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Cooling down a Pana cotta

by R. Grothmann

This is just a fun application. A Pana cotta (could be any other pudding) is cooling down. The measurements were

at 0 min : 66°
at 30 min : 53.5°
at 60 min : 44.6°

The measurements are somewhat inaccurate, since the Pana cotta was stirred, windows were opened etc. However, what is the room temperature, and when will the Pana cotta have reached 35°?

We use the following model for our cooling.

whera a is the temperature of the room.

>function f(t,a,b,c) := a+exp(b*t-c)

Enter the measurements.

>t=[0,30,60]; T=[66,53.5,44.6];
>plot2d(t,T,points=1); plot2d(t,T,add=1):

Now we want to interpolate the function. We use Nelder-Mead for this.

The following is the error.

>function delta (v) := totalmax(abs(T-f(t,v[1],v[2],v[3])))

It turns out, that we need a good starting point. So we use a guess for a, the room temperature, and fit b and c according to log(M-a) = bt-c.

>a=20; M=t'|-1; b=log(T-a)';
>w=fit(M,b); b=w[1], c=w[2]

-0.0104315825592
-3.82725856953

This does not quite interpolate.

>delta([a,b,c])

0.0927776426688

So we apply Nelder-Mead. The room temperature should be 22.5°.

>w=neldermin("delta",[a,b,c]), a=w[1]; b=w[2]; c=w[3];

[22.5972,  -0.0113226,  -3.77052]

Plot the result.

>plot2d("f(x,a,b,c)",0,240); plot2d(t,T,points=1,add=1):

Solve for the time. After 156 min the Pana cotta should have 30°.

>solve("f(x,a,b,c)",120,y=30)

156.207181302

The true room temperature was about 24°. Let us try to fix that with another measurement.

>t=t|85; T=T|40.2;
>w=neldermin("delta",[a,b,c]), a=w[1]; b=w[2]; c=w[3];

[28.2825,  -0.0138396,  -3.63589]

So we get another room temperature. Probably the method of measurement should have been more scientific.

>plot2d("f(x,a,b,c)",0,240); plot2d(t,T,points=1,add=1):

The predicted time is very sensitive to these errors, since it is close to the room temperature.

>solve("f(x,a,b,c)",120,y=30)

223.637373862

After 125 minutes, the predicted temperature is 34.6°.

>f(125,a,b,c)

35.0083063303

In reality, it was 35°. Let us add this to our data.

>t=t|125; T=T|35.0;
>w=neldermin("delta",[a,b,c]), a=w[1]; b=w[2]; c=w[3];

[28.4844,  -0.0139632,  -3.63019]

>plot2d("f(x,a,b,c)",0,240); plot2d(t,T,points=1,add=1):

>solve("f(x,a,b,c)",120,y=30)

230.201564176

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