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# Manual headloss calculation benchmark with MES software

In MiTS software, we can choose to use either Hazen-Williams Equation or Darcy-Weisbach Equation to calculate the head loss. Users can change the head loss formula used in the analysis at Options > Project Settings > Water Ret > Design > Headloss Formula

# Hazen-Williams Equation#

$$V\:=\:kC\left(\frac{D}{4}\right)^{0.63}S^{0.54}$$

where;

$$S\:=\:\frac{h}{L}$$

$$Q\:=\:\frac{V\pi D^2}{4}$$

## MiTS Software#

1. The output value from MiTS software is shown as per image below.

1. We also test the value given in our software with an online head loss calculator.
3. Pipe 1 is taken as an example, please refer image below:

1. As you can see that the result generated from MiTS software is equivalent to the one calculated by the online head loss calculator.
• Velocity = 0.3874 m/s
• Head Loss = 0.2031 m

## Manual Calculation #

1. We used formula below to calculate the head loss with known Discharge, Q (m3/s), Pipe Diameter, D (m), Pipe Length, L (m) and Hazen-William Coefficient, k of the material used.

$$V\:=\:kC\left(\frac{D}{4}\right)^{0.63}S^{0.54}$$

where;

$$S\:=\:\frac{h}{L}$$

$$Q\:=\:\frac{V\pi D^2}{4}$$

1. As the Q is known, we will need to rearrange the equation to find the value of V.

$$Q\:=\:\frac{V\pi D^2}{4}$$

$$V\:=\:\frac{4Q}{\pi D^2}$$

$$V\:=\:\frac{4\left(\frac{3.043}{1000}\right)}{\pi \left(\frac{100}{1000}\right)^2}$$

$$V\:=\:0.3874\:m/s$$

1. After obtaining the value for V, we will rearrange the Hazen-William Equation to get the value for S.

$$V\:=\:kC\left(\frac{D}{4}\right)^{0.63}S^{0.54}$$

$$S\:=\:\sqrt[0.54]{\frac{V}{kC\left(\frac{D}{4}\right)^{0.63}}}$$

$$S\:=\:\sqrt[0.54]{\frac{0.3874}{0.85\times 100\left(\frac{100/1000}{4}\right)^{0.63}}}$$

$$S\:=\:0.00342$$

1. Finally, we can calculate the value for h.

$$S\:=\:\frac{h}{L}$$

$$h\:=\:SL$$

$$h\:=\:0.00342\times 59.466$$

$$h\:=\:0.2034\:m$$

# Darcy-Weisbach Equation#

$$H\:=\:\frac{fLV^2}{2dg}$$

$$f\:=\:\frac{1.325}{\left[ln\left(\frac{e}{3.7D}+\frac{5.74}{Re^{0.9}}\right)\right]^2}f\:=\:\frac{1.325}{\left[ln\left(\frac{e}{3.7D}+\frac{5.74}{Re^{0.9}}\right)\right]^2}$$

$$Re\:=\:\frac{VD}{v}$$

where;

f = Darcy friction factor (Moody friction factor)

L = pipe length

V = velocity

D = pipe diameter

g = gravitational acceleration (9.806 m/s2)

e = surface roughness

Re = Reynolds Number

v = kinematic viscosity (water at 20oC = 1.003E-6)

## MiTS Software#

1. The output value from MiTS software is shown as per image below.

1. We also test the value given in our software with an online head loss calculator.
3. Pipe 1 is taken as an example, please refer image below:

1. Moody Friction Factor can be calculated using the online calculator here. Please refer image below:

1. As you can see the result generated from MiTS software is equivalent to the one calculated by the online head loss calculator:
• Velocity = 0.3874 m/s
• Head loss = 0.1202 m

## Manual Calculation#

1. Formula below is used to calculate the head loss with known length, L (m), velocity, V (m/s), pipe diameter, D (m), pipe roughness, e (mm) and kinematic viscosity, v (m2/s)

$$H\:=\:\frac{fLV^2}{2dg}$$

$$f\:=\:\frac{1.325}{\left[ln\left(\frac{e}{3.7D}+\frac{5.74}{Re^{0.9}}\right)\right]^2}f\:=\:\frac{1.325}{\left[ln\left(\frac{e}{3.7D}+\frac{5.74}{Re^{0.9}}\right)\right]^2}$$

$$Re\:=\:\frac{VD}{v}$$

1. Darcy friction factor is needed to calculate the head loss and Reynolds Number is needed to calculate Darcy friction factor. So we will find the Reynolds Number first.

$$Re\:=\:\frac{VD}{v}$$

$$Re\:=\:\frac{0.387\times 0.1}{1.003\times 10^{-6}}$$

$$Re\:=\:38268.793$$

1. Next, we need to find Darcy friction factor.

$$f\:=\:\frac{1.325}{\left[ln\left(\frac{e}{3.7D}+\frac{5.74}{Re^{0.9}}\right)\right]^2}f\:=\:\frac{1.325}{\left[ln\left(\frac{e}{3.7D}+\frac{5.74}{Re^{0.9}}\right)\right]^2}$$

$$f\:=\:\frac{1.325}{\left[ln\left(\frac{0.15}{3.7\times 100}+\frac{5.74}{38268.793^{0.9}}\right)\right]^2}$$

$$f\:=\:0.26384364\:≈\:0.264$$

1. Finally, we can calculate the head loss using Darcy-Weisbach formula.

$$H\:=\:\frac{fLV^2}{2dg}$$

$$H\:=\:\frac{0.264\times 59.466\times 0.387^{2}}{2\times 0.1\times 9.806}$$

$$H\:=\:1.1989\:≈\:1.20\:m$$

# Chezy-Manning Equation#

Note: Work in progress