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Background:

I am doing experiments trying to map the practical solubility of a certain chemical species called an "activator" solution. This solution is essentially a sequential mixture of water, $NaOH$ pellets and aqueous sodium silicate. In short (without getting into too much background), my results will help me determine if I can use certain chemical species in some very environmentally friendly materials science applications.

One of the parameters to keep track of is the temperature of solution which changes the solubility of the activator. I need to model this as I've found identical solutions respond differently when the final sodium silicate ingredient is added to the aqueous $NaOH$ solution at different temperatures. Here is some experimental data from the lab showing T vs t:

enter image description here

Essentially, there are 2 main "periods" during the experiment:

  1. The Initial Temperature Rise (i.e. from $T_i$ = 23.4 $^{\circ} C$ = ambient temperature $\rightarrow$ $T_{Max}$ = 97 $^{\circ} C$).
  2. The Cooldown Period (i.e. from $T_{Max} \rightarrow T_i$)

Here is a picture of the experimental setup showing 2 x ~200mL of activator in a conical flask. Both activators were made the same initially but on the left, the sodium silicate was added at ambient temperatures (which caused precipitation) as opposed to the right, where the sodium silicate was added at around 80% of $T_{Max}$.

enter image description here

Problem:

I'm wanting to model the T vs t for the system (similar to the experimental data above) so I can see the temperature profile of different activators with different chemical ratios. It seems, fundamentally, to start with an energy balance with:

$$\frac{dQ}{dt} = Q_{in} - Q_{out} + Q_{rxn}$$

Assuming no heat input, a known value for $Q_{rxn}$ (from a previous question of mine) and $Q_{out} = Q_{Conduction,Plate} + Q_{Convection,surroundings}$ I'm wondering the following:

  1. How should the system be modelled? is it good to model $Q_{out}$ with the heat equation and solve for T (as per pic below)? On that, how do I even solve for t? I have minimal knowledge of Fourier transforms.

enter image description here

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  • $\begingroup$ Apologies, I have contextualised $Q_{out} = Q_{Conduction,Plate} + Q_{Convection,surroundings}$ in an edit to address your second comment. Do you mean separating each section into approximated fitted functions and then adding the functions together? I am not sure how long it would take to get to $T_{Max}$ despite my model for the value itself. I also want a predictive model for other solutions I could make, I don't believe fitting functions to one experimental dataset will be the same for the others. $\endgroup$
    – Hendrix13
    Nov 24, 2022 at 8:02
  • $\begingroup$ I deleted my previous comments since I think they are not so helpful. I have just two questions: (1) you mix NaOH and silicate solution. The ensuing reaction is very fast on the timescale of measurement. That means heat production is like a pulse. Do you agree with this description? (2) Is the hot plate at the same T as the air? $\endgroup$
    – Buck Thorn ♦
    Nov 24, 2022 at 8:33
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    $\begingroup$ If (2) is true, and you assume surroundings are at constant T, then I don't think you need to separate conduction and convection losses etc. Just use Newton's law and keep it simple: en.wikipedia.org/wiki/Newton%27s_law_of_cooling $\endgroup$
    – Buck Thorn ♦
    Nov 24, 2022 at 8:35
  • $\begingroup$ (1) Sure for this situation, I wonder if a linear equation instead would be more fitting for solutions with larger volumes instead of a pulse though? On (2), that's great thank you! Although I'm having trouble validating the model using the equation as I can't seem to solve for the right r value properly (i.e with an example (x,y) = (t, T) = (t_max, T_Max) = (105, 97) with $T_{env} = 23.4$ and $T_{Max} = T(0) = 97$. When I try I get 0.0274 which makes it decay too quickly. Any help here? $\endgroup$
    – Hendrix13
    Nov 24, 2022 at 9:27
  • $\begingroup$ Because, as a bonus I'd like to (at least somewhat) accurately predict when the system cools to ambient temperature again. $\endgroup$
    – Hendrix13
    Nov 24, 2022 at 9:33

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