Calculating the thermal coefficient between the insulated interior of a telescope instrument package and the cold ambient air outside

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SUMMARY

This discussion focuses on calculating the minimum outside temperature that a heater can maintain within an insulated telescope instrument chamber using a digital PID thermostat. The relationship between heater power and temperature difference is linear under stable conditions, allowing for effective estimation through collected power-temperature data. Key parameters include the exterior surface area, thermal conductivity of insulation, insulation thickness, and convection coefficients both inside and outside the enclosure. The formula provided for calculating the internal temperature is based on surface energy balance principles.

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  • Understanding of thermal conductivity and insulation properties
  • Familiarity with PID thermostat functionality and operation
  • Knowledge of convection heat transfer principles
  • Basic mathematical skills for solving thermal equations
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  • Explore advanced heat transfer concepts, including convection and conduction
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Engineers, physicists, and hobbyists involved in thermal management of sensitive instruments, particularly in astronomical applications, will benefit from this discussion.

Erwinux
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I've built an insulated chamber to protect a sensitive instrument at freeze temperatures in the winter. The instrument is mounted on a telescope, so the heat inside the chamber will slowly dissipate in the ambient. A digital PID thermostat is used to keep the temperature at a safeguard level. The outside and inside temperature and the duty cycle of the thermostat are recorded on a remote server, so I know the power consumed at a certain difference of temperature, anytime.

How to calculate the outside minimum temperature the heater will maintain the safeguard temperature inside insulated chamber?
 
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A simple model will work pretty well here for steady state solutions. The power your heater makes will be proportional to the temperature difference it creates. This should be a linear relationship in normal stable systems (not true if you have things that change state like water freezing/melting) so twice the heat flow (power) will create twice the temperature difference. Also, the temperature difference won't depend much on the temperature values within "normal" temperature ranges.

So, you should be able to get a good estimate by collecting some power-temperature data and then extrapolate to maximum heater power and the minimum desired temperature.
 
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Heater Box.jpg


So you have something like that above.

I think you can apply surface energy balance to the interior/exertion walls to get in the ball park for a steady state solution. Its a bit hand wavy, but it shouldn't be that bad:

Take ## A_s ## to be the exterior surface area of the enclosure
## k ## is the thermal conductivity of the insulation
##L## is the insulation thickness
## h_{int}## is the average convection coefficient inside the enclosure maybe ## 10 \rm{ \frac{W}{m^2 K}}##
## h_{ext}## is the average convection coefficient outside the enclosure could be as high as ## 200\rm{ \frac{W}{m^2 K}}##
## q ## is the maximum output power rating for the heater

Then you get ## T_{int} ## from solving the following relation:

$$ q = A_s \frac{T_{int} - T_{\infty}}{ \frac{1}{h_{int}} + \frac{L}{k} + \frac{1}{h_{ext}} } $$

let me know if you have any questions.
 

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