Macro-Level Explanation of Fluid Heat Transfer

Question

Here's something I've been struggling with for a while.

Say you have a hot fluid and cold surface (or the reverse, so long as there is a temperature difference) where the fluid flows over the surface. I know from heat transfer class that the heat transfer rate goes as the heat transfer coefficient, which increases with relative velocity. Looking at the equations, this makes sense.

What I'm struggling with is a qualitative understanding on the macro level. To help demonstate my issue, here's another example

There are two long slabs each of a different temperature. They are drawn across each other at some velocity. The total heat transfer which occurs while the slabs are in contact increases as velocity decreases. This is the opposite of what would happen if one slab were a fluid.

Now we could change one slab to a fluid, then increase its viscosity until it behaves like a solid, so to me, my conclusion that they behave differently must be non-physical. Why?

Some ascii graphics for the second example:

+---------------------+
|    slab 1: v = x    |                                          FRAME 1
+---------------------+  +-----------------------+
                         |      slab 2: v = 0    |
                         +-----------------------+       

          +---------------------+
          |    slab 1: v = x    |                                FRAME 2
          +--------------+------+----------------+
                         |      slab 2: v = 0    |
                         +-----------------------+

                  +---------------------+
                  |    slab 1: v = x    |                        FRAME 3
                  +------+--------------+--------+
                         |      slab 2: v = 0    |
                         +-----------------------+              

Answer 1

I think that you are trying to compare two situations which actually are quite different. In the solid/solid contact case, the two solids are in contact with each other for a shorter amount of time as you go faster. If you made slab 2 very very long so that slab 1 is always in contact with it and then moved it at different speeds, but doing all of them for the same amount of time, then the one that moves the fastest would have the highest heat transfer (neglecting friction heating, etc.).

Why is this the case? For solids or fluids, the rate of heat transfer between to surfaces is directly proportional to the temperature difference between them, this is Fick's law $q=-k\frac{dT}{dx}$, where $k$ is just a proportionality constant.

So now think about a packet of fluid (or solid) moving across a solid surface. When it first comes in contact with the solid surface, the heat flux $q$ is high because the temperature difference $dT$ is high. As it absorbs (or loses) heat though, the temperature difference decreases and so $q$ decreases as well.

If you move slower, this effect becomes pronounced, and eventually the two surfaces can reach the same temperature, so there is no heat transfer between them. But if you move faster, $q$ remains high because any individual packet of liquid (or solid) is not in contact with the solid surface long enough for it to change its temperature much.

Hence, if you move a cool fluid quickly across a hot solid surface, the solid loses more heat than if the fluid is moving slowly.

For a solid/solid surface analogy, imagine a hot iron on an ice skating rink. From the perspective of the iron, if you move it slowly the iron won't lose as much heat because it iron will have time to heat the ice beneath it, perhaps melting or even boiling it depending on the energy output of the iron and how slowly you move it. From the ice's perspective, any portion of ice that is in contact with the iron absorbs a lot of energy from the iron because the iron is in contact with it for such a long time.

However if you move the hot iron across the ice rink surface very quickly, the opposite happens. The iron loses a lot of heat because it doesn't have time to heat up any individual portion of ice, it's constantly being refreshed with fresh cold ice. From the ice's perspective any individual location on the ice only sees the iron for a very short amount of time, so the ice doesn't melt, it barely heats up at all.