Time for heating to equilibrium with constant heat flux

[Komakech]

I have the following scenario and hope one of you can help me. I need to find an equation describing a heating process. A cylindrical metal of known dimensions and properties is heated from one side while the temperature of the other side is kept constant. I need to find the equation describing the temperature rise of the side being heated until it reaches equilibrium, so that from it, one can obtain a temperature time graph and know how long it takes to reach equilibrium.

Or rather does anyone know of an equation that can be used to describe how long a piece of metal being heated with constant heat flux takes to reach equilibrium temperature?

 

[chaoseverlasting]

Ok. If a constant amount of heat is supplied, then H=kA dt/dx=constant. Hdx=kA dt. Integrating from 0 to x and 0 to t. kA (t)=H(x). This would make it seem like the temperature varies linearly... with distance? Dunno... sorry can't be of much help...

 

[m2003]

I would suggest using the heat transfer equation, also known as Fourier's law, to describe the heating process. This equation states that the rate of heat transfer (Q) is equal to the product of the thermal conductivity (k), the cross-sectional area (A), and the temperature gradient (dT/dx). In this scenario, the temperature gradient would be the difference between the constant temperature on one side and the changing temperature on the other side.

To find the equation for the temperature rise over time, you can use the heat transfer equation in conjunction with the heat capacity equation (Q = mcΔT). This will give you an equation that relates the change in temperature over time (ΔT/Δt) to the heat flux, thermal conductivity, cross-sectional area, and heat capacity of the metal.

To determine the time it takes for the metal to reach equilibrium temperature, you can set the heat flux to zero (since the temperature on one side is being kept constant) and solve for the time at which the temperature rise reaches zero. This will give you the time it takes for the metal to reach equilibrium temperature.

I would also recommend considering any additional factors that may affect the heating process, such as the temperature of the surroundings, the material properties of the metal, and any heat losses due to convection or radiation. These can be incorporated into the equations to provide a more accurate description of the heating process.