Question

Air ows steadily in a thermally insulated pipe with a constant diameter of 6.35 cm, and an average friction factor of 0.005. At the pipe entrance, the air has a Mach number of 0.12, stagnation pressure 280 kPa, and stagnation temperature 825 K. Determine:

a. The length of pipe required to reach the sonic state
b. The static pressure and temperature at the exit if the pipe is 25 m long

Answer 1

Given :

D = 6.35 cm

`$\bar f = 0.005$`

`$P_s = 280 \ kPa$`

`$T_s= 825 K`

a). From fanno flow table (γ = 1.4)

At
`$M_1 = 0.12$` ,
`$\left((4 \bar f L_(max))/(D)\right)_1 = 45.408$`

At
`$M_2 = 1$` ,
`$\left((4 \bar f L_(max))/(D)\right)_2 = 0$`

∴
`$\left((4 \bar f L_(max))/(D)\right)_1 - \left((4 \bar f L_(max))/(D)\right)_2 = (4 \bar f L)/(D)$`

`$45.408 - 0 = (4 * 0.005 * L)/(0.0635)$`

`$45.408 = (4 * 0.005 * L)/(0.0635)$`

L = 144.17 m

b). If L = 25 m

`$(4 \bar f L)/(D)=(4 * 0.005 * 25)/(0.0635) = 7.874$`

From fanno flow , (γ = 1.4)

`$At, M_2 = 0.26 , (4 \bar f L)/(D) = 7.874$`

`$(P_s)/(P_1)=(T_s)/(T_1)^{(\gamma)/(\gamma - 1)} = (1+(\gamma-1)/(2)M_1^2)^{(\gamma)/(\gamma - 1)}$`

`$(280)/(P_1)=(825)/(T_1)^{(1.4)/(1.4 - 1)} = (1+(1.4-1)/(2)(0.12)^2)^{(1.4)/(1.4 - 1)}$`

`$(280)/(P_1)=\left((825)/(T_1)\right)^(3.5) =1.0101$`

`$P_1 = 277.2 \ kPa$`

`$T_1=822.63 \ K$`

`$(T_2)/(T_1)=(1+(\gamma -1)/(2)M_1^2)/(1+(\gamma -1)/(2)M_2^2)$`

`$(T_2)/(822.63)=(1+(1.4 -1)/(2)(0.12)^2)/(1+(1.4 -1)/(2)(0.26)^2)$`

`$(T_2)/(822.63)=(1.00288)/(1.01352)$`

`$T_2=814 \ K$`

`$(P_2)/(P_1)=(M_1)/(M_2)\left((1+(\gamma -1)/(2)M_1^2)/(1+(\gamma -1)/(2)M_2^2)\right)^(1/2)$`

`$(P_2)/(P_1)=(0.12)/(0.26)\left((1+(1.4 -1)/(2)(0.12)^2)/(1+(1.4 -1)/(2)(0.26)^2)\right)^(1/2)$`

`$(P_2)/(277.2)=0.459$`

`$P_2=127.27 \ kPa$`