In general, heat transfer describes the flow of heat (thermal energy) due to temperature differences and the subsequent temperature distribution and changes.
The study of transport phenomena concerns the exchange of momentum, energy and mass in the form of conduction, convection, and radiation. These processes can be described via mathematical formulas.
The fundamentals for these formulas are found in the laws for conservation of momentum, energy and mass in combination with constitutive laws, relations that describe not only the conservation but also the flux of quantities involved in these phenomena. For that purpose, differential equations are used to describe the mentioned laws and constitutive relations in the best way possible. Solving these equations is an effective way to investigate systems and predict their behavior.
Without external help, heat will always flow from hot objects to cold ones which is a direct consequence of the second law of thermodynamics.
We call that heat flow. In the early nineteenth century, scientists believed that all bodies contained an invisible fluid called caloric(a massless fluid thought to flow from hot to cold objects). Caloric was assigned properties, some of which proved to be inconsistent with nature (for instance it had weight and it could not be created nor destroyed). But its most important feature was that it was able to flow from hot bodies into cold ones. That was a very useful way to think about heat.
Thompson and Joule showed that this theory of the caloric was wrong. Heat is not a substance as supposed, but a motion at the molecular level (so called kinetic theory). A good example is rubbing our hands against each other. Both hands get warmer, even though initially they were at the same cooler temperatures. Now if the cause of the heat was a fluid, then it would have flowed from a (hotter) body with more energy to another with less energy (colder). Instead, the hands are heated because the kinetic energy of motion (rubbing) has been converted to heat in a process called “friction”5
${}^{5}$.
The flow of heat is happening all the time from any physical entity to objects surrounding it. Heat flows constantly from your body to the air surrounding you. Small buoyancydriven (or convective) motion of the air will continue in a room because the walls can never be perfectly isothermal as in theory. The only domain free from heat flow would have to be isothermal and completely isolated from any other system allowing heat transfer.
The cooling of the sun are the primary processes that we experience naturally. Other processes are the conductive cooling of Earth’s center and the radiative cooling of other stars1
${}^{1}$.
Heat Transfer is transmission of thermal energy due to a gradient in temperature.
Fourier’s law: Joseph Fourier (see Figure 3) published his book “Théorie Analytique de la Chaleur” in 1822.
In this book he formulated a very complete theory of heat conduction. He stated the empirical law that bears his name: the heat flux (indicated with q
$q$– a heat rate per unit area and can be expressed as QA)
$\frac{Q}{A})$, q(W/m2)
$q(W/{m}^{2})$, resulting from thermal conduction is proportional to the magnitude of the temperature gradient. If we name the constant of proportionality, k
$k$, that means
q=−kdTdx(1)
$$\begin{array}{}\text{(1)}& q=k\frac{dT}{dx}\end{array}$$
The constant, k
$k$, is called the thermal conductivity with the dimensions Wm∗K
$\frac{W}{m\ast K}$, or Jm∗s∗K
$\frac{J}{m\ast s\ast K}$.
Please keep in mind that the heat flux is a vector quantity! Equation (1) tells us that, if temperature decreases with x
$x$, q
$q$will be positive – it will flow in positive x
$x$direction. If T
$T$increases with x
$x$, q
$q$will be negative; it will flow in negative x
$x$direction. In either case, q
$q$will flow from higher temperatures to lower temperatures as already mentioned in the introductory paragraph. Equation (1) is the onedimensional formulation of Fourier’s law. The threedimensional equivalent form is:
q→=−k∇T
$$\overrightarrow{q}=k\mathrm{\nabla}T$$
where ∇
$\mathrm{\nabla}$indicates the so called gradient.
In onedimensional heat conduction problems, there is no problem to tell in which way the heat flows. For that reason, it is often convenient to write Fourier’s law in simple scalar form:
q=kΔTL(2)
$$\begin{array}{}\text{(2)}& q=k\frac{\mathrm{\Delta}T}{L}\end{array}$$
where L
$L$is the thickness in the direction of heat flow and q
$q$and ΔT
$\mathrm{\Delta}T$are both written as positive quantities. We just have to keep in mind that q
$q$always flows from high to low temperature1
${}^{1}$.
The thermal conductivity of gases can be understood with the imagination of molecules. These molecules move through thermal movement from one position to another position as can be seen in the picture below.
The internal energy of the molecules is transferred by impact with other molecules. Areas with low temperature will be occupied by molecules of high temperature and areas with high temperature will be occupied by molecules of lower temperature. The thermal conductivity can be explained with this imagination and be derived with the kinetic theory of gases:
T=23KNkB
$$T=\frac{2}{3}\frac{K}{N{k}_{B}}$$
which states that “the average molecular kinetic energy is proportional to the ideal gas law’s absolute temperature”6
${}^{6}$. For an ideal gas the thermal conductivity is independent from the pressure and increases with the root of the temperature.
This theory is pretty hard to understand for objects other than metals. And for fluids it is even more difficult because there is no simple theory. In nonmetallic components, heat transfers via lattice vibrations (Phonon). The thermal conductivity transferred by phonons also exists in metals but surpassed by the conductivity of the electron gas.
The low thermal conductivity of insulating materials like polystyrene or glass wool is based on the principal of the low thermal conductivity of air (or any other gases).
Material  Thermal conductivity W/(m.K)
$W/(m.K)$


Oxygen  0.023 
Steam  0.0248 
Polystyrene  0.0320.050 
Water  0.5562 
Glass  0.76 
Concrete  2.1 
Steel highalloyed  15 
Steel unalloyed  4858 
Iron  80.2 
Copper pure  401 
Diamond  2300 
Table 1: Thermal conductivity of different materials
Analogous definitions
Heat Transfer: Heat flux density ∝
$\propto $grad T (Thermal conductivity)
Diffusion: Partial current density ∝
$\propto $grad x (Diffusion coefficient)
Electric lead: Current density ∝
$\propto $grad Uel
${U}_{el}$(Electric conductivity)
Radiation describes the phenomenon of transmission of energy from one body to another by propagation through a medium. All bodies constantly emit energy by electromagnetic radiation. The intensity of such energy flux depends not only on the temperature of the body but also on the surface characteristics. If you sit in front of a camp fire, most of the heat that reaches you is radiant energy. Very often, emission of energy, or radiant heat transfer, from cooler bodies can be neglected in comparison to convection and conduction. Heat transfer processes happening at high temperature, or with conduction or convection suppressed by evacuated insulation, involve a significant fraction of radiation in general1
${}^{1}$.
The electromagnetic (EM) spectrum: This spectrum is the range of all types of electromagnetic radiation. Simply put, radiation is energy travelling and spreading out like photons being emitted by a lamp or radio waves. Other well known types of electromagnetic radiation are XRays, gammarays, microwaves, infrared light etc7
${}^{7}$.
Electromagnetic radiation can be seen as a stream of photons, each traveling in a wavelike pattern, moving at the speed of light and carrying energy. The difference between the different forms in the electromagnetic spectrum is the energy of the photons. It is important to keep in mind that if we talk about the energy of a photon, the behavior can either be that of a wave or that of a particle which we call the “waveparticle duality” of light.
Each quantum of radiant energy has a wavelength, λ
$\lambda $and a frequency, ν
$\nu $, associated with it. The relation between energy, wavelength, λ
$\lambda $and frequency, ν
$\nu $, can be written as wavelength equals the speed of light divided by the frequency, or
λ=cν
$$\lambda =\frac{c}{\nu}$$
and energy equals Planck’s constant times the frequency, or
E=h∗ν
$$E=h\ast \nu $$
where h
$h$is Planck’s constant (6,626070040∗10−34Js)
$(6,626070040\ast {10}^{34}Js)$.
The table below shows various forms over a range of wavelengths. Thermal radiation is from 0.11000 μm
$\mu m$.
Characterization  Wavelength 

Gamma rays  0.3 100 pm
$pm$

Xrays  0.0130 nm
$nm$

Ultraviolet light  3400 nm
$nm$

Visible light  0.40.7 μm
$\mu m$

Near infrared radiation  0.730 μm
$\mu m$

Far infrared radiation  301000 μm
$\mu m$

Microwaves  10300 mm
$mm$

Shortwave radio & TV  300 mm
$mm$
100 m $m$

Table 2: Electromagnetic wave spectrum
If radiation meets a body or a fluid it will be:
A body itself can also emit radiation
Q˙=QA˙+QT˙+QR˙
$$\dot{Q}=\dot{{Q}_{A}}+\dot{{Q}_{T}}+\dot{{Q}_{R}}$$
1=QA˙Q˙+QT˙Q˙+QR˙Q˙
$$1=\frac{\dot{{Q}_{A}}}{\dot{Q}}+\frac{\dot{{Q}_{T}}}{\dot{Q}}+\frac{\dot{{Q}_{R}}}{\dot{Q}}$$
1=αS+τS+ρS
$$1={\alpha}^{S}+{\tau}^{S}+{\rho}^{S}$$
where
αS:Absorptance
$${\alpha}^{S}:\text{Absorptance}$$
τS:Transmittance
$${\tau}^{S}:\text{Transmittance}$$
ρS:Reflectance
$${\rho}^{S}:\text{Reflectance}$$
Black Body:
$\phantom{\rule{1em}{0ex}}$αS=1
${\alpha}^{S}=1$$\phantom{\rule{1em}{0ex}}$
ρS=0
${\rho}^{S}=0$$\phantom{\rule{1em}{0ex}}$
τS=0
${\tau}^{S}=0$
Gray Body:
$\phantom{\rule{1em}{0ex}}$αS,ρS
${\alpha}^{S},{\rho}^{S}$and τS
${\tau}^{S}$uniform for all wavelengths.
White Body:
$\phantom{\rule{1em}{0ex}}$αS=0
${\alpha}^{S}=0$$\phantom{\rule{1em}{0ex}}$
ρS=1
${\rho}^{S}=1$$\phantom{\rule{1em}{0ex}}$
τS=0
${\tau}^{S}=0$
Opaque Body:
$\phantom{\rule{1em}{0ex}}$αS+ρS=1
${\alpha}^{S}+{\rho}^{S}=1$$\phantom{\rule{1em}{0ex}}$
τS=0
${\tau}^{S}=0$
Transparent Body:
$\phantom{\rule{1em}{0ex}}$αS=0
${\alpha}^{S}=0$$\phantom{\rule{1em}{0ex}}$
ρS=0
${\rho}^{S}=0$$\phantom{\rule{1em}{0ex}}$
τS=1
${\tau}^{S}=1$
.
PA=σT4
$$\frac{P}{A}=\sigma {T}^{4}$$
where σ
$\sigma $is the StefanBoltzmann constant which can be derived from other constants of nature:
σ=2π5k415c2h3=5.670373∗10−8Wm−2K−4
$$\sigma =\frac{2{\pi}^{5}{k}^{4}}{15{c}^{2}{h}^{3}}=5.670373\ast {10}^{8}\phantom{\rule{1em}{0ex}}W{m}^{2}{K}^{4}$$
For hot objects other than ideal radiators, the law is expressed in the form:
PA=eσT4
$$\frac{P}{A}=e\sigma {T}^{4}$$
where e
$e$is the emissivity of the object (e
$e$= 1 for ideal radiator). If the hot object is radiating energy to its colder surroundings at temperature Tc
${T}_{c}$, the net radiation loss rate takes the form:
P=eσA(T4−T4c)
$$P=e\sigma A({T}^{4}{T}_{c}^{4})$$
Due to the fourth power of the temperatures in the governing equation, radiation becomes a very complex, highlevel nonlinear phenomenon2
${}^{2}$.
Consider a convective cooling situation. Cold gas flows past a warm body as shown in the figure below.
The fluid forms immediately adjacent to the body a thin sloweddown region called a boundary layer. Heat is conducted into this layer, which vanishes and mixes into the stream. We call this process of carrying heat away by a moving fluid convection.
Isaac Newton (1701) considered the convective process and suggested a simple formula for the cooling:
dTbodydt∝Tbody−T∞
$$\frac{d{T}_{body}}{dt}\propto {T}_{body}{T}_{\mathrm{\infty}}$$
where T∞
${T}_{\mathrm{\infty}}$is the temperature of the oncoming fluid. This expression proposes that energy is flowing away from the body1
${}^{1}$.
The steadystate form of Newton’s Law of cooling defining free convection is described by the following formula:
Q=h(Tbody−T∞)
$$Q=h({T}_{body}{T}_{\mathrm{\infty}})$$
where h
$h$is the heat transfer coefficient. This coefficient can be denoted with a bar h¯¯¯
$\overline{h}$which indicates the average over the surface of the body. h
$h$without a bar it denotes the “local” values of the coefficient.
Depending on how the fluid motion is initiated, we can classify convection as natural (free) or forced convection. Natural convection is caused for instance by buoyancy effects (warm fluid rises and cold fluid falls). In the other case, forced convection causes the fluid to move by external means such as a fan, wind, coolant, pump, suction devices, etc.
Forced convection: The movement of a solid component into a fluid can also be considered as forced convection. Natural convection can create a noticeable temperature difference in a house or flat. We recognize this because certain parts of the house are warmer than others. Forced convection creates a more uniform temperature distribution and therefore comfortable feeling throughout the entire home. This reduces cold spots in the house, reducing the need to crank the thermostat to a higher temperature3
${}^{3}$.
Structural Heat Transfer Analysis is used when:
Coupled Heat Transfer Analysis (FluidSolid) used when:
Category  Structural Analysis (linear static)  Heat Transfer Analysis (steady state) 

Material properties  Young’s modulus(E)  Thermal conductivity(k) 
Laws  Hook’s law σ=E⋅dudx
$\sigma =E\cdot \frac{du}{dx}$

Fourier law q=−k⋅dTdx
$q=k\cdot \frac{dT}{dx}$

Degree of Freedom (DOF)  Displacement (u)  Temperature (T) 
Gradient of DOF  Stain ϵ
$\u03f5$
Stress σ $\sigma $

Temperature gradient (∇T)
$(\mathrm{\nabla}T)$

Similarities  Axial force per unit length: Q Crosssectional area: A Young’s modulus: E  Internal heat generation per unit length: Q Crosssectional area: A Thermal conductivity: k 
Table 3: Heat Transfer Analysis compared to Structural Analysis
Heat Transfer takes the energy balance of the studied systems into account. When investigating thermomechanical components, structural deformations, caused by the effects of thermal loads on solids can also be included. Simulating the stress response to thermal loads and failure is essential for many industrial applications. An example for an application is a thermal stress analysis of a Printed Circuit Board.
Conjugate Heat Transfer (CHT) simulations analyze the coupled heat transfer in fluids and solids. The prediction of the fluid flow while simultaneously analyzing the heat transfer that takes place within the fluid/solid boundary is an important feature of CHT simulations. One of the areas in which it can be used is for electronics cooling.
In theory, heat passes from a hot to a cold object. Conduction is the heat transfer from a hot to a cold object, that are in direct contact to each other. The thermal conductivity of the different objects decides how much heat in which time is being transferred. Examples include CFL light bulbs.
Convective Heat Transfer is the transfer of heat between two areas without physical contact. Convective currents occur when molecules absorb heat and start moving. As you can imagine, these effects are difficult to predict which is why high computing power is needed to obtain reliable results from a simulation. One application is the cooling of a Raspberry pi mother board.
Electromagnetic waves are the source of heat transfer through radiation. They usually play a role at high temperatures. The amount of heat that is emitted via radiation depends on the surface type of the material. A general rule is that the more surface there is, the higher the radiation is. An example application where simulation of radiation is used, is the simulation of laser beam welding.
Many materials and products have temperaturedependent characteristics which makes analyzing the impact of heat and ensuring thermal management of structures and fluids crucial in product development. The Heat Transfer Module of SimScale’s online simulation platform allows you to predict the airflow, temperature distribution and heat transfer. This involves convection, conduction and radiation to ensure the performance, endurance and energy efficiency of your designs.
1
${}^{1}$: http://web.mit.edu/lienhard/www/ahtt.html, opened April 2017
2
${}^{2}$: http://hyperphysics.phyastr.gsu.edu/hbase/thermo/stefan.html, opened April 2017
3
${}^{3}$: http://energyeducation.ca/encyclopedia/Forced_convection, opened April 2017
4
${}^{4}$: http://scienceworld.wolfram.com/physics/Blackbody.html, opened April 2017
5
${}^{5}$: https://www.thermalfluidscentral.org/encyclopedia/index.php/Heat, opened April 2017
6
${}^{6}$: https://en.wikipedia.org/wiki/Kinetic_theory_of_gases, opened April 2017
7
${}^{7}$: https://imagine.gsfc.nasa.gov/science/toolbox/emspectrum1.html, opened April 2017