Question 12.14: Effect of Heat Transfer on Flow Velocity Starting with the d...

It is to be shown that flow velocity increases with heat addition in subsonic Rayleigh flow and that the opposite occurs in supersonic flow.
Assumptions 1 The assumptions associated with Rayleigh flow are valid. 2 There are no work interactions and potential energy changes are negligible.
Analysis Consider heat transfer to the fluid in the differential amount of δq. The differential form of the energy equations can be expressed as

\delta q=d h_{0}=d\left(h+\frac{V^{2}}{2}\right)=c_{p} d T+V d V                   (1)

Dividing by c_{p} T and factoring out dV/V give

\frac{\delta q}{c_{p} T}=\frac{d T}{T}+\frac{V d V}{c_{p} T}=\frac{d V}{V}\left(\frac{V}{d V} \frac{d T}{T}+\frac{(k-1) V^{2}}{k R T}\right)                   (2)

where we also used c_{p}=k R /(k-1) . Noting that M a^{2}=V^{2} / c^{2}=V^{2} / k R T and using Eq. 7 for dT/dV from Example 12–13 give

\frac{\delta q}{c_{p} T}=\frac{d V}{V}\left(\frac{V}{T}\left(\frac{T}{V}-\frac{V}{R}\right)+(k-1) Ma ^{2}\right)=\frac{d V}{V}\left(1-\frac{V^{2}}{T R}+k Ma ^{2}- Ma ^{2}\right)                       (3)

Canceling the two middle terms in Eq. 3 since V^{2} / T R=k Ma ^{2} and rearranging give the desired relation,

\frac{d V}{V}=\frac{\delta q}{c_{p} T\left(1- Ma ^{2}\right)}                             (4)

In subsonic flow, 1-M a^{2}>0 and thus heat transfer and velocity change have the same sign. As a result, heating the fluid (\delta q>0) increases the flow velocity while cooling decreases it. In supersonic flow, however,  1 -M a^{2}<0  and heat transfer and velocity change have opposite signs. As a result, heating the fluid (\delta q>0) decreases the flow velocity while cooling increases it (Fig. 12–55).

Discussion Note that heating the fluid has the opposite effect on flow velocity in subsonic and supersonic Rayleigh flows.