A pipe 900 m long and 200 mm diameter discharges water to atmosphere at a point 10 m below th

A pipe 900 m long and 200 mm diameter discharges water to...

A pipe $900 \mathrm{~m}$ long and $200 \mathrm{~mm}$ diameter discharges water to atmosphere at a point $10 \mathrm{~m}$ below the level of the inlet. With a pressure at inlet of $40 \mathrm{kPa}$ above atmospheric the steady discharge from the end of the pipe is $49 \mathrm{~L} \cdot \mathrm{s}^{-1}$. At a point half way along the pipe a tapping is then made from which water is to be drawn off at a rate of $18 \mathrm{~L} \cdot \mathrm{s}^{-1}$. If conditions are such that the pipe is always full, to what value must the inlet pressure be raised so as to provide an unaltered discharge from the end of the pipe? (The friction factor may be assumed unaltered.)

Key Concepts

Bernoulli's Principle

Bernoulli's Principle explains energy conservation in fluid flow by relating pressure, kinetic, and potential energy along a streamline. In pipe flow problems, it is used to account for the pressure differences between inlet and outlet, taking into consideration elevation differences and the velocity of the flow, though corrections must be made to incorporate frictional losses.

Darcy-Weisbach Equation

The Darcy-Weisbach equation is a fundamental relation in pipe flow analysis that quantifies head loss due to friction. It relates the frictional loss to factors such as the pipe's length and diameter, the flow velocity, the fluid's properties, and a friction factor that encapsulates flow roughness and turbulence, making it essential for determining pressure requirements in piping systems.

Friction Losses

Friction losses in pipe flows represent the energy dissipated due to the viscous interaction between the fluid and the pipe walls. These losses are proportional to the flow velocity squared and the distance travelled through the pipe, and they play a crucial role in determining the overall pressure drop and the necessary driving pressures to maintain desired flow rates.

Conservation of Mass

The principle of conservation of mass, often applied through the continuity equation, states that the mass flow rate must remain constant in a steady flow system. When a branch or tapping is introduced into a pipe, the total flow at the inlet must equal the sum of the flows in the individual downstream segments, which is critical for analysing changes in system behavior.

Pipe Network Analysis

Pipe network analysis involves understanding how alterations such as adding taps or branches affect the flow distribution and pressure drops within the system. When a branch is introduced, the pipe experiences changes in local velocities and friction losses, requiring adjustments in driving pressures to ensure that specified discharges are achieved in each part of the network.

Transcript

00:01 Starting with our given information from the question, the inlet pressure p1 is equal to 40 kilopascals, which is equal to 40 ,000 pascales.

00:35 The outlet pressure, p2, is equal to the atmospheric pressure.

00:46 That diameter of pipe is 200 millimeters or 0 .2 meters.

01:01 The elevation change delta z is equal to 10 meters.

01:16 A pipe length is equal to 900 meters.

01:23 And the inlet flow rate, q1, is equal to 49 liters per second, or 0 .049 cubic meters per second.

01:43 First, we calculate the head loss.

01:53 Hl is equal to p1 row g plus delta z.

02:00 This is equal to 40 ,000 pascales over 1 ,000 kilogram per cubic meter, 9 .81 meter per second squared plus 10 meters, works out to 14 .077 meters.

02:20 Then, calculate the cross -sectional area of the pipe.

02:42 So the radius of the pipe is equal to diameter over 2, so that's 0 .2 meters over 2, so radius is 0 .1 meters.

02:54 And the area is equal to pi r squared, so that's pi 0 .1 meters squared works out to 0 .034 square meters and then we can calculate the inlet velocity using the formula b1 is equal to q1 over a q1 is 0 .049 cubic meter per second over area is 0 .0 314 square meters gives us 1 .56 meters per second using the diversity y spock equation isolate and solve for the friction factor so darcy westbock equation, hl is equal to f, l over d, v squared over 2g.

04:09 Rearranging, f is equal to hld2g over lv squared.

04:20 So this is going to give us 14 .077 meters, 0 .2 meters, 2, 2, 9 .81, meter percent.

04:32 Squared over 900 meters, 1 .56 meter per second squared works out to 0 .0252.

04:51 The flow rate in is equal to the flow rate out plus the flow rate in the middle, which is equal to 18 liter per second plus 49 liter per second or 6.

05:12 67 liter per second equivalent to 0 .067 cubic meter per second...

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