Calculating energy requirements to sustain temperature

[Low-Q]

TL;DR Summary

Heating an object that is surrounded by colder air temperature.

Hi.
I have 3D printer with a heated bed, made of a 400x300 mm aluminium, that is 4mm thick. The surrounding air temperature is about 10°C. When printing with different materials, it is required to keep the bed at different temperatures. What I experience is that the driver which controlls temperature works at a duty cycle of approx 40% for keeping the bed at 50°C when printing PETG. However, when printing ABS I need to increase temperature to at least 100°C, but even if the driver is constantly on, the bed never reach 100°C, but stops at 70-75°C.

So, to solve this problem, i built a cabinet from 5mm transparent acrylic plastic.
Now the driver works at approx 70% duty cycle at 100°C (not 100% sure if that is the correct duty cycle). The temperature inside the cabinet increase to 45-50°C.

My question is: How does one calculate how much energy is required to increase an objects temperature by 1°C when we consider that the surrounding air temperature is lower and constant?

I guess it has something to do with the area which is in contact with air, and the mass of the object, and I guess this question can be applied to how much energy must be added to heat a house at different temperatures at givet outdoor temperatures.

I'm 48 years old, so this is not homework questions.

Br. Vidar

 

[hutchphd]

Summary:: Heating an object that is surrounded by colder air temperature.

surrounding air temperature is lower and constant?

Do you mean the air inside the box surrounding the plate or the air outside the box?

 

[anorlunda]

Summary:: Heating an object that is surrounded by colder air temperature.

My question is: How does one calculate how much energy is required to increase an objects temperature by 1°C when we consider that the surrounding air temperature is lower and constant?

Yes it has to do with the area, but not the mass. The key is that energy is conserved. Power in = power out. Power out is the heat lost to the environment of the room. If you keep the enclosing box in place, the heat flow to each wall of the bod depends on the temperature difference in-out, the area, and the heat transfer coefficient of the material. Circulation of the air, both inside and outside the box also has an influence.

Wait for more answers. Some of the mechanical engineers in this forum may be able to give you more practical advice. Ping @russ_watters @Chestermiller @jrmichler

 

[hutchphd]

I'm hoping he can do a cooldown curve from a known initial temperature of the aluminum plate. From the (known) heat capacity of the Al and assuming linearity with temperature difference the efflux coefficient is trivial...but you need a simple system (i.e.the block).

 

[anorlunda]

Can you insulate the container?

 

[Lnewqban]

... My question is: How does one calculate how much energy is required to increase an objects temperature by 1°C when we consider that the surrounding air temperature is lower and constant?

I guess it has something to do with the area which is in contact with air, and the mass of the object, and I guess this question can be applied to how much energy must be added to heat a house at different temperatures at givet outdoor temperatures.

That depends mainly on the resistance of the envelope of the building (exterior walls, roof, windows and exterior doors) to the flow of heat from inside to outdoors (for heating) or from the exterior (air temperature and solar load) towards indoors (for cooling).
Thermal loads due to persons, devices, cooking, ventilation air, etc. should be considered in any calculation.

Please, see:
https://www.engineeringtoolbox.com/cooling-heating-equations-d_747.html

 

[jrmichler]

something to do with the area which is in contact with air, and the mass of the object

These are two separate things.

The thermal mass of an object is its mass times the specific heat of that material. It determines the amount of heat needed to change its temperature. Good search term to learn more is specific heat.

The surface area of an object affects the amount of heat needed to keep it at a temperature above the ambient temperature. The equation is heat = U * A * delta T, where:

heat = amount of heat lost in watts (or BTU per hour)
U = heat transfer coefficient
A = surface area
delta T = temperature difference between the object and the air

The heat transfer coefficient is a complex function of air velocity and object geometry. Estimating it occupies several chapters in a book on heat transfer. Building a house around the printer prevents the air from circulating through the room, so has two effects:

1) It reduces the U by slowing the air circulation, and
2) It surrounds the printer with warm air, which reduces the delta T

When heating up an object, the heat first goes into raising the temperature of the object. As the object gets warmer, more and more heat is lost to the air around it. At some point the heat lost to the air is equal to the heat added. That is the equilibrium temperature.

Be careful. If the air around the printer gets too warm, the electronics will stop working.