Data analysis procedure

The raw pressure and flow data were entered onto a spreadsheet in Microsoft Excel and the following steps were performed on the data:

4.4.3.1 Step 1: interpretation of the data

The raw flow and pressure data were plotted against time. Experimental data points were obtained by identifying stable sections of the flow and pressure graphs and then taking the average values over each of the ranges.

The flow and pressure data points were recorded in litres/min and bars, respectively. The units were then converted to SI units of 𝑚3/𝑠 and 𝑚 for the flow rate and pressure, respectively. The following conversions were used:

𝑄_{𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑}(𝑚3_{⁄ ) = 𝑄𝑠}

𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 (𝑙 𝑚𝑖𝑛⁄ ) × 1 60000⁄ .

Equation 4-1

ℎ_{𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑}(𝑚) = ℎ𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 (𝑏𝑎𝑟) × 10.1997

Equation 4-2

4.4.3.2 Step 2: determining the actual pressure in the test pipe

Since the pressure sensor of the PCAE is not located at the leak, the measured pressure data point values had to be adjusted in order to estimate the true pressure at the leak. This adjustment was done by accounting for (a) the elevation difference between the pressure sensor and the leak, (b) head losses due to friction between the pressure sensor and the leak, and (c) minor losses due to the various hydraulic components between the pressure sensor and the leak. The elevation correction was determined by measuring the height between the device’s pressure sensor and the leak. Since the device was on the floor (see Figure 4-6), the height from the floor to the pressure sensor of the device was measured first; thereafter the height from the floor to the leak was measured. The difference between them was 0.05 meters.

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The friction losses were calculated to account for friction loss between the PCAE’s pressure sensor and the entry point into the test rig. To do so, the first step was to calculate the Reynolds equation, given here,

𝑅 = 4𝑄 𝜋𝐷𝜐

Equation 4-3

Where 𝑄 is the flow rate, 𝐷 is the internal diameter, and 𝑣 is the kinematic viscosity. All flow was observed to be turbulent. Using the parameters given in Table 4-1, the friction factor, f, was calculated for each data point using the Haaland equation for turbulent flow, given here as: 1 √𝑓 = −1.8 log [( 𝑒 𝐷⁄ 3.7) 1.11 + 6.9 𝑅 ] Equation 4-4

Finally, to calculate the friction head loss, ℎ_𝑓, for each data point, the Darcy Weisbach equation was used: ℎ𝑓 = 𝑓𝐿𝑣2 2𝐷𝑔 = 𝑓 𝐿 𝐷 (𝑄 _𝜋𝐷2 4 ⁄ ) 2 2𝑔 Equation 4-5

Where ℎ_𝑓 is the friction head loss (m), 𝑄 is the measured flow rate (m3_{/s), 𝐷 is the internal} diameter of the pipe (m), 𝑓 is the friction coefficient, 𝑔 is the acceleration due to gravity (m/s2₎ and 𝐿 is the length pipe (m). The parameters used to calculate the friction head loss are given in Table 4-1.

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Table 4-1: Parameters used to calculate the friction head loss for each step

Parameter Value

Delivery line Internal diameter, 𝑫 (m) 0.0452

Length of delivery line, 𝑳 (m) 10

Acceleration due to gravity, 𝒈 (m/s2_{) 9.81}

Rubber hose pipe Absolute roughness, 𝒆 (mm) 0.05

Kinematic viscosity, 𝒗 (m2_{/s) 11.39x10}-7

The minor losses were calculated to take into account the effect of the various hydraulic components between the pressure sensor and the test rig. The hydraulic components are listed below in Table 4-2, with their respective minor loss coefficient obtained from Finnemore and Franzini (2009).

Table 4-2: Hydraulic components and their minor loss coefficients

Hydraulic components Minor loss coefficient, K

Ball valve 0.1

Adapter (Sudden Contraction) 0.3

Adapter (Sudden Expansion) 0.75

Rubber hose pipe bends 0.5

Straight connecters 0.2

The minor loss coefficients in Table 4-2 were added up and the minor loss for each data point was calculated, using the minor head loss equation given here:

ℎ𝑚 = ∑ 𝐾

(4𝑄 𝜋𝐷_⁄ 2₎2 2𝑔

Equation 4-6

Finally, once the measured pressure (h measured) from the pressure transmitter is obtained. The

adjusted pressure values were obtained by subtracting the elevation height, ∆z, friction head loss, h_f, and the minor head loss, h_m, from the measured average pressure, and the pressure at the leak could thus be established using the following equation:

ℎ_{𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑} = ℎ_{𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑}− ℎ_𝑓− ℎ_𝑚− ∆𝑧

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4.4.3.3 Step 3: determining leakage parameters for the power equation

The leakage parameters for the power equation were determined empirically. The flow rate (𝑄) and adjusted pressure (ℎ_{𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑}) values were plotted with a power equation fitted to the data. The power equation has a coefficient value, representing the leakage coefficient (𝐶) and an exponent value representing the leakage exponent (𝑁1), respectively. The N1 equation is given here:

𝑄 = 𝐶 ℎ𝑁1

Equation 4-8

In order to verify the power equation leakage parameters, Equation 4-8 is converted into an equivalent linear function by applying logs of base 10 on both sides of the equation. This results in the following linear expression:

𝐿𝑜𝑔10𝑄 = 𝐿𝑜𝑔10𝐶 + 𝑁1 𝐿𝑜𝑔10ℎ𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑

Equation 4-9

The slope and intercept of Equation 4-9 represents the 𝑁1 leakage exponent and the log of the leakage coefficient, respectively. Using the Microsoft Excel function of Slope and Intercept and the flow and adjusted pressure data, the slopes and intercept of the linear function in Equation 4-9 can be obtained. The 𝑁1 leakage exponent value is obtained directly from the slope of the linear equation; however, in order to obtain the leakage coefficient, the following formulation is required:

𝐶 = 10𝐿𝑜𝑔10𝐶

Equation 4-10

4.4.3.4 Step 4: calculating the effective leak area (𝑪

𝑨)

The effective leak area, C_dA, at each pressure step was calculated by re-arranging the orifice equation as follows:

𝑪_𝒅𝑨 = 𝑸

√𝟐 𝒈 𝒉𝒂𝒅𝒋𝒖𝒔𝒕𝒆𝒅

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Where 𝑪_𝒅 is the discharge coefficient, 𝑸 is the measured flow rate (m3_{/s), 𝒉}

𝒂𝒅𝒋𝒖𝒔𝒕𝒆𝒅 is the test pipe pressure (m), 𝒈 is the acceleration due to gravity (m/s2_{) and 𝑨 is the equivalent leak area} (m2_).

If the total effective leak area of the system is estimated at different pressures, the effective leakage area can be plotted against pressure and a linear function fitted to the data points. The intercept of this line with the effective area axis gives the effective initial leak area (𝐶_𝑑𝐴₀ 𝑜𝑟 𝐴′), and the slope of this line gives the effective head-area slope (𝐶_𝑑𝑚 𝑜𝑟 𝑚′).

4.4.3.5 Step 5: determining the leakage parameters for the modified orifice

equation

Since the leak sources are drilled into the pipe, the actual initial leak area can be physically measured and thus is known (𝐴_{0,𝑘𝑛𝑜𝑤𝑛}). Consequently, the discharge coefficient (𝐶_𝑑) can be estimated by dividing the obtained initial effective leak area (𝐶_𝑑𝐴₀), determined in step 4, by the known initial leak area (𝐴_{0,𝑘𝑛𝑜𝑤𝑛}) as follows:

𝐶_𝑑 = 𝐶𝑑𝐴0 𝐴_{0,𝑘𝑛𝑜𝑤𝑛}

Equation 4-12

4.4.3.6 Step 6: estimating the leakage flow rate

The leakage flow rates were estimated using the power equation and the modified orifice equation models. The leakage parameters for the power equation (𝑁1, 𝐶) were substituted into the power equation, and the leakage parameters for the modified orifice equation (𝐶_𝑑𝐴₀, 𝐶_𝑑𝑚) were substituted in the modified orifice equation. The leakage equations models are given as follows:

Power Equation:

𝑄 = 𝐶 ℎ𝑁1

Equation 4-13

Modified Orifice Equation:

𝑄 = √2𝑔[(𝐶𝑑𝐴0)ℎ0.5+ (𝐶𝑑𝑚)ℎ1.5]

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Once the leakage parameters are substituted into their respective leakage equation models, the flow rates, 𝑄, can be obtained and plotted for a range of pressure heads, i.e. ℎ = 0, 15, 20, 25, 30, 35𝑚. The leakage flow rates, 𝑄, obtained using the leakage equation models are plotted against pressure as continuous lines. The measured experimental flow rate data points are plotted against the measured pressure as discrete data points on the same flow vs pressure graphs. This is done to determine whether the measured flow and pressure data points correlate with the power equation and modified orifice equation flow rate models.

4.4.3.7 Step 7: determining the leakage number

Finally, a dimensionless leakage number (𝐿𝑁) is calculated for the leak. The leakage number (𝐿𝑁) represents the ratio between the expanding leakage area and the fixed leak area, and is expressed as follows:

𝐿𝑁= 𝑚ℎ

𝐴0

Equation 4-15

A simple equation was proposed by Cassa and van Zyl (2010) to convert between the leakage number (𝐿𝑁) and the leakage exponent (𝑁1):

𝐿𝑁 = 𝑁1 − 0.5 1.5 − 𝑁1 Equation 4-16 𝑁1 = 1.5 𝐿𝑁− 0.5 𝐿_𝑁+ 1 Equation 4-17