Chapter 16 : Compressible Flow

Example 16.1 page no : 859

In [1]:
import math 

# Variables
V1 = 300.; 			#m/s
p1 = 78.; 			#kN/m**2
T1 = 313.; 			#K
p2 = 117.; 			#kN/m**2
R = 287.; 			#J/kg K
y = 1.4;

# Calculations
#Let r1 = p1/rho1
r1 = R*T1;
V2 = math.sqrt(2*(y/(y-1)*r1*(1-(p2/p1)**((y-1)/y)) + V1**2/2));

# Results
print ("Velocity of gas at section 2  = %.3f")% (V2), ("m/s")

Velocity of gas at section 2  = 112.987 m/s

Example 16.2 page no : 861

In [1]:
# Variables
p1 = 35.; 			#kN/m**2
V1 = 30.; 			#m/s
T1 = 423.; 			#K
V2 = 150.; 			#m/s
R = 290.; 			#J/kg K
y = 1.4;

# Calculations
#Let r1 = p2/p1
r1 = R*T1; 
p2 = p1*(1-((V2**2/2-V1**2/2)*(y-1)/y/r1))**(y/(y-1));
print ("pressure = %.3f")% (p2), ("kN/m**2")

T2 = T1*(p2/p1)**((y-1)/y);
t2 = T2-273;
print ("temperature  = %.3f")% (t2), ("°C")

pressure = 32.014 kN/m**2
temperature  = 139.360 °C

Example 16.3 page no : 866

In [3]:
import math 

# Variables
SG = 0.8;
rho_oil = 800.; 			#kg/m**3
K_oil = 1.5*10**9; 			#N/m**2; crude oil
K_Hg = 27*10.**9; 			#N/m**2; Mercury
rho_Hg = 13600.; 			#kg/m**3

# Calculations and Results
C_oil = math.sqrt(K_oil/rho_oil);
print ("Sonic velocity of crude oil  = %.3f")% (C_oil), ("m/s")

C_Hg = math.sqrt(K_Hg/rho_Hg)
print ("Sonic velocity of Mercury  = %.3f")% (C_Hg), ("m/s")

Sonic velocity of crude oil  = 1369.306 m/s
Sonic velocity of Mercury  = 1409.005 m/s

Example 16.4 page no : 866

In [4]:
import math 

# Variables
T = 228.; 			#K
M = 2.;
R = 287.; 			#Jkg K
y = 1.4;

# Calculations
C = math.sqrt(y*R*T);
V = M*C*3600./1000;

# Results
print ("Velocity of the plane  = %.3f")% (V), ("km/h")

Velocity of the plane  = 2179.239 km/h

Example 16.5 page no : 868

In [5]:
# Variables
a = 40*math.pi/180; 			#Mach angle in radians
y = 1.4;
R = 287.; 			#J/kg K
T = 288.; 			#K

# Calculations
C = math.sqrt(y*R*T);
V = C/math.sin(a);

# Results
print ("Velocity of bullet  = %.3f")% (V), ("m/s")

Velocity of bullet  = 529.217 m/s

Example 16.6 page no : 868

In [6]:
import math 

# Variables
p = 88.3; 			#kN/m**2
T = 271.; 			#K
M = 40.*math.pi/180;
y = 1.4;
R = 287.; 			#J/kg K

# Calculations
C = math.sqrt(y*R*T);
V = C/math.sin(M);

# Results
print ("Velocity of the projectile  = %.3f")% (V), ("m/s")

Velocity of the projectile  = 513.360 m/s

Example 16.7 page no : 868

In [7]:
import math 

# Variables
h = 1800.; 			#m
T = 277.; 			#K
t = 4.; 			#s
y = 1.4;
R = 287.; 			#J/kg K

# Calculations
C = math.sqrt(y*R*T);
a = (math.cos(C/h*t));
V = C/math.sin(a)*3600/1000;

# Results
print ("Speed of the aircraft  = %.3f")% (V), ("km/h")

Speed of the aircraft  = 1785.959 km/h

Example 16.8 page no : 873

In [9]:
import math 

# Variables
R = 287.; 			#J/kg K
y = 1.4;
V0 = 1000.*1000/3600; 			#m/s
p0 = 78.5; 			#kN/m**2
T0 = 265.; 			#K

# Calculations and Results
C0 = math.sqrt(y*R*T0);
M0 = V0/C0;

ps = p0*(1+((y-1)/2*M0**2))**(y/(y-1));
print ("(i) Stagnation pressure  = %.3f")%(ps), ("kN/m**2")

Ts = T0*(1+((y-1)/2*M0**2));
print ("(ii) Stagnation temperature  = %.3f")% (Ts), ("K")

rho_s = ps*10**3/R/Ts;
print ("(iii) Stagnation density  = %.3f")%(rho_s), ("kg/m**3")

(i) Stagnation pressure  = 126.067 kN/m**2
(ii) Stagnation temperature  = 303.407 K
(iii) Stagnation density  = 1.448 kg/m**3

Example 16.9 page no : 874

In [10]:
import math 

# Variables
V0 = 1000.*1000./3600; 			#m/s
T0 = 320.; 			#K
p_atm = 98.1; 			#kN/m**2
p = 9.81; 			#kN/m**2
p0 = 98.1-p;
R = 287.; 			#J/kg K
y = 1.4;

# Calculations and Results
C0 = math.sqrt(y*R*T0);
M0 = V0/C0;

ps = p0*(1+((y-1)/2*M0**2))**(y/(y-1));
print ("Stagnation pressure  = %.3f")% (ps), ("kN/m**2")

Ts = T0*(1+((y-1)/2*M0**2));
print ("Stagnation temperature  = %.1f")% (Ts), ("K")

rho_s = ps*10**3/R/Ts;
print ("Stagnation density  = %.3f")% (rho_s), ("kg/m**3")

M = 0.8;

CF = 1+(M0**2./4)+((2-y)/24.*M0**4);
print ("Compressibility factor %.2f")% (CF)

#Note : Answers are slightly different because of rounding error.

Stagnation pressure  = 131.282 kN/m**2
Stagnation temperature  = 358.4 K
Stagnation density  = 1.276 kg/m**3
Compressibility factor 1.16

Example 16.10 page no : 875

In [12]:
import math 

# Variables
R = 287.; 			#J/kg K
y = 1.4;
p0 = 220.*10**3; 	#N/m**2
T0 = 300.; 			#K
V0 = 200.; 			#m/s

# Calculations and Results
C0 = math.sqrt(y*R*T0);
rho_0 = p0/R/T0;
print ("Stagnation pressure  = ")

print ("(i) Compressibility is neglected")
ps = (p0+rho_0*V0**2/2)/10**3;
print ("ps = %.3f")% (ps), ("kN/m**2")

print ("(ii) Compressibility is accounted for")
M0 = V0/C0;

ps = (p0+rho_0*V0**2/2*(1+M0**2./4+(2-y)/24*M0**4))/10**3;
print ("ps = %.3f")% (ps), ("kN/m**2")

Stagnation pressure  = 
(i) Compressibility is neglected
ps = 271.103 kN/m**2
(ii) Compressibility is accounted for
ps = 275.484 kN/m**2

Example 16.11 page no : 875

In [13]:
import math 

# Variables
p0 = 35.*10**3; 			#Pa
T0 = 235.; 			#K
ps = 65.4*10**3; 			#N/m**2
R0 = 8314.; 			#Nm/mole K
M = 28.;

# Calculations
R = R0/M;
rho_0 = p0/R/T0;
Va = math.sqrt(2*(ps-p0)/rho_0);

# Results
print ("Speed of the aircraft  = %.3f")% (Va), ("m/s")

Speed of the aircraft  = 348.159 m/s

Example 16.12 page no : 885

In [14]:
import math 

# Variables
p0 = 30.*10**3; 			#N/m**2
V0 = 152.; 			#m/s
y = 1.4;
rho_0 = 1.224; 			#kg/m**3
ps = p0+rho_0*V0**2/2;

# Calculations
rho_0 = 0.454; 			#kg/m**3
V0 = math.sqrt(2*(ps-p0)/rho_0);
C0 = math.sqrt(y*p0/rho_0);
M = V0/C0;
ccf = (1+M**2/4); 			#Compressibility correction factor
V = V0/math.sqrt(ccf); 			#True speed of aircraft

# Results
print ("True speed of aircraft  = %.3f")% (V), ("m/s")

True speed of aircraft  = 230.900 m/s

Example 16.13 page no : 886

In [2]:
import math 

# Variables
M = 3.; 			#Mach number
d = 0.2; 			#m
p_nozzle = 7.85; 			#kN/m**2
T_nozzle = 200.; 			#K
y = 1.4;

# Calculations and Results
A = math.pi/4*d**2;
p_res = p_nozzle*(1+((y-1)/2*M**2))**(y/(y-1));
print ("Reservoir pressure  = %.3f")% (p_res), ("kN/m**2")

T_res = T_nozzle*(1+((y-1)/2*M**2));
print ("Reservoir temperature  = %.3f")% (T_res), ("K")

Ac = A*M/((2+(y-1)*M**2)/(y+1))**((y+1)/2/(y-1));
print ("Throat area (critical)  = %.5f")% (Ac), ("m**2")

Reservoir pressure  = 288.352 kN/m**2
Reservoir temperature  = 560.000 K
Throat area (critical)  = 0.00742 m**2

Example 16.14 page no : 887

In [16]:
import math 

# Variables
R = 287.;   			#J/kg K
y = 1.4;
p_atm = 100.; 			#kN/m**2
p1 = 284. + p_atm; 			#kN/m**2
T1 = 297.; 	    		#K
D = 0.02; 		    	#m

# Calculations
A2 = math.pi/4*D**2;
rho_1 = p1*10**3/R/T1;
m_max = 0.685*A2*math.sqrt(p1*10**3*rho_1);

# Results
print ("Maximum flow rate  = %.3f")% (m_max), ("kg/s")

Maximum flow rate  = 0.283 kg/s

Example 16.15 page no : 888

In [17]:
# Variables
R = 287.; 			#J/kg K
y = 1.4;
p1 = 2500.*10**3; 	#N/m**2
T1 = 293.; 			#K
p2 = 1750.*10**3; 	#N/m**2

# Calculations
rho_1 = p1/R/T1;
V2 = math.sqrt(2*y/(y-1)*p1/rho_1*(1-(p2/p1)**((y-1)/y)));

# Results
print ("Velocity of air  = %.3f")% (V2),("m/s")

Velocity of air  = 238.812 m/s

Example 16.16 page no : 889

In [18]:
import math 

# Variables
R = 287.; 	    		#J/kg K
y = 1.4;
p_atm = 10.**5; 		#N/m**2
T1 = 293.    			#K
D2 = 0.025; 			#m
p1 = 140.*10**3; 		#N/m**2

# Calculations and Results
A2 = math.pi/4*D2**2;

print ("(i) Mass rate of flow of air when pressure in the math.tank is 140 kN/m2 (abs.)")
rho_1 = p1/R/T1;
p2 = 10**5; 			#N/m**2

m = A2*math.sqrt(2*y/(y-1)*p1*rho_1*((p2/p1)**(2/y) - (p2/p1)**((y+1)/y)));
print ("m = %.3f")% (m), ("kg/s")

print ("(ii) Mass rate of flow of air when pressure in the math.tank is 300 kN/m2 (abs.)")
p1 = 300.*10**3; 			#N/m**2
p2 = 10.**5; 			#N/m**2
rho_1 = p1/R/T1;

print ("The pressure ratio p2/p1 being less than the critical ratio 0.528, the flow in the nozzle will be sonic");

m_max = 0.685*A2*math.sqrt(p1*rho_1);
print ("m_max = %.3f")% (m_max), ("kg/s")

(i) Mass rate of flow of air when pressure in the math.tank is 140 kN/m2 (abs.)
m = 0.149 kg/s
(ii) Mass rate of flow of air when pressure in the math.tank is 300 kN/m2 (abs.)
The pressure ratio p2/p1 being less than the critical ratio 0.528, the flow in the nozzle will be sonic
m_max = 0.348 kg/s

Example 16.17 page no : 890

In [3]:
import math 

# Variables
p1 = 200.; 			#kN/m**2
V1 = 170.; 			#m/s
T1 = 473.; 			#K
A1 = 0.001; 		#m**2
R = 287.; 			#J/kg K
cp = 1000.; 		#J/kg K
y = 1.4;

# Calculations and Results
print ("(i) Stagnation temperature (Ts) and stagnation pressure (ps)")

Ts = T1+V1**2/2/cp;
print ("Ts = %.3f")% (Ts), ("K")

ps = p1*(Ts/T1)**(y/(y-1));
print ("ps = %.3f")% (ps), ("kN/m**2")

print ("(ii) Sonic velocity and Mach number at this section")

C1 = math.sqrt(y*R*T1);
print ("Sonic velocity  = %.3f")% (C1), ("m/s")

M1 = V1/C1;
print ("Mach number  =  %.3f")% (M1)

print ("(iii) Velocity, Mach number and flow area at outlet section where pressure is 110 kN/m2")
p2 = 110.; 			#kN/m**2
M2 = math.sqrt(2/(y-1)*((ps/p2)**((y-1)/y) - 1));
print ("M2 = %.3f")% (M2)

T2 = Ts*(p2/ps)**((y-1)/y);
C2 = math.sqrt(y*R*T2);
V2 = M2*C2;
print ("V2 = %.3f")% (V2), ("m/s")

A2 = (p1*A1*V1*T2/T1/p2/V2)*10**6;
print ("A2 = %.3f")% (A2), ("mm**2")


print ("(iv) Pressure (pt), temperature (Tt), velocity (Vt), and flow area (At) at throat of the nozzle")
Mt = 1.;
Tt = Ts/(1+(y-1)/2*Mt**2);
print ("Tt  = %.3f")% (Tt), ("K")

pt = ps*(Tt/Ts)**(y/(y-1));
print ("pt = %.3f")% (pt), ("kN/m**2")

Ct = math.sqrt(y*R*Tt);
Vt = Mt*Ct;

At = (p1*A1*V1*Tt/T1/pt/Vt)*10**6;
print ("At = %.3f")% (At), ("mm**2")

(i) Stagnation temperature (Ts) and stagnation pressure (ps)
Ts = 487.450 K
ps = 222.214 kN/m**2
(ii) Sonic velocity and Mach number at this section
Sonic velocity  = 435.949 m/s
Mach number  =  0.390
(iii) Velocity, Mach number and flow area at outlet section where pressure is 110 kN/m2
M2 = 1.055
V2 = 422.183 m/s
A2 = 617.168 mm**2
(iv) Pressure (pt), temperature (Tt), velocity (Vt), and flow area (At) at throat of the nozzle
Tt  = 406.208 K
pt = 117.392 kN/m**2
At = 615.673 mm**2

Example 16.18 page no : 893

In [20]:
import math 

# Variables
y = 1.4;
p1 = 26.5; 			#kN/m**2
rho_1 = 0.413; 			#kg/m**3
M1 = 2.;
R = 287.;

# Calculations and Results
M2 = math.sqrt(((y-1)*M1**2 + 2)/(2*y*M1**2 - (y-1)));
print ("Mach number M2 = %.3f")% (M2)

p2 = p1*(2*y*M1**2 - (y-1))/(y+1);
print ("p2 = %.3f")% (p2), ("kN/m**2")

rho_2 = rho_1*((y+1)*M1**2)/((y-1)*M1**2 + 2);
print ("density, rho_2  = %.3f")% (rho_2), ("kg/m**3")

T1 = p1*10**3/rho_1/R;
print ("T1 = %.3f")% (T1), ("K")

T2 = T1*((y-1)*M1**2 + 2)*(2*y*M1**2 - (y-1))/((y+1)**2*M1**2);
print ("T2 = %.3f")% (T2), ("K")

C1 = math.sqrt(y*R*T1);
V1 = M1*C1;
print ("V1 = %.3f")% (V1), ("m/s")

C2 = math.sqrt(y*R*T2);
V2 = M2*C2;
print ("V2  = %.3f")% (V2), ("m/s")

Mach number M2 = 0.577
p2 = 119.250 kN/m**2
density, rho_2  = 1.101 kg/m**3
T1 = 223.570 K
T2 = 377.275 K
V1 = 599.435 m/s
V2  = 224.788 m/s

Example 16.19 page no : 895

In [21]:
import math 

# Variables
M1 = 1.5;
p1 = 170.; 			#kN/m**2
T1 = 296.; 			#K
y = 1.4;

# Calculations and Results
print ("(i) Pressure, temperature and Mach number downstream of the shock")
p2 = p1*(2*y*M1**2 - (y-1))/(y+1);
print ("p2 = %.3f")% (p2), ("kN/m**2")

T2 = T1*((y-1)*M1**2 + 2)*(2*y*M1**2 - (y-1))/(y+1)**2/M1**2;
print ("T2 = %.3f")% (T2), ("K")

M2 = math.sqrt(((y-1)*M1**2 + 2)/(2*y*M1**2 - (y-1)));
print ("M2 = %.3f")% (M2)

strength = p2/p1 - 1;
print ("Strength of stock  = %.3f")% (strength)

(i) Pressure, temperature and Mach number downstream of the shock
p2 = 417.917 kN/m**2
T2 = 390.784 K
M2 = 0.701
Strength of stock  = 1.458