Dier Et Al. 2011.pdf

Flow Measurement and Instrumentation 22 (2011) 97–103

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Time resolved measurement of pulsating flow using orifices K. Doblhoff-Dier a,∗ , K. Kudlaty b , M. Wiesinger b , M. Gröschl a a

Institute of Applied Physics, Vienna University of Technology, Wiedner Hauptstrasse 8/134, A-1040 Vienna, Austria

b

AVL List GmbH, Hans-List-Platz 1, A-8020 Graz, Austria

article

info

Article history: Received 27 September 2010 Received in revised form 8 December 2010 Accepted 12 December 2010 Keywords: Flow metering Pulsating flow Orifice

abstract This article reviews the theoretical background of the measurement of pulsating flow using orifices as flow to pressure transducers, providing a synopsis of work done in this field. Special attention is paid to the temporal inertia and the applicability of expressions thereof given in the literature. Other factors influencing the measurement, such as changing flow profiles and the effect of connection tubes between the pressure sensor and the orifice are discussed. An experiment was performed to investigate the applicability of an equation taking reverse flow and temporal inertia into account for the measurement of pulsating flow with relative pulsation amplitudes around 1 and frequencies up to 50 Hz. It was found that the suggested equation may give tolerable results if the ratio of the pulsating part of the velocity to the angular frequency times orifice diameter is not too high. For high ratios, however, the results could not be explained by the suggested equation. © 2010 Elsevier Ltd. All rights reserved.

1. Introduction In many industrial applications, the accurate flow rate measurement of liquid or gas is important. Various different techniques have been developed to allow precise measurement under different circumstances. Well established devices for the measurement of steady flow include differential pressure flow meters, electromagnetic and ultrasonic flow meters, vortex shedding flow meters, hot wire anemometers, etc. The exact, time resolved measurement of pulsating flow, on the other hand, still poses great problems. Pulsating flow, however, often occurs in real life applications. An example from the automotive industry is the measurement of blow-by gases in engines, i.e. gases escaping from the pistons and piston rings via leaks. While accurate measurement of the mean flow can be achieved by installing damping devices, this deteriorates the time resolution. Hot wire anemometers can provide accurate results for pulsating flow up to extremely high frequencies [1], but their use is limited to clean flow [1,2]. Orifice plates can handle high temperatures and dirty flow up to a certain extent, but in the literature their use for the measurement of unsteady flow is often discouraged. Nevertheless, some effort has been made by various authors to investigate and establish the use of orifice plates for the measurement of pulsating flow. This paper is intended to provide a review of work done in this field and to outline the limits of applicability of existing

∗

Corresponding author. Tel.: +43 1 368 70 68; fax: +43 1 58801 13499. E-mail address: [email protected] (K. Doblhoff-Dier).

0955-5986/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.flowmeasinst.2010.12.001

theories. Furthermore, the paper is intended to summarize factors that may influence the measurement of pulsating flow using orifices. The last section of this paper deals with an experiment made in order to investigate the possible extension of existing theories to flow conditions beyond their theoretical limits of applicability. 2. Theory review 2.1. Background The flow rate measurement with orifices as transducers is based on the determination of the differential pressure between the upstream and the downstream side of the orifice. For steady flow, the measurement procedure is regulated in the corresponding ISO standards ISO 5167-1 and 5167-2 [3,4]. According to these standards, the mass flow rate qm is given by qm = 

CD 1 − β4

π  ε d2 2∆pρ1 , 4

(1)

where ∆p is the differential pressure between the upstream and the downstream pressure port and ρ1 is the upstream density. The variable β defines the ratio of orifice diameter d to tube diameter D. The discharge coefficient CD and the expansibility factor ε are given by empirical formulas, which depend on β , the Reynolds number, the location of the pressure taps, the upstream and the downstream pressure, and the adiabatic index. Typically, the value of CD lies around 0.6, that of ε is near to 1. In order to be able to appreciate the implications of pulsating flow, it is crucial to be aware of the origin of these equations.

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Fig. 1. Typical flow pattern through an orifice [5].

Although some of the following is common knowledge, it will be repeated here for reasons of clarity. Fig. 1 shows a typical flow pattern through an√orifice. With this flow pattern in mind, the scaling of qm with ∆pρ1 and all prefactors except for the discharge coefficient CD and the expansibility factor ε can be explained by assuming a one-dimensional, incompressible flow and neglecting the dissipation of energy. The derivation is based on the time independent Bernoulli equation

v12 2

+

p1

ρ

=

v22 2

+

p2

ρ

,

(2)

2.2. Pulsating flow through orifices As mentioned above, the equations given in the ISO standards ISO 5167-2 only apply in the case of steady, i.e. one-directional, non-pulsating flow. For unsteady flow, the common procedure is to use adequate damping systems [1,10]. This, however, hampers proper time resolution, as damping devices act as low pass filters. In the case of pulsating, i.e. time dependent flow, a potential derivation of the equation qm = qm (∆p) for incompressible fluids cannot be based on the time independent form of the Bernoulli equation any longer. Instead, the following form of the Bernoulli equation [11] should be considered ⇀

∂v 1 ⇀ ⇀ + v (∇ v ) + ∇ p = 0, ∂t ρ

(3)

which can be integrated along a streamline to give

v2 1 ∂v ds + + ∂t 2 ρ

∫

dp = const.

(4)

2.2.1. Temporal inertia Astonishingly enough, the first integral in Eq. (4) does not evaluate to zero in the case of pulsating flow, even if the pressure taps are situated close to the orifice and hence ds → 0. Several different expressions, replacing this integral for the case of an orifice can be found in the literature: The probably best known is that given in the ISO technical report ISO/TR 3313 [10]

∆p =

8(1 − β 4 ) CD2 π 2 d4 ρ

q2m +

which implies that the following replacement was made:

ρ

where v is the flow velocity and the subscripts 1 and 2 refer to the upstream and the downstream side of the orifice, respectively. The complete derivation can be found in various textbooks (e.g. [6]). The coefficient CD is typically reasoned to arise due to an effective cross section of the flow which is smaller than that of the orifice and due to frictional effects [2,1]. The coefficient ε is usually simply said to account for the compressibility of the fluid. Unlike the discharge coefficient CD [7], for which no theoretical expressions can be found, theoretical equations exist for ε [8,9], which show remarkable agreement with the empirical equations stated in Ref. [4].

∫

Fig. 2. Comparison of different models for the effective length Le : models by Mottram [12], Keyser [15], Gajan et al. [12], Johnston [14], Karal [13], and the model given in ISO/TR 3313 [10].1 (Ratio β = d/D, where d and D are orifice and tube diameter respectively.)

4Le

dqm

π d2 CC dt

,

(5)

∫

∂v ∼ 4Le dqm ds = , ∂t π d2 CC dt

Le ≈ d ,

(6)

where CC is the contraction coefficient. Le is typically referred to as effective length. An overview of several other, differing, models is given in Ref. [12], many more can be found for example in Refs. [13,14]. Although there are significant quantitative differences between the models, all expressions have the same dependency on the time derivative of the mass flow rate, dqm /dt. The differences can thus be reduced to a varying effective length Le . A comparison of the results for six different models is shown in Fig. 2. Most models show a decrease of Le towards zero for β → 1. This is reasonable, as we do not expect a pressure drop between two closely situated points if there is no obstruction in the flow path. In this light, it seems worthwhile reconsidering the validity of the widely accepted expression given in the ISO/TR 3313 [10] where Le is constant even for high values of β . Furthermore, it is highly interesting to note that the models by Karal, Johnston and Keyser et al., which already match closely, can be brought to near congruence with the model by Mottram, if CC is set to 1 in the latter. This could be explained by the fact that the expression by Mottram was obtained assuming viscous flow, whereas the expressions by Karal and Johnston were obtained neglecting viscosity. Therefore, it seems probable that the factor CC comes into play due to viscosity and should therefore be added to the expressions by Karal and Johnston. The literature says little about the range of applicability of the expressions shown in Fig. 2. Analyzing the readily accessible derivations by Karal and Johnston in more detail brought the following to light: Although the two authors used completely different approaches (Karal an analytic approach and Johnston a computational one) both made the same (implicit) assumptions. These assumptions are (1) the validity of linear approximations, (2) zero time-averaged flow and (3) non-viscous fluids. Karal assumed furthermore that the wavelength is long in comparison to the tube diameter. In linear approximations, the term proportional to v 2 in Eq. (4) is neglected. Assuming a plane wave and performing a Fourier transform of Eq. (3), it can be shown that this can only be done if

1 For this graph, it was assumed, that C is defined by C 2 = C 2 C 2 (1 − β 4 )/(1 − c v C D CC2 β 4 ), as in Ref. [10]. CD was set to 0.6 and Cv was assumed to be equal to 1. For the model by Gajan et al., the pressure taps were assumed to be situated right next to the orifice, such that this distance is negligible compared to the orifice diameter in the first case and to −D and D/2 in the second case. Setting CC to the values proposed in Ref. [12], does not make any significant difference for the given range of 0.4 ≤ β ≤ 0.7.

K. Doblhoff-Dier et al. / Flow Measurement and Instrumentation 22 (2011) 97–103

v ′ ≪ ω/k = c, where v ′ is the pulsating component of the velocity, ω is the angular frequency, k the wave number and c the speed of sound. In the case of an orifice, the (pulsating) flow is accelerated ⇀

⇀

through the orifice. This will increase the v (∇ v )-term compared to a plane wave and the velocity would have to be even smaller in order to allow neglecting this term. The following discussion should give and idea of the order of magnitude we may expect ⇀

⇀

for the v (∇ v )-term. We use the expressions for potential flow through an (unbounded) orifice given in Ref. [16] and restrict ourselves to the centerline, where the integrals can easily be ⇀

evaluated. In this case, v(x = 0) = v0 , and (∇ v ) has a maximum at x = 0 which evaluates to 2/dv0 . Assuming that the flow pattern is unchanged in pulsating flow and that the pulsating component of the velocity is given by v ′ (x, t ) = v ′ (x) · exp(iωt ), we find additionally to the above restriction: v0′ ≪ ωd/2, where v0′ is v ′ (x = 0) averaged over the cross section of the orifice. If the flow is bounded by a tube upstream of the orifice, the flow pattern would of course be changed, but from this appraisal, we can gain a first insight to the applicability of Eq. (6). The nonlinear impedance of an orifice, and hence the correct representation of Eq. (6) for high v2 , has been studied by various authors [17–19], but no analytical results or comprehensive quantitative values can be found. The issue of non-zero mean flow has been addressed by Peat [20]. He extended the theoretical work done by Karal [13] and performed an additional finite element analysis. Hence, he showed that a mean flow at a Mach number below 0.3 does not significantly change the results for the inductance,2 which corresponds to the effective length Le , found by Karal. This is an extremely important result, as it allows accounting for a mean flow simply by adding a resistive term to the total impedance, as was done in Refs. [10,12]. 2.2.2. Pulsating flow with flow reversals According to the ISO/TR 3313 [10], one limiting condition for the applicability of Eq. (5) is that no flow reversals may occur. Using a similar approach as shown in Ref. [6], but using Eqs. (4) and (6) instead of Eq. (2), this limitation can easily be explained. If v1 is the velocity on the left hand side of the orifice, and v2 that on the right hand side, then the mass flow rate can be approximated as qm = ρv1

π D2 4

= ρv2 CC

π d2 4

,

(7)

if the flow direction is from left to right. But if the flow direction reverses (i.e. qm → −qm ), the geometry shown in Fig. 1 is flipped and the last equation has to be replaced by qm = ρv2

π D2 4

= ρv1 CC

π d2 4

.

(8)

Hence, we obtain two different solutions for ∆p(qm ) depending on the flow direction. Together with the temporal inertia this leads to

∆p =

8(1 − β 4 ) CD2 π 2 d4 ρ

q2m sign(qm ) +

4Le

dqm

π d2 CC dt

.

99

a conference paper from 1981 [22]. In both cases the authors do not (necessarily) assume that Le is independent of v (or v ′ ). However, most sources either neglect the temporal inertia term or the sign(qm ) in the first term. This is less astonishing considering that linear approximations are used in general to obtain expressions for the temporal inertia: If flow reversal occurs due to flow pulsations, then the pulsating component v ′ of the velocity has to be of the same order of magnitude as the time-averaged velocity v and hence v ≪ ωd/2. In this case, however, the time average of the first term is much smaller than that of the second term. Errors in the evaluation of the second term would thus probably render the measurement of the mean mass flow impossible. Another point worth noting when considering reverse flow is that the flow rate has to go through zero in order to reverse. At low flow rates, however, measurement errors might occur. Indeed, for steady flow, the related ISO standard [4] requires the Reynolds number to be greater than 5000. 2.2.3. Further effects due to pulsations The arguable value of the effective length in Eq. (9) is not the only uncertainty in the determination of the differential pressure. In Ref. [12] Gajan et al. state that the first term of Eq. (9) has to be adjusted too, if strongly pulsating flow is considered. The authors of that paper present a graph implying that the ‘‘correct’’ term differs by approximately −7% from the conventional term for ∆p′rms /∆ps = 1, where ∆ps is the differential pressure that would be measured under steady flow conditions and ∆p′rms is the root mean square value of the fluctuating component of the differential pressure. This deviation is explained by the change of the mean flow profile under pulsating flow conditions. Flow profiles under pulsating flow conditions are extremely complex. While an established theory exists describing the flow field of laminar pulsating (oscillating) flow [23], which corresponds well with measurements [23–26], the regions of transitional and turbulent pulsating flow have been little investigated [26]. Many other authors also acknowledge the fact that pulsations may have an effect on the first term of Eq. (9). Most of them account for the variability in a changing discharge coefficient [27,12,10]. In Ref. [12] from 1992 the variability of the discharge coefficient is referred to as ‘‘the biggest gap in our knowledge’’. 2.3. Dynamic pressure measurement If the flow through the orifice pulsates, so will the differential pressure over the orifice, which has to be measured in order to assess the mass flow rate. For a correct dynamic pressure measurement, not only the pressure sensor itself has to have a fast response, but the whole system consisting of pressure sensor and tubes connecting the sensor with the orifice, referred to in the following as connecting tubes. Resonances may occur in the overall system, which can seriously distort the transfer function of the pressure sensor, namely standing waves in the connecting tubes and Helmholtz resonances in the complete system.

(9)

Note that this equation differs from Eq. (5) by a sign in the first term if flow reversal occurs. At least two references in which ∆p has the structural form given by Eq. (9) can be found, namely a patent from 2007 [21], and

2 In acoustic theory, the term impedance is often used in analogy to the impedance in electronics. Thereby, the impedance relates the differential pressure across a component t to the flow velocity or volume flow rate. As in electronics, the inductance is part of the impedance and is related to inertia.

2.3.1. Resonances The resonance frequency f0 = ωo /2π for standing waves is given by

ω0 = 2π

c 4l

,

(10)

where l is the length of the tubing [28]. Often used, simple expressions for the Helmholtz resonances can be found for example in Refs. [29,30]. However, these equations can only be expected to hold true if the internal volume V of the pressure sensor is large against the tube volume lA. Otherwise the expressions proposed

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by Bajsic et al. [31] should be used: f¯0 =

1 2π

 c

A lVeff

,

where Veff = V +

lA 2

,

(11) ω0 = 2π f¯o 1 − ξ 2 , 2µ , ξ= f¯0 ρ A where µ is the dynamic viscosity and ρ is the fluid density at the



inlet.3 It is important to design the pressure measurement system such that the system’s resonance frequencies given in Eqs. (10) and (11) are much higher than the expected pulsation frequency of the flow, as the resonance will otherwise distort the transfer function of the sensor. 2.3.2. Response time The response time of the system has to be much shorter than the time period of the pulsation in order to allow correct measurement. Otherwise, the signal will be low pass filtered. Assuming that the pressure sensor itself is ideal, the response time of a pressure measuring system is influenced by two factors: The Helmholtz resonator-behavior and the friction in the connecting tube. The response time [31] due to the former effect can be evaluated for ξ < 1 to give

ρA τ= . 4πµ

(12)

The latter effect can be understood as follows: In gas filled tubes, a mass flow will occur, if the pressure changes at the tube inlet. This mass flow continues until the pressure at the tube inlet and at the pressure transducer are equal. Due to friction at the walls, this adjustment takes time. The time which the system needs to reach 1/e of the final value may be considered as the response time of the system. Sinclair and Robins [32] derived a formula for the response time assuming laminar flow in the connecting tube. Their derivation is only valid if the pressure change is quasi-steady. Neglecting some terms of minor influence, the formula may be simplified to the form given in [28,33]. For air and the usual tubings with a diameter of about 2 mm and a length no longer than 20 cm, this response time is smaller than 0.001 s. For longer tubing and especially for smaller diameters this response time increases rapidly. 3. Experimental investigation Making use of the theoretical findings summarized above, an experimental investigation was performed in order to identify the magnitude of the measurement errors occurring in measurement of pulsating air flow with relative pulsation amplitudes v ′ /v around one and frequencies up to 50 Hz. 3.1. Calculation Following the example in Refs. [21,22], the theoretically derived Eq. (9) was generalized for the experimental work to A

ρ

q2m sign(qm ) + B

dqm

, (13) dt where A = A(qm ) and B = const. were treated as calibration function and calibration constant, respectively. A priori, this ansatz allows the effective length Le to take an unknown value. Furthermore, it is no longer necessary to use a standardized orifice with known CD . Once A and B are known, the time dependent mass flow rate qm = qm (t ) can be calculated from the differential pressure ∆p = ∆p(t ) by numeric integration. ∆p =

3 Note that Bajsic et al. state that their values for ξ are systematically too low as inlet and outlet losses are not taken into account.

Fig. 3. Measurement setup.

3.2. Setup All measurements were performed on a flow bench. As measurement device, an AVL 442 Blow By Meter, which has an inbuilt orifice, was used. The pipe diameter was chosen to be 7 cm. The device was adapted by equipping it with a fast response differential pressure sensor (Keller PD-23; range: −200–200 mbar), a thermocouple and an absolute pressure sensor (Keller PAA23; range 0–2 bar). The tubes connecting the differential pressure sensor with the orifice were about 20 cm long. Shorter tubings could not be used due to the necessary adaptations of the AVL 442 Blow By Meter to fast response pressure sensors. The differential and absolute pressure and the temperature were each recorded with a sample rate of 10 kHz using a Dewetron data acquisition system. As reference device, a Sensy Flow FMT700-P was used. Pulsations were generated with a ball valve. The setup of the measurement is shown in Fig. 3. 3.3. Method In order to determine A in Eq. (13), the butterfly valve was removed from the setup and the differential pressure ∆p was measured at several different mass flow rates between 0 and 300 kg/h. The calibration constant B was not determined from measurements, instead it was set to several different plausible values. These values were determined by B = 4Le /(π d2 CC ), using Le in the range of 0 to 1.4d as indicated by the values in Fig. 2. Using these values for B, the mass flow rate was calculated from the differential pressure using Euler’s method to solve the differential equation. As the total pressure tap was situated about 30 cm downstream of the orifice, the time series of the total pressure values was time shifted by ∆t = 1/c · 0.3 s. This is equivalent to assuming a plane wave propagating with the speed of sound c in flow direction. Furthermore the differential pressure signal was filtered with a high order low pass filter with a cut-off frequency of about 130 Hz. This was necessary due to distortions of the transfer function of the pressure measurement system at higher frequencies. The measurement error was defined as

 t2 error[%] =

t1

qorifice dt − m

 t2 t1

 t2 t1

qintake dt m

qintake dt m

,

(14)

where the intake air was measured with the Sensy Flow FMT700-P. Mean mass flow rates between 50 and 150 kg/h (corresponding to mean flow velocities of approximately 3.5–9 m/s), were measured at pulsation frequencies between 12.5 and 50 Hz.

K. Doblhoff-Dier et al. / Flow Measurement and Instrumentation 22 (2011) 97–103

101

Fig. 4. Typical waveforms obtained at different B-values.

Fig. 5. Measurement errors for several measurements at different B-values. The values on the x-axis correspond to the measurement numbers given in Table 1.

3.4. Results 3.4.1. Calibration For the flow regime considered, it proved to be sufficient to set A(qm ) constant. From a regression analysis, A was found to be approximately 300 000 1/m−4 ; the relative offset in qm of the calibration from the measurement was below 1% for mass flows above 50 kg/h and below 3% for smaller qm , where a residual offset of the differential pressure measurement plays a great role. An investigation of the transfer function of the steady flow measurements showed a resonance with a peak at approximately 165 Hz. Hence, as mentioned above, a low pass filter was applied in order to avoid strong distortions of the measured differential pressure. 3.4.2. Pulsating flow measurements The value of A was then used to infer the contraction coefficient CC , assuming Cv = 1, and hence to calculate reasonable values for B by comparing Eq. (13) with Eq. (9). Setting Le ≈ d, we thus find B ≈ 37.5 1/m−1 . As most theories predict Le to be smaller than d, the calibration constant B was set to the following values: B = {0, 10.5, 22.4, 37.5, 52.2} 1/m−1 . As small zero point offsets of the differential pressure sensor proved to have a high influence on the measurement and as zeroing alone did not sufficiently reduce this error, the offset was estimated from measurements in which the ball valve was in an open position. Using the procedure described in Section 3.3, the mass flow rate qm (t ) was calculated from the differential pressure measurements

taken during rotation of the ball valve. Some key figures of the resulting mass flow rate functions are shown in Table 1. Typical waveforms obtained for the mass flow rate are shown in Fig. 4. Fig. 5 shows the resulting measurement error as defined in Section 3.3. 3.5. Discussion Before discussing the measurement errors shown in Fig. 5, it is important to take a closer look at Table 1. The measured mass flow pulsations have a relative amplitude between approximately 0.5 and 2, depending on the mean mass flow rate and the pulsation frequency generated by the butterfly valve. The calculated form and hence the deduced pulsation amplitude depends on the value of B used in the evaluation: As is evident from Fig. 4, increasing B smoothes out the function and hence decreases the pulsation amplitude slightly. From Table 1, we can estimate the ratio of v ′ to ωd/2. In linear theory, this ratio should be well below 1, in the experiments performed, the ratios lie approximately between 1 and 20 as shown in Table 2. Hence, there is no guarantee that the linear approximations leading to Eqs. (9) and (13) are valid (see Section 2.2.1). After these preliminary considerations we will now discuss the actual measurement errors obtained from the measurements. At the lowest mass flow rates (measurements 1–4) the choice of the Bvalues has, as expected, the greatest influence on the measurement error. Unfortunately, these measurements are also most affected by an offset error of the differential pressure measurement. To give an example, at a flow rate of 45 kg/h an offset of 0.05 mbar

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Table 1 Key figures for the calculated mass flow rate in pulsating measurements. Number of measurements

Mean qm (kg/h)

Pulsation amp. (kg/h)a

Relative pulsation amp.

Frequency (Hz)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

46 40 48 45 76 67 92 82 97 97 144 123 108 128 200 160

70 70 40 40 100 80 50 50 150 100 70 50 200 100 100 70

1.5 1.7 0.8 0.9 1.3 1.2 0.5 0.6 1.5 1.0 0.5 0.4 1.9 0.8 0.5 0.4

12.5 25.0 37.5 50.0 12.5 25.0 37.5 50.0 12.5 25.0 37.5 50.0 12.5 25.0 37.5 50.0

a

Pulsation amplitudes were roughly estimated from the resulting functions qm (t ). Strictly speaking, they depend slightly on the value of B used in the evaluation.

Table 2 Approximate ratios of v ′ to ωd/2 for different measurements. (Values based on Table 1.) Measurement

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

v ′ :ωd/2

6

3

1

1

9

4

2

1

14

5

2

1

18

5

3

2

would lead to a measurement error of more than 5%. At a flow rate of 70 kg/h the measurement error due to the same differential pressure offset would only amount to approximately 2%. Hence, although the B-dependency of the first four measurements is most prominent, the measurement results are least reliable, especially considering the fact, that zero point offsets of the differential pressure measurement led to difficulties throughout all measurements. Therefore, only measurements numbered 5–16 will be considered in the following. If the linear theory, and hence Eq. (9), are valid, there should be an optimum B-value at which the measurement error is minimal. Guessing the ideal value from Fig. 5 is impossible as no strong trend can be recognized. In order to estimate the optimum Bvalue, the average of all errors at a certain B-value, the average of the absolute errors and the root mean square value of the errors are considered. These values are shown in Table 3. According to these values, the best guess for the optimal B-value probably lies between B = 22.4 1/m−1 and B = 37.4 1/m−1 . At this point, we would like to emphasize that any numeric values given for A and B here should not be regarded as universal constants. The constant A is known to depend on the orifice geometry and is given in the corresponding ISO standards for standardized orifices. The value of B depends on CC , d and the β -ratio of the orifice and hence again on the exact geometry (see Fig. 2 and Section 3.3). The A and B values given here are thus only valid for the setup investigated in this paper. For any other setup, we suggest to perform a similar experiment (preferably in the pulsation range that will be under consideration) to obtain suitable values for A and B. Even for our measurement setup the value of B is not highly reliable for several reasons: (1) the trend is not strong and (2) there are measurements (especially measurement number 9 and 13) for which the error is positive, independently of the B-value used in the evaluation. As the mean mass flow rates are high for these measurements it is not very likely that these errors arise from a zero point offset of the differential pressure. Taking a look at Table 2, another fact about these two measurements becomes obvious: In both cases the ratio v ′ to ωd/2 is extremely high. Hence, this error might be due to a nonlinear impedance of the orifice. If this is the case, though, it is likely that the value estimated for the ideal B-value is too low for the other measurements. This assumption can be explained by the fact that the measurement

error is correlated with the B-value as can be seen in Fig. 4. However, there might be other sources of error causing the positive errors of measurements number 9 and 13. Sparks showed for example in Ref. [34] that high pressure pulsations might occur at an orifice which are not necessarily due to velocity pulsations. This explanation seems less probable, though, as one would expect negative errors due to the square root effect in this case. A more likely explanation was already mentioned in Section 2.2.3: The decrease of the first term of Eq. (13) with increasing pulsation amplitude observed by Gajan et al., which they attributed to a changing flow profile, might have caused the anomalous results. This thesis is further confirmed by the following fact: reverse flow, which may significantly change the flow profile, only occurred in measurements number 1, 2, 5, 6, 9 and 13 (rel. pulsation amplitude >1, see Table 1). Ordering these measurements by increasing average flow rate gives the following sequence: 2, 1, 6, 5, 9, 13 (see again Table 1). Considering the measurements in this order, a significant trend from negative errors at low mean flow rates to positive errors at high mean flow rates can be observed for all B-values used, except for B = 52.2 m−1 (see Fig. 5). In contrast, the relative error decreases with increasing flow rate for all measurements without reverse flow, as would be expected. Finally, another word shall be said on the resonance found in the pressure measurement system. The resonance frequency found was about 164 Hz, which is approximately by a factor 2 smaller than the resonance frequency expected from Eqs. (10) and (11) if inserting the free-space sound velocity. It is, however, naïve to insert the free-space sound velocity as it is known [35,36] that the speed of sound is decreased in tubes. The observed resonance frequency is, however, even slightly smaller than that expected when using the relation between the free-space sound velocity and that in the tube given in Ref. [36]. This might be due to the flexibility of the tubings used. 4. Conclusion It has been clear long since, that the steady flow equation Eq. (1), relating the differential pressure with the mass flow rate through an orifice, may not be used if flow pulsations are present. As long as linear approximations are valid, various theoretical expressions can be found (see Eq. (5) and Fig. 2) which only differ

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103

Table 3 Averaged measurement errors for different B-values.

Average error Av. abs. error Root mean square

B = 0 1/m−1

B = 10.5 1/m−1

B = 22.4 1/m−1

B = 37.5 1/m−1

B = 52.2 1/m−1

−3.3

−3.0

−1.2

4.9 4.9

8.6 4.4

6.4 3.2

1.2 4.5 3.8

3.4 4.9 5.6

quantitatively within about one order of magnitude. It could be reasoned that these linear approximations are valid only if the ratio of the pulsating component v ′ of the velocity is small against the speed of sound c and if v ′ ≪ ωd/2. No quantitative expressions or theoretical derivations could be found in the literature if the linear approximation is violated. For the experiment, in which pulsations beyond the linear regime were measured, a theoretically motivated equation already used by other authors, was used for the measurement of the mass flow. It was found that this equation might yield fair results as long as the ratio v ′ to ωd/2 is no higher 10. For higher ratios, results were obtained that cannot be explained by the ansatz used. We hence concluded that in this case other effects, such as nonlinear impedance or effects observed by Gajan et al. [12] which they attributed to a changing flow profile, come into play. So far, it remains open if these effects can be quantized. Acknowledgements This work was done under a contract with AVL List GmbH. Thanks go to Dipl.-Ing. Franz Graf for providing information from preceding experiments and to Michael Sammer for his help with the mechanical setup. References [1] Benard CJ. Handbook of fluid flowmetering. 1st ed. The Trade & Technical Press Limited; 1988. [2] Fiedler E. Strömungs und Durchflußmesstechnik. Oldenbourg; 1992. [3] Durchflussmessung von Fluiden mit Drosselgeräten in voll durchströmten Leitungen mit Kreisquerschnitt—teil 1. Allgemeine Grundlagen und Anforderungen. EN ISO 5167-1. 2003. [4] Durchflussmessung von Fluiden mit Drosselgeräten in voll durchströmten Leitungen mit Kreisquerschnitt—teil 2. Blenden. EN ISO 5167-2. 2003. [5] Wilson MP, Teyssandier RG. The paradox of the vena contracta. Journal of Fluids Engineering, Transactions of the ASME 1975;366–71. [6] Merritt H. Hydraulic control systems. John Wiley and Sons; 1967. p. 40–41. [7] Reader-Harris MJ, Sattary JA, Spearman EP. The orifice plate discharge coefficient equation—further work. Flow Measurement and Instrumentation 1995;6(2):101–14. [8] Reader-Harris M. The equation for the expansibility factor for orifice plates. In: FLOMEKO’98. 1998. p. 209–14. [9] Buckingham E. Note on contraction coefficients of jets of gas. Bureau of Standards Journal of Research 1931;765–75. [10] Measurement of fluid flow in closed conduits—guidelines on the effects of flow pulsations on flow-measurement instruments. ISO/TR 3313. 1998(E). [11] Fluidmechanik Truckenbrodt E. Grundlagen und elementare Strömungsvorgänge dichtebeständiger Fluide, 4th ed. vol. 1. Springer; 2008. [12] Gajan P, Mottram RC, Hebrard P, Handriamihafi H, Platet B. The influence of pulsating flows on orifice plate flowmeters. Flow Measurement and Instrumentation 1992;3(3):118–29.

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