# How do I calculate the heat of a metal near a heat source?

What are all the factors that go into this calculation?

And how do this calculation?

I'm looking error be within $$\pm 3^\circ \text{C}$$.

What I think the factors are :-

Referring to the picture:

1. Distance from heat source
2. Heat of heat source
3. Heat being given off by metal
4. Heat being asorbed by metal
5. Volume of metal
6. Heat capacity of metal(not shown)

Other notes :- I think I have to know 2 and 1 before I can know 4.

And I have to know 5 and 6 before I can know 3.

• When you say "the heat of a metal" do you mean the temperature? It's unlikely that you can get a very accurate result (unless the setup is such that the temperature is "not far from ambient" in which case the 3°C tolerance is not so hard to hit). Air flow and emissivity (of source and metal object) are two significant terms you omitted, – Floris Sep 20 '17 at 5:00

By "heat of a metal" I'll assume that you mean either heat absorbtion or temperature.

There exist three types of heat transfer:

$$\dot Q=-A\kappa\frac{\Delta T}{\Delta x}$$

where $\dot Q$ is heat flow rate, $A$ (contact) area, $\kappa$ thermal conductivity (a material property) and $\Delta x$ the depth/distance between the points of temperature difference $\Delta T$.

• (Thermal) convection when a fluid flow (such as air blowing past your skin) carries energy away/towards the object, given by:

$$\dot Q=hA(T-T_\infty)$$

where the object with exposed surface $A$ has temperature $T$ and $T_\infty$ is fluid temperature. The $h$ is a heat transfer coefficient that depends very much on the situation - it is looked up or a specific correlation in each specific case is used.

$$\dot Q=\varepsilon \sigma A T^4$$

where $\dot Q$ is heat flow rate, $\varepsilon$ emissivity of the object, $\sigma=5.67\times 10^{-8}\;\mathrm{\frac W{m^2K^4}}$ the Stefan–Boltzmann constant, $A$ area and $T$ temperature.

Every object emits this energy. Meaning, your metal looses energy constantly through radiation. Luckily, the surroundings also radiate, some of which hits the metal. The relationship taking surroundings into account can then be written:

$$\dot Q=\varepsilon \sigma A (T^4-T_\infty^4)$$

where $T_\infty$ is the temperature of the surroundings/environment.

So, unless you have a specific situation with specific relationships, these general relationships are what you need to know the heat flow rate. Keep in mind that heat losses can be hard to keep track of, unless you insulate your project. Heat flow rate $\dot Q$ is of course heat transferred per second, $\dot Q =Q/t$, so you can easily find the heat delivered/lost $Q$ by knowing for how long $t$ it goes on.

If you then need the temperature of the object, you may use this general relationship between how much heat is absorbed and how the temperature rises:

$$Q=cm\Delta T$$

where $m$ is mass, $c$ the specific heat capacity and $\Delta=T_{final}-T_{initial}$ the temperature change.