A 70 percent efficient pump delivers water at 20^∘ C from...

A 70 percent efficient pump delivers water at $20^{\circ} \mathrm{C}$ from one reservoir to another 20 ft higher, as in Fig. P6.102. The piping system consists of $60 \mathrm{ft}$ of galvanized iron 2 -in pipe, a reentrant entrance, two screwed $90^{\circ}$ long-radius elbows, a screwed-open gate valve, and a sharp exit. What is the input power required in horsepower with and without a $6^{\circ}$ well-designed conical expansion added to the exit? The flow rate is $0.4 \mathrm{ft}^{3} / \mathrm{s}$

Key Concepts

Pump Efficiency

Pump efficiency is the ratio of the useful hydraulic work delivered by the pump to the total energy or input power supplied. This concept is crucial because it accounts for energy losses within the pump and the system, meaning that more input power is required than is actually delivered to move the fluid. It is fundamental in determining the actual energy consumption in pumping applications.

Hydraulic Power

Hydraulic power is the rate of work done in moving a fluid through a system, usually calculated as the product of the fluid’s weight flow rate and the effective elevation head. This concept links the mechanical work needed to overcome gravitational effects and frictional losses, and it forms the basis for calculating the necessary pump output before accounting for efficiency losses.

Minor Losses in Piping Systems

Minor losses represent the additional head losses incurred due to fittings, bends, valves, and abrupt changes in the flow path within a piping system. Unlike friction losses along straight pipes, these can be quantified using empirical loss coefficients. They are essential for accurate system design because they can significantly impact the total head that a pump must overcome.

Well-Designed Conical Expansion

A well-designed conical expansion is a geometric modification at the exit of a piping system that gradually enlarges the flow area, reducing sudden changes in velocity and minimizing turbulence. This design helps to lower the energy losses at the exit, thereby decreasing the overall head loss and reducing the pump input power required for a given flow rate.

Transcript

00:01 Here in this question as we can see in the given diagram we are going to calculate the input power required in horsepower with and without a six degree well designed conical extension now from properties of water at 20 degrees celsius density raw is equal to 1 .94 slug per 8 cube and dynamic viscosity, that is mu is equal to 2 .09 multiply by 10 to the power minus 5 flood per feet second and for galvanized iron roughness is equal to epsilon equal 0 .005 feet.

01:56 Now from the loss coefficient charts, the minor losses are loss of head due to the entrance entrance that is k the entrance is equal to one and lows of head due to 90 degree long radius elbows or two inch diameter wipe that is k elbows is equal to 0 .41 and loss of head due to screwed open gate valve that is that is k gate equal to 0 .16 and at last loss of head due to chart exit that is k sharp exit is equal to 1 .0 .0.

04:50 Now we are going to calculate the average velocity based on the discharge.

04:58 So average velocity based on the discharge that is capital v is equal to capital q divided by a.

05:20 Now, we know that we can write capital a as pi divided by 4 multiplied by d square.

05:34 Now, here on substitute the values, we get v equal to 0 .4 divided by pi divided by 4, multiply by 2 divided by 12 whole square so after dividing this we get v is equal to 18 .33 fifth per second so this is the average velocity based on the discharge now we need to calculate the renaud's number so the reynolds number is given as r .e.

06:33 Equal to row multiplied by capital v, multiplied by d of h divided by mu.

06:42 Now on substitute the values, we get 1 .94 multiplied by 18 .33 multiplied by, 2 divided by 12 divided by 2 .09 multiplied by 10 to the power minus 5.

07:11 Now after dividing this we get r .e.

07:17 Is equal to 2 .835 multiply by 10 to the power 5.

07:23 Now this is actually greater than the critical renal number which we know is equal to 4 ,000.

07:49 Therefore the flow will be of turbulent flow.

08:10 Now here we need to calculate the relative roughness ratio so the relative roughness ratio is given as epsilon divided by small d on substituting the values we get 0 .005 divided by 2 divided by 12 so the relative reference ratio that is epsilon divided by a small d is equal to 0 .003 now here we need the value for the fiction factor from the moody chart which is corresponding to the value of reynolds number and the relative roughness so from the moody chart we get f moody is equal to 0 .0266...

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