# Simultaneous heating and cooling

fishspawned
I am having trouble wrapping my head around a particular problem I am thinking about.

Consider that I have a heating source that I can describe in terms of either degrees Celcius / second or Joules / second, but essentially I am able to determine a fixed heating rate.

Let's say:
dT/dt = 5 degrees / second

If I have an object in a room with a specific ambient temperature, it would also be experience simultaneous cooling, which is determined [i assume] through Newton's Law of Cooling, namely:

T = (To - Troom)e-kt + Troom

Where k is a constant and Troom is the ambient temperature, also fixed.
The issue is how to combine these two things into one equation because with Newton's Law of Cooling, the assumption is that you start from a fixed initial temperature, To. But if it is being heated, then To is going to be changing as it gets warmed up, assuming the heating rate is larger than the cooling rate.

So I am hoping someone can offer some help in how this can be reconciled in some way - or is this simply regulated to hands on testing of the object to see what happens in the real world?

Mentor
The issue is how to combine these two things into one equation because with Newton's Law of Cooling, the assumption is that you start from a fixed initial temperature, To. But if it is being heated, then To is going to be changing as it gets warmed up, assuming the heating rate is larger than the cooling rate.
Write down the rate at which heat energy is transferred from the source to the object and from the object to the room at time t as a function of the temperatures at time t. That will give you a differential equation that can be solved for the temperature as a function of time.

(That Newton's Law of Cooling you cite is itself the result of solving that differential equation, for the special case in which no heat is being added to the object).

fishspawned
Nugatory,
So, if I have what you are saying right, I am starting with

dT/dt = -kT(t) + kTroom + c

where k is a constant for cooling and c is a constant for the heating rate [as it is not affected by the temperature difference]. This seems to show that the rate of cooling is proportional to the difference between the temp and the ambient temp at any given time.

which I think can be rewritten as

dT = [-kT(t) + kTroom + c]dt

OR
since i am really looking for how long this takes to get to a specific temperature...

dt = dT/[-kT(t) + kTroom + c]

so am i able to integrate from some starting temperature to a final desired temperature and then get how long it would take?

Mentor
Nugatory,
So, if I have what you are saying right, I am starting with

dT/dt = -kT(t) + kTroom + c

where k is a constant for cooling and c is a constant for the heating rate [as it is not affected by the temperature difference]. This seems to show that the rate of cooling is proportional to the difference between the temp and the ambient temp at any given time.

which I think can be rewritten as

dT = [-kT(t) + kTroom + c]dt

OR
since i am really looking for how long this takes to get to a specific temperature...

dt = dT/[-kT(t) + kTroom + c]

so am i able to integrate from some starting temperature to a final desired temperature and then get how long it would take?
Sure, as long as you know a value for k and as long as the desired temperature is not higher than the final steady state temperature.