Research Papers: Flows in Complex Systems

Obtaining Time-Varying Pulsatile Gas Flow Rates With the Help of Dynamic Pressure Difference and Other Measurements for an Orifice-Plate Meter

[+] Author and Article Information
A. Narain

e-mail: narain@mtu.edu

N. Shankar

Department of Mechanical
Engineering-Engineering Mechanics,
Michigan Technological University,
Houghton, MI 49931

Manuscript received May 25, 2012; final manuscript received November 17, 2012; published online February 22, 2013. Assoc. Editor: John Abraham.

J. Fluids Eng 135(4), 041101 (Feb 22, 2013) (19 pages) Paper No: FE-12-1264; doi: 10.1115/1.4023195 History: Received May 25, 2012; Revised November 17, 2012

Use of a conventional orifice-plate meter is typically restricted to measurements of steady flow rates. For any gas flowing within a duct in a pulsatile manner (i.e., large amplitude mass flow rate fluctuations relative to its steady-in-the-mean value), this paper proposes a new and effective approach for obtaining its time-varying mass flow rate at a specified cross section of an orifice meter. The approach requires time-varying (dynamic) pressure difference measurements across an orifice-plate meter, time-averaged mass flow rate measurements from a separate device (e.g., Coriolis meter), and a dynamic absolute pressure measurement. Steady-in-the-mean turbulent gas flows (Reynolds number ≫2300) with low mean Mach numbers (<0.2) exhibit effectively constant densities over long time-durations and are often made pulsatile by the presence of rotary or oscillatory devices that drive the flow (compressors, pumps, pulsators, etc.). In these pulsatile flows, both flow rate and pressure-difference fluctuation amplitudes at or near the device driver frequency (or its harmonics) are large relative to their steady mean values. The time-varying flow rate values are often affected by transient compressibility effects associated with acoustic waves. If fast Fourier transforms of the absolute pressure and pressure-difference measurements indicate that the predominant frequency is characterized by fp, then the acoustic effects lead to a nonnegligible rate of change of stored mass (associated with density changes) over short time durations (∼ 1/fP) and modest volumes of interest. As a result, for the same steady mean mass flow rate, the time variations (that resolve these density changes over short durations) of mass flow rates associated with pulsatile (and turbulent) gas flows are often different at different cross sections of the orifice meter (or duct). Together with the experimental measurements concurrently obtained from the three recommended devices, a suitable computational approach (as proposed and presented here) is a requirement for effectively converting the experimental information on time-varying pressure and pressure-difference values into the desired dynamic mass flow rate values. The mean mass flow rate measurement assists in eliminating variations in its predictions that arise from the use of turbulent flow simulation capabilities. Two independent verification approaches establish that the proposed measurement approach works well.

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Fig. 1

Proposed assembly of the orifice meter with instruments

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Fig. 3

The schematic shows small amplitude of density variations at a point in the flow field. The amplitude of ∂ρ/∂t (represented by the slope of the dotted line) is significant over the small time scale Δtc ≡ 1/fP of interest but is negligible if averaged over large times T ≫ 1/fP.

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Fig. 4

The geometry of a specific orifice meter. For the hardware, lOM1 = 127 mm. This is also the geometry of OM1 in Fig. 6.

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Fig. 5

(a) Five steady and one unsteady pressure-difference history curves are shown. These are used for assessing the adequacy of the models proposed in Refs. [3-7]. (b) For the unsteady pressure-difference history (over time interval ΔtUnsteady) marked in (a), the model Δpom(t)=k·Q(t)2 should lead to Q(t) values that lie on parabolic curves (such as the solid black curve for a representative k value). Instead reasonable estimates of the flow rate Q(t) values (obtained by CFD) plotted against the unsteady pressure-difference values lead to the nonparabolic dotted curve. For steady pressure-difference histories shown in (a), however, the corresponding steady flow rates (reasonably estimated by CFD) are adequately described by the solid black curve given by: Δpom-steady(t)=k·Qsteady2 (where k = 3.26847·1010 Pa·s2/m6 and ρ0 = 19 kg/m3). (c) For the unsteady pressure-difference history (over time interval ΔtUnsteady) marked in (a), the relationship between Δpom(t)-k·Q(t)2 and dQ(t)/dt should be approximately linear if the model Δpom(t)=k·Q(t)2+L·dQ(t)/dt is reasonable. Instead, reasonable estimates of the flow rate Q(t) and its derivative dQ(t)/dt values (obtained by CFD) are such that the resulting dotted curve is far from the best linear fit (with k taken from the steady curve fit of (b) and L = −39,100 Pa · s2/m4 being the slope of the solid line in (c)).

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Fig. 6

Experimental flow loop

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Fig. 7

The geometry of the orifice meter OM2 used in Fig. 6

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Fig. 2

A representative example of pressure-difference and mass flow rate values encountered in the pulsatile flows of interest

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Fig. 8

Time-varying values of measured pressure differences (across OM1 and OM2 in Fig. 6) ΔpOM1(t) and ΔpOM2(t) for run 3 in Tables 1 and 2

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Fig. 9

The fast Fourier transform representation of the differential-pressure data ΔpOM1(t) and ΔpOM2(t) in Fig. 8

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Fig. 10

(a) Velocity Vx (r, x, t) profiles as a function of r at xM′ = 0.12 m in Fig. 4 and time t = 0.8235 s for three different representative mesh sizes (with mesh 1, mesh 2, and mesh 3 described in the Appendix) associated with the quadrilateral elements used in the CFD solver. The incompressible CFD results are for run 3 in Tables 1 and 2. (b) Velocity Vx (r, x, t) profiles as a function of r at xM′ = 0.12 m in Fig. 4 and time t = 0.8235 s arrived at for four different time steps (with TS1, TS2, TS3, and TS4 described in the Appendix) used in the CFD solver.

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Fig. 11

(a) For run 3 in Table 1 and time t = tP marked on Fig. 8, the plot shows the streamline pattern within OM1 (see Fig. 4). Only half of the geometry (0 ≤ r ≤ R = 8.51 mm) that is symmetric with respect to the x-axis in Fig. 4 is shown. (b) At the cross-section location xB = 0.068 m in Fig. 4, the pressure variations pCFD (x, r, t) as a function of radius r are obtained from incompressible CFD simulations. As shown for three different time instants t = 1.0 s, 1.025 s, and 1.05 s, these variations are found to be nonuniform. (c) At the cross-section location xM′ = 0.12 m in Fig. 4, the pressure variations pCFD (x, r, t) as a function of radius, r, are obtained from incompressible CFD simulations. As shown for three different time instants t = 1.0 s, 1.025 s, and 1.05 s, these variations are found to be uniform.

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Fig. 12

For the representative solution associated with the pressure differences in Fig. 8 (run 3 in Tables 1 and 2), the values of m·CFD-I|OM1(t) and m·CFD-I|OM2(t) are shown above

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Fig. 13

Plots of m·CFD-I|OM1(t) for two different turbulence models (mod-1 and mod-2) are shown for run 3 in Tables 1 and 2

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Fig. 14

For the representative values of m·CFD-I|OM1(t) and m·CFD-I|OM2(t) in Fig. 12, the figure above shows the respective empirically corrected values m·Inc|OM1(t)=α·m·CFD-I|OM1(t) and m·Inc|OM2(t)=α·m·CFD-I|OM2(t) for turbulence model mod 1. In addition, the figure also shows m·Inc|OM1(t) for turbulence model mod 2.

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Fig. 15

The plot above shows FFT of the empirically corrected mass flow rates values m·Inc|OM1(t) and m·Inc|OM2(t) in Fig. 14 for turbulence model mod 1

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Fig. 16

For the representative run 3 in Tables 1 and 2, the plot above shows computed values of the nondimensional integral NI(t)

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Fig. 17

(a) This figure shows the time-varying values of pressure fluctuation at cross sections located by point L (pL'(t)≡pL(t)-p¯L) and M (pM'(t)≡pL'(t)-ΔpOM1(t)). These data are for run 4 of Tables 1 and 2. (b) The plot shows the FFT of pL'(t) and pM'(t) in (a).

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Fig. 18

The incompressible mass flow rate values of m·Inc|OM1(t) for run 6 and its compressibility corrected values m·L(t) and m·M(t), respectively associated with points L and M in Fig. 4, are shown

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Fig. 19

(a) The plot shows the mass flow rate m·M(t) at point M of OM1 in Fig. 4 for run 3 of Tables 1 and 2. These values are obtained from compressible flow models for two different thermal boundary conditions for the orifice-meter walls. One is an isothermal assumption and the other is an adiabatic assumption. (b) This plot shows, for run 3 in Tables 1 and 2, the comparison between m·stored(t) values obtained from the compressible flow CFD model and their values obtained from the proposed compressibility correction theory (Eq. (21)) for the incompressible CFD model (which has m·stored(t)=0). (c) This plot shows, for run 3 in Tables 1 and 2, the comparison between mass flow rate m·M-Comp(t) obtained from the compressible flow CFD model and m·M(t) obtained from a compressibility correction on the m·Inc|OM1(t) values—which are obtained from an incompressible CFD model.

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Fig. 20

This figure shows the geometry for the two devices that are merged together, with proper instruments, and placed between points L and O of Fig. 6




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