Consider a container with a base of L(m) by L(m) and a top rim of L(m) by W(m) as depicted in Figure 1. The container is H(m) high and can be assumed rigid. Initially the container is empty, after which it is being filled with 2 different fluids at rates f1(m3/s) and f2(m3/s), respectively. There is a circular whole of radius R(m) at the base through which the mixed fluids can flow out driven by gravity without any friction. The fluids both have the same constant density, ̑(kg/m3).
Figure 1: Filling/Draining of a container
(a) Find an expression that describes the volume of fluid inside the container, V(m), as a function of the fluid height, h(m). [5 marks]
(b) Derive an ODE describing the time evolution of fluid level, h, of the form
dh/dt + ̑1∑h = ̑2
Determine constants ̑1 and ̑2. [5 marks]
(c) Solving this ODE using an Implicit Euler approach results in the following non-linear equation:
̒1∑hn+1 + ̒2hn+1 = ̒3
Determine constants ̒1, ̒2 and ̒3. [5 marks]
(d) Derive an equation in the form ĥi+1 = F(ĥi), when the non-linear equation in c) is solved using Fixed Point Iterations, where F a function of ĥi (Note: ĥ = hn+1). [5 marks]
(e) Derive an equation in the form ĥi+1 = G(ĥi), when the non-linear equation in c) is solved using the Newton Raphson method, where G a function of ĥi (Note: ĥ = hn+1). [5 marks]