A 125 -mm-diameter pipeline conveying water at 10^∘ C contains 50 m of straight galvanized pipe,

step 1 In this question you have to find the pressure drop across the pump or delta p. So first, we can

A 125 -mm-diameter pipeline conveying water at 10^∘ C...

A 125 -mm-diameter pipeline conveying water at $10^{\circ} \mathrm{C}$ contains $50 \mathrm{m}$ of straight galvanized pipe, 5 fully open gate valves, 1 fully open angle valve, 7 standard $90^{\circ}$ elbows, 1 square-edged entrance from a reservoir, and 1 free discharge. The entrance conditions are $p_{1}=150 \mathrm{kPa}$ and $z_{1}=15 \mathrm{m},$ and exit conditions are $p_{2}=0 \mathrm{kPa}$ and $z_{2}=30 \mathrm{m} .$ A centrifugal pump is installed in the line to move the water. What pressure rise must the pump deliver so that the volume flow rate will be $Q=50 \mathrm{L} / \mathrm{s}^{\circ}$

Key Concepts

Flow Rate and Continuity

The flow rate is an essential parameter that describes the volume of fluid moving through a pipeline per unit time. The principle of continuity states that, for an incompressible fluid, the flow rate must remain constant along a streamline, which implies that flow velocity varies inversely with the cross-sectional area of the pipe. Accurately determining the flow rate is critical for computing velocity-related terms in friction and minor loss calculations, and it influences both the energy required by the pump and the overall design of the system.

Pump Head and Pressure Rise

In fluid systems, a pump is used to add energy to the fluid, compensating for energy losses due to friction and losses from minor components, as well as changes in elevation. The pump head or pressure rise is the increase in pressure provided by the pump, which is calculated based on the total energy required to overcome these resistances. This concept is key when sizing a pump to ensure that the desired flow rate is achieved while balancing the energy elevations and losses throughout the system.

Pipe Friction Losses

Friction losses arise from the resistance encountered by a fluid as it flows through a pipe due to the interaction with the pipe’s internal surface. These losses depend on factors such as the pipe’s length, diameter, roughness of the internal surface, the flow velocity, and the fluid properties. The Darcy-Weisbach equation is typically used to quantify these losses by introducing a friction factor that encapsulates the effect of these various parameters.

Conservation of Energy in Fluid Systems

This concept involves applying the energy equation (often known as Bernoulli’s equation) to fluid flow, which balances the different forms of mechanical energy along a streamline. It considers pressure energy, kinetic energy, potential energy due to elevation, and energy losses due to friction and other dissipative effects. In pipeline problems, the energy equation is modified to include pump work and losses from fittings, valves, and friction, thereby establishing the relationship between the inlet and outlet conditions of the fluid flow.

Minor Losses in Piping Systems

Minor losses refer to additional energy losses associated with components such as valves, elbows, bends, and entrances/exits in a piping system. Each of these fittings causes localized disturbances in the flow that translate into extra head losses beyond the frictional losses in the straight sections of pipe. These losses are quantified using loss coefficients that are determined empirically and are crucial for accurate energy balance calculations.

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