Sphere melting/freezing timescale-scaling (simplest approach)?

[bzz77]

Sphere melting/freezing timescale--scaling (simplest approach)?

Hello everyone:

I am new to melting/freezing-type behaviour. I would like to calculate approximate melting timescales for spheres (uniform composition) thrown into a hotter liquid. The spheres would initially be at a lower temperature than their melting point, so they'd have to heat up before melting. I'd like to stick with conduction.

I would like to know if there's a simple scaling-type method (that incorporates latent heat) for getting approximate melting timescales. From my reading so far, it looks these sorts of Stefan problems are usually approached with numerical methods, and I'd like to start with simple scaling-based approximations before diving into the numerical methods!

If anyone could describe how I'd go about doing a reasonable, but simplified timescale calculation for a melting sphere, I'd be very grateful! I don't know where to start.

 

[Mapes]

It's all about the assumptions.

For example, I'll assume that each immersed sphere has a uniform temperature $T(t)$ the entire time. In other words, conduction within the sphere is fast compared to convection between the sphere and the liquid. In other other words, the sphere is not too big, the thermal conductivity of the sphere material is not too small, and the convection coefficient is not too large. Calculate the http://en.wikipedia.org/wiki/Biot_number" to get a handle on whether this assumption makes sense.

I'll assume that the bath is large, so that the bath temperature $T_b$ is constant.

I'll assume that all material properties are temperature-independent.

Convection from the liquid will deliver a thermal power of $hA[T_b-T(t)]=h(4\pi r^2)[T_b-T(t)]$ to the sphere, where $h$ is the convection coefficient and $r$ is the radius. The energy that needs to be supplied to melt the sphere is $\rho V[c(T_m-T_0)+L]=\rho(4\pi r^3/3)[c(T_m-T_0)+L]$, where $\rho$ is the density, $c$ is the specific heat capacity, $T_0$ and $T_0$ are the initial and melting temperatures, and $L$ is the latent heat of melting.

Now I'll assume that the bath temperature is large compared to the melting temperature $T_m$ and that the melting temperature isn't much larger than the starting sphere temperature $T_0$, so that $T_b-T(t)\approx T_b-T_m$ and $c(T_m-T_0)+L\approx L$.

Then the time required to absorb this energy and melt the sphere entirely is

$$\tau=\frac{\rho r L}{h(T_b-T_m)}$$

This is just one example of a scaling analysis in heat transfer. Your assumptions may be different.