Solving Fluid Flow in Pipe: Mass Rate 4Kg/s, P1 700Pa, P2 600Pa

[Classic49]

1. I am trying to help my grandson with his homework and I have to determine the flow of a fluid in a pipe. I am told that mass flow rate is 4Kg/s; density p is 1012Kg/m³; Diameters D1 = 300mm, D2 = 400mm; Pressure P1 = 700Pa, P2 = 600Pa; Heights y1=0mm, y2=100mm

2. I have applied Bernoulli's Equation, Continuity Equation and Mass Flow Rate but get different results so am not sure where I am going wrong.

3. Putting values into the Bernoulli equation, we have:

700 + (1012 x 9.81 x 0) + (½ x 1012 x V1²) = 600 + (1012 x 9.81 x 0.1) + (½ x 1012 x V2²)

This simplifies to:

700 + 0 + (506 x V1²) = 600 + 992.772 +(506 x V2²), or
506 x V1² = 600 + 992.772 - 700 +(506 x V2²) = 892.772 +(506 x V2²)I need to determine V1 and V2 so I tried using the Continuity equation to establish a relationship between the two. From the gives diameters, Areas are A1 = 0.07065 m² , A2 = 0.1256 m²

Therefore V1 = (V2 x A2)/A1 =( V2 x 0.1256)/0.07065 = 1.78 x V2

Putting the value of V1 into the Bernoulli equation above, I get
506 x (1.78 x V2)² = 892.772 +(506 x V2²)
This gives me V2 = 0.902 m/s
From the Continuity Equation, V1 = 1.78 x V2 = 1.6 m/s

4. The piece of information I have not used is the Mass Flow Rate. If I use the information in the equation
Mass Flow Rate = Density (p) x Area (A1) x Velocity (V1), I get V1 = 4/(1012 x 0.07065) = 0.0559 and
V2 = 4/(1012 x 0.1256) = 0.0315 which are widely different from the values calculated above.

5. I would be grateful for some guidance as to where I am going wrong.

 

[SteamKing]

If you are trying to determine the mass flow rate of the fluid, that is given as 4 kg / s.

If you are trying to determine the velocity of the flow in the pipe, that can be worked out from the continuity equation, knowing the mass flow rate of the fluid (4 kg / s), the diameters of the pipe (300 mm and 400 mm), and the density of the fluid (1012 kg / m³).

The pressure info is a red herring. The fluid can be treated as incompressible. If you plug the values of flow velocity obtained from the Bernoulli equation into the continuity equation, you'll see that a much higher mass flow rate than 4 kg / s is obtained.

 

[Classic49]

Thank you for your reply. As I said, I am trying to calculate the flow into (V1) and out of (V2) the pipe.

I guess I am missing something in my understanding. I am able to plug all information into the Bernoulli equation except the 2 items I need to find - V1 and V2. The Continuity Equation gives me the relationship between V1 and V2 which I then feed into the Bernoulli Equation to give me values for V1 and V2 as shown in my workings in the first post at para 3.

What I cannot understand is why the Mass Flow Rate calculation gives me a different answer from the combination of the Bernoulli and Continuity calculations (my point 4). With regard to your 2nd paragraph, I do not understand how the mass flow rate and density tie into the Continuity Equation which I thought was simply equating the products of velocity and area anywhere in the pipe.

 

[Chestermiller]

Please provide the exact problem statement, and not your interpretation of it.

Chet

 

[SteamKing]

Thank you for your reply. As I said, I am trying to calculate the flow into (V1) and out of (V2) the pipe.

I guess I am missing something in my understanding. I am able to plug all information into the Bernoulli equation except the 2 items I need to find - V1 and V2. The Continuity Equation gives me the relationship between V1 and V2 which I then feed into the Bernoulli Equation to give me values for V1 and V2 as shown in my workings in the first post at para 3.

What I cannot understand is why the Mass Flow Rate calculation gives me a different answer from the combination of the Bernoulli and Continuity calculations (my point 4). With regard to your 2nd paragraph, I do not understand how the mass flow rate and density tie into the Continuity Equation which I thought was simply equating the products of velocity and area anywhere in the pipe.

The mass flow of the liquid in the pipe is what it is: 4 kg/s. This is not going to change even if the diameter of the pipe changes. Knowing the density of the fluid flowing allows you to calculate the volume of fluid flowing in the pipe in m³ / s, and knowing the diameter of the pipe allows you to calculate the velocity of flow which is consistent with the rate of mass flow, given the diameter of the pipe.

If you have a given constant quantity of liquid flowing into a pipe of a certain diameter of pipe, then the velocity of the liquid must increase if the diameter of the pipe decreases, just as the velocity of the liquid must decrease if the diameter of the pipe increases, in order to maintain a constant mass flow.

The continuity equation is simply a mathematical statement that the mass going into the pipe must be the same mass coming out, i.e., no mass is created or destroyed while it flows within the pipe.