+++
title = "Heat Transfer"
author = ["Dehaeze Thomas"]
draft = false
+++

Tags
:


## Conduction (diffusion) {#conduction--diffusion}

The _conduction_ corresponds to the heat transfer \\(P\\) (in watt) through molecular agitation within a material and is specified with:
\\[ P = \frac{\lambda \cdot A \cdot \Delta T}{L} \quad [W] \\]
with:

-   \\(\lambda\\) the thermal conductivity in \\([W/m \cdot K]\\)
-   \\(A\\) the surface area in \\([m^2]\\)
-   \\(\Delta T\\) the temperature difference in \\([K]\\)
-   \\(L\\) the length of the barrier in \\([m]\\)


## Convection {#convection}

The convection corresponds to the heat transfer through flow of a fluid.
It can be either _natural_ or _forced_.

The _forced convection_ \\(P\\) (in watt) can be described with:
\\[ P = h A (T\_0 - T\_f) \quad [W] \\]
with:

-   \\(h\\) the convection heat transfer coefficient in \\([W/m^2 \cdot K]\\).
    \\(h \approx 10.5 - v + 10\sqrt{v}\\) with \\(v\\) the velocity of the object through the fluid in \\([m/s]\\)
-   \\(A\\) the surface area in \\([m^2]\\)
-   \\(T\_0\\) the temperature of the object in \\([K]\\)
-   \\(T\_f\\) the temperature of the convecting fluid in \\([K]\\)

Note that clean-room air flow should be considered as forced convection.


## Radiation {#radiation}

_Radiation_ corresponds to the heat transfer \\(P\\) (in watt) through the emission of electromagnetic waves from the emitter to its surroundings is:
\\[ P = \epsilon \cdot \sigma \cdot A \cdot (T\_r^4 - T\_s^4) \\]
with:

-   \\(\epsilon\\) the emissivity which corresponds to the ability of a surface to emit energy through radiation relative to a black body surface at equal temperature.
    It is between 0 (no emissivity) and 1 (maximum emissivity)
-   \\(\sigma\\) the Stefan-Boltzmann constant: \\(\sigma = 5.67 \cdot 10^{-8} \\, \frac{W}{m^2 K^4}\\)
-   \\(A\\) the surface in \\([m^2]\\)
-   \\(T\_r\\) the temperature of the emitter in \\([K]\\)
-   \\(T\_s\\) the temperature of the surrounding in \\([K]\\)

The emissivity of materials highly depend on the surface finish (the more polished, the lower the emissivity).
Some examples are given in <tab:emissivity_examples>.

<a id="table--tab:emissivity-examples"></a>
<div class="table-caption">
  <span class="table-number"><a href="#table--tab:emissivity-examples">Table 1</a>:</span>
  Some examples of emissivity (specified at 25 degrees)
</div>

| Substance                  | Emissivity |
|----------------------------|------------|
| Silver (polished)          | 0.005      |
| Silver (oxidized)          | 0.04       |
| Stainless Steel (polished) | 0.02       |
| Aluminium (polished)       | 0.02       |
| Aluminium (oxidized)       | 0.2        |
| Aluminium (anodized)       | 0.9        |
| Copper (polished)          | 0.03       |
| Copper (oxidized)          | 0.87       |

<div class="exampl">

Let's take a polished aluminum plate (20 by 20 cm) at 125K (temperature of zero thermal expansion coefficient of silicon) surrounded by elements are 25 degrees (300 K):
\\[ P = \epsilon \cdot \sigma \cdot A \cdot (T\_r^4 - T\_s^4) = 0.36\\, J \\]

</div>


## Heat {#heat}

The _heat_ \\(Q\\) (in Joules) corresponds to the energy necessary to change the temperature of the mass with a certain material specific heat capacity:
\\[ Q = m \cdot c \cdot \Delta T \\]
with:

-   \\(m\\) the mass in \\([kg]\\)
-   \\(c\\) the specific heat capacity in \\([J/kg \cdot K]\\)
-   \\(\Delta T\\) the temperature different \\([K]\\)

<div class="exampl">

Let's compute the heat (i.e. energy) necessary to increase a 1kg granite by 1 degree.
The specific heat capacity of granite is \\(c = 790\\,[J/kg\cdot K]\\).
The required heat is then:
\\[ Q = m\cdot c \cdot \Delta T = 790 \\,J \\]

</div>

<a id="table--tab:specific-heat-capacity"></a>
<div class="table-caption">
  <span class="table-number"><a href="#table--tab:specific-heat-capacity">Table 2</a>:</span>
  Some examples of specific heat capacity
</div>

| Substance           | Specific heat capacity [J/kg.K] |
|---------------------|---------------------------------|
| Air                 | 1012                            |
| Aluminium           | 897                             |
| Copper              | 385                             |
| Granite             | 790                             |
| Steel               | 466                             |
| Water at 25 degrees | 4182                            |


## Heat flow {#heat-flow}

The heat flow \\(P\\) (in watt) is the derivative of the heat:
\\[ P = \cdot{Q} = \frac{dQ}{dt} = \frac{dT}{R\_T} = C\_T \cdot dT \\]
with:

-   \\(Q\\) the heat in [W]
-   \\(R\_T\\) the thermal resistance in \\([K/W]\\)
-   \\(C\_T\\) the thermal conductance in \\([W/K]\\)


## Bibliography {#bibliography}

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</div>