@forhaduzzaman42

# Topic : Mathematical Formula Teacher

## Message From You:

From the water works a straight, horizontal pipeline leads 10oC water to the city. The length of

the pipeline is 3km, the diameter 500 mm and roughness k=0,6 mm. The pressure in the city

water network is 8 bar (abs.). How big is the pressure needed at the water works to give a flow

of 1200 m3/h?

## Message From ChatGPT (models : gpt-3.5-turbo-16k-0613):

To calculate the pressure needed at the water works to give a flow of 1200 mÂł/h, we can use the Darcy-Weisbach equation, which relates the pressure drop in a pipe to the flow rate, pipe characteristics, and fluid properties.

The Darcy-Weisbach equation is given by:

ÎP = (f Ă (L/D) Ă (Ď Ă VÂ˛))/2

Where:

ÎP is the pressure drop (in Pa)

f is the Darcy-Weisbach friction factor

L is the length of the pipeline (in meters)

D is the diameter of the pipeline (in meters)

Ď is the density of the water (in kg/mÂł)

V is the velocity of the water (in m/s)

First, letâs convert the given values to the appropriate units:

Length of the pipeline = 3 km = 3000 m

Diameter of the pipeline = 500 mm = 0.5 m

Flow rate = 1200 mÂł/h

Density of water at 10Â°C = 999.7 kg/mÂł (approximately)

Pressure in the city water network = 8 bar (abs.) = 8 Ă 10^5 Pa

Now, we need to calculate the velocity of the water using the flow rate and the cross-sectional area of the pipe:

Velocity (V) = (Flow rate)/(Cross-sectional area)

Cross-sectional area = Ď Ă (D/2)Â˛

Substituting the values:

Cross-sectional area = Ď Ă (0.5/2)Â˛ = 0.19635 mÂ˛

Velocity (V) = (1200 mÂł/h) / (3600 s/h) / (0.19635 mÂ˛) â 0.1809 m/s

Next, we need to calculate the friction factor (f) using the Colebrook-White equation, which takes into account the roughness of the pipe:

1/âf = -2 Ă log10((k/3.7D) + (2.51/(Re âf)))

Where:

k is the roughness of the pipe (in meters)

Re is the Reynolds number

To calculate the Reynolds number (Re), we use the formula:

Re = (Ď Ă V Ă D) / Îź

Where:

Îź is the dynamic viscosity of water at 10Â°C (approximately 1.307 Ă 10^(-3) PaÂˇs)

Substituting the values:

Re = (999.7 kg/mÂł Ă 0.1809 m/s Ă 0.5 m) / (1.307 Ă 10^(-3) PaÂˇs) â 690,014

Now, we can solve the Colebrook-White equation iteratively to find the friction factor (f). We start with an initial guess for f and iterate until we reach a desired level of accuracy. Letâs assume an initial guess of 0.02 for f.

Using an iterative method, we find that the friction factor (f) is approximately 0.0198.

Now we can calculate the pressure drop (ÎP) using the Darcy-Weisbach equation:

ÎP = (f Ă (L/D) Ă (Ď Ă VÂ˛))/2

ÎP = (0.0198 Ă (3000 m) / (0.5 m) Ă (999.7 kg/mÂł Ă (0.1809 m/s)Â˛))/2 â 9,875 Pa

Finally, to find the pressure needed at the water works, we add the pressure drop to the pressure in the city water network:

Pressure at the water works = Pressure in the city water network + Pressure drop

Pressure at the water works = 8 Ă 10^5 Pa + 9,875 Pa â 8,098,750 Pa

Therefore, the pressure needed at the water works to give a flow of 1200 mÂł/h is approximately 8,098,750 Pa.