Problem 1. (Total 10 pts) You have a large tank of stationary air with stagnatio

Question

Problem 1. (Total 10 pts) You have a large tank of stationary air with stagnation temperature To 305 K and stagnation pressure po (which we will leave unspecified for now). Air flows out of the tank through the converging nozzle pictured below, into surroundings having pressure ps 101.3 kPa. The nozzle has exit area Ae 1.25-10-5 m2. (In this problem use the tables in the Appendix, using linear interpolation as necessary.) Ps Po, T (a) (4 pts) Your goal is to make the air temperature in the exit plane of the nozzle, which we'll call Te, as low as possible. What is that minimum value of Te? What value, or range of values, of the tank pressure, po, do we need to achieve that temperature? (b) (6 pts) Suppose that po is set so that the exit plane static pressure, Pe, is exactly equal to ps What's the momentum flow rate (in units of Newtons) of air out of the nozzle exit?

Explanation / Answer

a) For minimum temperature at exit Mach no. Should be equal to 1

Which means To/T =1

Where To is stagnation temperature in tank

T is temperature at exit

Now ,

To/T = 1+ [(y-1)× M^2]/2

At M=1

To/T = (1+y)/2

Where y is ratio of specific heats = 1.4

Substituting we get To/T = 1.2

Now To = 305 K (given)

T = 254.166 K

Assuming the process to be reversible adiabatic then

Pe/Po = (T/To)^ (y/(y-1))

Since exit pressure not given so taking it to be surrounding pressure = 101.3 kPa

Po = 191.755 kPa

Thus pressure in tank should be 191.755 kPa

b) momentum flow rate = Pe × A

Where A = 1.25 × 10^(-5) m^2 (given)

Momentum flow rate = 101.3 × 1.25 × 10^(-2) N

= 1.266 N

  

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