Mathematics

ASSIGNMENT

ENGINEERING MATHEMATICS 1 (BMFG 1313)

The assignment should be done in group of 4 students.

Submission Date: 8th December 2018 (Dr. Fudhail)

QUESTION 1

In mechanical and aerospace engineering, one of the applications of determining fluid flow

through pipes is the flow of liquids and gases through cooling systems. The resistance to flow

in such conduits is parameterized by a dimensionless number called the friction factor. For

turbulent flow, the Colebrook equation provides a means to calculate the friction factor is

0 = 1

√𝑓 + 2.0 log (

𝜀

3.7𝐷 +

2.51

𝑅𝑒√𝑓 )

where 𝜀 is the roughness (𝑚), 𝐷 is diameter (𝑚) and 𝑅𝑒 is the Reynolds number given by

𝑅𝑒 = 𝜌𝑉𝐷

𝜇

where ρ is the fluid’s density [𝑘𝑔 𝑚−3], V is its velocity [𝑚 𝑠−1], and μ is dynamic viscosity [𝑁𝑠 𝑚−2]. By using Newton Raphson Method, determine the friction factor 𝑓 for air flow through a smooth, thin tube where the parameters are 𝜌 = 1.23 𝑘𝑔 𝑚−3, 𝜇 = 1.79 × 10−5 𝑁𝑠 𝑚−2, 𝐷 = 0.005 𝑚, 𝑉 = 40 𝑚 𝑠−1 and 𝜀 = 1.5 × 10−6 𝑚. Note that friction factors range from about 0.008 to 0.08. The initial guess by Swamee-Jain equation

approximation is

𝑓 = 1.325

[ln ( 𝜀

3.7𝐷 + 5.74 𝑅𝑒0.9

)] 2.

Note: Use Microsoft Excel for the iterative calculation given that the convergence criteria, 𝜀 = 5 × 10−7.

(10 marks)

QUESTION 2

a) The volume of water left has been recorded within time interval [0,6] min when the

water flow from a draining hemispherical reservoir. Complete the following table using

forward difference approximation of 𝑂(ℎ) for t = 0 and backward difference

approximation of 𝑂(ℎ) for t = 6 and centered difference approximation of 𝑂(ℎ2) for t =1,2,3,4 and 5.

Time, t

(minute)

0

1

2

3

4

5

6

Volume, V

(m3)

36

30

24

18

12

6

0

𝑑𝑉

𝑑𝑡

(4 marks)

b) The volume of water left has also been recorded for each water level (deep), y. Complete

the following table using forward difference approximation of 𝑂(ℎ) for y = 12 and

backward difference approximation of 𝑂(ℎ) for y = 0 and centered difference

approximation of 𝑂(ℎ2) for y=10,8,6,4 and 2.

Water level, y

(m)

12

10

8

6

4

2

0

Volume, V

(m3)

4072

2545

2078

1244

586

155

0

𝑑𝑉

𝑑𝑦

(4 marks)

c) Calculate the rate of water level changing when the water is 8 m deep using parts (a)

and (b).

(2 marks)

<Thank you>