to calculate flow rate correctly? What help does standards like ISO 567 provide? ifferences between different version (99 to 003). Which other information's are essential in a given application? What influences the flow measurement result and how important is it? What uncertainties must we expect?
Introduction Flow is always measured indirectly Flow rate one of the most complex and difficult quantity to measure and to calibrate. No principle for direct measurement exists. Many different physical measurement principles all have advantages and drawbacks. Roundabout way via speed, rotation speed, frequency, phase difference, temperature difference, force, voltage etc. (thousands of patents exist!) But: All metering devices need a calibration in one form or another. Question: How should we calibrate meters for use in hot water (> 300 C, 80 bar) or steam? Introduction Standards for dp-meters Primary devices - measure pressure difference Δp produced by flow passing an obstruction and calculate the mass flow. Any obstruction can be used if the relation between q and Δp is well known. Orifice plates Nozzles Venturi tubes are standardized obstructions. But others exist as well. The standards tell detailed how to use these devices and what the limitations are. ISO 567 Part,, 3 and from 003 (revision from 99) ISO/TR 5377 contains complementing information to ISO 567 ASME MFC-3M, 00 (revision from 989) ASME PTC 6-00 Performance Test Codes (revision from 996)
Introduction 3 The Situation for Power Plants In the late 60-ties and 70-ties no robust flow measurement technique was available for the high temperature/pressure applications. The nuclear power industry is conservative and focused on safety. The thermal effect is limited to a licensed value. A flow measurement accuracy of % kept the output at 98 % of rated effect. dp-devices showed no stable behavior due to fouling. Over the years a lot of measuring technique has been exchanged. Introduction Thermal Power Increase MUR Measurement Uncertainty Recapture is a trial to lower measurement uncertainty (mainly in flow), which in turn can be used to increase the thermal output closer to the rated effect. How can measurement uncertainty be reduced? By calibration! And by sensing of eventual changes over time! Who can calibrate flow meters at 330 C, 80 bar, Re >0 7? Nobody! What to use for sensing? A second measurement principle! 3
Introduction 5 - ifficulties in Meter Calibration Gravimetric Method - (short traceability to kg, s, C) The water must be kept at constant pressure and temperature Large amount of water big vessels are needed iverter and weighing tank in closed system to avoid steam evaporation Volumetric Method - (longer traceability to m 3, ρ water, s, C) Pipe- or Ball prover (long big pipes working under high pressure Suitable material for ball or piston to tighten at high temperature/pressure (lubrication) Large positive displacement meter as master meter Flow is forced through the orifice We need a standard to clear up how pressure p, speed v, flow q m and flow area are interrelated and used to calculated mass flow from the pressure difference between point and with satisfactory measurement uncertainty? p p A A q m v v
Flow in pipe with change in cross area Assume continuity and incompressibility p q q m m = A = A u ρ u ρ p s p static pressure V m u u s V s m s V m u ue to friction within the medium and with the wall pressure falls along the pipe. u s > u p s < p Mass m and volume V of a flow element does not change if area changes Average speed increases, static pressure falls where the area is smaller. Measurement principle Incompressibility means density does not change ρ = ρ Continuity means the flow is the same every where in the pipe q m = q m tappings for Δp-measurement q m = A ρ u u d A q m = A ρ v q m u = A ρ u : pipe diameter d: orifice diameter A = π A d = C π Area at the smallest flow cross section not exactly known! 5
6 Bernoullis equation on energy conservation p 0 : total pressure in medium p = p 0 where u = 0 u ρ : dynamic pressure 0 p const u p u p = = ρ + = ρ + The sum of static and dynamic pressure is the same everywhere in pipe. ( ) u u p p p ρ = = Δ Replace the speed terms through geometric measures and insert. m m m q u q u u A q ρ π = ρ π = ρ = ρ π ρ = Δ m q p ρ π ρ = Δ m q p = π ρ Δ m q p Replace with d, introduce β, solve for q m Final equation for calculating q m ρ π ε β Δ = p d C q m ρ Δ π = p q m Exchange unknown with known d Introduce a discharge coefficient C relating the unknown diameter to the known orifice diameter d! Introduce an expansion coefficient ε for compressible media (gas, steam). To measure flow means to determine Δp and use it in the equation with all other constants and calculate q m. β = d
Why a discharge coefficient? C determines the relation between the real flow through a primary device and the theoretical possible flow. For a given primary device this dimensionless parameter is determined using an incompressible fluid. With fixed geometry C only depends on the actual Reynolds number. In a way C can be regarded as a calibration constant for a primary device. Generally, different installations, that are geometrically equivalent and sense the same flow conditions and are characterized by the same Reynolds number, renders the same value for C. But: Re varies with pressure, temperature, viscosity and flow. And so does C. It is not easy to analyse the effect of different parameters on the flow. Why an expansibility factor ε expresses the different compressibility of different fluids (gases, steam), i.e. molecules are, depending on their form more or less compressed when passing the orifice. ε Water and other liquids are considered incompressible ε =. This makes calculations much simpler. The size of ε depends on the pressure relation and the isentropic coefficient κ, i.e. the relation between a relative change in pressure and the corresponding relative change in density. κ is a property that is different for different media and varies also with the pressure and temperature of the medium. For many gases there are no published data for κ. The standard recomends to use c P /c V. 7
What does the standards treatise on? Allowed variations for different measures (pipe diameter, orifice diameter, distance to upstream disturbances). Tolerances for all measures. How orifices must be installed. How thick the orifice can be and how much it may buckle. Which tolerances in angle are allowed How sharp or round the edges have to be. Which upstrem surface roughness the pipe must underpass. How the pressure tappings may be shaped and where they must be placed. How the tappings must be arranged in detail (diameter of drilled hole, smoothness of pipe wall, distance to possible flow disturbances etc.) Orifice plates (part of ISO 567) 8
Isa - och Venturi nozzles (part 3 of ISO 567) : 50 500 mm β: 0,3 0,8 Re : 7 x 0 x 0 7 för β < 0, Re : x 0 x 0 7 för β > 0, : 65 500 mm β: 0,36 0,775 Re :,5 x 0 5 x 0 6 Venturi tubes (part of ISO 567) All specified measures have to follow certain tolerances (not at least the surface roughness) and must be verified by measurement at several places. Production by a) cast b) machined c) welded Cylindrical Entrance section Pressure tappings a) : 00 800 mm β: 0,3 0,75 Re : x0 5 x0 6 b) : 50 50 mm β: 0, 0,75 Re : x0 5 x0 6 c) : 00 00 mm β: 0, 0,7 Re : x0 5 x0 6 Flowdirection Conical converging section Cylindrical throat section Conical diverging section 9
The standard gives advice by installation examples : conical diameter increase : totally open valve 3: place for orifice : 90 elbow As long as certain distances (in multiples of ) between the orifice (or other dp-devices) and upstream disturbances are kept, the uncertainties stated in the standard are valid. No extra uncertainties need to be added. A table for lowest distance to different installation disturbances are given in a table. Table for critical distances to upstream disturbances 0
Measurement of pressure and differential pressure In flow computers or intelligent transmitters the density for the known liquid or gas is calculated from equations of state if pressure, temperature and gas composition is known. The measurement of static pressure p and differential pressure Δp should be separated. A correct Δp is of outmost importance. Pressure i triple T form pipe To be observed in Δp-measurement Zero point adjustment Clogged pressure tappings/lines Pressure lines must be easily drained or ventilated No disturbances for flow close to the pressure tappings Pressure transmitter for Δp C = 0,596+ 0,06β + + ischarge coefficient C for orifice plates Purpose: defines the relation of jet flow diameter to orifice diameter and distance to pressure tappings. ( 0,088 + 0,0063 A) 0L 7L ( 0,03 + 0,080 e 0,3 e ) ( 0, A),, 3 0,03 ( M 0,8 M ) β 0,6 β β 3,5 8 6 0 Re 6 0 β + 0,0005 Re 0,3 0,7 β β d β = relation between orifice and pipe diameter 0, 8 9000 β A function of β and Re A M function of β and L M Re = L β ata on pipe orifice plate medium ata on upstream pressure tappings ata on downstream pressure tappings = Reader-Harris Gallagher Equation
Standards are updated from time to time ISO 567 (99) & ASME MFC-3M (00) ischarge coefficient ISO 567 (99) C = 0,596 + 0,06β + + ( 0,088 + 0,0063 A) 0L 7L ( 0,03 + 0,080 e 0,3 e ) ( 0, A),, 3 0,03 ( M 0,8 M ) β ε = 0,6 β 8 6 0 β + 0,0005 Re 0,3 6 3,5 0 β Re Reader-Harris Gallagher equation ( ) κ 8 p 0,35 + 0,56 β + 0,93 β * ε = ( 0,+ 0,35 β ) β β p 0,7 C= 0,5959+ 0,03β + 0,0900L β Expansibility factor, 8 0, 80β + 0,009β ' 3 ( β ) 0,0337L β Stolz equation Δp κ p,5 6 0 Re 0,75 ifferent flow result with new version from 003 ifference in air flow calculation using different versions of ISO 567 ISO 567 Flow rate Expansion coefficient ischarge Coefficient 99 003 iff [%] 99 003 iff [%] 99 003 iff [%] 887,69 893,5 0,65 0,9908 0,9909 0,008 0,6007 0,606 0,63 Air 0 ºC 6,5 68,6 0,553 0,987 0,988 0,00 0,5990 0,6067 0,53 783,36 79,7 0, 0,9673 0,9673-0,00 0,5987 0,6039 0,3 07,9 5,05 0,350 0,957 0,95-0,03 0,59838 0,60069 0,38 86,00 869,87 0,676 0,9909 0,99 0,008 0,60053 0,6057 0,667 Air 0 ºC 0, 7,7 0,580 0,987 0,988 0,00 0,59966 0,60305 0,570 70,95 79,00 0,68 0,9673 0,967-0,00 0,598888 0,607 0,70 7,6 5,6 0,377 0,957 0,95-0,03 0,59856 0,60099 0,0 8,0 87,09 0,73 0,9909 0,99 0,008 0,6009 0,605 0,70 Air 50 ºC 50,09 57,3 0,67 0,987 0,988 0,00 0,599936 0,6036 0,607 66, 63,37 0,505 0,967 0,967-0,00 0,59935 0,609 0,507 0,77 0, 0,3 0,957 0,953-0,03 0,598736 0,60 0,6 The difference is predominantly caused by the discharge coefficient. The change in C decreases with increasing flow
Changes in flow rate calculation according to ISO 567 between 99 and 003 With the same conditions the latest version of ISO 567 translates the differential pressure Δp to 0, - 0,3 % higher flow rate qm 0,35 ifference in mass flow calculation qm Increase in qm from 99 to 003 [%] 0,30 0,5 0,0 0,5 0,0 0,05 Orifice plate d: 3,9 mm :,7 mm β: 0,55 low flow rate 60 kg/min high flow rate 79 kg/min 0,00 0 50 00 50 00 50 300 350 00 Water temperature [ C] Change in discharge coefficient between versions Steam at bar and ~88 ºC Δp [Pa] qm [kg/s] C C iff [%] 99 003 573 0,50 0,59890 0,6007 0,30% 5030 0,753 0,59835 0,59973 0,3% 993,058 0,5979 0,59885 0,6% 503,300 0,5977 0,59838 0,% The change in discharge coefficient decreases with increasing flow rate. 3
How are a dp-devices often used? The primary flow device is bought with a belonging discharge coefficient C (one value). calculated to a standard for a specified application (medium, nominal flow rate, pressure, temperature Re ) or calibrated in a similar reference condition Re Actual flow q m (act) is calculated in relation to nominal flow rate q m (nom) using the square root dependency from Δp q (act) = m Δp(act) q Δp(nom) m (nom) For a different application a new relation q n Δp must be applied What must be known to calculated flow rate correctly? How correct does the simplified model work? q (act) = m Δp(act) q Δp(nom) m (nom) Change in indicated Δp Change in Calculated flow qn Error in simplified calculation Example for orifice plate In water at 90 ºC Δp = 0, bar (0000 Pa) β = 0,736 Re= 50 000 + % +0 % 0,5 % -,86 % 0,00 % 0,03 % -0% -5, % -0,06 % Conclusion: The formula works well if only Δp changes Question: What happens if temperature, viscosity or media pressure varies?
Not registered process parameters changes lead to measurement errors Example: Hot water flow at 90 ºC Increase in temperature ΔT=3 ºC (influences directly and indirectly) viscosity -3,7 %, density -0, %, Re +3,5 %, C -0,0 % FLOW RATE Δq m -0, % Increase in viscosity Δμ =3 % (influences indirectly) Re -,89 %, C + 0,0 %, FLOW RATE Δq m +0,0 % Increase in density Δρ=0,3 % (influences directly) Re +,9 %, C - 0,00 %, FLOW RATE Δq m +0,5 % Errors do not add statistically! What more information is needed? qm calculated from Δp if everything else is known C π q m = ε d Δp β ρ Calculation parameters for mass flow q m : Pressure difference Δp ensity of medium ρ Orifice diameter d iameter relation β =d/ Expansibility factor ε ischarge Coefficient C Except β all other factors change with pressure and/or temperature Measured with differential pressure transducer T P P Measurement computer or intelligent tansmitter 5
How are the variables and parameters interrelated? Not relevant for liquids Isentropic exponent of media Static pressure up / down stream κ p p β ε Orifice diameter pipe diameter d β Pipe diameter, coefficient of expansion ynamic viscosity of medium prel. flow estimation (T) μ q m Geometry of pressure tappings up/down stream Orifice diameter, coefficient of expansion ensity of medium at process conditions Re L L measured C β d(t) Δp ρ (T,p) q m Necessary knowledge about process conditions How much does viscosity change with temperature? How much does density change with temperature? How much does density change with pressure (steam)? How much does temperature change with pressure (steam)? Viscosity and density of water as a function of temperature Important for calculating Reynolds number Re ischarge coefficient C (and Expansion coefficient ε) Viscositet [mpa s],,,0 0,8 0,6 0, Viscosity of water as a function of temperature Temperature [C] 00 000 900 800 700 ensity of water as a function of temperature 0, 600 0,0 0 50 00 50 00 50 300 350 00 Temperature [C] 500 0 50 00 50 00 50 300 350 00 ensity [kg/m3] For hot water these relations are reasonably well known - IAPWS 6
Reference calculation for testing - simulating saturated steam Iterative calculation of flow rate Assume C =0,6 Calculate qm Calculate Re Calculate C Calculate qm Calculate Re Calculate C Calculate qm Calculated Process parameters Reference flow Set upstream pressure Set downstream pressure How is flow measurement affected? Unrecognized process Temperature change +,5 ºC at 90 ºC parameter change: + 3 ºC at 00 ºC C π q m = ε d Δp β Temperature influences flow rate q m directly indirectly ρ C = f(re; β qm ;L) Re = π μ Change in ensity and Viscosity 90 +,5 ºC -0,0 % -,7 % 00 + 3 ºC -0,38 % -,5 % Orifice/pipe d: 3,9 mm :,7 mm β: 0,550 Δp = is assumed constant Flow changes at high low flow 90 +,5 ºC -0,05 to -0,06 % 00 + 3 ºC -0,8 to -0,9 % Effect on flow rate due to an unrecognized temperature increase. 7
How is flow measurement affected? Unrecognized process Viscosity change: + % or + % parameter change: at 90 and 00 ºC C π q m = ε d Δp β ρ Viscosity influences flow rate indirectly qm C = f(re; β;l) Re = π μ Δp = is assumed constant Orifice/pipe d: 3,9 mm :,7 mm β: 0,550 Flow changes at 90 ºC high low flow Δμ =% 0,003 to 0,005 % Δμ=% 0,006 to 0,00 % Flow changes at 00 ºC Δμ =% 0,00 to 0,007 % Δμ =% 0,008 to 0,0 % Both flow rate and temperature have marginal influence How is flow measurement affected? Unrealized error in pipe diameter +0,5 mm or + mm at 90 ºC at 00 ºC C π Δ = 0,5 mm,7 % of Δ = mm,3 % Δ = 0,5 mm Δβ -,6 % Δ = mm Δβ -,9 % Orifice/pipe d: 3,9 mm :,7 mm β: 0,550 Pipe diameter influences flow rate indirectly q = ε d Δp ρ ) m C f(re ; ;L β q = β β = m Re = π μ Δp = is assumed constant Flow changes at 90 ºC high low flow Δ=0,5 mm -0,5 to -0,6 % Δ = mm -0,9 to -0,5 % Flow changes at 00 ºC Δ=0,5 mm -0,9 to -0,5 % Δ = mm -0,83 to -0,96 % The flow rate and temperature influences marginal d 8
How is flow measurement affected? Unrealized error in orifice diameter d +0,05 mm or +0, mm at 90 ºC at 00 ºC C π Δd = 0,05 mm 0, % of d Δd = 0, mm 0,3 % of d Δd = 0,05 mm Δβ -0, % Δd = 0, mm Δβ -0,3 % Orifice/pipe d: 3,9 mm :,7 mm β: 0,550 Orifice diameter influences flow rate indirectly q = ε d Δp ρ ) m C = f(re ; ;L β directly β β = Δp = is assumed constant Flow changes at 90 ºC high low flow +0,05 mm -0,73 to -0,7 % +0, mm -0,98 to -0,950 % Flow changes at 00 ºC +0,5 mm -0,7 to -0,73 % + mm -0,96 to -0,97 % d The flow rate and temperature influences marginal Uncertainty in discharge coefficient C (ISO 567) Venturi tubes: a) casted 0,7 % U(C) depends on b) machined,0 % production method c) welded,5 % Venturi nozzles. U( C) = (, +,5 β )% β=0,736:,6 % ISA nozzles Orifice plates If <7, mm U U(C) = (0,7- β) % for β 0, to 0, U(C) = 0,5 % for β 0, to 0,6 U(C) = (,667β 0,5) % for β 0,6 to 0,75 U(C) = + 0,9 ( 0,75 β),8 % 5, If moreover β > 0,5 och Re < 0 000 ( C) = 0,8% if β 0, 6 0,8 % ( C) = ( β 0, )% if β 0, 6 β=0,736:,07 % U > = 70 mm β = 0,3 = 60 mm β = 0, 0,6 0,5 % 0,5 % 0,5 0,75 % +0,08 % +,38 % +0,5 % 9
Uncertainty in expansibility factor ε Uncertainty depends on geometry and pressure conditions. With β = 0,736 (d=73,6 mm / =00mm) ΔP = 5000 Pa ( 0,5 bar); P = 500 Pa (,5 bar absolute) κ =,399 (for air at 50 C). 8 Δp Venturi tubes U() ε = ( + 00 β ) %,6 % p p Venturi nozzles U() ε = Not ( + 00 relevant β ) % p,6 % when measuring water flow Δp ISA nozzles U() ε = % 0,6 % p Δp κ p 8 Δ () % Orifice plates U ε = 3,5 0,336 % Problems with direct temperature and density measurement Fluid temperature and density should be measured 5 5 downstream from orifice plate in order not disturb flow. Reynolds no Re needing this information apply however for upstream conditions. For very accurate measurements (especially non ideal gases) the temperature drop over the orifice must be calculated: ΔT = μ Δϖ JT Ru T Z β ( C ) μ JT = Δϖ = p cm,p T p Joule Thomson coefficient R u : universal gas constant T: absolute Temperature p: static pressure in medium c m,p : molar heatcapacitet at const. p Z: compressibility factor β ( C ) Cβ + Cβ Δp Pressure drop over orifice β: diameter ratio C: discharge coefficient Δp: pressure difference,9 Δϖ β Δp 0
What uncertainties then must we expect in measurement? U(model) 567 U(C) U(ε) U(β) U(d) ifference to experimental experience U(Δp) U(T) U(P) Measurement of process variables Way and equipment To perform the calculations (iterative approach) U(κ) U(μ) U(ρ) Model of calculating process parameters Reproducibility of measurement Installation induced errors Flatness Concentricity istance to disturbances Tolerances Continuous control Quality of Δp-device Analysis of the measurement uncertainty according to GUM for an idealized situation using GUM-Workbench U(qm) Model equation in GUM Workbench Test situation for air at 50 ºC Orifice plate with β=0,736
Variable declaration Estimation of all uncertainty components
The total uncertainty budget Uncertainty contributions at idealized conditions Process media: Air at 50 C! =00 mm, d=73,6 mm β=0,736 ρ=,098 kg/m 3, μ:,97*0-5 Pa s, κ=,399 P=,038 bar, Δp=7,53 mbar Important uncertainty contributions: o ischarge coefficient U(C): 0,73 % - at best 0,5 % (,5 %) o ensity U(ρ ): % (5,7 %) o iameter U(d): 0, mm of 73,6 mm (0, % ) (3,9 %) o iameter U(): 0,5 mm of 00 mm ( 0,5 %) ( %) o ifferential pressure U(Δp): 0, % of reading (, %) o Pressure U(P,P ): 0, % of reading (, %) of calculated flow rate qm=0,376 ± 0,007 kg/s (, %) With iteration process qm=0,376 kg/s 3
Uncertainty contributions at idealized conditions Process media: Water at 50 C! =00 mm, d=73,6 mm β=0,736 ρ=988, kg/m 3, μ: 5,85*0-5 Pa s, P=,038 bar, Δp=50 mbar Important uncertainty contributions and their contribution to relative importance: total uncertainty ischarge coefficient U(C): 0,73 % - at best 0,5 % (57, %) iameter U(d): 0, mm of 73,6 mm (0, % ) (9, %) iameter U(): 0,5 mm of 00 mm ( 0,5 %) (6,5 %) ifferential pressure U(Δp): 0, % of reading (5,8 %) Temperature U(T): 3 ºC (, %) ensity U(ρ ): kg/m 3 (0, %) (0, %) All others contributions below 0, % Calculated flow rate qm = 6,6 ± 0,59 kg/s (k=) ±0,96% Installation induced systematic effects add linearly to uncertainty About measurement quality Standard ISO 567 specifies a measurement uncertainty for C och ε if all demands are fulfilled concerning: production (according to tolerances) installation (with respect to disturbances) usage (within the limits of the standard) The standard expects to examine the orifice plates on a regular basis, to verify pipe measures (roundness, dimension, roughness), to have good knowledge of the process conditions of the medium (i.e. pressure fluctuation, single-phase, pressure > liquid evaporation pressure, temperature > condensation point of gases, τ=p /p 0,75). At low gas flow and large temperature differences between medium and surrounding insulation is necessary. Any violation of ideal flow conditions must be explored. Swirl and non-symetric flow profiles must be avoided. Flow conditioners are highly recommended.
Total measurement uncertainty If all process parameters were known without any error: The best measurement uncertainty (orifice plates) according to standard amounts to: 0.7 % on a 95 % confidence level. The uncertainty in C, the dominating contribution can possibly be decreased by testing. But other contributions from installation can easily generate larger uncertainty contributions. Orifice plates -- their + and - Simple and cheap production (especially for big diameters) Cheap in installation No moving parts - robust Works as primary element no calibration!? Good characterized behavior Simple function control Suits most fluids and gases even extreme conditions Available in many dimensions and a large flow range Less dynamic flow range for fixed diameter Large pressure loss Non linear output, complex calculation - needs urgent calculation help Process parameters must be known for flow calculation Problems with different phases or flow variations Can be sensitive for wear and disposition Very sensitive for disturbance and installation caused effects To achieve the measurement uncertainties indicated by ISO 567 many conditions must be fulfilled 5
Limitations for Orifice Plates med ISO-567 Pipe diameter : Reynolds no Re : Medium: Flow: Pressure tappings: 50 00 mm >350 turbulent flow is demanded No upper limit but there is no proof single phase liquid, steam or gas steady, no fast changes or pulsations corner-, flange eller & / tappings no support for other tappings by the standards Pressure measurement: Static pressure absolute units efinition: p upstream, p downstream Pressure difference: Δp =p -p only for specified tappings Geometry: β=d/, d orifice diameter, pipe diameter d >,5 mm; 0, β 0,75 To measure flow traceable with Δp devices we need. Transfer model between differential pressure Δp measurement and the momentary flow rate qm.. Precise definition of the selected primary device with the related discharge coefficient. Use the same standard ISO 567. 3. esign and production tolerances of all geometric measures that determine the Δp q m relation.. Clear advice for installation to keep disturbing effects below a significant error level. 5. Additional contributions to uncertainty due to violation of point. - quantified tolerances for distance to certain perturbations. 6. Consider process parameter changes in calculation. 7. A routine to ensure that original conditions will be preserved in future! 8. A proven method to determine the discharge coefficient C at extreme temperatures and pressure. 6