9.1. Venturi meter example

One of the other applications of Bernoulli equation is the venturi meter. How would you find the flow Q given the pressure drop between point 1 and 2 and the diameter of the 2 sections? You may assume the head loss is negligible.

A Venturi meter is a device that measures the flow rate of a fluid by using the principle of pressure differences. The flow rate (Q) can be calculated based on the pressure drop between two points along the meter, usually denoted as point 1 and point 2. You will need to know the diameters of the two sections of the meter and the pressure readings at these points.

Consider the Venturi meter as a funnel used in cooking. Picture how the liquid flows faster when it passes through the narrow end of the funnel. Just like how pressure changes as water moves through the funnel, the Venturi meter detects changes in pressure as fluid flows through its varying diameters.

The first thing that we can do is, we apply at points at 2 points we apply the Bernoulli’s equation. So, we take the pressure p 1 p 2 on one side z 1 = z 2, so that will be p 1 / gamma - p 2 / gamma = V 2 square / 2g - V 1 square / 2g.

To apply Bernoulli's equation across two points in a Venturi meter, we compare the pressure differences and velocities at those two points. Since the heights are the same (z1 = z2), we can simplify the equation to relate the pressures (p1 and p2) to the velocities (V1 and V2). The equation helps us understand how pressure reduction in the narrow segment results in increased fluid velocity.

Imagine two garden hoses connected inline. When you pinch one hose, the water flows faster through the pinched section. Similarly, the Venturi meter works by changing the flow area, affecting the speed and pressure of the fluid as it moves through.

As you see the point 1 and 2 the datum is the same that is why we were able to cancel this 1 out and Q = velocity into area V 1 A 1 = V 2 A 2.

In a Venturi meter, the continuity equation states that the product of the cross-sectional area of the flow (A) and the velocity (V) must remain constant along the flow path (A1V1 = A2V2). By knowing the areas and applying Bernoulli's equation, we can find the flow rate (Q) by relating the velocities and areas at both points.

Think of a crowded subway station where people (fluid) are entering and exiting through two doors (areas). If more people try to use a narrow door (smaller area), they will move faster than those using a wider door. Similarly, fluid in a Venturi meter speeds up when passing through a narrower section.

Normally, we know d 1 and d 2. So, there is a relationship obtained between V1 and V2. So, putting V1 in terms of V2, so we can instead of V 1 obtained from here, we can put here and therefore, we can get this equation here, you know.

Given the diameters d1 and d2 of the Venturi meter, we can derive the relationship between V1 and V2 using the continuity equation. This relationship allows us to express one velocity in terms of the other, ultimately leading us to calculate the discharge (Q), which takes into account the changes in pressure and areas.

Imagine a group of kids (fluid) running down a playground slide (venturi). If two kids start at the top and move down through different levels (areas) of the slide, their speed changes based on the level they're on. By comparing their speeds and the sizes of the slides, you can see how changing areas affects their speeds and overall flow.

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