How to calculate the change of heat between metal heater and air? I have a basic system, let's say a cube of air of size $1m \times 1m \times 1m$. I place a metal heater next to it, for simplicity its size will be $0.1m \times 0.1m \times 0.1m$ (so the contact area would be $0.1m \times 0.1m$). How can we calculate the change of air's temperature?
Some important rules:


*

*Heater's Work is a constant and known amount.

*Air's volume is $1m \times 1m \times 1m$.

*Contact area is $0.1m \times 0.1m$.

*Heat doesn't escape nor come any other way than from the heater.

*Heater is a metal, eg. alluminium. Air is an actual Earth air and not a perfect gas.



We can use the first law of thermodynamics to calculate heater's heat:
$$Q = ΔU + W, \quad \text{can we say} \quad \Delta U = 0?$$
For air temperature change we can use this formula:
$$\Delta t = Q/c_h m$$
where:


*

*$c_h$ - specific heat

*$m$ - air mass


Saying we know $c_h$ and $m$ we now need $Q$. How can we calculate the $Q$ knowing $Q$ of heater?
PS. (or $W$ if we don't know $Q$). I'm sorry for my incomprehension and any mistakes, as I am a beginner.
 A: A first approximation can be obtained by means Lumped Systems Analysis, with the following assumptions:


*

*heater operates at constant temperature $T_H$

*air temperature $T(t)$ increases over time but is uniform over the $1\ \mathrm{m} \times 1\ \mathrm{m} \times  1\ \mathrm{m}$ domain. There are no temperature gradients in the domain

*heat transfer from heater to air proceeds via convection and radiative transfer only


The convective heat ($Q_C$) flow from the heating element is given by Newton's cooling law:
$$\frac{\mathrm{d}Q_C}{\mathrm{d}t}=hA[T_H-T(t)]\tag{1}$$
where $h$ is the (convective) heat transfer coefficient, $A$ the surface area of the heater and $T(t)$ the air's temperature (as a function of time).
The radiative heat ($Q_R$) flow from the heating element is given by Stefan-Boltzmann Law:
$$\frac{\mathrm{d}Q_R}{\mathrm{d}t}=A\sigma\epsilon[T_H^4-T(t)^4]\tag{2}$$
The total heat ($Q$) flowing from the heating element is obtained by combining $(1)$ and $(2)$:
$$\frac{\mathrm{d}Q}{\mathrm{d}t}=hA[T_H-T(t)]+A\sigma\epsilon[T_H^4-T(t)^4]\tag{3}$$
Also:
$$\mathrm{d}Q=mc_v\mathrm{d}T(t)\tag{4}$$
where $c_v$ is the air's heat capacity (at constant volume) and $m$ is the total mass of air.
With $(4)$ into $(3)$:
$$mc_v\frac{\mathrm{d}T(t)}{\mathrm{d}t}=hA[T_H-T(t)]+A\sigma\epsilon[T_H^4-T(t)^4]\tag{5}$$
$(5)$ is a first order differential equation (DE), which is separable in the variables $T(t)$ and $t$. 
If $T_0$ is the initial temperature of the air (at $t=0$) then integrate the DE between $(0,T_0)$ and $(t,T(t))$ to get the temperature evolution in time, $T(t)$.
