Newton's Law of Cooling - ode45

Hello:

I am trying to write a code using a function to plot the ratio of time intervals between the following measurements:

T_o=100 →initial temperature in Celsius T_a=ambient temperature in Celsius T_1=80 →temperature in Celsius after cooling for 5 minutes T_2=65 →temperature in Celsius after cooling for 5 minutes t=5 →time left to cool M=25 →final temperature

My answer: T_a=20 degrees Celsius →temperature in the kitchen

I am trying to develop matlab code that can examine the ration R between the time intervals.

Using T(t)=80-15e^5k, I attempted to translate this into a function.

Defining the function:

function y = func5(T, k)
y = 80 - 15*exp(5k);
end

I have only been using matlab for a few weeks and any assistance would be greatly appreciated.

V/R jb

[JB]

Interesting. How would it look if I didn’t integrate and needed to use ode45?

[Rakib-Al- Hasan]

A person places $20,000 in a savings account which pays 5 percent interest per

annum, compounded continuously. Find

(a) the solution of the dollar balance in the account at any time

(b) the amount in the account after three years.

Answer 1

[Star Strider]

I am not certain what you want to do. You already have the integrated differential equation, so ode45 is likely unnecessary here.

This would be my approach:

% % % b(1) = T_a,  b(2) = k
Tfcn = @(b,t,T_0) (T_0 - b(1)).*exp(b(2).*t) + b(1);

tv = [  0;  5; 10];
Tv = [100; 80; 65];

p = fminsearch(@(b) norm(Tv - Tfcn(b,tv,Tv(1))), [10; 1]);      % Estimate Parameters

Final_T = Tfcn(p,15,Tv(1));                                     % Temperature @ 15 Minutes
Final_t = fzero(@(t) Tfcn(p,t,Tv(1)) - 25, 1);                  % Time To Reach 25°C

time = linspace(min(tv),Final_t);
Tfit = Tfcn(p,time,Tv(1));

figure
plot(tv, Tv, 'p')
hold on
plot(time, Tfit, '-r')
hold off
grid
xlabel('Time (min)')
ylabel('T (°C)')

[Star Strider]

To use ode45, it becomes a bit more involved, because you need to integrate the function within an objective function to use in your optimization routine (here, fminsearch). I put this entire code inside a test function I use for these purposes. You can do that, or save the ‘NLC’ function to a separate ‘.m’-file on your MATLAB search path. The file must be named ‘NLC.m’:

tv = [  0;  5; 10];
Tv = [100; 80; 65];

      function Y = NLC(b,t)
          Ta = b(1);
          K  = b(2);
          T0 = b(3);
          nlc = @(t,y,b) K.*(y-Ta);                               % Newton’s Law Of Cooling (ODE)
          [T,Y] = ode45(@(t,y)nlc(t,y,b), t, T0);              
      end

p = fminsearch(@(b) norm(Tv - NLC(b,tv)), [10; 1; Tv(1)]);      % Estimate Parameters

time = linspace(min(tv),50);                                    % Time Vector
Tfit = NLC(p,time);                                             % Calculate Temperatures
t25 = interp1(Tfit, time, 25);                                  % Time To T = 25°C

figure
plot(tv, Tv, 'p')
hold on
plot(time, Tfit, '-r')
plot(t25, 25, 'gp')
hold off
grid
xlabel('Time (min)')
ylabel('T (°C)')

fprintf('T_init   = %7.2f\nT_end    = %7.2f\nK        = %7.4f\nTime 25° = %7.2f\n\n', p([3 1 2]), t25)

The estimated parameters:

T_init   =  100.00
T_end    =   20.00
K        = -0.0575
Time 25° =   48.19

The plot looks similar, and the estimated parameters are the same.

For a more detailed exploration of this and similar applications, see: Monod kinetics and curve fitting (link), Solving Coupled Differential Equations (link), and Parameter Estimation for a System of Differential Equations (link).

[JB]

Thank for your help and professionalism. This will certainly help me understand Newton's Law of Cooling through the lens of Matlab functionality. I am beginning to really like Matlab.

[Star Strider]

As always, my pleasure!

With MATLAB, scientific computing is extremely efficient. The more you work with it, the more you will like it.

Answer 2

[Aj Barayang]

When an object with an initial temperature To is placed in a substance that has a temperature Ts, according to Newton’s law of cooling, in t minutes it will reach a temperaturuje T(t) using the formula where k is a constant value that depends on properties of the object. For an initial temperature of 100 C and k = 0.6, graphically display the resulting temperatures from 1 to 10 minutes for two different surrounding temperatures: 50 C and 20 C. Use the plot function to plot two different lines for these surrounding temperatures.