# Heat transfer in mechanical systems

## Engineering Fundamentals

#### Introduction

Hotspots in the construction such as actuators which dissipate heat affect e.g. the dimensional stability of the construction. In special environments such as vacuum and cryo, special care for heat abduction is needed. This sheet provides some insight in relation to this subject.

#### Conduction (diffusion)

Heat transfer through molecular agitation within a material and is specified with the typical thermal conduction.

$P=\frac{\lambda\cdot A\cdot\Delta T}{L}\left[W\right]$

$\lambda$
$A$
$\Delta T$
$L$

Thermal conductivity $\left[W/mK\right]$
Surface area $\left[m^2\right]$
Temperature difference $\left[K\right]$
Length of barrier $\left[m\right]$

#### Convection

Heat transfer through flow of a fluid. 2 Types: Natural and forced convection. Forced convection can be described with:

$P=hA\left(T_o-T_f\right)\left[W\right]$

$h$

$A$
$T_o$
$T_s$

Convection heat transfer coefficient$\left[W/m^2K\right]$
$h\approx10.5-v+10\sqrt v$ with $v\ \left[m/s\right]$ velocity of object trough fluid
Surface area $\left[m^2\right]$
Temperature of object $\left[K\right]$
Temperature of convecting fluid $\left[K\right]$

• In vacuum convection is negligible
• Cleanroom air flow must be considered as forced convection

Heat transfer through the emission of electromagnetic waves from the emitter to its surroundings.

$P=\varepsilon\cdot\sigma\cdot A\cdot\left({T_r}^4-{T_s}^4\right) \left[W\right]$

$\varepsilon$
$\sigma$
$A$
$T_r$
$T_s$

Emissivity $\left[-\right]$
Stefan-Boltzmann constant $\sigma=5,67\cdot{10}^{-8}\frac{W}{m^2K^4}$
Surface area $\left[m^2\right]$
Temperature of emitter $\left[K\right]$
Temperature of surrounding $\left[K\right]$

• Radiation increases with increasing temperature

#### Contact heat transfer

Critical in contact heat transfer is contact area dependent on clamp force, surface roughness, environment, cleanliness, humidity, etc. In other words, it cannot be calculated analytically but must be tested and results are based on statistics.

• ‘minimal/perfect contact area’ can be calculated with: $A=\frac{\sigma_{0.2}}{F}$

#### Heat

Energy necessary to change the temperature of a mass with certain material specific heat capacity:

$Q=m\cdot c\cdot∆T$

$m$
$c$
$\Delta T$

Mass $\left[kg\right]$
Specific heat capacity $\left[J/kgK\right]$
Temperature difference $\left[K\right]$

#### Heat flow

$P=\dot{Q}=\frac{dQ}{dt}=\frac{dT}{R_T}=C_T\cdot dT$

$P$
$Q$
$R_T$
$C_T$

Power, heat flow $\left[W\right]$
Heat $\left[J\right]$
Thermal resistance $\left[K/W\right]$
Thermal conductance $\left[W/K\right]$

#### Total thermal resistance

Analog to stiffness and electrical resistance:
In parallel: $R_t=\left(\sum_{1}^{n}\frac{1}{R_i}\right)^{-1}$
In series: $R_t=\sum_{1}^{n}R_i$

#### Emissivity

Emissivity is the ability of a surface to emit energy through radiation relative to a black surface at equal temperature. Maximum emissivity is $\varepsilon=1$ (the black surface) and no emissivity is: $\varepsilon=0$.

• Emissivity increases with increasing temperature
• Emissivity decreases with reflectiveness
MaterialTypical ε* [-]@ Temp [°C]
Platinum (polished) / Silver (polished)0.00525
Gold (highly polished)0.015100
Stainless steel (polished)0.0225
Aluminum (polished)0.0225
Copper (polished)0.0325
White ceramic (Al2O3)0.9093
Human skin0.9837
Quartz (glass)0.9021

*Just for indication please verify with other rescources before using

#### Dissipation

Irreversible heat transfer, typical the loss of power in an electrical resistor.

$P=I^2R=V^2/R\left[W\right]$

$I$
$V$
$R$

Current through resistor $\left[A\right]$
Voltage drop across resistor $\left[V\right]$
Electrical resistance $\left[\Omega\right]$

Special case: Piezo dissipation
$P=f\cdot C\cdot V^2\cdot lossfactor$

$f$
$C$
$V$
$lossfactor$

Operating frequency $\left[Hz\right]$
Piezo capacitance $\left[F\right]$
Voltage $\left[V\right]$
empirical: $\approx30\%$

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