Why is maximum turbojet thrust at nozzle exit pressure equal to ambient pressure?

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extreme534
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Hello,

Im wondering if an analytical proof exists that shows why maximum thrust in an ideal turbojet is obtained when the nozzle exit pressure is expanded to the ambient pressure. As in the equation..

F = mdot*(Ue-Ua) + Ae*(Pe-Pa)

my intuition tells me that we'll have to take a derivative with respect to Pe and set it to zero, but the expression gets a little complicated to perform by hand.
 
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That equation alone isn't enough to tell you why that is the case, since as you vary Pe and Pa, your Ue (exit velocity, I'm assuming) will also vary based on the nozzle parameters. It turns out if you work out the equations for a converging-diverging nozzle (supersonic nozzle), the maximum thrust is obtained when the exit pressure is equal to the ambient pressure - further expansion doesn't gain you enough velocity to offset the pressure drag, and less expansion doesn't give you enough pressure thrust from the high exit pressure to offset the loss in momentum available.