CMPUT 340 Introduction to Numerical Methods Department of Computing Science University of Alberta

Lab 2: Systems of Linear Equations & Assignment 2

Introductory exercises. 
We will follow up the lecture material with a set of simple Matlab exercises. This will involve working with matrix representation of systems of linear equations, writing loops, using matrix inversion, Gaussian elimination and other techniques for solving systems of linear equations. (Example 1, Example 2, Example 3)
 
Recommended Matlab background: "Loops" section from the UNH Matlab tutorial. (excluding the O.D.E. part)

Part II.

Exercise 1: Solve problem 2.3 from handout (Heath, p. 100, a copy will be given to you in class).

Assignment: Heat flow balance problem. 
This assignment is a practical application of the systems of linear equations for solving the heat flow balance problem. The general setting of the task is as follows. In order to keep a house warm, we use additional heating (especially during the winter!). This extra warmth, provided by the indoors heat sources (furnace, electrical appliances, etc.) escapes the house, even when all of the doors and windows are firmly shut, due to a heat transfer mechanism called "conduction". The state of heat flow balance is reached when the amount of heat generated by the indoors heat sources gets equal to the amount of heat that leaves a house due to conduction through walls, windows, etc.
 
In order to complete this assignment you will need to:

1. Formulate the heat flow balance equations for a simple 2-room house given the heat conduction properties of the materials it is made of (e.g., the heat conductivity of glass windows, wooden doors, etc.)
2. Solve for the furnace heat needed to balance-heat all rooms evenly at temperature T₀
3. Solve for the steady-state temperature in the rooms of the house given the power of the furnace, P_F
4. Perform similar calculations for a real house with a larger number of rooms (find a layout on the internet, be creative and make up one, or just use the place where you live!).

Notes: 

1. The amount of heat Q lost due to conduction can be calculated using a simplified version of the Fourier equation: where is the difference between outside and inside temperature, and k is a thermal conductivity of a given material via a given area. The following diagram illustrates the process:
    
   
2. The plan of a 2-room house is depicted on the diagram below. where T₁ and T₂ are the temperatures in room 1 and 2, T₃ the outside temperature, P₁ and P₂ are the indoors heat sources (such as furnace, heaters, appliances, etc.).
    
   
3. The steady state of heat flow balance is reached when the heat generated by the indoors heat sources is equal to the heat lost due to conduction: (these equations written for each room would constitute the system of linear equations you'll need to solve).
    
   Hint: count all home appliances as heat generating devices (e.g. fridge 200W, lamp 100W, TV 100..200W, etc.)
    
   
4. The thermal conductivity k is to be computed as follows: where A is the area of a component (a wall, door, window), R - insulative value (e.g., see table 1), and c_u - units of measurement correction coefficient.
    
   
5. Don't worry if the above formulae don't make much sense as of now - we'll do a simple example in during the lab.

Submission details:

When to submit: by Feb 11th, 11:00am (updated on Feb. 7)
What to submit: For Part II - the log of matlab commands or a script for soving the system (see "diary" command) with an explanation of your approach (optional). For Part III - a short (2-4 pages) report of your solution of the heat balance problem.
You are expected to hand in both electronic and hard copy versions of your work. Please put a hard copy in the drop-off box while submit your electronic version as "try c340 a2 [assignment_file.tar]". An example of the contents of the assignment file is given below:

• problem-2.3.txt - your Matlab solution of problem 2.3 from the handout
• house.m/house.log - the Matlab script or log of your commands that you used to evaluate and solve the systems of linear equations for the heat flow balance problem.
• report.ps/report.doc - write-up for your solution to the heat flow balance problem.

Updates and additional information:

1. Heat flow calculation example.
2. Thermal conductivity parameters of some other materials.
3. Online and downloadable units of measurement converters: Online converter , Downloadable converter (Win32, eval. version), yet another online converter.
4. Approximate Wattage parameters of various home appliances: table 2