Formulate a solution to the diffusion equation using an

Question

Part -1:

Objectives

1. Develop an appropriate approach to solve an engineering problem.

2. Apply numerical techniques to analyse a system represented by ordinary and/or partial differential equations.

3. Apply numerical techniques to analyse a non-linear system.

4. Develop a computer simulation program to assist in the analysis of an engineering problem.

5. Evaluate the solutions to an engineering problem using a general-purpose numerical/simulation software package.

Question One "Forearm Model"

Task Description

A simple biomechanical model of the forearm lifting a mass, M is shown in Figure 1 (Maroti, Berjes and Tolgyesi 1988). The model considers the rotation of the hand and forearm for small angles of rotation θ about the X-Y plane

By balancing the moments about the pivot point "O", the arm dynamics for small oscillations θ about the static-equilibrium position can be written as:

(M + m/3)l²θ^.. + K_ta²θ^. + k_baθ + (M + m/2)glcosθ = 0

where K_t is the constant for the triceps, and k_b is the constant for the biceps.

Consider the rotation of the hand in the X-Y plane described by the angle θ as shown in Figure 1 where the

• Mass of the object is M = 5 kg
• Mass of the forearm is m = 1.5 kg
• Length of the forearm is l = 0.27 m
• Force provided by the biceps is F_b N
• Force provided by the triceps is F_t N

• Spacing between the pivot point "O" and F_b (or F_t ) is a = 0.04 m

• Acceleration due to gravity is g = 9.81 m/s²

You are asked to consider the situation where the person is holding the weight and suddenly lowers their arm. At the time where the arm passes the angle θ = 0 radians (t = 0) the forearm is rotating at 0.1 rad/s downwards therefore θ^.(0) = -0.1 rad/s and θ(0) = 0 rad. Solve equation (1.1) to obtain the dynamics of the arm responses for the following cases:

• K_t = 1 x 10³ Ns/(m.rad) and kb = 2 x 10³ N/rad;

• Kt = 2 x10 Ns/(m.rad) and kb = 2 x 10 N/rad;

• Kt = 4 x10 Ns/(m.rad) and kb = 2 x10³ N/rad;

The force exerted by each muscle is calculated from:

F_b = -k_bθ

F_t = -k_tαθ^.

Develop a Matlab script(s) utilising the ode solvers in order to solve for the angle θ of the arm for the system described. Calculate and plot the angle θ and the force exerted by the bicep muscle ( Fb ) over 2 seconds for each of the three cases. Comment on the effect of different triceps muscle strengths on the oscillation of the mass and the peak force exerted by the muscles.

Part -2:

Objectives

1. Develop an appropriate approach to solve an engineering problem.
2. Apply numerical techniques to analyse a system represented by ordinary and/or partial differential equations.
3. Apply numerical techniques to analyse a non-linear system.
4. Develop a computer simulation program to assist in the analysis of an engineering problem.
5. Evaluate the solutions to an engineering problem using a general-purpose numerical/simulation software package.

Question Two "Hydraulic Press"

Background

A device has been constructed in order to apply compression to an object using a hydraulic press. The device as shown in Figure is controlled by sliding a control lever from left to right a distance of x metres. This actuation results in compression of the object by a distance of y metres.

The Task

You have been asked to predict the behaviour of this system using a computer model. In order to do this you have selected SIMULINK due to its ability to handle complex systems.

a) Develop a Simulink model which describes both the hydraulic press and control valve.

b) Use Simulink to model the load displacements for an activation force of F_v = 10 N.

c) Plot the movement of the control lever/control valve and determine how long it will take the operator to move this control valve assuming the activation force (F_v) is a constant 10 N.

d) Plot the following quantities over the time t = 0 to 100 seconds:

i. the load displacement, y

ii. the force exerted by the hydraulic press on the object being compressed

iii. the flow rate of hydraulic oil, q

e) From your results determine the maximum movement (y), maximum force applied by the hydraulic cylinder and peak oil flow rate.

f) Using the model, analyse the behaviour of the system if the seals inside the hydraulic piston were deteriorated to a point that the leakage coefficient increased to k_L = 50 x10^-12 m /(Pa.s)

Part -3: Question "Diffusion of Benzene"

The distribution and movement of a substance (solute) within another substance (solvent) is governed by advection and diffusion. For an example, consider a small amount of food dye (solute) added to a large tank of water (solvent). This dye will eventually spread evenly through the entire tank of water via a combination of these two mechanisms.

Advection describes the bulk movement of the solute which will in turn result in dispersion of the solute. Stirring the water would result in bulk movement of the water and rapid spread of the dye through the entire tank.

Diffusion instead refers to the slow spread of the solute through the substance without bulk movement of the solvent. This mechanism would dominate if the water in the tank was kept completely still and the dye left to slowly diffuse.

The Problem

A ship is docked in the Chicago Ship Canal, Figure. Suddenly the ship has started to leak benzene in the canal from a side pipeline. The leak maintains a concentration of benzene at 0.020 mg/L at the point of the leak. Due to the diffusion and advection process the benzene starts to spread in the canal in the direction of the water flow in the canal.

The equation that governs the diffusion of the benzene in the moving canal water can be described as follows:

∂c/∂t + u ∂c/∂x = D∂²c/ ∂x²

where:
c is the benzene concentration at a particular point in the canal water (mg/L)
u is the velocity of the water in the canal (m/s)
D is the diffusion coefficient (m²/s)
x is the distance measured from the side pipeline (m)
t is the time (sec)

The velocity of water in the canal is 0.5 m/s and the water diffusion coefficient is 3.0 m²/s. Assume that at x= 100 m far from the side pipeline (the source of the leak), the concentration c of benzene in the canal is zero.

The Task

Your task is to develop a numerical model using the finite difference approach to approximate the solution of the diffusion equation.

The specific tasks are as follows:
- Formulate a solution to the diffusion equation using an explicit method
i) Develop a Matlab script file implementing this explicit method
- Formulate a solution to the diffusion equation using an implicit method
i) Develop a Matlab script file implementing this implicit method
- Investigate the stability of both the explicit and implicit solution schemes
- Use both of the models above to:

i) Plot the concentration of the benzene after 5, 10 and 20 sec within the domain x = 0 -100 m.

ii) Compare the variation of the benzene concentration at x = 10, 20 and 30 metres form the source of the leak after 20 sec.

iii) Comment on the validity of the assumed boundary condition of c = 0 at x = 100 m.
- Discussion and review,

Some points to cover in addition to discussing your results:

i) Highlight the assumptions and limitations of your model.