Background information
I am trying to model the thermal efficiency of a Cogeneration engine based on two Ordinary Differential Equations (ODE)s, based Newton’s law of cooling. These equations were originally developed by a group of scientists working for the international project (if interested, see COGEN-SIM!) The equations are as follows: $$ [MC]_{eng}\frac{dT_{eng}}{dt} = K_{eng} ({T_{eng}-T_{cw,o}})+K_{loss} ({T_{eng}-T_{amb}}) $$ $$ [MC]_{cw}\frac{dT_{cw,o}}{dt} = [mc_{p}]_{cw} ({T_{cw,i}-T_{cw,o}})+K_{eng} ({T_{eng}-T_{cw,o}}) $$
Description of the equations
First equation: the thermal energy transferred from the engine ($eng$) to the the ambient ($amb$)
Second equation: the thermal energy transferred from the engine ($eng$) to cooling water ($cw$)
Problem statement
In their case, the models were developed and calibrated based on the experiments which followed the modelling part of the project. The error and $R^2$ were calculated and finally the model was validated.
Unlike them (them rich boys of science), I am not a part of international consortium and the research group that I am working does not have fund or connections to buy a Cogeneration engine to conduct experiments.
Problem: I do not know the exact solution - I don't have any measurements of $T_{eng}$ or $T_{cw,o}$.
Aim: To solve the mentioned equations without knowing the exact solution.
Questions
Considering the case above, does the stated aim make sense? In other words, is it reasonable (scientifically/mathematically sound) to solve ODE without knowing the exact solution? If yes, how am I going to understand that the solution has converged? What kind of assumptions should I make?
I am planning to use RK4 from the Runga-Kutte methods family to solve the aforementioned equations. RK4 requires the boundary and step-size as inputs to estimate the solution. Now I am not quite sure how the boundary and the step-size contributes to the solution: The result of my model is consisted of the electricity and thermal output of the cogeneration unit for every minute of the day (1440 minutes/day). Therefore, for every minute I need to calculate $T_{eng}$ and $T_{cw,o}$. Let's assume that I have step-size of 1 ($h=1$) and my boundary limit is 20 ($x_{final}=20$).
2.1. Is the following statement correct?: "The solver will estimate the $T_{eng}$ and $T_{cw,o}$ after 20 iterations for every minute of the day".
2.2. What are the parameters to be considered when the boundary limit and the step size are being chosen?