How Long Until a Heated Iron Handle Becomes Too Hot to Touch?

[MisterX]

Homework Statement

Problem 1.60. A frying pan is quickly heated on the stovetop to 200 C. It has an iron handle that is 20 cm long. Estimate how much time should pass before the end of the handle is too hot to grab with your bare hand. (Hint: The cross-sectional area of the handle doesn't matter. The density of iron is about 7.9 g/cm3 and its specific heat is 0.45 J/g-C).

For iron [itex]k_t = 80 \frac{W}{m\cdot K}[/itex]

Homework Equations

[itex] \frac{Q}{\Delta t} = -k_t A \frac{dT}{dx}[/itex]

The Attempt at a Solution

So I might consider a little section at the end of the handle with length d which is receiving heat.

[itex]m = \rho A d[/itex]

[itex]T_{end} = \frac{Q_{end}}{c \cdot m} = \frac{Q_{end}}{c \rho A d}[/itex]
[itex]c \rho A d T_{end} = Q_{end}[/itex][itex] \frac{ c \rho A d \Delta T_{end}}{\Delta t} = -k_t A \frac{dT}{dx}[/itex]
The area cancels

[itex] \frac{ c \rho d \Delta T_{end}}{\Delta t} = -k_t \frac{dT}{dx}[/itex]

But we still don't know what is [itex]\frac{dT}{dx}[/itex], which presumably depends upon time. There is also that d still there.

Note that we are asked to derive the heat equation in a later problem, so I'm assuming I'm not supposed to use heat equation for this problem, but perhaps I am wrong. (I have already derived the heat equation from the Fourier Law of Heat Conduction).

I supposed I could assume d = dx = 20 cm, and dT = T - 200, with the initial condition for T being at room temperature and solve that differential equation. Is that what I'm supposed to do?

This problem is 1.60 from Schroeder Thermal Physics. It's not coursework or homework, as I am doing this independently, but I like you to treat it as if it were.

 

[Chestermiller]

You use the term heat equation. Is this shorthand for the "transient heat conduction equation," or is it something else. What exactly do you mean by the heat equation?

 

[MisterX]

What is named the heat equation in my book:

[itex] \frac{\partial T}{\partial t} = K\frac{\partial ^2 T}{\partial x^2} [/itex]

Due to this problem being before where the above equation is introduced, I am thinking I am not supposed to use this equation to, for example, solve the temperature at every point on the handle as a function of time.

 

[Chestermiller]

What is named the heat equation in my book:

[itex] \frac{\partial T}{\partial t} = K\frac{\partial ^2 T}{\partial x^2} [/itex]

Due to this problem being before where the above equation is introduced, I am thinking I am not supposed to use this equation to, for example, solve the temperature at every point on the handle as a function of time.

Well, there are approximate ways to solve the transient heat conduction equation using a numerical approach similar to the one you were beginning to set up. The lowest order of these is to put a grid point at the center of the handle and assume that this is representative of the average temperature of the handle. The heat flux at the pan end of the handle would be approximated as k(200-T)/(L/2). The heat flux at the far end would be zero. The rate of change of the average handle temperature would be calculated from ρC_pLdT/dt=k(200-T)/(L/2). This would give you a very rough approximation to the transient temperature variation (probably enough to answer your question). You could subdivide the handle into smaller sections, but then you would be solving for the temperatures at two locations. This gives you a rough idea of how transient heat conduction problems can be solved numerically.

Chet

 

[Nickie]

I would approach this problem by first acknowledging that the Fourier Heat Conduction Law is a fundamental principle in thermodynamics that describes the transfer of heat through a material. It states that the rate of heat transfer is proportional to the temperature gradient and the cross-sectional area, and inversely proportional to the material's thermal conductivity. This law can be derived from the heat equation, which describes the change in temperature over time in a given material.

In this specific problem, we are asked to estimate the time it takes for the end of an iron handle to reach a temperature that is too hot to touch with bare hands. To do so, we can use the Fourier Heat Conduction Law to set up a differential equation that relates the temperature at the end of the handle to the temperature gradient along its length.

We can start by considering a small section at the end of the handle with length d that is receiving heat from the stovetop. The mass of this section can be calculated using the density of iron and the cross-sectional area. We can then use the formula for heat capacity to find the change in temperature at the end of the handle, assuming that the heat transfer is only occurring in this small section.

Next, we can use the Fourier Heat Conduction Law to relate the change in temperature to the temperature gradient along the handle. This allows us to set up a differential equation that describes the change in temperature over time at the end of the handle. Solving this differential equation will give us the time it takes for the end of the handle to reach a temperature that is too hot to touch.

It is important to note that the assumption of constant cross-sectional area in this problem is valid because the handle is a solid object and the change in temperature is relatively small. However, in more complex systems, this assumption may not hold and the heat equation should be used instead.

In conclusion, as a scientist, I would approach this problem by using the principles of thermodynamics and the Fourier Heat Conduction Law to set up a differential equation that relates the temperature at the end of the handle to the temperature gradient along its length. This approach ensures that we are using fundamental principles to solve the problem and not just relying on assumptions or approximations.