Geyser Model Problem

This problem was suggested by a friend who claimed one could make money with it for applications. This is my effort it is a typical engineering problem.

As essentially the same ideas our solution may be much more refined and more accurate and as reliable method. I solve the highly simplified problem.

One will have to read my forum topic www )?  where I explain the engineering and problem formulation in detail and the following blog scans accompany this forum Blog Scans )?

The scans will follow soon.

Description of our Geyser Problem

Mathematical Model and Considerations

Specific internal heat (capacity) as energy is a physical property of the material (water now), internal heat is directly related to temperature as a linear function. In this case it means the equations in heat and in temperature are essentially the same, because T(t) = C.E(t) is the internal temperature of the geyser with C a constant at a given time t, and let,

E(t) = E(0) +  E1(t)   ( E2(t) + E3(t) )
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Where the total internal heat of the contents is E(t) and internal temperature T(t), the events described are,

E1 is the result of step functions controlled either manually, or automatically determined of a thermostat, depending on constant values switching the heating element switched on and off with minimum and maximum values of temperature.

E2 is from a discontinuity on the internal heat energy of the water, as manually replaced and depends on volume exchanged and internal temperature, difference between ambient and T, also manually or automatic.

E3 is of warmth leakage and dissipation as Newton's cooling law a linear function of the temperature difference, internal and ambient but ideally the geyser is insulated and as such then zero.

In our model the last term due to dissipation may hopefully probably well be ignored since the tank should be insulated and not losing heat.

Please note that if the ambient temperature is given and as a (known) function of time but not a constant, it affects the whole calculation and things become considerably more complicated.
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Note that the respective E1, E2 and E3 functions each also involve E as the content temperature T itself, the formula is not an explicit equation. It is important to realise this for the algebraic equation, that if so the Laplace Transform would have to be solved.

Obviously one cannot really assume the ambient temperature as constant, it usually would be much rather as variable and periodic in nature, and also each term also depends on T itself and not as manual but automatic, we have an implicit equation to solve the best way should then be the Laplace transform.

The transform of E , fs = L{ Et }is a algebraic equation that could be solved for fs, the function Et of t would be the inverse transform and our desired solution (very poor notation, see the diagrams).
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Note that with time there would be loss of information due to approximations and inaccuracy being present, and a regular built in self-correction is necessary, for applications there has to be a simple (such as manual re-set) mechanism for instance as e.g. completely re-filling and starting again as a time zero, taken as in a new initial state of equilibrium.

Calculations

If I take the ambient temperature as constant, as well as the other heat exchanges all as manual, the internal temperature is an explicit function i.e. a formula of only time t, and the answer a routine matter, however without the the given assumptions and approximations thus (it is important to realise in practice the terms all depend on T itself too), the equation for E we will have obtained is not explicit.

If we can solve for the Laplace Transform fs as of this algebraic equation, and then obtain the function Et as the inverse transform of f then E our desired solution, where

Ls { E(t) } = f(s)

The scans on my blog )? explain our model's constants and variables, terminology given labels and names, special functions and Laplace Transforms.

For the Laplace we need the transform of a constant, of E1 as step-functions, and E2 as a (Dirac-delta) distribution and for E3 the linear function. The formulas may be looked up in tables and will be given on the scans.

So that, as a routine matter we might solve the algebraic equation f(s) = f1(s) + f2(s) + f3(s)  for f(s) and apply the inverse transform to obtain the function E(t).

It is important to realise that in applications you will have to have some kind of  periodic re-setting or recovery mechanism.

The much simplified case and calculations are provided on the scan with result,

T(t) = C.E(t)  with  C = 1Mc

More details of the mathematics may be left to the reader if interested, it should really be straightforward at this stage.