Finding the temperature of a box cooled by a peltier device

[Azzeral]

Homework Statement

I need to find the temperature of the air in a sealed box after 15minutes, that has been cooled by a Peltier device. The set up is as follows:

The peltier device is between 2 aluminium blocks of the same Area but depth of 1cm. The aluminium block that is receiving heat from the peltier device is connected to a heat sink. The other aluminium block is connected to a larger aluminium block that is inside the cool box (there is a hole big enough for them to be in contact). The larger block blocks off the hole and has the same depth as the other blocks. A heat sink is also attached to larger aluminium block.

The thermal conductivity of: aluminium is 237 W m^-1 K^-1, heat sink 201 W m^-1 K^-1. Area of peltier is 16 cm^2
Depth of heat sink is 7cm
Area of heat sink is 64cm^2
The temperature of the pelter device (cool side) stays about constant at 279K for 15 minutes (after 15 minutes its 281K)

Homework Equations

ΔQ/Δt = k A (T_{hot} - T_{cold})/depth

The Attempt at a Solution

So far I have assumed that T_hot is just room temperature in this case let that be (301K).
I have also assumed that each object starts at room temperature

so for going from the peltier device to the small aluminium block:
ΔQ1/Δt=237*(1.6*10^-3)*(301-279)/0.01=834.24

but for the small aluminium block to the larger one:
ΔQ2/Δt=237*(1.6*10^-3)*(301-T_small block)/0.01

Then heat sink ΔQ2/Δt=201*(6.4*10^-3)*(301-T_large block)/0.07

but I have no idea how to find T_s,b and T_l,b and how I can use this to find the temperature of the air in the box

 

[anathema]

after 15 minutes.

Hi there! It seems like you have made some good progress in your calculations. To find the temperature of the air in the box after 15 minutes, you will need to consider the heat transfer between the larger aluminium block and the air in the box. This can be done by using the equation for heat transfer through convection, which is:

Q = h A (T_{block} - T_{air})

Where Q is the heat transfer rate, h is the convective heat transfer coefficient (which you can assume to be 10 W/m^2K for air), A is the surface area of the larger block that is in contact with the air, T_{block} is the temperature of the block, and T_{air} is the temperature of the air in the box.

You can also use the equation you mentioned, ΔQ/Δt = k A (T_{hot} - T_{cold})/depth, to find the temperature of the larger block after 15 minutes, since you already know the heat transfer rate (ΔQ/Δt) and the thermal conductivity (k) of the block. From there, you can use the equation for heat transfer through conduction (Q = k A (T_{hot} - T_{cold})/depth) to find the temperature of the air in the box after 15 minutes.

I hope this helps and good luck with your calculations!