How to calculate the change of heat between metal heater and air?

Question

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.

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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.

Answer 1

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, $T(t)$.