Numerical Methods

Question 1:

A tank with a volume of 1000 litres contains dissolved salt at a concentration of 1.0 kg/litre. At a certain point in time (t=0 min), an input stream of the dissolved salt is fed to the tank at a rate of 50 L/min resulting in an equal outflow. The salt content of the input stream is variable and follows a cycle represented by:

C_salt (t) = 0.1 [ 2 + cos (tΠ/5) kg/L

Derive the ODE representing this system and solve for the concentration of the outlet stream as a function of time. Plot this until equilibrium is established.

Question 2:

A radiator heats a 5(L) x 4(W) x 7(H) metre room and delivers heat at a rate of kJ/s (kW) and a 100 W blower circulates the heated air throughout the room such that it can be assumed all the air in the room is at the same temperature. Heat losses from the room are dependent upon the indoor/outdoor temperature differential and can be estimated through a heat transfer coefficient:

Q_loss = hA (T - T_∞) where h = 0.75 W/m² K

If the initial temperature of the room is 5^oC determine how long it will take the air temperature to reach 20^oC if:

a) the outside temperature (T_∞) is 10^oC.
b) the outside temperature is -20^oC
c) what is happening for your answer in part b)?

Assume the air density and heat capacity are constant and 1.15 kg/m³ and 1005 J/kg K respectively.

Question 3:

A copper rod, 1 cm in diameter and 30 cm in length, is attached to a hot surface. The rod is well insulated except at the very end, which is submersed in a constant temperature bath.

a) Derive and solve the equation that describes the temperature profile in the rod. Assume the hot end is at a temperature, Thot, and the cold end is at Tcold. State all assumptions.

b) If Thot = 120^oC and Tcold = 5^oC, plot the temperature profile in the rod. The thermal conductivity, density and heat capacity of copper are: k = 401 W/mK, ρ = 8933 kg/m³ and Cp = 385 J/kgK respectively.

c) Now assume, that the exposed end of the rod is cooled by forced convection in air which is at 20^oC. The heat transfer coefficient is h = 50 W/m²K. Re-derive and solve the equation and plot the new temperature profile.

d) Now, assume that the rod is a tube connected to a tank of pure helium (partial pressure in tank is 1 atm, i.e. p_He = 1 atm). If the diffusion coefficient (He into air) is D = 7.2x10^-5 m²/s, derived and solve the equation for this case and plot the helium concentration profile along the tube.