Calculating heat loss to environment while an object is being heated up using Newton's Law of Cooling

Question

I was experimenting with Newton's Law of Cooling lately, and I was wondering, how would one go about calculating heat loss to environment, while an object is being actively heated? Let's say I had a boiler, whose constant of cooling k I know. I measure the temperature at t=0 - it's equal to the ambient temperature (A). I then measure the temperature some time later (at time t), it's equal to T(t).

I know the mass of water, its specific heat, ambient temperature, final temperature, time passed, as well as the constant k. I'm trying to work out a concept so I'm completely disregarding radiated heat, air heating up etc.

I set up this differential equation, trying to find energy lost to environment:

dT/dt = m*c*dT/dt - k(T-A)

Where m*c*dT could be rewritten as dQ. But then I realized I messed up the units and I have no idea how to fix it (and I only know basic calculus).

Would appreciate some help.

Answer 1

For an object that is heated internally with a constant power $P$ and losing heat convectively to its environment, the heat balance in an infinitesimal time interval is:

$$\text{Heat generated = Heat retained + Heat lost}$$

or:

$$P\mathrm{d}t=Mc\mathrm{d}T(t)+hA\left[T(t)-T_A\right]\mathrm{d}t$$

Divide both sides by $\mathrm{d}t$ and make a substitution:

$$\Theta=T(t)-T_A$$

That gets:

$$P=Mc\frac{\mathrm{d}\Theta}{\mathrm{d}t}+hA\Theta$$ $$Mc\Theta'+hA\Theta=P$$ $$\Theta'+\alpha \Theta=\frac{P}{Mc}$$

This a 1st order ODE, separable. Solve it to find $T(t)$.

and I only know basic calculus

You can use Wolfram alpha (solution link) to solve the ODE.

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Another interesting point to make is, what is the maximum temperature $T_{\infty}$ the object will reach?

Well, at some point the power $P$ will equal the convection losses. The temperature can then no longer increase:

$$P=hA(T_{\infty}-T_A)$$

$$\Rightarrow T_{\infty}=T_A+\frac{P}{hA}$$