(1) Water Level Oscillations In A Surge Tank

The following figure depicts a reservoir R supplying a turbine T via a cylindrical pipe of length L and diameter D. A surge tank S is located just upstream of the turbine as shown.

 

 

HR (consant) and HS denote the water levels in R and S respectively.

QT (constant) and Q denote the volumetric flow rates in the turbine and pipe respectively.

The water level difference H(t) = HS HR and flow rate Q(t) are governed by the coupled first-order nonlinear ordinary differential equations (describing conservation of mass and Newton’s Second Law)

 

H˙ =  1  (Q  QT )

Q˙ =   g A (H c |Q| Q) (g = 9.81 m/s2)

 

where AS and A denote the cross-sectional areas of the surge tank and pipe respec- tively, and c is an empirical parameter used to model the local and friction head losses between R and S.

With the set of parameter values and initial (t = 0) conditions allocated from the table* (all in SI units) use Euler’s Method (with time-step h = 0.5s) to estimate the time taken for the water level in the surge tank to reach its maximum value.

 

 abcde
L100.0000100.0000100.0000100.0000100.0000
D2.25001.95002.75003.00002.5000
c0.01000.01000.02000.02000.0200
AS20.000020.000015.000025.000020.0000
QT10.000010.000010.000010.000010.0000
H(0)15.000010.000010.000012.000015.0000
Q(0)20.000015.000025.000030.000025.0000

 

 

Given that the system reaches an equilibrium state (i.e. stops changing as t ), write down general formulæ giving the equilibrium values

H = lim H , Q = lim Q .

 

t→∞

t→∞

[30 marks]

 

* The last digit of your student number determines which column of data values to use:

 

last digitcolumn
0 , 1a
2 , 3b
4 , 5c
6 , 7d
8 , 9e

 

(2) Solving Ax = b Using LU Decomposition

Given the matrix A and vector b allocated from the table below (use your student number as in part (1))

  • Find the LU decomposition ofA.

 

  • Usethis decomposition to solve the corresponding system of equations Ax b.

 

[20 marks]

 

 Ab
 

a

 2 0 2 

2 1   1 

4 1 4

 0 

1

0

 

b

 1   1 0 

1 1 2 

1 5 5

 1 

1

2

 

c

 1 1 1 

1 2   1 

2 1 6

 2 

3

3

 

d

 1 2   1 

2  3 0 

1 2 0

 3 

2

1

 

e

 1 0 1 

2 1 4 

0 1 1

 1 

5

1