Flow rate and mass flow rate

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Byron Bailey
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1 EEN-E1040 Measurement and control of energy systems Flow measurements / 14 Sep 2017 WELCOME! v. 01 / T. Paloposki Flow rate and mass flow rate Consider the system shown here 1

2 Volume flow rate through cross-section A A: = = yields = = Mass flow rate through cross-section A A: = = = yields = = = 2

3 Here it was assumed that: a) The flow is steady. b) The velocity is uniform. c) The density is constant. Everywhere within the conduit Things get more complicated if these assumptions are not satisfied. Today, we mostly consider the case where assumptions a) and c) are satisfied. Note that we normally consider flow rate through some finite cross-sectional area (through a conduit, through an opening, etc.) Thus, flow rate is an integrated quantity (not a point quantity, such as, e.g., flow velocity). 3

4 Non-uniform velocity profile Case 1: Fully developed pipe flow in a straight pipe with a constant cross-section laminar flow entrance region fully developed flow turbulent flow laminar flow turbulent flow Source: Denn (1980) 4

5 Non-uniform velocity profile Case 2: Geometrically more complex flow geometries a bend in the conduit a fan in the conduit Practical example: SBI test (SBI = Single Burning Item) location of velocity profile measurement Source: Hietaniemi & Baroudi (2002) / VTT Research Notes

6 12 SBI Test Apparatus Measurement of velocity profile Average values from all measurements VTT / 13 Jan SBI Test Apparatus Measurement of velocity profile Positive x traverse from edge to center VTT / 13 Jan Velocity [m/s] 6 Velocity [m/s] 8 3 Horizontal traverse Vertical traverse Radial distance from centerline [mm] Radial distance from centerline [mm] What now? We want to use the equation = But the velocities might be different at different points in our system. So what velocity value should be used in the calculation? 6

7 How to cope with non-uniform velocity profiles? Option 1: One-shot technique Option 2: Mathematical averaging Option 3: Physical averaging One-shot technique Measure velocity in one point (which is supposed to be somehow representative). Use a correction factor to obtain average velocity. Good in cases where the velocity profile is relatively neat. May produce very bad results if the velocity profile is messy. 7

8 Mathematical averaging Measure velocity at several points. Calculate average velocity mathematically. High accuracy can be achieved. Measurements can be slow and tedious. Physical averaging Use a device which carries out the averaging for you. More of this later today. 8

9 Measurement of local flow velocity: the pitot tube (pitot-static tube) Flow direction Cross-section A A Static pressure Stagnation pressure 9

10 Mathematical analysis The Bernoulli equation: total pressure = = constant dynamic pressure static pressure stagnation pressure Apply to pitot tube: = (neglect gravity term) pressure at probe tip flow velocity at static holes (assumed to be equal to flow velocity w) pressure at static holes flow velocity at probe tip ( = 0 ) 10

11 Solving from the Bernoulli equation, we obtain the flow velocity = 2( ) = 2 Pressure difference [Pa] Pitot tube output signal Velocity [m/s] 11

Possible sources of error in pitot tube measurements: Probe tip positioning Misalignment Viscous effects (low Reynolds number flow) Compressibility effects (high Mach

12 Possible sources of error in pitot tube measurements: Probe tip positioning Misalignment Viscous effects (low Reynolds number flow) Compressibility effects (high Mach number flow) Nonuniform velocity profile effects Turbulence effects Extensions of the pitot tube technique: Multi-hole probes (here a five-hole probe) Bi-directional probe 12

13 Now it is time to get back to physical averaging: Averaging pitot Venturi tube Flow nozzle Orifice plate Measuring ring Averaging pitot Static pressure Stagnation pressure 13

14 Venturi tube Pressure taps Nozzle Throat Flow direction Diffuser Flow nozzle Pressure taps Nozzle 14

15 Orifice plate Pressure taps Orifice plate Measuring ring Pressure taps 15

16 The mathematical analysis of these devices is normally based on the Bernoulli equation. However, the geometry is more complicated than in the case of a pitot tube, and pressure losses also need to be accounted for. Thus, the equations are typically of the type = = 2( ) Coefficient to be specified Flow area (to be specified) Vortex shedding flow meter Endress & Hauser: 16

17 That s all for today, folks. Good luck and enjoy! 17

Measurements using Bernoulli s equation

An Internet Book on Fluid Dynamics Measurements using Bernoulli s equation Many fluid measurement devices and techniques are based on Bernoulli s equation and we list them here with analysis and discussion.