3.3.2. Second Example: Double Exit Elbow

Now let us go to the second example 2 where we have the water flows through a double exit elbow. Okay this is elbow type of concept, where we have a V 1 velocity and V 2 velocity. One is 5 meter per second another is 10 meter per second. Inside the volume of the elbow is 1 meter cube, okay the volume of this. What are the vertical and horizontal force component of from air and water on this elbow if pressure in the free jet is atmospheric and the unit weight of the water will be 1000 kg per meter cube.

In this example, we are dealing with a scenario where water flows through a specially designed elbow, which is a type of bend in a pipe. Here, two different flow velocities are mentioned: V1 at 5 meters per second and V2 at 10 meters per second. The inside volume of the elbow is given as 1 cubic meter. The task is to determine the vertical and horizontal forces acting on the elbow due to water and air. Understanding the pressure conditions is crucial as the pressure in the free jet is atmospheric pressure, and we know the density of water (1000 kg/m³).

Imagine water flowing through a flexible hose that turns at an angle. Depending on how fast the water flows in different sections of the hose, it will exert forces on the walls of the hose. Similarly, in this example, the elbow shape of the pipe influences the flow, affecting how forces are distributed on it.

Flow classification: Two dimensional, Steady flow, Turbulent, Incompressible.

For this problem, it is important to classify the type of flow based on its characteristics. The flow is considered two-dimensional because we are analyzing it in a plane. It is steady flow since the flow conditions do not change over time. The flow is classified as turbulent indicating that it has chaotic property changes, and it is incompressible since the density of the water remains constant throughout the flow.

Think of driving a car on a straight road versus a winding, hilly road. On the straight road, the conditions (speed, direction) remain steady—similar to steady flow. If the road suddenly spirals down, it mirrors turbulent flow where the conditions change with twists and turns.

So if we have this fixed control volumes, first we will find out that the pressures. The pressures at the inlet is given is 25,000 Pascal. At the outlet is pressure is atmospheric pressure. That what is there. So you have the free jet at these two locations where pressure is the atmospheric pressures. At these two point you have a pressure equal to atmospheric pressure, but at this point I have the pressure which is 25,000 Pascal, Newton per millimeter square.

In analyzing the elbow, we first need to determine the pressure values at different points within the control volume. The inlet pressure is specified as 25,000 Pascals (which is a measure of force per unit area), while at the outlets, the pressure is atmospheric, meaning it is the pressure exerted by the weight of air above us. This distinction is vital as it determines how forces act on the elbow structure.

Consider how the pressure in a balloon changes. When you blow air into the balloon, the pressure inside increases (similar to the inlet pressure of 25,000 Pa). However, once you let go, the air exits and the balloon returns to atmospheric pressure, just like the water exiting the elbow to the outside atmosphere.

Then we will apply mass conservation equations that let us apply mass conservation equations to compute it what could be the velocity at the section 3. So we are applying the mass conservation equations for these control volumes that to compute what could be the velocity at this point.

The principle of conservation of mass states that the mass flow rate entering a control volume must equal the mass flow rate exiting the control volume. In our elbow, we apply this principle to find the velocity at a certain point in the elbow. The continuity equation helps us relate the inflow and outflow velocities and areas, allowing us to compute unknown velocities based on known data.

Imagine water flowing through a funnel. As it moves from a wide opening to a narrow tube, it speeds up. This speed-up is because of mass conservation: the amount of water must stay the same, but it has to move fast to fit through the smaller space.

Applying RTT, sum of forces should equal to the rate of change of momentum flux within the control volume is equal to the net outflux of momentum flux passing through these control surface.

In the context of fluid dynamics, momentum conservation is a key concept. We apply the Reynolds Transport Theorem (RTT) to relate the change in momentum within the control volume to the forces acting on the elbow. Essentially, we are equating the net forces acting on the fluid to the change in its momentum, allowing us to compute the forces resulting from the fluid flow.

Think of a swimming pool. When a swimmer pushes off the wall, they propel themselves forward. The force they exert on the wall equals the change in momentum they experience. Similarly, in our elbow, the water's momentum change interacts with the elbow's structure, exerting forces.

So if I apply this part, then F x will come it to the 9539 Newton. Similar way the force component acting on this y direction that one will be the 7720 Newton.

By applying the conservation equations for momentum, we find the forces acting in both the horizontal (Fx) and vertical (Fy) directions on the elbow. The calculations yield values of 9539 Newtons for the horizontal force and 7720 Newtons for the vertical force, providing insight into the resultant forces exerted on the elbow due to the flowing water.

Consider a pizza chef tossing a pizza dough into the air. The force they apply can be thought of in two directions: upward (to get it off the table) and outward (to spread it). In our case, the elbow experiences similar dual directional forces from the water flow.

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