Condensation In Automotive Headlamp

Excerpt from the Proceedings of the 2006 Nordic COMSOL Conference

Automotive Headlamp - Analytic Solution and Measurements of Condensation inside a Headlamp Erik Preihs General Motors Europe Abstract: Increasing styling features for automotive headlamps speed up the focus of understanding condensation at inner surfaces. Dew point higher than the inner surface temperature can be a cosmetic problem since transparent optical plastics are commonly used in combination with a black background leading to visibility of dew. In this paper, a suggested analytical approach of dew point analysis is brought forward. The analytical solution is based on a temperature map consisting of heat transfer, natural convection and radiation. An empirical expression for relative humidity is then implemented based on the calculated temperature map. Further a Magnus-Tetens formula for dew point is used for the area inside the headlamp with temperatures between 0° and 60° Celsius. The physical foundation is to understand weather the calculated inner surface temperature is lower than the analytical dew point inside the volume. Comparison between analytical results and measured temperatures, relative humidity and dew point calculated based on Magnus-Tetens formula show realistic results. Keywords: natural convection, heat transfer, radiation, relative humidity, dew point,

process of understanding the basic physics and assess changes to the product based on these models will therefore be of more and more importance.

2. Governing equations Thermal properties are needed as a numerical solution for a geometry field to form a base for two scalar expressions; relative humidity and dew point temperature. 2.1 Thermal field The concept in this paper for approaching the physics of condensation inside a headlamp has been to use the bi-directional coupling between Navier-Stokes equation, and the heat transfer equation, giving four dependent variables • • •

Pressure, p Velocity field components, u and v Temperature, T

Navier-Stokes equations: The incompressible Navier-Stokes equations, (1), consist of a momentum balance and a mass conservation and incompressibility condition. The equations are: ∂u + ρ (u ⋅ ∇)u = −∇p + η∇ 2u + F ∂t ∇ ⋅u = 0

ρ

1. Introduction Customer driven requirement to have a clear headlamp without fogging, even in severe weather conditions, intense the focus to understand the physics of condensation at inner surfaces. At the same time product cost and time to market drive the process of virtually assessment for customer requirements. Full vehicle test on road tracks and in climate wind tunnels add cost to the product that in some cases are hard to motivate from a customer perspective. The

with the following variables: F - volume force. u - velocity field. ∇ - vector differential operator. p - pressure. η - dynamic viscosity. ρ - fluid density.

(1)

Excerpt from the Proceedings of the 2006 Nordic COMSOL Conference

Heat Equation: The heat equation, (2), is based on energy conservation saying that the change in energy is equal to the heat source minus the divergence of the diffusive heat flux: ρc p

∂T + ∇ ⋅ (−k∇T + ρc pTu ) = Q ∂t

(2)

Cp is the heat capacity of the fluid, and ρ is fluid density. The expression within the brackets is the heat flux vector. Q is a source term. The heat flux vector contains a diffusive and a convective term. The velocity field, u comes from the incompressible Navier-Stokes equation. This concept give a one-way multi physics coupling from the fluid field to the energy transport by convection. Both equations are well documented and used extensively and in this case implemented into the Multiphysics software 3.2 Internal radiation, describing the heat transfer via electromagnetic waves between two surfaces is of interest inside a headlamp. Two equations define the heat flux with surface-to-surface radiation condition, (3). The first equation defines the heat flux through the boundary. The two first terms on the right side of this equation are the same as for the heat flux condition without radiation. The third term on the right side defines the radiative heat flux − n ⋅ (−k∇T ) = qo + h(Tinf − T ) + ε (G − σT 4 )

(3)

(1 − ε )G = J o − εσT 4

where ε - Surface emissivity, G - Surface irradiation, (arriving heat flux) σ - Stefan-Boltzmann constant. The second equation, (4) relates the surface irradiation to the surface radiosity, (heat flux leaving) (1 − ε )G = J o − εσT 4

where

(4)

J0 - Surface radiosity expression, and G - Surface irradiation 2.2 Relative humidity To be able to study moisture inside the headlamp volume a new variable for relative humidity, RH, was introduced, (5-8). A parameter was also needed to define the humidity between 0 and 100% at the inlet ventilation hole - inletrel_H2O. Parameters inside the previously solved thermal model were therefore set up in a scalar expression according to: RH relative _ Humidity =

cinlet cekvivalent _ H 2O

cinlet = inlet rel _ H 2 0 *

cekvivalent _ H 2O =

pref R ⋅T

pref R ⋅ Tinlet

⋅ exp(

inlet rel _ H 2 0 = 1

(5)

⋅ exp(−

− dH vap R

dH vap R

1 1 ⋅( − ) T Tref

1 1 ⋅( − ) T Tref

(6)

(7) (8)

2.3 Dew point Dew point is a function of relative humidity and temperature. A high relative humidity indicates that the dew point is closer to the current air temperature. If the relative humidity is 100%, the dew point will be equal to the current temperature. Given a constant dew point, when the temperature increases the relative humidity will decrease. A formula, (9-10), to calculate the dew point in degree Celsius within ±0.4 °C was brought forward in the work by Barenbrug [1]. It is valid for: 0 °C < T < 60 °C 0.01 < RH < 1.0 0 °C < Td < 50 °C

Excerpt from the Proceedings of the 2006 Nordic COMSOL Conference

where T - Temperature in degrees Celsius RH - Relative humidity Td - Dew point temperature The equation are: Td =

b ⋅ α (T , RH ) a − α (T , RH )

(9) Figure 2. Mesh distribution

where α (T , RH ) =

a ⋅T + ln( RH ) b +T

(10)

and a = 17.27 b = 237.7 °C

3. Method

Clear technical steps can be identified and are exemplified with figure 3 to arrive at a calculated dew field. The first step is to produce thermal properties. These values are used in step number two for empirical estimation of relative humidity. Temperature is reused and the third and final step is derived by the scalar expression for dew. Step 1

Step 2

Step 3

1) Natural Convection

The method for estimating reasonable values for dew point temperatures at a surface was gained by a coupled thermal field. A two dimensional geometry, cut from a headlamp, was modeled as solids with relevant properties and with boundaries for convection, conduction and radiation, figure 1.

A) Temperature Map 2) Heat transfer 3) Radiation

B) Empirical RH Calculation

Empirical Dew Point

Figure 3. Schematic overview for calculation steps

A stepwise solver strategy was used to get reasonable convergence stability by alternatively solve the thermal equations alone and thereafter the flow equation alone. The next step was to start reusing the result of the previous solution as starting point for the combined equation system.

4. Experimental results Figure 1. Geometry

Mesh size, in many cases less than a millimeter was used and thicknesses were treated as solids. Figure 2 exemplifies mesh distribution.

The experimental done to support the twodimensional concept calculations aimed at providing reasonable values for relative humidity and temperature in time domain for a dynamic driving cycle during rain. A chip was applied in the gap between outer lens and inner bessel together with a digital thermocouple.

Excerpt from the Proceedings of the 2006 Nordic COMSOL Conference

Figure 4 document a number of temperatures inside and outside of the headlamp together with estimated dew point between inner bessel and outer lens for a full vehicle driving cycle. 150

Temp inside headlamp between lens and bessel Dew point

130

Outer Lens Temperature

20

Relative humidity

Temperature, (C)

Vehicle speed

110 90

15

70 10

50 30

5 10 0

18:00

Relative Humidity, (%) and Vehicle speed (kpm)

25

5. Numerical results Solving only the heat equations without the bi-bidirectional coupling, with boundaries tuned towards measurements, give an evenly distributed thermal field far from a realistic scenario for how a head lamp physically functions, figure 7.

-10

19:12

20:24

21:36

22:48

00:00

01:12

Figure 4. Temperatures and dew for full vehicle

The circle show the start of condensation i.e. where the lens temperature drop below calculated dew point. Temperatures measured inside and at the outer shell of the headlamp for this particular time slice were used as boundaries for the twodimensional calculations.

Figure 7. Stationary temperature profile

A coupled thermal field with a pressure difference between inlet and outlet ventilation holes will typically have an extensively diffuse heat pattern due to the interaction of convective velocity field u, figure 8.

Figure 5 and figure 6 exemplifies the size and location of the condensation pattern at the time stamp 21:10 and 21.36 in figure 4.

Figure 8. Stationary temperature profile Figure 5. Condensation pattern at T=21:10

Figure 6. Condensation pattern at T=21:36

The coupled field was used to implement the scalar expression for relative humidity, figure 9, typically produce high levels of humidity in the tip of the headlamp well in line with the results in figure 5 and figure 6 when the inlet boundary had a high relative humidity value specified.

Excerpt from the Proceedings of the 2006 Nordic COMSOL Conference

then 10 °C. The result will be production of water at the inner surface.

6. Discussion

Figure 9. Stationary relative humidity profile

The next physical step will be to produce a dew temperature field and the results can be seen in figure 10 together with a line plot in figure 11.

Error estimations of measured data was derived to understand if the results is of relevance and also to set the numerical results into perspective since measured temperatures was used as parametric values for outer boundaries. Figure 12 document error bars for measurement based dew point, (upper bars only to enhance readability), and temperature named TD1, inside the headlamp at the tip, (lower bar to enhance readability). Uncertainty will be in the vicinity of 1-1.5 degree C when adding up the equipment and empirical expression. Data point between 13.55 and 15:30 state that the inner temperature is higher than the dew point and clearly no condensation will occur since lower and upper error bars never meet. This was also visually the case since nothing could be seen at the lens.

Figure 10. Stationary dew temperature profile 100

22 TD2 Dew Point

90

TD1 80

RH

70

18

60 50

16

40 14

30

Typical start point for condensation

12

10 13:26

Relative Humidity [%]

20

Temperature Degree C

Figure 11 document a line plot starting at the inner surface of the lens going inwards towards the tip of the Bessel, (black line in figure 10).

13:55

14:24

14:52

15:21

15:50

16:19

16:48

20 10 0 17:16

TIME

Figure 12. Typical measurement errors for equipment and empirical dew point estimation given by Barenbrug, [1]

Figure 11. Stationary temperature profile

It was chosen to plot dew temperature versus temperature field. The result indicates that the dew temperature at the inner surface, (start point for the line plot), for this particular load case has a temperature higher than 10 °C. At the same data point the temperature is plotted well lower

The circle indicates the start point for condensation. An uncertainty to whether there is a built up of a water layer at an inner surface or if there is a transition from an inner surface back to the air even though most of the data point are likely to produce dew will naturally be present. The measurement method is still considered to state condensation.

Excerpt from the Proceedings of the 2006 Nordic COMSOL Conference

All efforts are based on one stationary solution. Normal evaluation of the performance of any optics in the vehicle industry will be a driving cycle. There will therefore be a need to do accumulative stationary solutions for a number of driving situations or even better to make time dependent simulations with the boundaries specified as measured temperature profiles. This is of course not an option seen as realistic today but rather something to strive towards. Some further understanding is also needed about how to implement different formulations for dew point calculations since the used Magnus-Tetens expression only is valid between 0° and 60° Celsius. It is therefore not possible to use this model outside this temperature interval. Other empirical formulations could be implemented to cover a wider temperature range with the mind set for minus degrees. When the dew point temperature and air temperature are equal, the air is said to be saturated. Dew point temperature is not ever greater than the air temperature. Therefore, if the air cools, moisture must be removed from the air and this is accomplished through condensation. This process results in the formation of tiny water droplets that can lead to the development of fog and even frost. The presented model will produce results where dew point temperatures are higher than the temperature. The energy potential in between these two temperatures will likely be possible to develop further and use in an effort as an estimate of the gradient for the amount of water vapor produced. The optical material used in today’s headlamps will also need further attention since the model in this paper not covers radiation for transparent materials. Experimental work suggests a need for further work in this area to get higher numerical accuracy. Normally a glass bulb generates heat of several hundred degree C at the glass surface. The assumption in this work at this point has been convective flow. The dew point calculation might benefit from a local turbulent model around the bulb that interacts with a convective model even though the temperature interval

between 0 to 60 ° Celsius are expected to be lesser effected. The need for this kind of complexity increase is today unknown and some initial work can be valuable to understand if this is of industrial relevance.

7. Conclusions It was showed that dew point temperature fields could be achieved and compared with air temperature with a reasonable accuracy for a two dimensional geometry. It was showed that dew field occurrence and spread are reasonable compared to measured data and documented pictures of a head lamp when measured temperatures are applied to the boundaries. It is concluded that a 3D activity will be the next step to in depth correlate this concept together with measured temperatures even though the degree of freedom for the model might be unreasonably large. Modeling of optical plastic parts i.e. transparent surfaces that will receive radiation heat need to be addressed more extensively. Never the less dew point calculations based on the given concept has the possibility to contribute towards a more robust design for head lamp condensation.

8. References 1. Barenbrug, A.W.T., Psychrometry and Psychrometric Charts, 3rd Edition, Cape Town, S.A.: Cape and Transvaal Printers Ltd., 1974

9. Acknowledgements General Motors Europe and I would like to thank Peter Georén, a former Comsol employee, who made the foundation for this work. Peter’s ability to listen and discuss the physics made us go forward with this model in combination with experimental work.