Fluid mechanics - Water flowing through a pipe

In summary, the conversation discusses a problem involving the calculation of mass flow rate using different methods. The first method involves solving for the average velocity using the equation mass flow rate = density * average velocity * area. The second method involves finding the average velocity by taking the integral of velocity over the integral of 1. However, it is pointed out that in the expression for mass flow rate, the average velocity should be calculated over the cross-sectional area of the pipe, not just the radius. The conversation then delves into a discussion about why this is the case and the role of different intervals of radius in the calculation of average velocity.
  • #1
theBEAST
364
0

Homework Statement


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The Attempt at a Solution


I am interested in why my method is flawed:
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Alright I know how to do this problem. Essentially you solve mass flow rate = rho * double integral of u dA. HOWEVER, I decided to do this differently. I know that I can solve for the average u from the equation mass flow rate = rho * u_avg * A. I also know that the average u is just the integral of u over the integral of 1. I tried this and I get 1.528 m/s which is clearly not one of the answers. Why does my method not work?
 
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  • #2
In the expression: ##\dot{m} = \rho u_{avg}A##,
##u_{avg}## needs to be the average over the cross-sectional area of the pipe, not the average over the radius.
 
  • #3
TSny said:
In the expression: ##\dot{m} = \rho u_{avg}A##,
##u_{avg}## needs to be the average over the cross-sectional area of the pipe, not the average over the radius.

Hmmm, according to the equation doesn't u only depend on r? In other words u is constant for some circular ring r = k?
 
  • #4
Consider two intervals of radius, one near the center of the pipe and one out at the edge. Suppose the interval dr is the same for both. There are more patches of area in the same interval dr out near the edge. So, the value of the velocity out at the edge contributes more to the average over the cross–section than the value near the center.
 

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  • #5


Your method is not taking into account the variation of velocity within the pipe. The average velocity you calculated is only valid for a uniform flow, which may not be the case in this problem. In fluid mechanics, it is important to consider the velocity profile and how it changes along the flow path. Therefore, using the mass flow rate equation is a more accurate approach as it takes into account the velocity distribution within the pipe.
 

1. How does the flow rate of water through a pipe affect the pressure?

The flow rate of water through a pipe is directly proportional to the pressure. This means that as the flow rate increases, the pressure will also increase. This is because the water molecules are moving at a faster rate, creating more collisions with the walls of the pipe and therefore increasing the pressure.

2. What factors affect the flow rate of water through a pipe?

The flow rate of water through a pipe can be affected by several factors, including the diameter of the pipe, the viscosity of the water, and the length and roughness of the pipe. The flow rate can also be affected by external factors such as temperature and the presence of any obstacles or bends in the pipe.

3. What is the relationship between the velocity of water and the cross-sectional area of the pipe?

The velocity of water flowing through a pipe is inversely proportional to the cross-sectional area of the pipe. This means that as the cross-sectional area of the pipe increases, the velocity of the water will decrease, and vice versa. This relationship is described by the continuity equation, which states that the product of velocity and cross-sectional area remains constant.

4. How does the density of water affect its flow through a pipe?

The density of water does not directly affect its flow through a pipe. However, it does play a role in determining the pressure and velocity of the water. As the density of water increases, the pressure also increases, and the velocity decreases. This is because denser water molecules have more mass, requiring more force to move them through the pipe.

5. Can the flow rate of water through a pipe be controlled?

Yes, the flow rate of water through a pipe can be controlled through various methods such as changing the diameter of the pipe, using valves or pumps, or altering the properties of the fluid (e.g. temperature or viscosity). By controlling the flow rate, we can also control the pressure and velocity of the water.

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