Modeling thermal mirror deformation

12 Aug 2006, Arjan te Marvelde

This article explores mirror deformation that may appear during cool-down and use, by means of various mathematical models. Apartial differential equation (PDE) solver is used to model the transient of temperature gradient inside a disk. As a first approximation, an algebraic formula is also used, which however assumes a plate of infinite area. Finally, the effect on the mirror shape is analyzed.

Mathematical models

Infinite plate approximation

This approximation can be used very well to estimate best case cooling times of various form factor mirrors. The inaccuracy that has to be taken into accountis the fact that the environment temperature in reality cannot be considered constant. This would only be the case when the plate is surrounded by a material with a heat capacity much higher than that of the plate itself: water cooling would be the best match.

Consider a plate of thickness -L<x<L, and infinite in y and z directions. The heat conductivity of the plate material is given by λ, the specific mass by ρ, and its specific heat by C_p. The environment temperature is kept continuously at T_e.

Boundary and start conditions for this probem are:
for

Now, due to heatflow, the temperature will develop in time according to:

Where:
and

As can be seen in the equation, the temperature distribution will at all times have the shape of a cosine between -π/2 and +π/2. The time behaviour is exponential, towards the asymptote of T'=0, or T=T_e.

Note 1: The factor 4/π is disturbing the initial phase (i.e. T(0,0) is larger than T₀) but this can be seen as an artifact due to the compensation of the initial distribution which clearly cannot be described with a cosine. The function therefore only is accurate for larger values of t.

Note 2: When the temperatures on either side of the plate are different, a linear gradient can be added to the cosine term. T₀ should be taken as average of both surface temperatures.

Heatflow PDE solution

A better model involves solving the heatflow PDE by means of a PDE solver program, such as FlexPDE or MSC NASTRAN. Such models allow many more parameters to be tweaked, and thus to simulate a more realistic thermal process. The model can work with any shape (such as a circular disk) and various cooling mechanisms can be tried out.

The PDE that determines the temperature gradients inside the disk is the heatflow equation:
for all directions.

Here again the heat conductivity of the plate material is given by λ, the specific mass by ρ, and its specific heat by C_p. The temperature T_x is dependent on location in 3 dimensions, and so is its gradient.

The heat loss due to convection is proportional to the local temperature difference:

The environment temperature T_e is considered to be constant, equal to the ambient air. The heat transfer coefficient may be location dependent, for example when a fan is directed to the center part of the disk, and includes effects like boundary layer etc.

There will also be heat radiation, which is described by the law of Wien:

Here ε is the emissivity of the surface, assuming so called 'grey body radiation'. This is an engineering approximation which averages out the spectral dependency of the radiation. The constant σ is Stefan Boltzmanns', which has the value 4.67 10^-8.

The equivalent radiative nightsky temperature T_s is approximately 200K. Due to the intermediate atmosphere this is much higher the cosmic background radiation (3K). The value will depend on the state of the atmosphere, for example pollution and water content.

The boundary conditions for the solution of the heatflow are determined by the outward flux of the heat on the disk surfaces, i.e. the heat loss as given above, as a function of location. Start of simulation assumes a continuous temperature over the entire disk, and hence the solution of the PDE effectively is the step response of the system.

Deformation

Apart from mechanical stress, which can be modeled with tools like PLOP, the shape of the mirror is determined by the temperature distribution inside the glass. When all temperature gradients are zero, the thermally induced deformation can be ruled out. However, even when a mirror is in thermal equilibrium there may be significant differences throughout the mirror, caused by various environmental factors. The mirror quality experienced at the eyepiece will ultimately be dependent on how small the temperature differences can be kept.

Approximation of deformation due to temperature differences in the disk is based on integration of temperature gradient and coefficient of expansion. One case is considered specifically: the linear temperature gradient in axial direction, from the warm back to the cold front of the mirror. This gradient causes the backside to be more expanded than the front side, resulting in a curve that at least deepens the sagitta and possibly also changes the correction.

The linear gradient has effect only in radial direction, i.e. the disk diameter is larger on the warmer side. This difference in diameter Δ will presumably become visible as a curving with a certain radius R.

The equation that can be derived for calculating R is:
reducing to:

The sagitta s of this curve is given by:

Note 1: This derivation is for a specific case, other temperature distributions will be a bit less obvious to convert into deformation.

Results

Some material properties:

As a basis for the calculations the following four mirrors are considered:

Approximations

For the best case approximations the temperature distribution formula for x=0 simplifies to:

The factor in the exponent determines the minimum time it takes until the core of the mirror is cooled down to within a certain range from T_e.

Heatfow PDE

The mirror is mounted in a telescope, with forced air cooling from the back (H=100 W/m².K), natural convection on the front (H=10 W/m².K) and radiation from an Aluminized front surface with SiO overcoat (ε=0.04). Assume a starting temperature of 293K (normal indoor temperature) and an ambient temperature of 283K.

When these values are plugged in the FlexPDE model, the folowing results:

Here curves a, b and c respectively reflect the center, the back surface and the front surface temperature, measured on the optical axis.
What can be concluded from these graphs is that the cooling depends foremost on thickness and on material. Also the side that receives forced airflow has a cooling rate which is to a large extent independent from the thickness. Initially it is this side that cools fastest, but it is bypassed by the other surface temperature after a while. The last thing that can be concluded is that a residual temperature gradient remains when equilibrium has been reached, front surface being colder than the back. This of course is due to radiation heatloss, and lack of a forced airflow.

Values for the temperature difference range from 0.2K for mirrors 1..3 and 0.3K for mirror 4. When also the front surface is supplied with flowing air the differences reduce to 0.06K and 0.08K respectively. When no forced convection is used at all, the disks take a lot mor time to equilibrate, but the residual temperature gradient is only 0.1K and 0.2K respectively.

Note 1: The radiative heatloss is taken as worst case. When the mirror is enclosed in a tube the equivalent temperature of the nightsky should be raised considerably, since a large part of the view from the mirror (the tube) is now at T_e, or slightly below that.

Note 2: Since the secondary mirror has a more free view on the sky, it will be more prone to the radiative cooling. Also, the secondary turns its glass side towards the sky, which has a large emissivity. For that reason it would be advisable to construct the spider in such a way that the glass is mostly hidden.

Deformation

The values found for the residual temperature gradient are a bit worrysome, especially when they are converted into actual deformation. The changes in sagitta for the four mirrors, depending on where forced air cooling is used:

Note 1: In the calculations the deformation is assumed to be spherical, as per the derivation above. Hence the effect on image quality will be much less than the figures suggest. In reality the gradient will have a different form, presumably the center of the front surface will be colder since in general it sees a larger part of the sky. The resulting deformation is then nolonger spherical, but will move towards a hyperboloid. In such case a startest will show overcorrection.

Final remarks

• The simulations clearly suggest that the effect of radiative cooling of the mirror front surface cannot be ignored. This will especially be true for open structure OTAs, where the mirror is exposed to a significant part of the nightsky.
• A solution that not only removes a boundary layer but also minimizes the residual temperature gradient, is to use a forced air flow over the front surface.
• It is yet unclear how much the mirror deformation will affect the star image at the eyepiece. As long as it is axially symmetric, parts can be compensated by refocusing, and other parts will become visible as zonal defects.