20.3. Application of Linear Momentum Equations

If you look it that most of the times also we have a valve to control the flow okay which is a gate valve okay, it is a gate type of systems. If you rotate it this valve closes the waters okay. And it can have a total open or half closed and all these conditions to regulate the flow to regulate the flow we have a these systems. Now if you look it if I am to draw the streamlines, how it happens it.

In fluid dynamics, controlling flow is essential. A gate valve is a common tool used to manage the flow of fluids in a pipeline. By opening or closing this valve, we can regulate how much fluid passes through. When the valve is completely open, maximum fluid flow occurs. However, if it is partially opened, the flow becomes turbulent, which can complicate the flow patterns. Drawing the streamlines gives a visual representation of how fluid flows around obstacles, which is crucial for understanding fluid movement.

Think of the gate valve like a faucet. When you open the faucet fully, water rushes out quickly, similar to a fully open valve. If you turn the faucet just halfway, the water flow slows down, causing changes in how the water spouts and splashes. This change in flow helps us understand how valves influence pipeline systems.

Now if you look it similar way we have a the globe valve which is more control valve systems. The flow comes here, then rotate it and this valve it goes off and flow goes like this. So we can look it now in terms of vortex formations here, the vortex formations here, and once flow goes it also could have the vortex formations also here.

A globe valve functions as a more intricate control mechanism compared to a gate valve. As the globe valve is adjusted, it modifies how the fluid flows, potentially causing vortex formations. These vortices indicate turbulent flow patterns, which can lead to energy losses within the system. Understanding the differences between valve types allows engineers to select the appropriate valve based on specific applications and the required control over fluid motion.

Imagine running your hand through water; the way your hand swirls the water mimics how vortex formations occur in fluid systems. If your hand (representing a valve) is moved or rotated, it changes the flow, creating eddies or swirls that demonstrate how vortices can form, leading to energy losses in pipelines.

Now coming to this, I am just applying the linear momentum equations for these control volumes okay. So if you have this is the control volumes. So you may have the shear stress is acting on this okay, this the shear stress part, okay. But we are neglecting this part, we are neglecting the shear stress part.

Linear momentum equations help in calculating the forces that act upon a control volume in a fluid system, focusing primarily on pressure differences rather than shear stresses. As the shear stress is usually negligible in such equations, the task becomes simpler by equating pressure force differences with momentum fluxes. This simplification makes it easier to analyze and predict how fluids behave under various configurations and conditions.

Think of pushing a swing; your push relates to your momentum, which affects how high the swing goes. Similarly, linear momentum equations relate the forces acting on a control volume to how the fluid moves within that volume, making management of fluid systems (like swings in motion) easier.

Now I will applying Bernoulli’s equations along the straight line, okay I am just applying the Bernoulli’s equation along the straight line. When you apply the Bernoulli’s equations, this line is horizontal. This streamline is horizontal, so you do not have a z components.

Bernoulli's equation relates pressure, velocity, and height in fluid flows. When we apply it along a streamline, particularly in horizontal flows, we focus only on pressure and velocity head, ignoring elevation changes. Understanding how energy losses are quantified in this manner aids engineers in evaluating flow efficiency and predicting pressure variations throughout a system.

Imagine a water slide: when a person goes down, they may speed up (increasing velocity) while losing potential height. Keeping track of the speed and pressure felt at different points relates to how we use Bernoulli’s equations to evaluate the energy and behavior of fluid flow.

This is what we have derived lot. So this is a rate of change of the momentum flux is equal to the, the pressure difference between the, the pressure force difference between this p and p and that what if you rearrange it you will get it this part okay.

Calculating energy losses in a fluid system requires understanding how momentum flux changes with pressure differences. By rearranging equations concerning pressure and momentum flux, engineers can derive essential relationships that help predict how energy dissipates through valves and fittings in a system, which is critical for effective design and efficiency.

Think of the energy loss while pouring syrup onto pancakes. The pressure you exert to pour the syrup will determine how fast and how far it goes—similar to how changes in pressure affect fluid flow through pipelines and the accompanying energy losses.

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