# Time dependent heating of a conductor

I have a conductor that's being heated with current. I want to know the temperature of the conductor

$$P_{\text{generated}}=W_{\text{conduction}}+W_{\text{convection}}+W_{radiation}$$

Where generated heat is equal to dissipated heat. I got those equations to have only $T_{\text{conductor}}$ to be unknown from $\Delta T=\left(T_{\text{conductor}}-T_{\text{ambient}}\right)$ and with that I got result for $T_{conductor}$ when it's in steady state.

My question is how do I get this all time dependent so I can see how long it would take the conductor to get to steady state and what is the temperature of conductor at any time?

Thanks

In general we could use Newton's cooling law for this problem.

An object with surface area $A$ and heat transfer coefficient $h$ will lose thermal energy acc.:

$$\frac{dQ}{dt}=-hA(T(t)-T_0),$$

with $\frac{dQ}{dt}$ the heat flux (loss, in $\mathrm{W}$), $T(t)$ temperature evolution of the object and $T_0$ the ambient temperature.

But you're also providing power as heat energy $P$.

The heat balance is thus:

$$P-hA(T(t)-T_0)$$

Assume now that the object has mass $m$ and specific heat capacity $C_p$, then for each infinitesimal increment in heat content $dQ$ there is an increment in temperature $dT(t)$:

$$dQ=mC_pdT(t)$$

Or divided by $dt$:

$$\frac{dQ}{dt}=mC_p\frac{dT(t)}{dt}$$

We can now create the following identity:

$$P-hA(T(t)-T_0)=mC_p\frac{dT(t)}{dt}$$

Which is a first order differential equation with separable variables.

Substitute: $u=P-hA(T-T_0)$, then $du=-hAdT$, so:

$$u=-\frac{mC_p}{hA}\frac{dT}{dt}$$

For ease of notation set $\alpha=\frac{hA}{mC_p}$, so:

$$u=-\frac{1}{\alpha}\frac{du}{dt}$$

Or $\ln{u}=-\alpha t + C$, with $C$ an integration constant.

The determined integral is obtained by substituting back and applying boundary conditions $0,T_0$ and $t,T$:

$$\ln[{\frac{P-hA(T-T_0)}{P}}]=-\alpha t$$

$$\frac{P-hA(T-T_0)}{P}=e^{-\alpha t}$$

$$\large{T=T_0+\frac{P}{hA}(1-e^{-\alpha t})}$$

Note that for $t \to \infty$, $T=T_0+\frac{P}{hA}$, which is the point where heat losses precisely match power input $P$.

No matter how you've calculated your value of $T_{conductor}$, this principle should be applicable to your problem.

This of course assumes the object is homogeneous in temperature, a reasonable assumption for small objects. For larger objects, temperature gradients $\frac{dT}{dr}$ would arise and Fourier's Law would have to be used, as basis for the derivation.

• With this, can i put put my current to vary with time, let's say it pulses, and see how much would my conductor would heat up and cool down between pulses? Commented Dec 8, 2015 at 15:35
• @SRC90: absolutely. During parts of the cycle where current is 'off', set $P=0$ and reintegrate the fourth equation (DE) with appropriate boundaries.
– Gert
Commented Dec 8, 2015 at 15:50