Guidance sought with numerical ODE solution

[groston]

The following image is from an old heat transfer book - my problem is quite similar. Note that for this discussion, only equation 4-6 need be considered.
[PLAIN]http://www.pairofdocs.net/images/eq.jpg

Consider the case when the liquid is being added to the tank at a constant rate, and without loss of generality, assume that the temperature of the liquid is higher than that of the container.

Equation 4.6 can be simplified for the purpose of my question to this:

$$K = -m\dot{T_{1}}$$

We know from physics that the temperature of the liquid will never be lower than the temperature of the container. However, if the value for m is sufficiently small, then $$\dot{T_{1}}$$ times the time step for the numerical integrator (dt) will yield a result such that:

$$T_{1}-\dot{T_{1}}dt<T_{2}$$

and this is clearly not possible. How can this happen? Quite simply by using a RK routine which automatically adjusts the step size. (And, needless to say, when this happens, the routine blows up.)

It occurred to me that one work-around is to not allow the value of m be reduced to less than some value, perhaps

$$m=\frac{K dt}{T_{1}-T_{2}}$$

Does this seem reasonable? Can you suggest another approach to solving this problem?

 

[jvicens]

Thank you for sharing your problem with us. It seems to me that your proposed work-around of setting a minimum value for m could be a valid solution. However, I would also suggest considering the physical implications of such a limitation. It is important to ensure that the chosen value for m still accurately reflects the heat transfer process in the system.

Another approach that you could consider is adjusting the time step (dt) in your numerical integrator. By decreasing the time step, you can ensure that the temperature difference between T1 and T2 remains within a reasonable range. This may be a more flexible solution as it allows for smaller changes in m, rather than imposing a fixed limitation.

Additionally, you could also explore using a different numerical integration method that is more stable for small time steps. This could help to prevent the routine from blowing up.

I hope these suggestions are helpful in solving your problem. Good luck with your research!
Scientist