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Gas velocity at nozzle exit

Helium gas enters an adiabatic nozzle steadily at 500 degrees C and 600 kPa with a low velocity, and exits at a pressure of 90 kPa. What is the highest possible velocity of helium gas at the nozzle exit?

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Consider a small mass element of the gas as it moves with the flow. Suppose the flow velocity is specified as a function of position and time as v(x,t). If at time t_{0} the mass element is at position x_{0}, then after a small time interval dt has passed it will be at position:

x_{0} + v dt.

The change in velocity of the mass element is thus:

v(x_{0} + v(x_{0}, t) dt, t + dt) - v(x_{0},t) = [dv/dt + v dot grad v] dt

Here dv/dt is the partial derivative w.r.t. time and

v dot grad v = v_{x} dv/dx + v_{y} dv/dy + v_{z} dv/dz

v_{i} is the i-th component of v and the derivatives are partial derivatives.

So, we see that the acceleration is given by the so-called convective derivative:

dv/dt + v dot grad v

According to Newton's second Law, ...

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