23.2. Limitations of Bernoulli's Equation

Bernoulli equations can be applied for unsteady flow, but the simplified derivations what you use it, those are for steady flow. That means, there is no time component is there.

Bernoulli's equation is generally derived and most effectively used in scenarios where the flow is steady, meaning that the fluid properties at any given point do not change over time. While it can be applied to unsteady flow, using it in such cases requires careful consideration of how the fluid dynamics change with time, which complicates the situation. For steady flows, parameters like velocity and pressure remain constant at particular points, allowing for easier calculations.

Think of steady flow like a river flowing calmly at the same speed over time. You can measure how high the water is and how fast it's flowing without worrying about changes. Unsteady flow, on the other hand, is like a river during a flood: the water levels and speeds are constantly changing, which complicates calculations.

Remember it, this equation is most frequently used, also misused equations. There are two solutions available for us, one is mass conservation and the other is energy conservation.

Bernoulli’s equation is often misunderstood or incorrectly applied by engineers and students. While it's a powerful tool for calculating fluid dynamics, its assumptions must be satisfied, such as incompressibility and the absence of viscosity. Misapplication often happens when energy losses, mass transfer, or pressure changes due to shock waves are not accounted for, resulting in inaccurate results. Recognizing that there are different forms of fluid conservation — both mass and energy — is crucial for accurate use.

Imagine a busy highway where cars are moving at the same speed. If you simply count the cars to determine traffic flow without considering different road conditions (like traffic jams or road construction), you would miscalculate the 'flow' of cars. This is similar to the misuse of Bernoulli's equation when not all conditions are considered.

For most of the fluid flow problems in civil engineering and mechanical engineering, where flow Mach number is less than 0.3, we can consider it as incompressible flow because the density variation will be less than 5%, which we can neglect.

One of the key assumptions of Bernoulli's equation is that the flow is incompressible. This means that the fluid's density remains constant, which holds true for most liquids when the flow speed is relatively low (Mach number < 0.3). If the fluid's density changes significantly, as it does in high-speed gas flows, then Bernoulli's equation may not be applicable without adjustments. In such situations, engineers must consider compressibility effects to get accurate predictions of flow behavior.

Consider air being blown slowly through a straw. The air moves at low velocity, and its density remains fairly constant — this is incompressible flow. If you start blowing really hard, the air compresses, and the density changes. For practical applications like plane aerodynamics, ignoring this change leads to errors.

The most important assumption is that the frictionless flow cannot be applied near to solid surfaces. Whenever the fluid goes near the solid, it creates a velocity gradient and shear stress, leading to viscous effects.

Bernoulli's equation assumes that the flow is frictionless, meaning that it does not account for any energy losses due to friction between the fluid and solid surfaces. Near solid boundaries, like the walls of a pipe, the fluid experiences drag and viscosity effects, causing velocity to change and losing energy, which violates Bernoulli's assumptions. Thus, the equation cannot be used accurately in these regions without modifications to account for these losses.

Think of ice skating on a smooth surface versus trying to skate next to the wall. In the middle of the rink, where there is no friction, you glide effortlessly. But when you get close to the wall, the friction slows you down, making it harder to maintain your speed. Similarly, fluids flowing near surfaces will lose energy due to friction.

We cannot apply Bernoulli's equation in mixing zones, as there is no shaft work because of the presence of either pumping or turbines that take energy from or add energy to the system.

In areas where fluids mix or interact, the assumptions of Bernoulli's equation break down because energy from one fluid can affect the other. Centrifugal pumps and turbines introduce energy into the system, or extract from it, creating conditions where Bernoulli's assumptions do not hold. These situations require the use of extended or modified forms of Bernoulli's equations, which consider energy contributions from these sources.

Imagine mixing two different colored paints. When stirred together, they change color and consistency. You can't just apply the rule for one color to predict the outcome because the mixing alters the characteristics. In fluid systems, pumps and turbines affect the flow similarly, complicating straightforward applications of Bernoulli.

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