Jay_ said:
Post me the link. I need to complete this without much more delay. :-/
Hi Jay. Unfortunately, I misspoke. I did not bookmark the link, and now I can't find it again. Sorry.
Back to heat transfer in flow over a flat plate, and its relationship to heat transfer in axial flow over a cylinder. Imagine in the case of a cylinder that you increase its radius until it is infinite. Then you end up with a flat plate. So you know that a flat plate is going to be a limiting case of a cylinder. All the "action" in heat transfer and momentum transfer in axial gas flow over a cylinder occurs in very close proximity to the surface; this is the so-called boundary layer region. Outside the boundary layer, the temperature and velocity variation in the air are unaware that the cylinder is there. If the boundary layer is very thin compared to the radius of the cylinder, the curvature of the cylinder is unimportant in determining the heat transfer through the boundary layer, and the system can be treated as a flat plate.
Even though the car is moving through stagnant air, if the car velocity is constant, it doesn't matter whether you regard the car as moving and the air stagnant, or the car stationary and the air blowing backwards over the exhaust. This is just a change in inertial reference frame. But, it makes it much easier to analyze what is happening. So if air is blowing backwards axially along the tailpipe, the leading edge of the tailpipe is at the muffler. The boundary layer builds up in thickness from the muffler (leading edge) toward the back of the tailpipe, in proportion to the square root of distance backward. Since heat has to be conducted through the boundary layer to the fresh air in the free stream, this causes the heat transfer coefficient to decrease with distance measured backwards from the muffler. This can now be recognized as something very similar to what happens in heat transfer to a gas flowing along a flat plate, starting at the leading edge.
The question of the validity of using the correlation for flow over a flat plate to describe flow and heat transfer over a cylinder hinges on the thickness of the thermal boundary layer compared to the radius of the cylinder. If this ratio is low compared to unity, then it is valid to use the flat plate approximation. Let's see how this plays out in our situation. The equation for the heat flux through the boundary layer is given by:
q= k\frac{ΔT}{δ}
where δ is the boundary layer thickness and ΔT is the wall temperature minus the free-stream air temperature. This is the equation we used in freshman physics for heat conduction through a slab of thickness δ. The only question is, "what is the boundary layer thickness δ?" Now, from our heat transfer correlation, we also know that:
q = h ΔT
If we combine these two equations, we get:
h=\frac{k}{δ}
Now, the local Nusselt number for flow over a flat plate from our correlation is given by:
Nu_x=\frac{hx}{k}
where x is the distance from the leading edge of the plate (i.e., in our situation, the distance measured backwards from the muffler). If we combine these two equations, we obtain:
δ=\frac{x}{Nu_x}
So, for x = 10 cm, we calculate δ≈0.5 cm. This compares with a cylinder radius of 3.5 cm, so the ratio of the boundary layer thickness to the cylinder radius is only ≈ 0.15. This validates the use of the flat plate approximation for our situation.
There are lots of factors that can affect the accuracy of the heat transfer coefficient we calculate, and the uncertainties are large (over and above neglecting the curvature of the tailpipe surface). The biggest one is the effect of the muffler. Its presence prevents the existence of a sharp leading edge to the tailpipe. There is also heat from the muffler entering the free-stream air that blows backwards over the tailpipe, and this affects the temperature difference ΔT. There can also be a circulation zone (stagnation zone) that forms in the wake of the muffler which will affect both the air flow and the heat transfer.
To model the heat transfer coefficient very accurately, one would have to construct a computational fluid dynamics (CFD) model of the flow and heat transfer that includes air flow over both the muffler and the exhaust pipe. This is well beyond the scope of what your assignment calls for. Therefore, we will have to live with the inaccuracies, and do the best we can with the correlations we have (using our best judgement). We have gotten about as much as we can out of this approach. That's why I was suggesting earlier some calibration experiments which measure both the exhaust gas temperature out of the tailpipe (or in the muffler) and the surface temperature of the exhaust pipe.
Chet
