Mathematical modelling and simulation - provide a sketch of

Question

Question 1:

The Universal Flow Principle applies to all systems that involve the flow of mass or energy. Re-pose the following equations into the Universal Flow Principle and find the driving force and the equation for the flow resistance.

a. The flow of thermal energy from a hot vertical wall to the cool air in contact with it is given by the formula:

q = A (0.21k/L).[(βgρ²C_pL³)/μk]^0.4 (T_s - T_a )^1.4, where q is the heat transfer rate, A is the surface area, L is the vertical height of the surface, β is the bulk thermal expansivity modulus of the air, g is gravitational acceleration, T_a is the absolute temperature of the air, Ts is the absolute temperature of the surface, k is the thermal conductivity, C_P is the specific heat of the air, ρ is the density of air and μ is the viscosity of the air.

b. N = DA(C_i - C_o)/L where N is the mole diffusion rate, D is the diffusion coefficient, A is the cross sectional area, C_i is the molar concentration at the inside surface, C_o is the molar concentration at the outside surface and L is the thickness of the membrane.

Question 2:

A tank with a volume V is supplied with compressed air at a rate that is a complicated function of the tank pressure, m^·_i = f(P). At the same time pressurised air is withdrawn from the tank at a mass flow rate that depends on the exit line flow resistance, R = a(P - P_e)^b, where a and b are constants, P is the instantaneous tank pressure, and Pe is the instantaneous usable pressure. Let C represent the mass storage capacity of the tank.

Provide:

a. A sketch of the problem.

b. A sketch of the analogous flow network using a capacitor symbol to indicate tank mass storage.

c. Set up a model for simulating the gas pressure, P versus time t. Assume the perfect gas law, PV = mGT, where G is the universal gas constant holds.

Question 3:

A lightweight steel cable positions the printing head of an old computer printer. The printing head weighs W and rides on three horizontal sliders that provide a combined resistance of FS = Cv^a where C and a are sliding parameters, and v = dx/dt is the horizontal linear velocity of the head. The instantaneous horizontal forces exerted on the head by the cable are F_r and F_l in opposite directions.

a. Provide a sketch of the problem.

b. Draw a free body diagram of the problem

c. Assume consistent units throughout, set up an appropriate model to simulate head position x versus time, t.

Question 4:

A mathematical model has been described by an engineer into the following differential equation:

3x - dx/dt + t = 0

x(0) = 10

a. Demonstrate an Euler method simulation of x versus t with a tabular algorithm using Δt = 0.1 and 0.0 ≤ t ≤ 0.3.

b. Demonstrate a 4th-order Runge Kutta method simulation of x versus t with a tabular algorithm using Δt = 0.1 and 0.0 ≤ t ≤ 0.3.