IT1005 Introduction to Programming with MATLAB
Lab 7 – Nonlinear equations
The objective of this lab is to step you through important concepts in solving nonlinear
equations using MATLAB. We will do this by focusing on two specific applications in
different fields on study – urban planning and chemical engineering. The first problem
will require you to tackle a problem of estimating populations for urban planning. This
problem will get you going on the basics of root finding. In the second problem, we will
try to generate pressure-volume phase diagrams by solving the R-K equation of state over
a range of pressures and temperatures. The main objective here is to get you to use your
programming skills in conjunction with in-built MATLAB functions such as fsolve.
This lab is worth a total of 25 marks, Pr.1 = 10, Pr. 2 = 15
Released: Thursday, October 10, 2013 Deadline: Friday, October 18, 2013
Your solutions to this lab must be compiled into a word document, and submitted to
IVLE on or before the deadline, as usual.
NOTE: Those in Tuesday lab groups (4A, 5A+5B, 6A+6B, 3A+3B) will be affected
by the Hari Raya Haji public holiday on Tuesday 15 Oct 2013.
Affected students (6*18+28 ~ 140 students) have three options:
a. Attend any Monday (14 Oct) or Wednesday (16 Oct) lab groups.
b. Attend extra class (in case you cannot attend any of the Monday/Wednesday groups):
Wednesday 16 Oct 2013, 12-2pm @ I3-3-46 (with Liu Xinyan) or I3-3-47 (with Ong
Chee Chin).
c. Do Lab 7 on your own. We will not record attendance on Week 09.
Problem 1 Urban planning and population estimates
Accurate population estimates form the underlying quantitative basis for urban planning
and development efforts for any city such as Singapore. For example, transportation and
traffic engineers might find it necessary to determine separately the population growth
trends of the central areas with significant commercial activity and those of nearby
suburban areas (eg. Tanjong Pagar GRC versus Tampines GRC).
Let us consider one such case. The population of a certain urban area (Pu) is declining
with time according to the following empirical formula obtained from census estimates:
Pu (t) = Pu,max e −ku t + Pu,min
In contrast, the population of a connected suburban area (Ps) is growing according to the
following quantitative estimate:
€
Ps (t) =
Ps,max
⎡ Ps,max ⎤ −k t
1+ ⎢
−1⎥e s
⎣ P0
⎦
In the above formulas, Pu,max, Pu,min, ku, Ps,max, P0 and ks are empirically derived
parameters from census data and urban planning targets set by the city.
The parameter €
values are as follows: Pu,max = 75,000, Pu,min = 100,000, ku = 0.045/yr,
Ps,max = 300,000, P0 = 10,000 and ks = 0.08/yr.
(a) In this problem, you are required to determine the time (in years) and
corresponding values of the two populations when the suburban area is 20% larger
than the urban area.
You will first need to take the given two formulas, and formulate a single new equation
of the type f (x) = 0 in accordance with the objective given above. How do you do it?
What is ‘x’ here?
An appropriate solver to use would be ‘fsolve’. Experiment with several initial guesses.
What do you find? Briefly discuss your findings (2-3 sentences)
(b) You can also verify your answer graphically. How? Try doing this using one
MATLAB plot.
Problem 2 Phase diagrams and equations of state
We have briefly discussed in class how ‘equations of state’ can be used to describe the
pressure-volume-temperature behavior of a gas (you will see much more of this in year 2
thermodynamics). The Redlich-Kwong (R-K) equation of state, is a more sophisticated
description of such behavior than the ideal-gas law (Pv = RT).
The R-K equation of state is given by:
where R = universal gas constant [=0.518 kJ/(kg K)], T = absolute temperature (K), P =
absolute pressure, measured in kilo Pascals (kPa), and v = the volume of a kg of gas
(m3/kg).
The parameters a and b for methane (CH4) are calculated as follows:
where PC = ‘critical’ pressure of the gas = 4580 kPa, and TC = ‘critical’ temperature of the
gas = 191 K (More about this in CN1111!).
(Warm-up exercise) We can use the idea of ‘anonymous functions’ to solve the
above equation for v, given the values of T and P. Please refer to the posted
supplementary notes on anonymous functions, and make sure you understand this
nice ‘trick’.
Now, in this problem, you are required to generate a ‘phase diagram’ for methane. In
other words, you are required to generate a GRAPH of volume (v) versus pressure (P), at
a fixed temperature T.
(a) Use MATLAB to generate a graph of volume versus pressure. The graph is required
at T=400 K, and for pressures between 10,000 kPa and 250,000 kPa. Let there be a
spacing of 10,000 kPa between pressures on the graph.
[HINT] (i) You will have to use ‘fsolve’ inside a loop, and (ii) a useful initial guess for
all cases is v = 0.01 (I estimated this using the ideal gas law…).
(b) Repeat (a), but this time, plot two additional curves on the same graph, corresponding
to T = 600 K, and T = 800 K, for the same pressure range. Preferably plot the three curves
in different colors, so you know which one corresponds to what temperature. [HINT] (i)
An additional loop is required, and (ii) use initial guesses as above.