Question

Carbon dioxide flows steadily at 1 kg/s through a device where the pressure, specific volume, and velocity of the gas are tripled according pVn = C. The inlet conditions are p1 = 200 kPa, v1 = 0.4 m3/kg, and v1 = 100 m/s. Determine (a) n; (b) the initial and final temperatures; (c) the change of enthalpy; (d) the power; (e) the heat.

Answer 1

Solution;

Given;

$\dot{m}=1kg/s$

$p_1=200kPa$

$V=0.4m^3/kg$

$v_1=100m/s$

$p_2=3p_1=600kPa$

$V_2=3V_1=1.2m^3/kg$

$v_2=3v_1=300m/s$

(a)

$p_1V_1^n=p_2V_2^n$

$\frac{200}{600}=(\frac{1.2}{0.4})^n$

$\frac13=3^n$

$3^{-1}=3^n$

Therefore;

$n=-1$

(b)

From;

$PV=mRT$

$M_{co_2}=44$

Hence;

$R=\frac{8.314}{44}=0.18895$

Therefore;

$T_1=\frac{pV}{mR}=\frac{200×0.4}{1×0.18895}=423.38K$

$T_2=\frac{p_2V_2}{mR}=\frac{600×1.2}{1×0.18895}=3810.53K$

(c)

$\Delta h=c_p\Delta T$

$c_p=0.756kJ/kgK$

$\Delta h=0.756×(3810.53-423.38)$

$\Delta h=2560.69kJ/kg$

(d)

Power = Change in Kinetic Energy

$P=\frac12m(v_2^2-v_1^2)$

$P=\frac12×1(300^2-100^2)$

$P=40000W$

$P=40kW$

(e)

Heat

Assuming the device is horizontal;

$z_1=z_2$

From steady flow energy equation;

$Q=\Delta h+\dot{m}(\frac{v_1^2-v_2^2}{2000})+W$

By direct substitution;

$Q=2560+40+40=2640.69kW$