Finding fluid flow with Numerical Method

In summary, the Fanning friction factor (f) is a dimensionless quantity dependent on the Reynolds number (Re) that describes friction in fluid flow in pipes. A formula, known as the von Karman equation, can be used to predict f for turbulent flow. For Reynolds numbers less than 2500, the flow is considered laminar and the Fanning friction factor can be calculated using the equation f = 16/Re. To determine the friction factor for Re = 10 000 within 5% relative error, the Newton-Raphson method can be used by first calculating g(f) = 1.74* ln ( Re* sqrt(f)) - 0.4 - 1
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Homework Statement


For fluid flow in pipes, friction is described by a dimensionless number, the Fanning friction factor (f). The Fanning friction factor is dependent on a number of parameters related to the size of the pipe and the fluid, which all can be represented by another dimensionless quantity, the Reynolds number (Re). A formula that predicts f given Re for turbulent flow is the von Karman equation,

1/sqrt(f) = 1.74* ln ( Re* sqrt(f)) - 0.4

Typical values for the Reynolds number for turbulent flow are 10 000 to 500 000 and for the Fanning friction factor are 0.001 to 0.01. For Reynolds numbers less than 2500 fluid flow type is called laminar flow and the Fanning friction factor is given by f = 16/Re. Use a numerical method to determine the friction factor for Re = 10 000 within 5% relative error.

Homework Equations





The Attempt at a Solution


our function is g(f) = 1.74* ln ( Re* sqrt(f)) - 0.4 - 1/sqrt(f)

how can i solve this problem?
 
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  • #2
You want to find f such that g(f) = 0. Looks like a good candidate for the Newton-Raphson method.

1. Start with an initial guess f0
2. Calculate g and g' = dg/df at f0
3. Update your estimate f1 = f0 - g/g'
4. Repeat steps 2-3 until the solution is obtained within the desired accuracy
 
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Related to Finding fluid flow with Numerical Method

1. How do numerical methods help in finding fluid flow?

Numerical methods use mathematical models and algorithms to solve complex equations that describe fluid flow. These methods can handle a large number of variables and provide accurate solutions for different boundary conditions.

2. What are the advantages of using numerical methods for fluid flow?

Numerical methods provide a cost-effective and time-efficient way to analyze fluid flow compared to traditional experimental methods. They allow for simulation of different scenarios and can handle complex geometries and boundary conditions.

3. What are the limitations of numerical methods for fluid flow?

Numerical methods rely on simplifications and assumptions in the mathematical models, which can lead to inaccuracies in the results. They also require a significant amount of computational resources and can be sensitive to input parameters.

4. How do you validate the results obtained from numerical methods for fluid flow?

Validation of numerical methods for fluid flow involves comparing the results with experimental data or analytical solutions for a given scenario. It is also important to perform sensitivity analyses and validate against other numerical methods.

5. Can numerical methods be used for all types of fluid flow?

Yes, numerical methods can be applied to different types of fluid flow, including laminar, turbulent, and multiphase flows. However, the accuracy and complexity of the models may vary depending on the type of flow and boundary conditions.

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