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How to calculate effect of 4 heaters on 1 sensor when heating aluminium slab

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Cas Giepmans
Cas Giepmans on 24 Nov 2017
Edited: Cas Giepmans on 24 Nov 2017
I'm currently struggling with a problem concerning the heating of a big aluminium plate. It's a big slap of about 18kg of aluminium, which is divided into four quadrants. Each quadrant has a couple of heater elements (just metal blocks with heating coils) which are all controlled by a PID controller (which I'm tuning). Every quadrant also has a temperature sensor.
I've tried modelling this in Simulink. The difficulty for me lies in how to combine the different transfer functions of the relationships between each quadrant of heater and temperature sensor. Let's take the first quadrant as an example. Each quadrant of heaters has an effect on the temperature of quadrant 1 when activated. I have identified these relations using MATLAB and previously measured data. For these TF's input is a number between 0 and 0.75 (duty cycle of heaters), output is the temperature of the quadrant (aiming for 200 degrees C).
How can I accurately model the temperature of quadrant 1-4, knowing I have the TF's of all heater-sensor relations concerning quadrant 1? Can I simply add up all the outputs? In the image: H11 = TF of Q1 heaters on Q1 sensor, H21 = TF of Q2 heaters on Q1 sensor etc.

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