Question

Suppose that upon entering the Earth's atmosphere, the surface of a meteorite has been heated to the melting point and the molten material is carried away by the flow. It is of interest to calculate the rate at which melting removes material from the meteorite. For this purpose, consider the following problem. The surface of a semi-infinite half-space moves downward into the half-space with constant velocity \(V\), as indicated in Figure \(4.39 .\) The surface is always at the melting temperature \(T_{m},\) and melted material above the instantaneous surface is removed from the problem. Assume that the surface of the half-space is melted by a constant heat flux \(q_{m}\) into the half-space from above the surface. Assume also that far from the melting surface the temperature is \(T_{0}\); that is, \(T \rightarrow T_{0}\) as \(\zeta \rightarrow \infty\). Find the temperature distribution in the half-space as a function of time \(T(\zeta, t),\) and determine \(V\) in terms of \(q_{m}\) and the thermodynamic properties of the rock. Account for the latent heat \(L\) required to melt the material.

Short answer

The temperature distribution is found using \( T(\zeta, t) = T_0 + (T_m - T_0) \text{ erf }(\frac{\zeta}{\sqrt{4\alpha t}}) \), and velocity \( V \) is \( \frac{q_m}{\rho L} \) in a steady state.

Step-by-step solution

Set up the Problem

We need to find the temperature distribution, \( T(\zeta, t) \), in the semi-infinite body, and the velocity \( V \) at which the surface moves. The key is to consider the heat conducted into the semi-infinite body and the heat required to melt the material. Assume the heat conduction follows one-dimensional heat conduction principles with constant properties.

Use the Heat Equation

For a semi-infinite body, the governing one-dimensional heat conduction equation is \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial \zeta^2} \), where \( \alpha \) is the thermal diffusivity \( \alpha = \frac{k}{\rho c_p} \). This equation must be solved with the boundary and initial conditions.

Define the Boundary and Initial Conditions

1. At \( \zeta = 0 \), the temperature is constantly at the melting temperature: \( T(0, t) = T_m \).2. Far from the surface, the temperature remains at the initial temperature: \( T(\zeta, t) \to T_0 \) as \( \zeta \to \infty \).3. Initially, the entire material is at uniform temperature \( T_0 \).

Apply the Stefan Condition at the Melting Interface

The Stefan condition gives the relationship between the velocity \( V \) and the heat flux \( q_m \), accounting for latent heat \( L \):\[ q_m = \rho V L + k \left. \frac{\partial T}{\partial \zeta} \right|_{\zeta=0} \]Where \( k \) is the thermal conductivity. This balances the heat conducted into the semi-infinite domain and the heat used for melting.

Solve for the Temperature Distribution

Knowing that the problem satisfies a semi-infinite domain under initial conditions, the solution can be expressed using the similarity variable \( \eta = \frac{\zeta}{\sqrt{4\alpha t}} \). The temperature distribution thus follows:\[ T(\zeta, t) = T_0 + (T_m - T_0) \text{ erf }\left(\eta\right) \]

Calculate Velocity from Heat Flux

Substitute the boundary condition into the Stefan condition to solve for \( V \). Integrate the error function to get the gradient at \( \zeta=0 \):\[ V = \frac{q_m - k\frac{\partial T}{\partial \zeta}|_{\zeta=0}}{\rho L} = \frac{q_m - k\frac{T_m - T_0}{\sqrt{\pi \alpha t}}}{\rho L} \] As \( t \to \infty \), the velocity \( V \) reaches a steady state under constant \( q_m \).

Key concepts

Heat Conduction

Heat conduction is a process where heat energy transfers from the warmer part of a material to the cooler part without any movement of the material itself. In the case of meteorite melting, this concept is crucial. Here, heat is conducted from the surface of the meteorite, which is at the melting temperature, into the cooler interior of the meteorite.
This scenario can be modeled using the heat conduction equation:

• \( \frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial \zeta^2} \)

where \( T \) is the temperature, \( t \) is time, \( \zeta \) is the depth into the meteorite, and \( \alpha \) is the thermal diffusivity.
Heat conduction in this context needs a steady flow to melt the meteorite effectively, as the heat moves from the surface deeper into the meteorite, changing the temperature distribution over time.

Latent Heat

Latent heat refers to the energy required for a substance to change its phase without changing its temperature. In the context of a melting meteorite, it’s the amount of energy needed to turn the solid rock into a liquid form.
In our problem, this comes into play when the solid rock melts due to the heat flux \( q_m \). The energy needed to turn the solid meteorite material into liquid without increasing its temperature is called latent heat \( L \).
Latent heat is vital because it determines how much energy is consumed in the phase change. This is crucial for understanding the meteorite's melting rate, as this energy subtraction from the total heat available affects how much further heat is conducted into the semi-infinite body.

Thermal Diffusivity

Thermal diffusivity \( \alpha \) is a material-specific property that measures how quickly heat spreads through the material. It's calculated as:

• \( \alpha = \frac{k}{\rho c_p} \)

where \( k \) is the thermal conductivity, \( \rho \) is the density, and \( c_p \) is the specific heat capacity of the meteorite.
This property combines the effects of thermal conductivity and heat capacity, indicating how effectively a material can conduct thermal energy relative to storing it. Higher thermal diffusivity means the material can conduct the heat quickly through itself, which implies that the meteorite surface will cool rapidly as the heat is distributed through the rock body.
The understanding of thermal diffusivity helps predict how the interior of the meteorite will warm up when exposed to the same heat flux continuously.

Stefan Condition

The Stefan condition is a principle used to relate phase changes involving latent heat with heat transfer. It gives us a way to calculate the velocity \( V \) of the moving interface where melting occurs. In our scenario, this interface is the surface of the meteorite.
The Stefan condition is expressed by the equation:

• \( q_m = \rho V L + k \left. \frac{\partial T}{\partial \zeta} \right|_{\zeta=0} \)

This equation states that the heat flux \( q_m \) used for melting is split between the latent heat used to melt the material and the conductive heat past the melting interface.
Using this condition involves balancing the energy to find \( V \), which indicates how fast the surface melts into the meteorite. Without considering this, one might miscalculate how fast the meteorite melts or fails to explain how certain properties affect the rate of melting.