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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.

YepNope
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  • You would also need to take into account the heat capacities of the boiler materials that get heated up in addition to the water contained in the boiler. – Bob D Nov 16 '21 at 22:24
  • You should study the answer to this question: https://physics.stackexchange.com/questions/677258/a-room-with-a-heater-in-contact-with-walls-as-a-heat-sink/677327#677327 It deals with the same type of problem: something is internally heated while losing convection heat to the env, (Newton's law of cooling) Also, don't used that wretched $k$ thingy. Use the full Law of cooling instead. – Gert Nov 16 '21 at 22:52
  • Cooling correction – Farcher Nov 16 '21 at 23:23

1 Answers1

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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.

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:

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

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

Gert
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