Heat Addition

The equations governing the infinitesimal changes in a constant area flow due to the addition of a small amount of heat are:

Let us estimate the relative magnitudes of the various effects for small flow velocity u (nondimensionally, for small Mach number.) For example, from (1) we see that the absolute change in density is much larger than the absolute change in velocity ,while from (2) it is seen that the absolute change in pressure is much smaller than the absolute change in velocity, hence even less compared to the change in density. The thermodynamic changes are pretty much isobaric.

This contrasts with isentropic flow where pressure and density changes ar similar in magnitude. This is not isentropic flow; the fluid has not enough inertia at low velocity to support significant pressure differences, but density differences are not restricted by inertia.

For constant pressure, in (3) the variation in h is due to the density variation, and since the changes in velocity are small compared to the density changes, (1), almost all of the heat in (3) goes into h, heating the flow, and very little into kinetic energy.

We conclude that for low Mach numbers for the flow with heat addition:

• The flow is approximately isobaric.
• Almost all the heat goes into raising the enthalpy, hence the temperature, of the gas. This will raise the speed of sound.
• The temperature increase is reflected in a corresponding decrease of density.
• The corresponding expansion of the fluid volume forces the fluid to go faster to preserve mass, (1).
• However, the increase in velocity is small compared to the increase in speed of sound.

So why does the Mach number go up? The problem is that a small Mach number is a lot more sensitive to changes in velocity than to changes in density. In particular, the relative change in Mach number is equal to the relative change in velocity minus the relative change in speed of sound:

According to continuity (1), the relative change in velocity is equal and opposite to the relative change in density. On the other hand, since a is proportional to the square root of temperature, the relative change in speed of sound is equal to only half the relative change in temperature, or, since temperature and density are inversely proportional, to minus half the relative change in density.

The bottom line is that while the absolute change in velocity is small, the relative change is twice the relative change in speed of sound, so the increase in velocity dominates the Mach number. Note that the velocity change is driven by the temperature-induced expansion of the fluid.