Greetings Marco, I have taken the liberty of combining your scripts into one Mathematica Manipulates. Basic initial conditions with ranges:

• Coffee 80 ?C
• 50 to100?C
• Milk 5?C
• 1?C to ambient
• Ambient /room temperature 20?C
• 0?C to 30?C
• Length of experiment in minutes 30
• 1 minute to 100 minutes
• Time to add milk 1 minute
• 1 minute to 100 minutes
• Volume of coffee 200ml
• 100ml to 500ml
• Volume of milk 20 ml
• 0 ml to 50ml

Code:

Manipulate[
 {
  sols = NDSolve[{
     D[Temp[t], t] == -0.1 (Temp[t] - ambient),
     Temp[0] == beginTemp,
     WhenEvent[
      t > addMilk,
      Temp[t] -> (coffee*Temp[t] + milk*milkTemp)/(coffee + milk)]},
    Temp,
    {t, 0, experimentTime}
    ];
  Plot[
   Temp[t] /. sols, {t, 0, experimentTime},
   PlotRange -> {{0, experimentTime}, {0, beginTemp + 10}},
   AxesLabel -> {"Time (minutes)", "Temperature  ?C"},
   LabelStyle -> Directive[Bold, Medium],
   ImageSize -> {{400, 600}}]
  },
 {{ambient, 20, "Ambient temperature"}, 0, 30, 0.5, 
  Appearance -> "Labeled"},
 {{experimentTime, 30, "Time of Experiement(minutes)"}, 1, 100, 0.5, Appearance -> "Labeled"},
 {{beginTemp, 80, "Coffee start temperature"}, 50, 100, 0.5, Appearance -> "Labeled"},
 {{milkTemp, 5, "Milk start temperature"}, 1, ambient, 0.5, Appearance -> "Labeled"},
 {{addMilk, 1, "Time to add milk(minutes)"}, 1, 100, 1, Appearance -> "Labeled"},
 {{coffee, 200, "Coffee volume (ml)"}, 100, 500, 10, Appearance -> "Labeled"},
 {{milk, 20, "Milk volume (ml)"}, 0, 50, 5, Appearance -> "Labeled"},
 ControlPlacement -> Left
 ]

Dear Marco,

very nice - thanks for sharing! And a good thing you supplied the measured data.

For me the most interesting question is: How well does Newtons "simple model" agree with reality anyway? So I extracted the exponential part of the "early milk data" and made a fit:

ClearAll["Global`*"]

data = Import[
   "<your path>  CoffeMilkGood.csv"];
dataclean = Table[Join[{data[[k]][[1 ;; 6]]}, data[[k]][[7 ;; 9]]], {k, 1, Length[data]}];
data2plotearlymilk = Select[Transpose[{dataclean[[All, 1]], dataclean[[All, 3]]/1000.}], #[[2]] > 0 &];

coolingData = {AbsoluteTime[#1], #2} & @@@ data2plotearlymilk[[32 ;;]];
t0 = AbsoluteTime[coolingData[[1, 1]]];
coolingData = Delete[{(#1 - t0)/3600, #2} & @@@ coolingData, 395];

model = First[temp[t] /. DSolve[{D[temp[t], t] == -\[Kappa] (temp[t] - tempEnv), temp[0] == temp0}, temp[t], t]]

params = FindFit[coolingData, model /. temp0 -> 54.812`, {\[Kappa], tempEnv}, t]

coffeeTemp[t_] = model /. params /. temp0 -> 54.812`

The agreement seems to be remarkably good:

I guess these differences are well within the accuracy of measurement. The only real discrepancy is concerning the temperature of the environment: You measured 25°C and from the fit one gets 27.94°C (quite hot anyway, we have winter after all). Can this be explained by some measurement issues - or is this actually showing the limitation of the model?

Best regards -- Henrik

Dear Marco,

thank you for your detailed response! Clearly the temperature for Aberdeen in winter is too high - as it is in Germany. At the moment I am at a conference in Austria (near Schladming), and it is much too warm as well - even without hot coffee. No promising perspectives for winter sport ...

I definitely do not want to initiate any extra work for anybody! Obviously I was a bit sleepy when I wrote my post: Of course the hot cup is warming up its immediate surrounding if no special measures against are taken. So I think the seemingly higher temperature in this surrounding does make sense here, and - as final result for me - we obviously see that Newtons approach works quite well.

Best regards -- Henrik

- Congratulations! This post is now a Staff Pick! Thank you for your wonderful contributions. Please, keep them coming!

We have also split your post into a separate discussion as it won our editorial selection. The original discussion still has a comment linking here. Thank you!