How Is Temperature Calculated in a Heated Copper Block?

[Froskoy]

Hi,

Suppose a copper block is heated on one side so that one end is at 800K. Given the dimensions of the copper block, is there a way of calculating the temperature of a point in the block distance x from the heated end after a given time?

With many thanks,

Froskoy.

 

[Andy Resnick]

This is a standard problem exhaustively treated in, for example, Mikhailov and Ozisik's "Unified analysis and solutions of heat and mass diffusion".

 

[technician]

If it can be assumed that there are no heat losses from the sides of the block then the equation for rate of heat flow is:
ΔQ/Δt = k A Δθ/Δx Δθ/Δx is the temperature gradient

 

[Froskoy]

Thanks! Your replies were really useful.

 

[sardar]

Hello Froskoy,

Thank you for your question. I can definitely provide some insight into the thermal gradient in copper.

The temperature of a point in a copper block can be calculated using the thermal conductivity of copper and the distance from the heated end. The thermal conductivity of copper is a measure of how well the material conducts heat, and it is typically given in units of watts per meter per Kelvin (W/mK).

To calculate the temperature, we can use the equation:

T(x) = T(0) + (T(h) - T(0)) * (x/L)

Where T(x) is the temperature at a point x from the heated end, T(0) is the temperature at the heated end (800K in this case), T(h) is the temperature at the other end of the block (assuming it is at room temperature, around 293K), and L is the length of the block.

This equation takes into account the thermal conductivity of copper and the distance from the heated end. It is important to note that this equation assumes steady-state conditions, meaning that the temperature distribution in the block has reached a constant state after a given amount of time.

To calculate the temperature at a specific time, we would also need to consider the rate of heat transfer and the specific heat capacity of copper. These factors can be incorporated into the equation through the use of a time constant, but the calculation would become more complex.

I hope this information helps. If you have any further questions, please don't hesitate to ask.

Best,