The main heat transfer mechanism in the automotive engineering is convection. It is based in three types, forced, free and mixed convections. An important point in vehicle aerodynamics is the thermal boundary layer, because its knowledge allows to the engineer where to put the aerodynamic devices. The Rayleigh-Bernard convection is a parametric flow used in wind tunnels (WT), turbulent channels and turbulent layers. It is a simple flow that contains the essential physics of the phenomena.

Heat transfer mechanism: Conduction

The first heat transfer physical process is the conduction. This is characterized by the heat flow along a solid body. The heat is transferred from a body which is at a higher temperature, consequently high energy, to a body with lower temperature. Hence, the conduction heat transfer equation describes what really matters to this heat transfer method.

q = – λ∙ΔT

Where the minus signal is present, because the heat transfer occurs from the most energetic body to the lower one. The energy conducted is rather proportional to the temperature delta. λ is a proportionality coefficient. The law defined for conduction is the Fourier law and it states that:

q = – λ∙∇T

Is the opposite, λ is the thermal conductivity, which allows to quantify the heat exchange in terms of the temperature gradient. Another efficient approach to account the effects of the conduction is using the internal energy equation, which is now given by:

ρ∙c∙(∂T/∂t) = ∇∙(λ∇T) + q_g“‘

Where c is the heat capacity, q_g“‘ is the source sink/volume, which is an additional term due to internal sources of heat, also called sink.

Heat transfer mechanism: Radiation

The radiation is not the main heat mechanism in the automotive industry, because it usually has an effect when the thermal gradient is very high. It is considered a local heat transfer, which there are particles that transfer the energy, or it can occur in the vacuum, which is described by a non-iterative process. The heat emission is described by the Stefan-Boltzman law, which is described by:

E_n = σT⁴ [W/m²]

Where σ is the Stefan-Boltzman constant, which is given by:

σ = 5.670∙10^-8 [W/(m²∙K⁴)]

An example of radiation heat transfer is the situation of heat soak. For example, when a car after hours of operation is suddenly stopped.

Heat transfer mechanism: Convection

It is the main heat transfer mechanism for automotive applications. However, this is not a physical mechanism, because the heat is transferred due to the flux or flow of particles, that transmits heat to other ones along the flow. Actually, in convection what makes the energy transfer is the conduction and/or radiation of the particles inside the flow. Hence, the convection is so important for automotive design, because a flow acts as an intermediate mean for the heat transfer. Basically, the convection heat transfer can occur by three approaches, the forced, the natural and the mixed convection.

The forced convection mechanism is dependent of an external agent, which is more influent than the temperature delta. For instance, it could be a car that is moving in the environment. The natural convection is more connected to the temperature difference and its influence in the density fluctuations, these motivates some flow, because the high density part goes down, while the low density particles tend to go up. This is called buoyancy, the motions are caused by the temperature field through the variation in density. Hence, the motion is driven by buoyancy, and this is driven by the temperature variation. The third category is called mixed convection, it combines buoyancy effects and external agents. For instance, a car which is moving, the flow motivated by the car also experiments an external agent, engine and disc temperature. Therefore, the forced convection usually is the flow in a boundary layer, which is around the car. If the car, the engine and the wheels are hot, the boundary layer around them creates a convection that transfer the heat from these parts to the wake. Hence, the flow and the motion is due to the fact that wheels are rotating and warming. The convection heat transfer is described by:

q = h∙ΔT

Which is called Newton’s law of cooling. Actually, this is more a modeling than a law, because convection mechanism itself is not a physic principle, it uses this to model the heat transfer as proportional to the temperature difference, given by ΔT. In addition, a proportionality coefficient is given by h, also called heat transfer coefficient. This parameter is important for engineering applications since it describes the efficiency of the heat transfer. Supposing that a hot wall with a specified temperature is immersed in a free stream with a determined temperature, thus ΔT will be given by the difference between the hot wall and free stream temperature. Hence, the flux is parametrized in function of ΔT, while h defines how hard is the heat transfer. It is important that, since convection is not a physical property, the heat transfer coefficient is not a property of the mechanism itself. Actually, the heat transfer coefficient is a property of the fluid thermophysical properties and the fluid velocity. In the motorsport field, the heat transfer coefficient is the optimization target for the CAD design. However, h is usually given by Watt per Kelvin (W/K) and this makes difficult to compare different cases and flows. In these situations, it is better to use the Nusselt number, which is a normalization for the heat transfer coefficient.

Nu = h∙L/λ

Its physical meaning is how efficient is the convection heat transfer relative to the conduction one. For instance, a radiator with two walls at different temperatures, T_cold and T_hot. If there is no air flow through the radiators, the heat transfer will mainly occurs by conduction. However, in case of vehicle movement and/or radiator fan activation, there is a forced motion and an imposed pressure gradient, thus in addition to conduction there is also convection, which increases the heat transfer from the bottom to the top. This increase is what is measured by the Nusselt number. When Nu is equal or near to 1, it means that the heat transfer is governed by the conduction mechanism. On the other hand, if Nu is equal or near to 0, it means that the heat transfer is mostly due to the convection mechanism. Nu basically measures the governance of the convection on the flow. When the heat transfer is dominated by conduction, usually when there is no-slip condition, the heat transfer is given by:

q = – λ∙(∂T/∂n)|_n=0

However, this occurs in points of the flow at the surface or very near to it, thus it is possible to apply non-slip conditions. On the distance from the top, the buoyancy effect increase or the pressure gradient can occur, depending on the situation. In this case, the heat transfer equation accounts the heat transfer coefficient, which results in:

h = (-λ∙∂T/∂n|_n=0)/ΔT

Since the Nusselt number is already known, it is possible to update it according to the distance from the top wall, that is:

Nu = (-∂T/∂n|_n=0)/(ΔT/L)

Now, the Nusselt number is a ratio between convection and conduction mechanisms.

Figure 2 illustrates what was described in the previous paragraph. When there is only conduction, the heat transfer is basically dependent of the temperature gradient, which is given by ∇T = ΔT/L. For this reason the heat transfer occurs linearly. This occurs only in conduction, or barely, in laminar conditions.

q = λ∙∇T → q = λ∙ΔT/L

Again, Nu measures the ratio between conduction and convection mechanisms. In this case, it admits a non-linear condition (Figure 2) when near the wall, then it becomes constant when far from the wall. However, the transition between near the wall and far from it occurs in a slope much higher than conduction slope (Figure 2). Hence, the heat transfer when there is a flow is drastically improved.

Thermal convection

The evolution equation is similar to the conduction one, but it is accounted the internal energy. Now, there is a flow described by:

ρ∙c∙[(∂T/∂t) + u∙∇T] = ∇∙(λ∙ΔT) + q_g“‘

At the right side the process is basically conduction, but added by the source of sink q_g“‘. However, at left side there is the convection due to the velocity field. As already mentioned, convection is not a physical property, it only occurs where there is a flow, but it is an interaction between a field with associated energy and a conventional velocity field. Hence, what rule the heat exchange is the evolution equation. The interesting point about that is, that the only way to improve the field is increasing the velocity field, and as already described, this can be done by buoyancy effects or/and external agents. However, the evolution equation can not be the incompressible Navier-Stokes Equations (NSE), because in general, if there is temperature, there is density variation, which is not allowed in incompressible equations. Hence, it is said that there is a heat transfer coupled with the flow motion. The momentum equation and the mass conservation take into account that the density can change. In addition, it is accounted the effects of buoyancy, ρgz.

ρ/t + ∇∙(ρ∙u) = 0

ρ[(∂u/∂t) + u∙∇u] = – ∇p + ∇∙(µ∇u) + ρgz

In some cases, buoyance is changed by the isostatic pressure. When the case is about fluids. The problem with NSE and the evolution equation is the amount of equations and the amount of unknowns. For instance, there is an equation for heat transfer, an equation for mass and three equations for momentum, thus five equations. The unknowns are the temperature field, the density ρ, the pressure p, the viscosity µ and the three velocity components. Therefore there are seven unknown for five equations. To solve this problem it is added two more equations, there are the equation of state, also known as the perfect gases equation given by p = ρRT. The second equation is to consider the law regarding the viscosity variation in function of the temperature, which is given µ = µ^0.7. Hence, with these two the number of equations equal the number of unknowns. Summarizing, as c and λ are constants, the unknown are:

∂u/∂t, µ, ∂ρ/∂t, ∂T/∂t, q_g“‘, ρgz

Thus, it is introduced two new equations:

p = ρRT

µ = µ^0.7

References

• This article was based on the lecture notes written by the author during the Industrial Aerodynamics course attended at Dallara Accademy.