Derive a relationship between velocity, pressure, elevation and density along the pipe.

Task 1:

The company is interested in using a widening pipe to transport fluid from location A to another location B. The pipe will be full with the flowing fluid during operation. The area of the pipe cross section at location A is A and this section is at an elevation z. It is expected that the fluid will flow with velocity v and pressure p. It is assumed that the system will have geometry and parameters similar to the one presented in Fig 1.

You are required to consider the rate of change of momentum from location A to B, the forces acting due to the effect of surrounding pressure and changes in elevation and employ calculus to:

Derive a relationship between velocity, pressure, elevation and density along the pipe.
What assumption(s) did you make on the fluid properties?
If the pipe is to be horizontal and the flow of the fluid is reversed such that at location B, the velocity, pressure and diameter are 1.5 m/s,138 kPa and 0.15 m, respectively. You are required to:

Determine the pressure the fluid will possess at another location of the pipe where the diameter is 100 mm assuming that there is no loss of energy.
{Hint:

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Fig 1

Task 2

Given that you need a pump to support the flow described above and the pump is to be driven by a steel shaft loaded as shown in Fig 2. It is anticipated that there is a point of elastic-plastic boundary along the section AB located at P where the shaft integrity is critical. If the shaft has modules of rigidity G = 80 GPa, yield stress τyield = 160 MPa, You are required to:

Derive an expression for the torque TAB in terms of P.
Determine the distance P which is the elastic-plastic boundary.

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Fig 2

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