• Carleton University

• MAAE 2300

LAB 3: Uncertainties in fluid flow rate Measurement 1
Lab 3: UNCERTAINTIES IN FLUID FLOW RATE MEASUREMENT

Dhairya Soni, 101129894
Carleton University-Fluid Mechanics (MAAE 2300)
[email protected]
Group L03-B
LAB 3: Uncertainties in fluid flow rate Measurement 2
SUMMARY
Fluid flow rate measurement is a vastly utilized technique in Aerospace and Mechanical
engineering. Depending on factors such as degree of accuracy and type of application, the choice
of flow meter devices varies as shown in this experiment. Each device provides a different level
of accuracy. The aim in this experiment is to measure and evaluate fluid flow rate measurements
uncertainties in a water closed-loop experimental setup. The five devices used are: The Venturi
Meter; Orifice Meter; Rotameter; Turbine Flow Meter; and an Ultrasonic Meter. Uncertainty
analysis of these are performed considering uncertainty associated with measurement of the flow
rate, as well as uncertainty associated with the flow meter instrument. In addition to the
uncertainty analysis, we learn how to establish error propagations in measurements and error
propagation methods which are essential techniques to be applied in any experimental and
computational work.
NOMENCLATURE
Patm = Atmospheric Pressure (Pa): 101325 Pa
ρAir = Density of Air (kg/m3): 1.225 kg/m3
ρWater = Density of Water (kg/m3): 1000 kg/m3
Cd = discharge coefficient
Re = Reynold’s number
q = flow rate
V = flow velocity
A = Cross-sectional area
g = acceleration of gravity 9.81 m/s2
rrms = root mean square
ω = angular velocity
ui,q = Bias error/instrument error
qj- q´ = mean value subtracted from value at point i
um,q = Standard deviation
utotal,q = total uncertainty
qact = actual flow rate
LAB 3: Uncertainties in fluid flow rate Measurement 3
FLOW ANALYSIS
For the Venturi Meter:
Figure 1: Venturi Meter diagram
 Velocity and pressure at the throat section and inlet section are in relation via the
Bernoulli Equation:
P1 + 1/2 ρ V1 = P2 + 1/2 ρ V2 = Const.
 Since uniform velocity profiles are assumed and ρ being the density of the fluid, q being
the flow rate and A1 and A2 are cross-sectional areas of section 1 and 2 respectively we
get:
q = V1 A1 = V2 A2
 Simplifying and combining equations 1 and 2 we obtain:
qactual = Cd (π / 4) D2 2 [ 2 (p1 - p2) ]1/2 / [ ρ (1 - d4) ]1/2
 Cd is the discharge coefficient valued at 0.95, D2 is the diameter at the throat section, D1
is the diameter at section 1, and d = D2 / D1 is the diameter ratio. P1-P2 becomes the
change in pressure.
For the orifice meter:
LAB 3: Uncertainties in fluid flow rate Measurement 4
Figure 2: Orifice Meter diagram
 Similar equation used for the Venturi Meter are used for the Orifice meter as well.
 As an analogy, V1 and P1 are the velocity and pressure as shown in section 1 of the
venturi meter. V2 and P2 are the velocity and pressure as shown in section 2 of the
venturi meter.
 The Cd value for the orifice meter is 0.65
qactual = Cd (π / 4) D2 2 [ 2 (p1 - p2) ]1/2 / [ ρ (1 - d4) ]1/2
For the Rotameter:
Figure 3:Rotameter diagram
LAB 3: Uncertainties in fluid flow rate Measurement 5
 For subsonic fluid flow, the incompressible Bernoulli Equation, relating fluid flow
velocity, V, pressure, P, and height of the fluid, h, can be applied:
V1 2 / (2 g) + h1 + P1 / (ρ g) = V2 2 / (2 g) + h2 + P2 / (ρ g) = Const. (6)
 where g is the gravity acceleration (9.81 m/s2). Applying this equation to a streamline
travelling up the vertical tube of the rotameter, yields:
P1 - P2 = ρ g h2 - ρ g h1 + V2 2 ρ / 2 - V1 2 ρ / 2 (7)
Or
ΔP = ρ g Δh + (ρ / 2) V2 2 [ 1 – (V1 / V2)2 ]
 Over here, Δh = h2 - h1 where subscript 1 and subscript 2 represent the position before
the float and right at the balanced point (top of the float) respectively.
V1 / V2 = A2 / A1
 Here, A2 = A2 - A1 is the annular area between the float and the tube wall. Using
equation (9), the velocity V2 can be substituted in equation (8), giving us:
ΔP = ρ g Δh + (ρ / 2) (q / A2)2 [ 1 – (A2 / A1)2 ]
 Whereas the pressure drop is mostly resulting from the weight of the float.
ΔP ≈ Ŵfloat / Afloat = (Ŵfloat - Bfloat) / Afloat = Vf (ρf - ρ) g / Af
 The discharge coefficient is introduced to account for the viscosity of the fluid and
combining equation 10 and 11 we finally obtain:
LAB 3: Uncertainties in fluid flow rate Measurement 6
q = Cd A2 [ 2 g ( Vf (ρf - ρ) / Af - ρ Δh ) ] 1/2 / [ ρ (1 - (A2 / A1)2 ) ]1/2
For the turbine flow meter:
Figure 4: Turbine flow Meter
 The value for the radius at the roots of the blade a and the value of the radius of the
turbine r are both given in the appendix. The width of the blades is c and the distance
between the blades is s. The velocity of the incoming flow, V1, will cause the turbine to
rotate at an angular velocity ω.
rrms ωi = V1 tan β ⇒ ωi / V1 = tan β / rrms
rrms = [ (a2 + r2 ) / 2 ]1/2
 As the flow passes the turbine, the velocity flow changes hence producing a torque and to
counteract the torque a Drag force is established allowing us to calculate the total net
torque on the entire system. A combination of equations and the discharge coefficient aids
us to find the actual flow rate q:
T = ρ A V1 (a2 + r2) (ωi - ω) / 2 = ρ A V1 (rrms )2 (ωi - ω) ⇒ ω = ωi – T / [ρ A V1 (rrms )2]
FD = ρ V1 2 CD S / 2 ≈ 0.074 Re-0.2 ρ V 2 S
T = n [ (a + r) / 2 ] FD sin β = 0.037 Re-0.2 n (a + r) ρ V1 2 S sinβ
q = CD V1 A = ω (rrms )2 A2 / [rrms A tanβ - 0.037 Re-0.2 n (a + r) S sinβ]