Heat Transfer Rate to Surroundings of Heating Battery

Question

I think this is probably a trivial question, but I am trying to understand how to do a thermodynamic calculation for heat transfer. I want to make a basic resistive heater by shorting a battery, but I want to do it in such a way that the power that goes into heating the battery will be dissipated into the surroundings at a rate such that no heat accumulates in the battery. I have no quantities of resistivity, voltage, or environment temperature, and I am just looking for a general process/equation to use for approximate convective heat transfer.

I think I know how to calculate the rate of heat accumulation in the battery. I do not, however, know how to go about finding the rate of heat transfer to the surroundings given:

Rate of Q_batt, Environment Temp, and any approximate constants for convection through air and the metal battery.

I would appreciate any help understanding that general process! Thank you.

Answer 1

Presumably the power dissipation $\dot{Q}$ inside the battery can be Ohmically modelled using only the current $I$ and the battery's internal resistance $R$.

So now we have two phenomena acting at once:

1. heat input due to Ohmic heating,
2. heat loss to the environment due to convective losses (radiative losses can be ignored at these low temperatures).

It's safe to assume that after some time a regime of steady state is achieved where all quantities are constant.

We can then thermodynamically model this in two ways.

1. use Fourier's Heat Equation but with a load term:

$$\frac{\partial T}{\partial t}=\alpha \nabla^2T+\dot{Q}$$

which in steady state reduces to:

$$\alpha \nabla^2T+\dot{Q}=0$$

Depending on geometry this can still be mathematically very demanding.

1. use Newton's law of cooling:

The law describes convection losses $\dot{Q_{con}}$ to the environment. In steady state we have:

$$\dot{Q_{con}}=\dot{Q}$$

where:

$$\dot{Q_{con}}=k A \left(T_{bat}-T_{env}\right)$$

where $k$ is the convection heat transfer coefficient, $A$ the surface area of the battery, $T_{bat}$ the battery's temperature (presumed uniform) and $T_{env}$ the environment's temperature.

From here $T_{bat}$ can then be estimated.