2.1. Estimation of Power Required

Static head is 110 – 95, which gives us a static head of 15 meters. The velocity in pipe 1 before the pump can be calculated as V1 = Q / A1. Given Q is 20 liters per second (0.02 m³/s) and the diameter D1 is 0.15 m, we calculate the area A1 as π/4 * D1², resulting in a velocity V1 of 1.132 meters per second.

The static head is the height difference between the water levels, which dictates how much pressure is available for the pump to move the water. Here, we find that the height difference is 15 meters. To calculate the velocity of water flowing in a pipe, we first determine the cross-sectional area of the pipe using the formula for the area of a circle. With a flow rate (Q) given in liters per second, the area derived from the diameter is converted to meters. Finally, we find the flow velocity (V1) by dividing the flow rate by the area.

Imagine you’re trying to pour water through a garden hose. The size of the hose (diameter) determines how fast water flows out (velocity). If you squeeze the hose (decrease the diameter), the water shoots out faster, similar to how the pipe's diameter affects the flow rate.

The major loss in pipe 1 can be calculated using the formula hf1 = f1 L1 / D1 * V1² / 2g, where f1 is the friction factor, L1 is the length of the pipe, and g is the acceleration due to gravity. With f1 given, L1 is 20 meters, D1 is 0.15 meters and V1 is known, we calculate hf1 to be 0.174 meters. Additionally, the minor loss due to inlet can be calculated as hl_minor = 0.5 * V1² / 2g leading to 0.033 meters.

Head loss is the energy loss due to friction as water flows through a pipe. We calculate it using the Darcy-Weisbach equation, which incorporates factors like the length of the pipe, its diameter, and the friction factor (a measure of how slippery the pipe is). We also add minor losses due to fittings or changes in flow direction, which can affect the overall pressure. Here, we computed both major and minor losses.

Think of riding a bike against the wind; the longer you ride (length of the pipe), the more effort (energy loss) is needed to maintain speed. Similarly, if the path is rough (high friction), you will notice it requires more energy to keep moving, analogous to head loss in a pipe.

For pipe 2, the major loss is calculated as hf2 = f2 L2 / D2 * V2² / 2g. Using the given values, the head loss comes out to be 7.975 meters. The total head loss from both pipes equals 0.174 + 0.033 + 7.975 + 0.160, which results in a total of 8.342 meters.

We repeat a similar head loss calculation for the second pipe using the same methodology. Total head loss is crucial because it informs us of how much energy is expended due to friction in both sections of the piping system. The summation of losses from both the major (due to pipe length and flow) and minor losses (due to fittings or bends) gives the total energy loss in the system.

Imagine traveling through two different terrains. The first (pipe 1) is flat and smooth (less friction), while the second (pipe 2) is hilly and rough (more friction), needing more fuel (energy) to cover the same distance. The total fuel consumed (total head loss) gives you an idea of the efficiency of both terrains.

The pump is required to overcome both the static head and the total head loss, which totals to Hf = 15 + 8.342 = 23.342 meters. The power delivered by the pump is calculated as Power = γQHf, where γ is the weight density of water. The answer is 4.57 kilowatts.

The pump must generate enough pressure to lift the water against gravity (static head) and against the friction (head loss). The total head is the sum of these two pressures. We calculate the power by considering the weight density of the water and the flow rate, multiplying these by the total head, which gives us the energy required by the pump over time.

Imagine a workout routine where you have to lift weights (gravity) while also running on a treadmill (friction). The total effort you expend to lift the weights while running at the same time is similar to the pump needing to exert pressure to both overcome gravity and friction in the water supply system.

To find the pressure at the suction side of the pump, we apply Bernoulli's equation considering head loss. Using the known static head and losses, we find that Ps/gamma = -5.272 meters (gauge), leading to an absolute pressure of 46.29 kilopascals.

Bernoulli's equation helps us understand the relationship between pressure, velocity, and height in fluid flow. We rearranged the equation to isolate the pressure at the pump's suction side, factoring in the height (static head), velocity (kinetic energy), and losses. The resultant pressure gives us insight into how the pump is operating relative to atmospheric conditions.

Think of a straw in a soda. When you suck on the straw, the pressure inside it drops; this drop draws soda up. Similarly, we're checking how much pressure the pump needs to work against the natural atmospheric pressure to successfully lift water through the system.

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