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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Chapter 12: Nozzle"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 12.1: area_and_Mach_number.sce"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clear;\n",
"clc;\n",
"disp('Example 12.1');\n",
"\n",
"// aim : To determine the\n",
"// (a) throat area\n",
"// (b) exit area\n",
"// (c) Mach number at exit\n",
"\n",
"// Given values\n",
"P1 = 3.5;// inlet pressure of air, [MN/m^2]\n",
"T1 = 273+500;// inlet temperature of air, [MN/m^2]\n",
"P2 = .7;// exit pressure, [MN/m^2]\n",
"m_dot = 1.3;// flow rate of air, [kg/s]\n",
"Gamma = 1.4;// heat capacity ratio\n",
"R = .287;// [kJ/kg K]\n",
"\n",
"// solution\n",
"// given expansion may be considered to be adiabatic and to follow the law PV^Gamma=constant\n",
"// using ideal gas law\n",
"v1 = R*T1/P1*10^-3;// [m^3/kg]\n",
"Pt = P1*(2/(Gamma+1))^(Gamma/(Gamma-1));// critical pressure, [MN/m^2]\n",
"\n",
"// velocity at throat is\n",
"Ct = sqrt(2*Gamma/(Gamma-1)*P1*10^6*v1*(1-(Pt/P1)^(((Gamma-1)/Gamma))));// [m/s]\n",
"vt = v1*(P1/Pt)^(1/Gamma);// [m^3/kg]\n",
"// using m_dot/At=Ct/vt\n",
"At = m_dot*vt/Ct*10^6;// throat area, [mm^2]\n",
"mprintf('\n (a) The throat area is = %f mm^2\n',At);\n",
"\n",
"// (b)\n",
"// at exit\n",
"C2 = sqrt(2*Gamma/(Gamma-1)*P1*10^6*v1*(1-(P2/P1)^(((Gamma-1)/Gamma))));// [m/s]\n",
"v2 = v1*(P1/P2)^(1/Gamma);// [m^3/kg]\n",
"A2 = m_dot*v2/C2*10^6;// exit area, [mm^2]\n",
"\n",
"mprintf('\n (b) The exit area is = %f mm^2\n',A2);\n",
"\n",
"// (c)\n",
"M = C2/Ct;\n",
"mprintf('\n (c) The Mach number at exit is = %f\n',M);\n",
"\n",
"// End"
]
}
,
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 12.2: increases_in_pressure_temperature_and_internal_energy.sce"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clear;\n",
"clc;\n",
"disp('Example 12.2');\n",
"\n",
"// aim : To determine the increases in pressure, temperature and internal energy per kg of air\n",
"\n",
"// Given values\n",
"T1 = 273;// [K]\n",
"P1 = 140;// [kN/m^2]\n",
"C1 = 900;// [m/s]\n",
"C2 = 300;// [m/s]\n",
"cp = 1.006;// [kJ/kg K]\n",
"cv =.717;// [kJ/kg K]\n",
"\n",
"// solution\n",
"R = cp-cv;// [kJ/kg K]\n",
"Gamma = cp/cv;// heat capacity ratio\n",
"// for frictionless adiabatic flow, (C2^2-C1^2)/2=Gamma/(Gamma-1)*R*(T1-T2)\n",
"\n",
"T2 =T1-((C2^2-C1^2)*(Gamma-1)/(2*Gamma*R))*10^-3; // [K]\n",
"T_inc = T2-T1;// increase in temperature [K]\n",
"\n",
"P2 = P1*(T2/T1)^(Gamma/(Gamma-1));// [MN/m^2]\n",
"P_inc = (P2-P1)*10^-3;// increase in pressure,[MN/m^2]\n",
"\n",
"U_inc = cv*(T2-T1);// Increase in internal energy per kg,[kJ/kg]\n",
"mprintf('\n The increase in pressure is = %f MN/m^2\n',P_inc);\n",
"mprintf('\n Increase in temperature is = %f K\n',T_inc);\n",
"mprintf('\n Increase in internal energy is = %f kJ/kg\n',U_inc);\n",
"\n",
"// there is minor variation in result\n",
"\n",
"// End"
]
}
,
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 12.3: throat_area_and_degree_of_undercooling.sce"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clear;\n",
"clc;\n",
"disp('Example 12.3');\n",
"\n",
"// aim : To determine the \n",
"// (a) throat and exit areas\n",
"// (b) degree of undercooling at exit\n",
"// Given values\n",
"P1 = 2;// inlet pressure of air, [MN/m^2]\n",
"T1 = 273+325;// inlet temperature of air, [MN/m^2]\n",
"P2 = .36;// exit pressure, [MN/m^2]\n",
"m_dot = 7.5;// flow rate of air, [kg/s]\n",
"n = 1.3;// polytropic index\n",
"\n",
"// solution\n",
"// (a)\n",
"// using steam table\n",
"v1 = .132;// [m^3/kg]\n",
"// given expansion following law PV^n=constant\n",
"\n",
"Pt = P1*(2/(n+1))^(n/(n-1));// critical pressure, [MN/m^2]\n",
"\n",
"//velocity at throat is\n",
"Ct = sqrt(2*n/(n-1)*P1*10^6*v1*(1-(Pt/P1)^(((n-1)/n))));// [m/s]\n",
"vt = v1*(P1/Pt)^(1/n);// [m^3/kg]\n",
"// using m_dot/At=Ct/vt\n",
"At = m_dot*vt/Ct*10^6;// throat area, [mm^2]\n",
"mprintf('\n (a) The throat area is = %f mm^2\n',At);\n",
"\n",
"// at exit\n",
"C2 = sqrt(2*n/(n-1)*P1*10^6*v1*(1-(P2/P1)^(((n-1)/n))));// [m/s]\n",
"v2 = v1*(P1/P2)^(1/n);// [m^3/kg]\n",
"A2 = m_dot*v2/C2*10^6;// exit area, [mm^2]\n",
"\n",
"mprintf('\n The exit area is = %f mm^2\n',A2);\n",
"\n",
"// (b)\n",
"T2 = T1*(P2/P1)^((n-1)/n);//outlet temperature, [K]\n",
"t2 = T2-273;//[C]\n",
"// at exit pressure saturation temperature is\n",
"ts = 139.9;// saturation temperature,[C]\n",
"Doc = ts-t2;// Degree of undercooling,[C]\n",
"mprintf('\n (b) The Degree of undercooling at exit is = %f C\n',Doc);\n",
"\n",
"// There is some calculation mistake in the book so answer is not matching\n",
"\n",
"// End"
]
}
,
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Example 12.4: velocities_and_areas.sce"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"clear;\n",
"clc;\n",
"disp('Example 12.4');\n",
"\n",
"// aim : To determine the \n",
"// (a) throat and exit velocities\n",
"// (b) throat and exit areas\n",
"\n",
"// Given values\n",
"P1 = 2.2;// inlet pressure, [MN/m^2]\n",
"T1 = 273+260;// inlet temperature, [K]\n",
"P2 = .4;// exit pressure,[MN/m^2]\n",
"eff = .85;// efficiency of the nozzle after throat\n",
"m_dot = 11;// steam flow rate in the nozzle, [kg/s]\n",
"\n",
"// solution\n",
"// (a)\n",
"// assuming steam is following same law as previous question 12.3\n",
"Pt = .546*P1;// critical pressure,[MN/m^2]\n",
"// from Fig. 12.6\n",
"h1 = 2940;// [kJ/kg]\n",
"ht = 2790;// [kJ/kg]\n",
"\n",
"Ct = sqrt(2*(h1-ht)*10^3);// [m/s]\n",
"\n",
"// again from Fig. 12.6\n",
"h2_prime = 2590;// [kJ/kg]\n",
"// using eff = (ht-h2)/(ht-h2_prime)\n",
"\n",
"h2 = ht-eff*(ht-h2_prime); // [kJ/kg]\n",
"\n",
"C2 = sqrt(2*(h1-h2)*10^3);// [m/s]\n",
"\n",
"// (b)\n",
"// from chart\n",
"vt = .16;// [m^3/kg]\n",
"v2 = .44;// [m^3/kg]\n",
"// using m_dot*v=A*C\n",
"At = m_dot*vt/Ct*10^6;// throat area, [mm^2]\n",
"\n",
"A2 = m_dot*v2/C2*10^6;// throat area, [mm^2]\n",
"\n",
"mprintf('\n (a) The throat velocity is = %f m/s\n',Ct);\n",
"mprintf('\n The exit velocity is = %f m/s\n',C2);\n",
"mprintf('\n (b) The throat area is = %f mm^2\n',At);\n",
"mprintf('\n The throat area is = %f mm^2\n',A2);\n",
"\n",
"// End"
]
}
],
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{
"text": "MetaKernel Magics",
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