A nozzle for an incompressible, inviscid fluid of density rho = 1000 kg/m3 consi

Question

A nozzle for an incompressible, inviscid fluid of density rho = 1000 kg/m3 consists of a converging section of pipe. At the inlet (lie diameter is Di = 100 mm, and at the outlet the diameter is D0 = 20 mm. The nozzle length is L = 500 mm, and the diameter decreases linearly with distance x along the nozzle. Derive and plot the acceleration of a fluid particle, assuming uniform flow at each section, if the speed at the inlet is Vi = 5 m/s. Plot the pressure gradient through the nozzle, and find its maximum absolute value. If the pressure gradient must be no greater than 5 MPa/m in absolute value, how long would the nozzle have to be?

Explanation / Answer

incompressible flow .. so according to mass conservation v A= constant given A is function of D^2 and D= kx + c ; x=0 D =100 x= 500 D = 20 k = -0.16 implies v(A)= v+ dx.dv/dx (A+dx.dA/dx) dA/dx = 2 (A/d) so Adx. dv/dx = - v dx dA/dx or dv/dx = -v 2A/d A/d = cd=c(kx+c) v= v1 e^(-0.08k1x^2 + 500 k1 x +c) at x=0 v=v1 implies c = 0 ; at x = 500 v= 125 so k1 = 1.4 *10^-5 dv/dt = accleration = -V^2*2A/d = -v^2*2pi*d

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