Hi!
I'm having a hard time figuring out if flow rate decreases when water goes up a vertical pipe. I know that at some point when the water can't be pushed any higher due to P=pgh, the water stops flowing from the pipe. I also believe that due to continuity, all sections of the pipe should have the same flow rate no? I have seen many people calculate the second velocity for the Bernoulli equation using the continuity equation A1V1=A2V2 assuming constant flow rate. Is that really practical in real life though? Wouldn't a pump have to have a variable output or something to maintain the same flow rate while pumping water in a vertical pipe higher and higher? I guess I'm a little confused on the subject and I really hope someone can clear it up for me. Thank you for your help!!
If the pipe has constant cross sectional area and if the fluid is incompressible (water is NOT squishy) then the velocity is the same at every section of the pipe since there is no place to store water along the pipe. That is what continuity means. Yes, A1V1 = A2V2 but since A2 = A1 then V2 = V1.
Now of course if the pipe is higher, the velocity will go down unless the pump works harder. That does not change the A1V1 = A2 V2 so V2=V1. It is just a different V at both point 1 and point 2 along the pipe. In other words less water flows through the higher pipe, everywhere along the pipe.
Hi there! I'd be happy to help clarify your confusion.
When water flows through a vertical pipe, the flow rate may or may not decrease depending on the conditions. Let's break it down step by step:
1. Flow rate: The flow rate refers to the volume of water passing through a particular point in the pipe per unit of time. It is commonly measured in liters per second or cubic meters per hour.
2. Pressure and height: As water moves up a vertical pipe against gravity, its potential energy increases due to an increase in height. According to the principle of hydrostatic pressure, the pressure exerted by a column of liquid increases with depth. The pressure at any given depth is given by the equation P = pgh, where P is the pressure, p is the density of the water, g is the acceleration due to gravity, and h is the height.
3. Continuity equation: The continuity equation states that the mass flow rate of a fluid remains constant at different points along a streamline, assuming no loss or gain of fluid. In other words, the product of cross-sectional area and velocity is constant. Mathematically, A1V1 = A2V2, where A represents the pipe's cross-sectional area and V represents the velocity of the water.
Now, let's address your concerns:
- Yes, according to the continuity equation, the flow rate remains the same at all sections of a horizontal pipe. However, when dealing with a vertical pipe, things can change.
- As water moves up a vertical pipe, the increase in height translates to an increase in pressure, as mentioned earlier (P = pgh). This increased pressure leads to a decrease in velocity. Due to the relationship between velocity and flow rate (Q = AV), the decrease in velocity will result in a decrease in flow rate.
- The Bernoulli equation can be used to analyze the energy changes along a streamline. While it includes the continuity equation (A1V1 = A2V2), it also considers other factors such as pressure and height. By applying the Bernoulli equation, we can determine the velocity at different points along the pipe, but it doesn't necessarily mean the flow rate remains constant.
- In real-life scenarios, maintaining the same flow rate while pumping water higher in a vertical pipe would require additional measures. A pump with a variable output or a system of pumps at different heights may be needed to maintain the desired flow rate against the increasing pressure due to the height.
To summarize, as water moves up a vertical pipe, the flow rate can decrease due to the increase in pressure and height. While the continuity equation is useful for analyzing flow rates in horizontal pipes, additional considerations are required for vertical pipes. Real-life scenarios often require pumps or systems with variable outputs to maintain a consistent flow rate against increasing pressure.