A marine terminal for unloading bulk wheat from a ship is...

A marine terminal for unloading bulk wheat from a ship is equipped with a vertical pipe with a nozzle at $A$ which sucks wheat up the pipe and transfers it to the storage building. Calculate the $x$ -and $y$ -components of the force $\mathbf{R}$ required to change the momentum of the flowing mass in rounding the bend. Identify all forces applied externally to the bend and mass within it. Air flows through the 14 -in.-diameter pipe at the rate of 18 tons per hour under a vacuum of 9 in. of mercury $\left(p=-4.42 \mathrm{lb} / \mathrm{in} .^{2}\right)$ and carries with it 150 tons of wheat per hour at a speed of 124 ft/sec.

Key Concepts

Momentum Conservation

This principle states that the net force acting on a control volume is equal to the rate of change of momentum of the fluid within that volume. In the context of fluid flow through bends or curves, the difference in momentum flux between the entrance and the exit of the control volume must be balanced by the external forces acting on the fluid. It is fundamental for analyzing and determining the force components required to change the direction of the flow.

Momentum Flux

Momentum flux is the rate at which momentum is carried by the flowing mass through a unit area and is typically calculated by multiplying the mass flow rate by the velocity of the fluid. It is crucial in understanding how changes in velocity or direction of the fluid lead to corresponding changes in momentum, which in turn require forces to be applied to redirect the flow.

Pressure Forces

Pressure forces arise from differences in pressure acting on different parts of the fluid in a system. When analyzing a flow through a bend or a curved section, differences in static pressure at the boundaries of the control volume can also contribute to the net force. Accounting for these forces is essential to correctly apply the conservation of momentum and to understand the full force balance on the system.

Control Volume Analysis

This analysis technique involves selecting a fixed region in space (a control volume) through which fluid flows, and then applying conservation laws (mass, momentum, and energy) to that region. It allows for the systematic accounting of inflows, outflows, and the effects of external forces, making it a powerful tool to study the forces needed to change the momentum of a flowing fluid in applications such as piping systems and bends.

Transcript

00:01 So here on this problem we are given this equipment that sucks wheat from the bottom by using the nozzle and then delivered it to the storage tank this one is storage tank and we have to calculate the forces required to change the momentum of the flowing mass in the rounding bed so initially the mass was flowing in this direction and then after the bend it is flowing like this we have to calculate this force required to change this momentum this is what we are given in the question and the given input data is that the diameter of the pipe is 14 inch and the mast for rate for a we are given to be as 18 tons per hour and mast for rate for water we are given to be as 150 tons per hour now we will be first making the free board diagram of this section first let you make it here so this is a diagram and the inlet velocity and the exit velocity we are also given v1 is equals to v2 so v1 and v2 are same and its value we are given to be as 124 feet per second now to solve this problem how we can use equations of motion in the y and the x direction this one is our direction and this one is our x direction but before applying these equation will be first converting the units mass flow rate for air we are given to be as 18 tons per hour but we'll be converting into slugs per second so 18 tons so it will be multiplied with 2000 divide by 32 .2 and to convert into second we have to multiply it with 3 ,600.

01:59 So from here we'll get the units into slugs per second for m -dash a and it comes out to be as 0 .3106 slugs per second similarly we can calculate we can convert m dash w so m dash w will be equals to 150 times 2000 divide by 32 .2 times 3 ,600 so we'll get this value to be nearly 2 .58 slugs per second now we can apply the equations of motion in the x and the y direction so first it will be applying in the x direction.

02:43 So sigma fx will be equals to m dash delta vx.

02:49 So in x direction we have only one force that is rx.

02:54 So rx will be equals to m dash we have to take it for both m dash a plus m dash what this wheat m dash a plus m dash wheat times the final velocity minus the initial velocity...

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