Hi DH,
D H said:
- Low thermal conductivity, high heat capacity. This is just delaying the inevitable. The vehicle still fries, it just happens later when the heat pulse finally reaches the aluminum skin. The heat capacity needs to be low, too.
- Low thermal conductivity, low heat capacity. This is the only winning combination. The low conductivity delays and spreads out the heat pulse. The low heat capacity means that the aluminum skin of the vehicle won't heat up all that much when the heat pulse finally does reach the skin.
I guess I don’t understand how the heat capacity enters into this. Let’s look at the steady state condition first, then consider how that differs from a transient condition.
For a steady state condtion, the kinetic energy of the shuttle is converted to heat energy in the air at the outer surface of the tiles, and this heat flows into the tiles. Let’s call this heating Q
k due to it being the conversion of kinetic energy to heat. Under steady state conditions, this heat flows away from the surface of the tiles in 2 ways.
Radiation heat transfer: Q
r = e C (T
h4 – T
c4) A
Where e is emissivity of the hot surface, C is the Stefan-Bolzmann constant, A is area and Th and Tc are the temperatures of the hot tile surface and the cold surface respectively, the cold surface being the environment. Note that regardless of what thermal conductivity or heat capacity the tiles have, the heat rejected from the surface of the tile is only a function of the temperature.
Conductive heat transfer: Q
c = k A dT / s
Where k is the thermal conductivity of the tile, A is area, dT is the difference in temperature between the hot outer surface and the inner surface of the tile and s is thickness of material. This heat is the heat entering the airframe, so the airframe must absorb that heat. More on this in a moment. Note again that heat capacity of the insulation doesn’t enter this heat transfer equation.
We can neglect convective heat transfer away from the tiles because that’s actually Q
k which is the heat going into the tiles.
We can also neglect any change in the temperature of the airframe if we assume that the insulation system is designed such that the increase in the airframe temperature is negligable. In other words, the airframe is a heat reservoir or lump mass who’s temperature doesn’t change substantially during the shuttle’s decent. The heat influx from the thermal conductivity through the tile insulation certainly adds to the thermal energy the airframe has as a function of the airframe’s heat capacity. But the overall heat capacity of the airframe is large and if we also assume the conduction of heat away from the inner surface of the tile insulation is high, we can safely model the airframe as a lump mass.
From the above, we can apply conservation of energy to the heat transfer. In order for there to exist a steady state:
Q
k = Q
r + Q
c
So for steady state heat transfer, the heat capacity of the material does not concern us. What we can glean from this analysis is that for two materials with identical thermal conductivity and emissivity but different heat capacity, the temperature T
h will be the same and by default, the airframe temperature will be the same also since that is our heat sink. Note also that for a steady state condition, the thermal gradient between the hot outer surface of the tile and the airframe is linear, which again is independent of the heat capacity of the material. The next step is to examine what changes due to there being a transient.
We can break up the transient into the heating up phase and the cooling down phase. During the heating up phase, the difference is that there exists a nonlinear temperature gradient between the hot, outer surface and the cool airframe. The inside of the tile heats up at a rate depending on the heat capacity. During the heating up of the tile, as we look at the temperature on the outer surface, the temperature will drop quickly and flatten out to a temperature near the initial temperature of the tile. As time goes by, heat added by thermal conductivity raises each ‘layer’ (dx) of the tile as a function of its heat capacity, the lower the heat capacity, the more quickly the temperature profile reaches steady state. The higher the heat capacity, the slower the temperature profile reaches steady state. But at no time does the thermal gradient exceed the steady state profile. Each layer inside the insulation is at or below the temperature reached during steady state, the only difference being how quickly it reaches steady state. Remember, we’re comparing two tiles with equal emissivity and equal thermal conductivity, one having a higher heat capacity than the other.
The cooling down phase is slightly different. It starts with the linear temperature profile and as the outer surface cools down, the inner layers remain at the steady state profile until the heat can be removed from the outer surface. Heat still flows into the airframe from the tile and this heat flux is a linear function of the change in temperature dT at the inner surface of the insulation. In other words, the higher the rate of change of temperature at the inner surface of the insulation (dT), the higher the heat transfer rate at that surface. Note this is true throughout the transient and the steady state condition. The slope of the line, being the temperature profile, at the inner surface is proportional to the heat being transferred into our lump mass airframe.
So during this cool down phase, the hotter, outer layers of the insulation cool down first. This cooling results in the outer layers of the insulation getting cooler while the inner layers are largely unaffected. You can imagine a bell shaped curve being set up where before, there was a linear one from a high temperature to a low one. The shape of this bell curve will always be below the linear line, so the rate of heat transfer into the airframe at the inner surface will only drop over time as it cools. The rate at which it cools is only a function of the heat capacity of the insulation. The higher the heat capacity, the slower this curve will develop and the more heat energy the insulation retains over time. Although a higher heat capacity will mean that during cool down, the airframe will absorb more heat, this amount of heat is negligable. But also consider that a higher heat capacity will mean that during the heating phase, the amount of heat transfer to the airframe will be less. So assuming these two phases are roughly the same length of time, there is no change in the total amount of energy transmitted to the airframe.
I don’t see any way for heat capacity to significantly alter the total amount of thermal energy transferred to the airframe. The airframe (lump mass) is not significantly heated during reentry so any slight differences between the warm up and cool down transients can’t be noticeably. Getting back to the OP:
cepheid said:
I came across an example figure in my first-year physics textbook depicting a tile of the same material used for the heat shield on the space shuttle. The tile is hot enough to be glowing red, and yet a person is holding it by the edges. The caption explains that this is due to the "extremely small thermal conductivity and small heat capacity of the material."
I think what the book is referring to when it talks about the heat capacity, is the ability of someone to hold this tile shortly after we remove the source of heat. I don't think it has anything to do with the function of the tile on the shuttle.
