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CEG2002 NUMERICAL METHODS
ASSIGNMENT
Your work must be submitted (as a single pdf file) to CANVAS/ASSIGNMENTS by
This assignment contributes 10% of the final mark for the course.
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
R
L
Hs
S
T
От
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
Q
=
As
(Q-QT)
−9 (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.
CEG2002
1
2023-24 (1) With the set of parameter values and initial (t = 0) conditions allocated from
*
=
0.5s)
the table (all in SI units) use Euler's Method (with time-step h
to estimate the time taken for the water level in the surge tank to reach its
maximum value.
a
b
с
d
e
L
100.0000
100.0000
100.0000
100.000
100.0000
D
2.2500
1.9500
2.7500
3.0000
2.5000
с
0.0100 0.0200
0.0200 0.0200
As
20.0000
20.0000 15.0000
25.000 20.0000
10.0000
4000 01
QT 10.0000 10.0000
H(0) 15.0000 10.0000
Q(0)
20.0000
15.0000
0000 01
25.0000
12.0000
30.0000
10.0000
15.0000
25.0000
[50 marks]
=
(H, Q)
(2) The MATLAB script on page 3 solves the differential system for z(t)
using the ODE solver ode45 (4th -order Runge-Kutta scheme with adaptive time
stepping).
Save this script in an m file and check that you can get it to run. It should
plot graphs of H(t) and Q(t).
Change the parameter values (L, D etc.) to match those you used in part (1),
and compare your hand calculations and estimates with results obtained from
MATLAB. Don't expect to get exact agreement; ode45 is much more accurate.
Investigate how your system behaves as t → ∞, and how this behaviour de-
pends of the value of c. Can you predict the long time (t → ∞) values of H
and Q? What happens if c = 0? Can you give a physical explanation for this?
[50 marks]
* The last digit of your student number determines which column of data values to
use:
CEG 2002
last digit column
0,1
a
2.3
"
4, 5
6,7
"
b
C
d
89
e
2
2023-24 % surge tank model
% ode system integrated using ode45
function system-surge
clear
global g As A LC QT
g=9.81;
L=50.0;
D=1.0%;
c=0.01;
As=10.0;
QT=20.0;
HO=25;
Q0=10;
A=(pi*D^2)/4;
% pipe area
zO=[HO,QO];
tspan [0,100];
% initial values
% time interval
[t, z]=ode45 (@rhs, tspan, z0);
plot(t,z)
% ++
function f=rhs(t,z)
global g As A LC QT
f=zeros (2,1);
f(1)=(z (2)-QT)/As;
f(2)-((g*A)/L)*(z (1)+(c*z (2) *abs (z (2))));
return
CEG 2002
3
2023-24