Nitrogen enters a steady-flow heat exchanger at 150 kPa, 10°C, and 100 m/s, and it receives heat as it flows through it. Nitrogen leaves the heat exchanger at 100 kPa with a velocity of 200 m/s.

a) Determine the Mach number of the nitrogen at the inlet and the amount of heat received if the exit Mach number is 0.51.

A) Mach number at Inlet; M1 = 0.292

B) amount of heat received; Q_in = 105.59 KJ/kg

Explanation:

A) The formula for the speed of nitrogen at the inlet is;

C1 = √kRT1

Where;

k is a constant = 1.4

R is the gas constant of nitrogen R = 296.8 J/kg. K

T1 is the temperature at inlet = 10°C = 10 + 273 K = 283K

Thus;

C1 = √ (1.4 * 296.8 * 283)

C1 = 342.9 m/s

Formula for mach number at inlet is;

M1 = V1/C1

We are given V1 = 100 m/s

Thus; M1 = 100/342.9

M1 = 0.292

B) We are given mach number at exit; M2 = 0.51

Also; V2 = 200 m/s

Thus; M2 = V2/C2

C2 = V2/M2 = 200/0.51

C2 = 392.16 m/s

Now, we know that;

C2 = √kRT2

So, 392.16 = √ (1.4 * 296.8 * T2)

392.16² = 415.52*T2

T2 = 392.16²/415.52

T2 = 370.11 K

Now formula for steady flow balance equation is;

Q_in = = cp (T2 - T1) + [ (v2) ² - (v1) ²]/2

Where Q_in is the amount of heat energy received and cp is the specific heat of nitrogen, = 1.040 KJ/kg. K

Plugging in the relevant values;

Q_in = 1.04 (370.11 - 283) + [ (200) ² - (100) ²]/2] (1KJ/Kg/1000m²/s²)

Q_in = 90.59 + 15

Q_in = 105.59 KJ/Kg