NUMERICAL METHODS NEW ASSIGNMENT

(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.

H_R (consant) and H_S denote the water levels in R and S respectively.

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

The water level difference H(t) = H_S H_R 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 − Q_T )

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

where A_S 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.

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:

(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]