0
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ 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. $\endgroup$
    – Bob D
    Commented Nov 16, 2021 at 22:24
  • $\begingroup$ You should study the answer to this question: physics.stackexchange.com/questions/677258/… 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. $\endgroup$
    – Gert
    Commented Nov 16, 2021 at 22:52
  • $\begingroup$ Cooling correction $\endgroup$
    – Farcher
    Commented Nov 16, 2021 at 23:23

1 Answer 1

1
$\begingroup$

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:

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

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

$\endgroup$
2
  • $\begingroup$ What if I don't know the power or if it's not constant? $\endgroup$
    – YepNope
    Commented Nov 17, 2021 at 8:45
  • $\begingroup$ If $P \neq \text{constant}$ but $P=f(t)$ then the ODE can still be solved, if $f(t)$ is reasonably well-behaved. $\endgroup$
    – Gert
    Commented Nov 17, 2021 at 16:26

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.