Chapter 2

A normal shock is in a Mach \(2.0\) flow. Upstream gas temperature is \(T_{1}=15^{\circ} \mathrm{C}\), the gas constant is \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) and \(\gamma=1.4\). Calculate (a) \(a^{*}\) in \(\mathrm{m} / \mathrm{s}\) (b) \(u_{2}\) in \(\mathrm{m} / \mathrm{s}\) (use Prandtl's relation) (c) \(a_{\mathrm{t}}\) in \(\mathrm{m} / \mathrm{s}\) (d) \(h_{2}\) in \(\mathrm{kJ} / \mathrm{kg}\)

A supersonic tunnel has a test section (T.S.) Mach number of \(M_{\mathrm{T} . s}=2.0\). The reservoir for this tunnel is the room with \(T_{\text {room }}=15^{\circ} \mathrm{C}\) and \(p_{\text {room }}=100 \mathrm{kPa}\). The test section has two windows (each \(10 \times 20 \mathrm{~cm}\) ). Calculate (a) the speed of sound in the test section (b) the force on each glass window Assume \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \gamma=1.4\).

An adiabatic constant-area duct has an inlet flow of \(M_{1}=0.5, T_{1}=260^{\circ} \mathrm{C}, p_{1}=1 \mathrm{MPa}\). The average skin friction coefficient, \(C_{f}=0.005\) and the duct cross-section is circular with inner diameter \(d=10 \mathrm{~cm}\). Assuming the gas is perfect and has properties \(\gamma=1.4\) and \(R=286.8 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), calculate (a) mass flow rate \(m\) (b) \(L_{\max }\) to choke this duct at the exit (c) \(p_{\mathrm{t}}\) loss in the duct at \(L=L_{\max }\)

A frictionless, constant-area duct flow of a perfect gas is heated to a choking condition. The rate of heat transfer per unit mass flow rate is \(500 \mathrm{~kJ} / \mathrm{kg}\). Assuming inlet Mach number is \(M_{1}=3.0\) and \(c_{p}=1.004 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\), calculate (a) inlet total temperature \(T_{\mathrm{t}}\) (b) percentage static pressure rise in the flow

In an adiabatic, constant-area flow of a perfect gas, the inlet conditions are \(p_{1}=100 \mathrm{kPa}, \rho_{1}=1 \mathrm{~kg} / \mathrm{m}^{3}\), and \(u_{1}=100 \mathrm{~m} / \mathrm{s}\). At a downstream station the gas is at \(200 \mathrm{~m} / \mathrm{s}\). Assuming the medium is air, calculate the corresponding static pressure and density, \(p_{2}\) and \(p_{2}\), respectively. $$ \left[R_{\text {air }}=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \gamma_{\text {air }}=1.4\right] $$

In a frictionless, constant-area flow of a perfect gas, the inlet conditions are \(\rho_{1}=100 \mathrm{kPa}, \rho_{1}=1 \mathrm{~kg} / \mathrm{m}^{3}\), and \(u_{1}=100 \mathrm{~m} / \mathrm{s}\). At a downstream station the gas is at \(200 \mathrm{~m} / \mathrm{s}\). Assuming the medium is air, calculate the corresponding static pressure and density, \(p_{2}\) and \(p_{2}\), respectively. $$ \left[R_{\text {air }}=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \gamma_{\text {air }}=1.4\right] $$

Consider a one-dimensional adiabatic flow of a perfect gas with friction. Air enters a circular duct of \(20-\mathrm{cm}\) diameter at Mach \(0.5\) and an average wall friction coefficient of \(C_{f}=0.005\). If the exit total pressure drops to \(85 \%\) of the inlet value, i.e., \(p_{\mathrm{t} 2}=0.85 P_{\mathrm{t}}\), calculate the length of the duct, \(L\).

A perfect gas flows in a well-insulated, constant-area pipe with friction at \(M_{1}=2.0\). For an average wall friction coefficient of \(C_{f}=0.004\), calculate (a) the maximum length of the pipe that can transmit the flow (b) total pressure loss at this length Assume the pipe cross-section is rectangular and has the dimensions of \(10 \times 20 \mathrm{~cm}\).

Air enters a frictionless, constant-area pipe at \(p_{1}=\) 60 psia., \(T_{1}=500^{\circ} \mathrm{R} /\) and \(M_{1}=0.6 .\) If heat is transferred to the air in the pipe at \(q=300 \mathrm{BTU} / \mathrm{lbm}\) of air, calculate (a) the exit Mach number (b) static and total pressure and temperature at the exit, \(p_{2}, T_{2}, P_{12}, T_{2}\) (c) the critical heat flux \(q^{*}\) that will thermally choke the pipe. $$ c_{p}=0.24 \mathrm{BTU} / \mathrm{lbm} .{ }^{\circ} R, \gamma=1.4 $$

A constant-area frictionless pipe is thermally choked. The inlet total temperature and the heat flux are \(T_{\mathrm{u}}=650{ }^{\circ} \mathrm{R}\) and \(q^{*}=500 \mathrm{BTU} / \mathrm{lbm}\), respectively. Calculate (a) the inlet Mach number \(M_{1}\) (b) exit total temperature \(T_{12}\) $$ c_{p}=0.24 \mathrm{BTU} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{R}, \gamma=1.4 $$

If we neglect the friction in a jet engine combustion chamber and assume the flow through the burner may be modeled as a Rayleigh line flow, calculate the combustor exit Mach number if \(M_{3}=0.2, T_{13}=900{ }^{\circ} \mathrm{R}\). The fuel heating value is 18,400 BTU/lbm (of fuel) and the fuel-to-air ratio is \(2 \% .\) Assume \(\gamma=1.33\) and \(c_{p}=0.27 \mathrm{BTU} / \mathrm{lbm} .{ }^{\circ} \mathrm{R}\) and neglect the added mass of the fuel.

Air enters an insulated duct with a constant area with an average wall friction coefficient of \(C_{f}=0.004\). The inlet Mach number is \(M_{1}=2.0\). There is a choking length \(L_{1}^{*}\), for this duct that corresponds to the inlet Mach number \(M_{1}\). For any duct longer than the choking length \(L_{1}^{*}\), a normal shock appears in the duct. Assuming the length of the duct is \(10 \%\) longer than \(L_{1}^{*}\), i.e., \(L=1.1 L_{1}^{*}\), calculate (a) the shock location along the duct \(x_{s} / D\) (b) percentage total pressure loss in the longer duct (c) percentage loss of fluid impulse Assume \(\gamma=1.4\).

A scramjet combustor has a supersonic inlet condition and a choked exit. The combustor flow area increases linearly in the flow direction, as shown. The inlet and exit flow conditions are $$ \begin{aligned} M_{1} &=3.0 \\ p_{1} &=1 \mathrm{bar} \\ T_{1} &=1000 \mathrm{~K} \\ A_{1} &=1 \mathrm{~m}^{2} \\ M_{2} &=1.0 \\ A_{2} &=1.4 \mathrm{~m}^{2} \\ \gamma &=1.4, R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K} \end{aligned} $$ The total heat release due to combustion, per unit flow rate in the duct, is initially assumed to be \(15 \mathrm{MJ} / \mathrm{kg}\). If we divide the combustor into three constant-area sections, with stepwise jumps in the duct area, we may apply Rayleigh flow principles to each segment, as shown. The heat release per segment is then \(1 / 3\) of the total heat release in the duct, i.e., \(5,000 \mathrm{~kJ} / \mathrm{kg}\). As the exit condition of a segment needs to be matched to the inlet condition of the following segment, we propose to satisfy continuity equation at the boundary through an isentropic step area expansion, i.e., \(p_{1}, T_{\mathrm{t}}\) remain the same and only the Mach number jumps isentropically through area expansion. If we march from the inlet condition toward the exit with the assumed heat release rates, we calculate the exit Mach number \(M_{2}\). Since the exit flow is specified to be choked, then we need to adjust the total heat release in order to get a choked exit. Calculate the critical heat release in the above duct that leads to thermal choking of the flow.

A subsonic diffuser flow is steady and adiabatic with inlet Mach number \(M_{1}=0.6\). Total pressure loss in the diffuser is \(2 \%\) of the inlet total pressure, i.e., \(p_{t 2} / p_{t 1}=0.98\). The diffuser area ratio is \(A_{2} / A_{1}=1.50\). Assuming that the flow in the diffuser has a uniform exit flow, calculate (a) the exit Mach number \(M_{2}\) (b) the static pressure recovery in the diffuser \(C_{\mathrm{PR}}\) (c) the (internal) force acting on the diffuser, i.e., \(F_{x, \text { wall }}\), nondimensionalized by the inlet static pressure and area, i.e., \(p_{1} A_{1}\)

A supersonic combustion is modeled as Rayleigh flow. The inlet Mach number is \(M_{1}=3.0\), the total temperature and pressure at the inlet are \(T_{\mathrm{tl}}=1,500 \mathrm{~K}\) and \(p_{\mathrm{t} 1}=\) \(100 \mathrm{kPa}\), respectively. Calculate (a) minimum \(q\) to choke the flow at the exit (b) exit temperature, \(T_{2}\) (c) if the fuel is hydrogen with a heating value of \(120,000 \mathrm{~kJ} / \mathrm{kg}\), calculate the fuel-to-air ratio, \(f\). Assume \(\gamma=1.4\) and \(c_{p}=1,004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\)

A scramjet combustor is modeled as a frictionless flow of a perfect gas in a constant-area duct with heating (Rayleigh flow). The inlet Mach number to the burner is \(M_{1}=2.8\) with \(\gamma=1.4\). Critical heating of the gas (i.e., by burning fuel) in the combustor achieves a sonic exit state. Assuming the gas is calorically perfect, calculate (a) the nondimensional critical heat flux, \(q_{1}^{*} / c_{p} T_{\mathrm{t}}\), to achieve thermal choking of the flow (b) the percentage static pressure rise, \(\Delta p / p_{1}\)

A duct with square cross-section is shown. The inlet flow conditions are known to be: \(M_{1}=0.3, p_{1}=150 \mathrm{kPa}\), \(T_{1}=300 \mathrm{~K}, \gamma=1.4, R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). The duct is insulated but the flow is frictional, with the wall friction coefficient \(C_{i}=0.005\). Assuming the flow is steady, calculate (a) the mass flow rate, \(m\) in \(\mathrm{kg} / \mathrm{s}\) and \(\mathrm{lbm} / \mathrm{s}\) (b) the hydraulic diameter of the duct in centimeter and feet (c) the exit Mach number, \(M_{2}\) (d) the percentage total pressure loss, \(\left(\Delta p_{1} / p_{\mathrm{t}}\right) \times 100\), across the duct

Air enters a constant-area scramjet combustor at Mach 3.2. Assuming the flow is frictionless and the combustion is simulated as external heating of air, use Rayleigh flow to calculate \(q^{*}\) and the fuel-air ratio that is needed to thermally choke the scramjet combustor. The fuel heating value is assumed to be \(118,500 \mathrm{~kJ} / \mathrm{kg}\) of fuel.

Consider an insulated constant-area duct. The average wall friction coefficient is \(C_{1}=0.005\). Assuming the length-to-(hydraulic) diameter ratio for the duct is \(L / D_{\mathrm{h}}=\) \(9.9\), the gas is perfect with \(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and the inlet flow conditions are: \(M_{1}=2.0, p_{1}=20 \mathrm{kPa}\) and \(T_{1}=650 \mathrm{~K}\). Calculate (a) inlet total temperature in \(\mathrm{K}\) (b) inlet total pressure in \(\mathrm{kPa}\) (c) exit Mach number, \(M_{2}\) (d) percentage total pressure drop in the duct (e) the minimum \(L D_{h}\) for this duct to choke (keep inlet at Mach 2.0) (f) nondimensional entropy rise, \(\Delta s / R\)

An inviscid perfect gas flows through a constant-area duct that is subject to heat transfer. The ratio of heat transfer to mass flow rate is \(q=250 \mathrm{~kJ} / \mathrm{kg}\). The inlet condition to the duct is: \(M_{1}=0.5, p_{1}=100 \mathrm{kPa}\) and \(T_{1}=600 \mathrm{~K}\). Assuming that gas properties are: \(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kg}\). \(\mathrm{K}\), calculate (a) inlet total pressure, \(p_{\mathrm{t} 1}\) in \(\mathrm{kPa}\) (b) inlet gas speed, \(u_{1}\) in \(\mathrm{m} / \mathrm{s}\) (c) exit Mach number, \(M_{2}\) (d) percentage total pressure drop in the duct

In an adiabatic flow of a perfect gas in a constantarea duct with friction, we have the inlet condition as \(M_{1}=\) \(2.0, p_{1}=25 \mathrm{kPa}, T_{1}=250 \mathrm{~K}\). The duct length-to-(hydraulic) diameter ratio is \(L D_{h}=10.262\) and the average friction coefficient at the wall is \(C_{f}=0.005\). Assuming gas properties are \(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), calculate (a) exit Mach number, \(M_{2}\) (b) total temperature of the gas at the exit, \(T_{12}\) in \(\mathrm{K}\) (c) total pressure of the gas at the exit, \(p_{12}\) in \(\mathrm{kPa}\)

Consider the flow of perfect gas in a duct with \(T_{2} / T_{1}=1.01, p_{2} / p_{1}=1.038\) and \(M_{2} / M_{1}=0.88\). Assuming gas properties are \(\gamma=1.4\) and \(R=287 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), calculate (a) the density ratio, \(\rho_{2} / \rho_{1}\) (b) the velocity ratio, \(V_{2} / V_{1}\) (c) the duct area ratio, \(A_{2} / A_{1}\)

In a Fanno flow problem with a supersonic inlet Mach number of \(M_{1}=2.0\), a normal shock appears at \(M_{x}=\) 1.2. Assuming the average friction coefficient in the pipe is \(C_{t}=0.005\) and \(\gamma=1.4\), calculate (a) \(L_{\mathrm{x}} / D\) (b) \(L / D\) (c) percentage total pressure loss, \(\Delta p_{\mathrm{t}} / p_{\mathrm{u} 1}\)

In an adiabatic flow of perfect gas in a constant-area duct with friction, the inlet is supersonic with Mach number \(M_{1}=2.0 .\) A normal shock appears in the duct where local Mach number is \(M_{x}=1.5\). Assuming the average friction coefficient in the duct is \(C_{f}=0.005\) and \(\gamma=1.4\), calculate (a) \(L_{\mathrm{x}} / D\) (b) \(L D\) (c) \(L_{x} / L\)

Consider the flow of air (perfect gas) in a constantarea duct with heat transfer. Air enters the duct at a supersonic Mach number, \(M_{1}=2.6\). The gas properties are: \(\gamma=1.4=\) constant and \(c_{p}=1004 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). Assuming the fluid is inviscid, and the rate of heat transfer per unit mass flow rate is \(q=\) \(572 \mathrm{~kJ} / \mathrm{kg}\), calculate (a) exit total temperature, \(T_{\mathrm{t} 2}(\mathrm{~K})\) (b) exit Mach number, \(M_{2}\) (c) exit static pressure, \(p_{2}\), in \(\mathrm{kPa}\) (d) total pressure loss, \(\Delta p_{1} / p_{\mathrm{tl}}(\%)\) (e) critical heat transfer rate to choke the inlet flow,