Question

Polynomial Curve Fitting

Often we can assume that certain thermodynamic properties are constant for the process under study. However if there is a large change in temperature during a thermodynamics process this assumption can lead to significant errors; a typical example would be combustion in an engine. In such cases tables can be used. However, tables are awkward to use for computer calculations; therefore, a polynomial equation is often used in computer programs that simulate thermodynamics processes.

As an example of how to apply polynomial equations in a computer program, we will consider the specific heat capacity at constant pressure, c_p, of carbon dioxide. The standard value used for CO₂ is the value at 300 K:

c_p = 0.846 kJ/kg·K

But for large temperature change the following polynomial is used

Cp/R = a1 + a2T + a3T^2 + a4T^3 + a5T^4

where the gas constant for CO₂ is R = 0.1889 kJ/kg·J.

The coefficients, a_i, depend on the type of gas and the temperature range. For CO₂, we have:

ai	300 ≤ T ≤ 1000	1000 < T ≤ 3000
a1	0.24007797e+01	0.44080e+01
a2	0.87850957e-02	0.309817e-02
a3	-0.6607878e-05	-0.123925e-05
a4	0.20021861e-08	0.227413e-09
a5	0.63274039e-15	-0.155259e-13

Programming:

You will write two MATLAB functions and a MATLAB script that will drive a menu:

CO2: Function c_p for CO₂: There will be one input to this function: a temperature in Kelvin. There will be one output from this function: the c_p for CO₂ based on the above polynomial at the specified temperature. So the function will accept one value for temperature and return one value for c_p. Do not use the element-by-element multiplication for array (.*) instead use the scalar multiplication (*).

CpRange: Function range of c_p: There will be three (3) inputs to this function: a minimum temperature, a maximum temperature and temperature increment—these will define an array of temperatures. This function will call CO2 to find each value of c_p for the given value of the temperature. The function will return the array of temperatures and specific heat capacities. Since the output is an array, what program structure should you use?

Script for the menu: This script will run the menu. When the script is run the user is give a choice of 3 items:

C_p for a given temperature

A plot of C_p vs. T.

Quit

The menu should run on an infinite loop and only terminate when item (3) is selected. If item (1) is select, the user will be prompted to enter a temperature and this will be sent to CO2; then T and c_p will be displayed with proper units in the MATLAB Command Window. If item (2) is selected the user will be prompted to enter a minimum and a maximum temperature and a temperature increment; this data will be sent to CpRange and the returned data will be used to produce a plot. You can call MATLAB’s plot function from the menu. Since this is a menu, this is a selection structure. For a menu selection structure it is usually most effective to use the switch/case structure.

Make sure you check the temperature data entered at the menu to ensure it is in the proper range. If the value(s) entered is outside of the range of validity (as specified in the table above), the user should be sent a warning message and the menu should run from the beginning again.

Answer 1

Answer:

code:

function cp=CO2(t)
R=.1889;
if t>=300 && t<=1000
cp=0.24007797e+01+0.87850957e-02*t-0.6607878e-05*t^2+0.20021861e-08*t^3+0.63274039e-15*t^4;
elseif t>1000 && t<=3000
cp=0.44080e+1+0.309817e-02*t-0.123925e-05^t^2+0.227413e-09*t^3-0.155259e-13+t^3;
end
cp=cp*R;
end

%%%%%%%%%%%%%%%%%%function CpRange%%%%%%%%%%%%%%%%%%%%%%

function [t,cp]=CpRange(tmin,tmax,inc)
inc=round(inc,3);%rounding to 3 decimal points
t=tmin:inc:tmax;
cp=zeros(size(t));
for i=1:length(cp)
cp(i)=CO2(t(i));
end
end

%%%%%%%%%%%%%%%SCRIPT%%%%%%%%%%%%%%%%%%%%%%%

i=1;
while i==1
fprintf('1. Cp for a given temperature\n2. A plot of Cp vs.T.\n3. Quit\n')
ch=input('Enter choice: ');
switch ch
case 1
t=input('Enter temperature value: ');
if t<300||t>3000
warning('Temperature must be between 300 to 3000 K!!')
  
else
cp=CO2(t);
fprintf('Temperature = %.2f K Cp = %.2f kJ/kg·K\n',t,cp)
end
case 2
tmin=input('Enter minimum temperature value: ');
if tmin<300||tmin>3000
warning('Temperature must be between 300 to 3000 K!!')
else
tmax=input('Enter maximum temperature value: ');
if tmax<300||tmax>3000
warning('Temperature must be between 300 to 3000 K!!')
else
inc=input('Enter temperature increment: ');
[t,cp]=CpRange(tmin,tmax,inc);
plot(t,cp);
xlabel('Temperature (K)')
ylabel('Specific Heat Capacity (kJ/kg·K)')
title('C_p vs t')
end
end
case 3
i=0;
end
end

OUTPUT

File d View nset Tools Desktop Window Help 1.02 ture must be betwees 300 3000 K 0 96 0 92 2-00 p 207620890-16 kg valaer 200 be betveen 30 3000 K 0 84 Temperature (K)