20.2. Derivations of Energy Losses

Let us come for derivations of energy losses and the pressure and velocity distributions of having a systems where two pipes we are enlarging it from the smaller diameter to bigger diameters. If you have that conditions and you can draw the velocity distributions. The velocity distribution it depends upon the type of the flow. If I have a laminar flow, the velocity distribution is different. If I have a turbulent flow velocity distributions is different.

This chunk discusses the initial concepts of energy losses in pipe flow systems. In particular, when fluid flows through pipes of varying diameters, the velocity distribution is affected. If the flow is laminar (smooth and orderly), it behaves differently compared to turbulent flow (chaotic and irregular). Understanding these distributions is critical for analyzing energy losses. A laminar flow generally offers less energy loss than turbulent flow due to the less chaotic motion of the molecules in the fluid.

Think of a calm river (laminar flow) versus a fast-moving rapids (turbulent flow). In the calm river, water flows smoothly without much disturbance. In contrast, the rapids create numerous waves and turbulence which would suggest higher energy loss due to friction with surfaces and chaotic interactions among water molecules.

Now coming to this, I am just applying the linear momentum equations for these control volumes. 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. Just we are assuming the pressure difference between these is equate with a momentum flux.

This section discusses the application of linear momentum equations to control volumes in a flow system. It highlights how shear stress, which usually contributes to energy loss, is neglected in some cases to simplify calculations. Here, the pressure difference is equated with momentum flux, allowing us to relate pressure changes to fluid momentum, which is important in energy loss analysis.

Imagine blowing air through a narrow straw (control volume). The pressure change you feel when you start blowing is similar to the pressure difference in the fluid. As you blow harder, the momentum (speed) of air increases, resulting in a change in pressure at the end of the straw, just like how momentum changes apply to fluid flows in pipes.

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. You have pressure head component and the velocity head component.

This part details the application of Bernoulli's equation, which describes the principle of energy conservation in fluid flows. By considering the horizontal streamline, we can analyze how pressure and velocity heads contribute to energy transfer across sections of flow. The concept simplifies calculations by assuming no vertical elevation changes, focusing solely on pressure and velocity influences on energy losses.

Consider a garden hose that's slightly elevated near the nozzle. When you apply pressure at the handle, water flows quickly out of the nozzle, decreasing pressure inside the hose. Bernoulli's principle helps us understand how the energy conservation between pressure and flow speed works, showcasing how energy is transformed rather than lost.

So the flow is come from the one section to sections two where through the flow there is energy losses, that is what we quantify. So because of that we call it is a modified Bernoulli’s equation which is nothing else. You are equating the energy, you know that how much of energy losses has happened.

In this segment, the focus shifts to quantifying energy losses due to flow transitions from one section to another. This is encapsulated in the modified Bernoulli’s equation, which takes into account these losses by including terms that represent the energy dissipated within the system. Understanding this quantification is vital for managing energy within fluid systems.

Imagine riding a bike on a flat road and then going uphill. The effort expended to climb that hill represents energy loss due to gravity. Similarly, when fluid moves through pipes, energy is dissipated, requiring us to account for that loss to maintain efficiency, akin to adjusting our pedaling effort to cope with the incline.

The same way if you have a gradually contractions happening it or gradual enlargement that you can compute this values. For gradual contraction or for gradual enlargement, the friction factor K is given by...

This chunk introduces how to calculate friction factors (K) for pipes experiencing gradual changes in diameter. The friction factor is a crucial parameter as it significantly influences the total energy losses in a pipe flow. Understanding how to calculate and apply this friction factor helps in designing efficient piping systems and predicting energy losses when the diameter changes.

Think about drinking from a funnel. As the liquid passes through the narrower end (gradual contraction), it experiences more friction, just like a pipe. The same principle applies in wider pipes where gradual enlargements can also affect flow characteristics. By knowing how to calculate these changes, we can better manage fluid flow.

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