digital-brain/content/zettels/heat_transfer.md

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

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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

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 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.

Table 1: Some examples of emissivity (specified at 25 degrees)

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

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): \[ P = \epsilon \cdot \sigma \cdot A \cdot (T_r^4 - T_s^4) = 0.36\, J \]

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 \([K]\)

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: \[ Q = m\cdot c \cdot \Delta T = 790 \,J \]

Table 2: Some examples of specific heat capacity

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

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