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###### 日期：2020-11-25 10:45

ECMM171P Programming for Engineers

Assignment 1 - Friction Factors

Hand-in date: 18:00 Friday 27th November 2020

This assignment will count for 30% of your total grade in the course.

Remember that all work you submit should be your own work. This assignment is not

meant to be a team exercise. The University treats plagiarism very seriously.

Ensure that your program runs without any error on the Matlab versions R2019a and

later. Additionally, ensure that you pay careful attention to the instructions and marking

scheme below, particularly in regards to defining specific functions, commenting your code

You should submit your source code by 18:00 Friday 27th November 2020

If you have more general or administrative problems please e-mail me. Always include

the course number (ECMM171P) in the subject of your e-mail.

As a fluid flows through a pipe, it encounters resistance according to several factors, including

the diameter and wall roughness of the pipe, the velocity and viscosity of the fluid, and whether

the fluid is in a laminar or turbulent state. However, the relationship between these variables is

typically complex and nonlinear, meaning that we require computational techniques to predict

pressure losses – a requirement in a range of engineering design scenarios.

The purpose of this assignment is to create a predictive engineering tool to calculate the socalled

friction factor of a pipe flow, thereby providing an estimate of pressure loss per unit

length.

1 Theory

Consider a circular pipe of diameter D and length L, through which a fluid of density ρ is flowing

at a mean flow velocity U.

where the dimensionless quantity fD is referred to as the Darcy friction factor. The precise

value of the friction factor, however, is difficult to determine analytically for turbulent flows.

Its calculation depends on:

The Reynolds number

Re = ρUD

μ

where μ is the dynamic viscosity of the fluid. Re dictates whether the fluid is in a turbulent

or laminar state: if Re < 2040 the flow can be assumed to be laminar, or if Re ≥ 2040

we assume it is turbulent.

For turbulent flows, the pipe’s relative roughness, r = /D, where  is the pipe’s effective

roughness height.

1

1.1 Calculating fD

If the flow is laminar, then the relationship for the friction factor is straightforward and can be

derived from the Poiseuille (laminar) flow profile, giving,

fD =

64

Re

On the other hand, if the flow is turbulent, the relationship is more complex. In this case, fD

can only be found by solving the nonlinear Colebrook-White equation.

Clearly, this relationship cannot be solved analytically. However, by rearranging this equation,

we can see that it can instead be turned into a root-finding problem; i.e. find fD such that the

function C(fD) = 0,

This is where your program will come into play! Your task is to write a Newton-Rapshon solver

that will solve this root finding problem.

1. Write two functions called colebrook and colebrook deriv that evaluate C(fD) and the

derivative of C(fD) with respect to fD, respectively. These functions should include:

Inputs: the friction factor fD, the Reynolds number Re, and the relative roughness /D.

Outputs: the value of C(fd) and of the derivative of C(fD) respectively.

2. In order to find the root of a function f(x) using the Newton-Raphson algorithm, we

first require the function’s derivative f

0

(x) and an initial starting point x0,

which after enough iterations usually gives a good approximation to a root of f.

In this task, you should write a function called newton, which will perform NewtonRapshon

using your colebrook and colebrook deriv functions to solve for a root of the

C(fD) function.

In the Newton-Raphson algorithm you should check at each step whether the sequence

has converged to a limit, this can be done by calculating |xn ? xn?1| < tol, where tol

is your desired tolerance. You should also include a variable controlling the maximum

number of iterations in case the algorithm does not converge.

Inputs: initial value, tolerance, max iterations

Outputs: the root, number of iterations the algorithm run.

2

103 104 105 106 107 108 109

Reynolds number: Re

10-2

10-1

Friction factor fD

Moody chart for laminar/turbulent pipe flow

Laminar

Smooth

e/D=1e-2

e/D=1e-3

e/D=1e-4

e/D=1e-5

e/D=1e-6

Turbulent transition

Requested parameters

Figure 1: Moody diagram for a range of relative pipe roughness factors.

NOTE: Depending on how you solve the problem your newton function will likely require

some extra parameters which may be needed by the colebrook function. You are free to

3. Finally, write a script called moody which calls the Newton-Raphson function to find the

friction coefficients and plots a Moody diagram similar to the one shown in Figure 1.

Moody diagrams visualise the friction factor for laminar and turbulent regimes across

a range of different pipe roughness factors and Reynolds numbers, on a log-log scale.

Typically a specific value of /D is compared against a range of reference values. In

particular, your Moody diagram should show:

the laminar friction factor for 500 ≤ Re ≤ 2500;

the turbulent friction factors given by the Colebrook equation for 2×103 ≤ Re ≤ 108

for the reference values of /D = 0 (i.e. a smooth pipe), 10?6, 10?5, 10?4, 10?3 and10?2.

the laminar-turbulent transition point at Re = 2040.

an optional list of points, shown as markers on the graph, that are read by your

script.

Your script should read a text file (found on ELE) which will contain values of: Pipe

diameter D, Fluid velocity U in m·s

?1

, Dynamic viscosity μ in P a·s, Fluid density kg·m?3

,

and Pipe roughness  in mm, and then highlight in the Moody diagram the locations where

they correspond. Finally it should output in a new text file called pressure loss.txt

the friction factor and the pressure loss per unit length (?p/L) for the given parameters.

Your program should produce a Matlab figure, but also export it into a .png format.

Please format the figure in a similar way as in the example shown in Figure 1 including

labels, and markers for the desired points (as provided by the text file).

NOTE: In order to check that your programs are working correctly, you can use the

following parameters and compare the friction factor with your program outputs:

3

4 SUBMISSION

Pipe diameter [m]: 1

Fluid velocity [m/s]: 1

Fluid density [kg/m?3]: 1000

Fluid viscosity [Pa s]: 0.0010518

Pipe roughness [mm]: 3

Reynolds number is: 9.5075e+05

Calculated friction factor: 2.6273e-02

3 Marking scheme

Your submission will be assessed on the following criteria:

Fully working implementation of the colebrook and colebrook deriv functions.

[10%]

Fully working implementation of the newton method. [30%]

Fully working implementation of the moody script which calls the functions. [30%]

Correct implementation of the text input/output in your script and correctly formatted

graph. [15%]

Discretionary marks for good programming technique, structure, testing and commenting.

[15%]

Details regarding the specific criteria that are used in the above can be found in the module

handbook on ELE. You should attempt all of the above routines; incomplete solutions

will be given partial credit. Note that these routines will all be marked independently,

and you can still receive full credit for complete solutions even if the preceding parts are

not fully implemented.

4 Submission

You should compress all your Matlab files and your Moody diagram in a single .zip file,

and submit using the online Turnitin link on the ELE page.

4