12.12.1 Liquid Propellant Thrust Chambers

Liquid propellant rockets usually operate for extended periods, that is, typically for several minutes, and thus expose the combustion chamber walls to intense heating. The combustion products reach (near stoichiometric level) temperatures of ∼3000–4700 K depending on the propellant combination (Table 12.1). All modes of heat transfer, that is, convection, conduction, and radiation exist in a rocket combustor and contribute to the combustor wall heating. The products of combustion then expand in a convergent–divergent nozzle to produce thrust. Although expansion of the combustion gases in the nozzle convert their thermal energy into kinetic energy, extended sections of the nozzle downstream of the throat are still exposed to excessive gas temperatures and thus need to be cooled. Fortunately, the liquid propellant onboard offers a cooling capacity that is tapped for the so-called regenerative cooling purposes of the thrust chamber walls. The cooling capacity is due to the phase change, that is, from liquid to vapor, known as latent heat of vaporization. Figure 12.31 shows a definition sketch of the regenerative cooling scheme in a liquid propellant rocket. Note that we switched the chamber designation of temperature, pressure, and so on from using the subscript “c” (for chamber) to “g” for (hot) gas. Since we are cooling the walls of the thrust chamber, we need the subscript “c” for the coolant.

As we discussed heat transfer in Chapter 10for turbine cooling, the steady-state heat transfer from the hot combustion gases to the coolant takes place convectively over the combustor inner wall, across a hot gas film, radiatively due to the volume of hot combustion gases to the wall, conductively through the combustor wall, and convectively over a coolant film on the coolant channel side. The net one-dimensional heat flux to the thrust chamber wall is the sum of the convection as well as the radiative heat transfer according to

(12.67)

The radiative heat transfer from the volume of hot gas to the solid wall in the combustor is governed by

(12.68)

where _g is the emissivity of the gas (equal to 1 for blackbody radiation and less than 1 for graybody radiation), σ is the Stefan-Boltzmann constant, which is 5.67 × 10^{− 8} W/m²K⁴ and T_g is the absolute gas temperature. Although the Stefan−Boltzmann constant is very small (due to 10⁻⁸ term), the temperature contribution to radiation heat transfer grows exponentially with temperature, that is, T⁴_g. Therefore, according to Sutton and Biblarz (2001), radiation accounts for 3–40% of the combustion chamber heat flux for the gas temperature range of 1900−3900 K. This is a reminder that we may not neglect radiative heat transfer unless the gas temperature is low, namely, ∼700 K. The presence of solid particles in the rocket combustion chamber increases the emissivity _g in Equation 12.68. The upper value of 1, that is, blackbody radiation, may be chosen for high concentration of solid particles in the gas stream. The convective heat transfer is governed by the Newton’s law of cooling according to

(12.69)

where h_g is the gas-side film coefficient, T_aw is the adiabatic wall temperature (on the gas side), and T_wg is the gas-side wall temperature. Fourier’s law of heat conduction governs the heat conduction through the wall,

(12.70)

where k_w is the thermal conductivity of the wall material (a property of the wall), T_wc is the wall temperature on the coolant side, and t_w is the wall thickness (a design parameter). Note that the minus in Fourier heat conduction law is absorbed in the parenthesis, that is, the heat transfer is from the combustor toward the coolant side. Now, combining these sources of heat flow in a steady-state (one-dimensional) rocket thrust chamber-cooling problem, we can write

(12.71)

We may eliminate the gas and coolant sidewall temperatures, T_wg and T_wc, from the heat flux equation to get

(12.72)

Let us examine the parameters in Equation 12.72. The stagnation temperature of the gas often approximates the adiabatic wall temperature T_aw as we encountered in turbine cooling section. This approximation, that is, T_aw ≈ T_tg, is appropriate in applying to the rocket thrust chamber cooling problem as well due to the approximations that are inherent in heat transfer solutions. Coolant temperature T_c is also known since the stored liquid temperature in the insulated tank is known and is adjusted for the transmission line connecting the pump and the thrust chamber heat transfer. Approximate expressions for film coefficients h_g and h_c are also known from the heat transfer literature related to the long tubes/channels. The emissivity of gas _g in the presence of (high concentration of) solid particles may be approximated as 1, that is, treating the particle-laden gas as a perfect blackbody radiator. Again this approximation, in the context of other heat transfer approximations, is acceptable in the preliminary design phase. The thickness of the combustor wall t_w and wall thermal conductivity k_w are design choices.

The film coefficient on the gas side h_g is empirically estimated for heat transfer in tubes and is related to Reynolds number and Prandtl number according to

(12.73)

where D_g is the (local) diameter of the thrust chamber, k_g is the gas thermal conductivity, ρ_gv_g is the average gas mass flow rate per unit area in the combustion chamber, μ_g is the gas coefficient of viscosity, and c_pg is the specific heat of the gas at constant pressure. The left-hand side of Equation 12.73 is the Nusselt number; the parentheses on the RHS are Reynolds and Prandtl numbers, respectively. Since the fluid density and properties μ_g, k_g, and c_pg are functions of gas temperature, the question of “which gas temperature” in the combustion chamber should we use to estimate these parameters arise. There is no unique answer to this question; however, for engineering approximations an “average film” temperature T_f is used as an acceptable reference temperature, that is,

(12.74)

The coolant side film coefficient (for the coolant in the liquid state) is related to the Reynolds and Prandtl numbers on the coolant side following

(12.75)

where is the coolant average specific heat and D_c is the coolant passage hydraulic diameter. We may express the film coefficient on the coolant side in terms of on coolant mass flux, ρ_cv_c, and specific heat:

(12.76)

This expression clearly shows that for a given coolant Reynolds number and Prandtl number, the coolant film coefficient h_c increases with coolant mass flux and the coolant specific heat. We may thus tailor the cross section of the coolant passage in critical areas such as the nozzle throat to increase the local mass flux, ρ_cv_c. Also, we may choose a liquid fuel with high specific heat. For example, hydrogen has a very high specific heat compared with other rocket fuels and is thus a coolant of choice for regeneratively cooling rocket thrust chambers. Table 12.4 from Sutton and Biblarz (2001) show some physical properties of some liquid propellants.

In general, we have to integrate the steady-state heat transfer Equation 12.71 along the thrust chamber (axis). We may divide the thrust chamber into cylindrical sections of small axial length Δx where all the heat transfer parameters are assumed constant. By further assuming axisymmetric condition per cylindrical sections, we can march along the axis from the combustion chamber out toward the nozzle exit. The overall heat transfer to an axisymmetric thrust chamber is the integral of the heat flux along the wall, which is thus the sum of the segments according to

(12.76)

The overall heat transfer causes the temperature of the coolant to rise according to

(12.77)

where T₁ represents the entrance temperature of the coolant into the jacket, and T₂ is the bulk coolant temperature as it (leaves the cooling jacket and) enters the injector. As a design choice, we may limit the bulk coolant exit temperature to below the boiling point, that is, T₂ < T_{boiling-point}. Table 12.5 lists the heat transfer characteristics of some liquid propellants (from Sutton and Biblarz, 2001). The pressure in the cooling jacket has a strong influence on the boiling point temperature of the fuel. For example, hydrazine boils at 387 K for a pressure of 0.101 MPa (i.e., 1 atm), whereas it boils at 588 K (an increase of 201 K) if the pressure is raised to 6.89 MPa. The cooling jacket pressure is to be provided by a turbopump (or a pressure) feed system and the higher pressure ratio will then require multistage pumps and very heavy turbopump feed system.

Also note that the critical heat flux, that is, the maximum heat transfer rate per unit area, in the thrust chamber theoretically occurs at the throat where the flow passage area is at a minimum. Experimental evidence of the peak heat flux near the throat is well documented in the literature, for example, Witte and Harper (1962). To lower the wall temperature on the gas side, we have to increase the coolant side film coefficient h_c, which according to Equation 12.76 behaves as

(12.78)

The design requirement on T_wg at the throat dictates the following characteristics on the coolant passage side:

• High mass flux ρ_cv_c or since the liquid density is nearly constant, high coolant speed v_c
• Small diameter D_c cooling tubes.

The coolant speed at the throat may be as high as ∼15−20 m/s. The hydraulic diameter of the cooling tubes near the throat scales in millimeters. The most effective throat cooling in regeneratively cooled thrust chambers combines these two effects. Therefore, the cooling jacket is composed of small-diameter tubes with a tapered cross-sectional area at the throat with relatively high coolant speed.

In very small liquid propellant thrust chambers, where either insufficient coolant flow is available or insufficient (throat) surface area is available for cooling, ablative materials are used. Also, in cases where the motor is pulsed as in vernier rockets, passive cooling technique, such as ablative cooling, is used. An ablative material directly transitions from the solid-to-gaseous phase while absorbing latent heat of vaporization in the process. Additionally, the emergence (and the flow) of the (ablative) gas on the surface acts as a cooling layer that protects the surface.

12.12.2 Cooling of Solid Propellant Thrust Chambers

Solid propellant rockets may use several (nonregenerative) methods to protect the thrust chamber walls (or motorcase) against the heat loads in the combustor. The purpose of an inhibitor is to inhibit grain burning where the designer wishes to protect the thrust chamber walls (or as a means of controlling thrust-time or vehicle dynamic behavior). The purpose of an insulating layer, that is, the one with low thermal conductivity, next to the wall is also to protect the wall from excessive heating. In addition to these two techniques, solid propellant grain designer may place a cooler burning grain in the outer shell next to the wall to protect the motorcase. However, this is an expensive manufacturing proposition seldom used. Since ablative materials have been extensively used for thermal protection with success during the Apollo era, and graphite has emerged as a viable high temperature material, designers use ablative materials and graphite as throat inserts (or for motorcase insulation) in solid propellant rockets today with confidence. The use of graphite offers the added advantage of erosion resistance (as compared with ablative materials that decompose), thus maintaining the throat (cross-sectional flow) area nearly constant as a function of burn time. Detailed engineering design considerations in solid rocket motors, including practical thermal management approaches are presented by Heister (1995), which is suggested for further reading.

12.14.1 Multiphase Flow in Rocket Nozzles

In our analysis so far, we have treated the nozzle fluid to be in the gaseous state. There are at least three sources for liquid or solid particle presence in a rocket nozzle flow. The first is the propellant itself, where solid particles are intentionally embedded or dispersed to increase the rocket performance through higher specific impulse. The second source is the combustion, which may lead to soot and particulate formation in the products of combustion. The third source is the appearance of liquid droplets, that is, condensation, in the exhaust stream due to static temperature drop in the gases. The very large area expansion ratios bring about very large temperature and pressure expansion ratios, which result in the condensation of a fraction of the exhaust stream. It is the impact of these solid or liquid fractions (present in the nozzle gases) on rocket performance that we address in this section.

The thermal and dynamic interaction of the two phases in a fluid, say solid and gas, pose complex fluid mechanics and heat transfer problems. However, in four limiting cases, we may arrive at closed-form solutions and shed light on the effects of multiphase flow on rocket performance.

These four limiting cases are:

1. Solid particles reach the same temperature as the gas at the nozzle exit, the so-called thermal equilibrium case
2. Solid particles are fully accelerated and reach the same speed as the gas at the nozzle exit, that is, they reach momentum equilibrium
3. Solid particles maintain their temperatures, that is, the case of zero thermal interaction or heat transfer between the two phases
4. Solid particles are not accelerated by the gas and thus maintain their negligible momentum at the nozzle exit.

Note that an underlying assumption in these four limiting cases is that the solid fraction present at the nozzle inlet remains unchanged, that is, frozen, in the nozzle and appear in whole at the nozzle exit. The four cases noted above address only the fate of these particles in heat and momentum transfer with the gas. To derive the fundamental equations of two-phase flow, we return to a slab concept where we apply the (one-dimensional) conservation principles to a slab of fluid. The fluid and flow parameters on the two sides of the slab are related to each other via a linear relationship using the Taylor series expansion of a continuous function. Figure 12.36 shows the definition sketch of a slab in a two-phase fluid flow with zero friction and heat transfer (i.e., reversible and adiabatic condition) at the wall.

By stipulating a constant mass flow rate for the gas and the solid phases in the fluid, we get

(12.81)

(12.82)

Note that the density of the solid is the mass of the solid contained in a unit volume of the gas–solid mixture and is not to be confused by the density of the solid itself. The gas density is the mass of the gas per unit volume of the mixture, as expected. By conserving the mass flow rate of solid and gas individually, through Equations 12.81 and 12.82, we conserve the collective mass flow rate in the nozzle. The conservation of momentum applied to the slab gives

(12.83)

Note that the last term on the RHS of the Equation 12.83 is the pressure force exerted by the sidewall on the fluid (in the x-direction). Also the “momentum” in quotation mark is used as the shorthand for “time rate of change of momentum.” Momentum equation simplifies to

(12.84)

We may divide both sides of Equation 12.84 by the flow area A to get

(12.85)

The steady energy equation written for a (calorically) perfect gas and solid phases in a flow is the balance between the net flux of power (in fluid and solid phases) and the rate of the external energy exchange through heat transfer and mechanical shaft power, that is,

(12.86)

The energy equation simplifies to

(12.87)

The fraction of solid particles in the mixture of two-phase flow may be given a symbol X defined as

(12.88)

In terms of this parameter, that is, solid mass fraction X that constitutes a constant in our problem based on our frozen-phase assumption, we may write the energy equation as

(12.89)

We replace the solid particle density−velocity term ρ_sV_s by

in the momentum equation to get

(12.90)

If we divide Equation 12.89 by (1 − X) and substitute Equation 12.90 for ρ_gdV_g, we get

(12.91)

Case 1

The solid and gas are in thermal equilibrium, that is, dT_s = dT_g = dT, therefore, Equation 12.91 reduces to

(12.92)

We may consider two scenarios for the solid particles momentum behavior in the nozzle flow. First, if the solid particles are accelerated by the gas to achieve the same speed as the gas, we get V_s = V_g, therefore, Equation 12.92 simplifies to

(12.93a)

This equation readily integrates for a perfect gas where ρ_g = p/R_gT to the following form:

(12.93b)

Also Equation 12.89 for V_s = V_g = V and T_s = T_g = T gives

(12.94)

which may be integrated from the chamber condition (with V_c ≈ 0) to the nozzle exit, V₂

(12.95)

In terms of pressure ratio, nozzle exit velocity is

(12.96)

Since solid particles have attained the same speed as the gas, the ratio of specific impulse with and without solid particulate flow is the ratio of the two exhaust speeds, namely,

(12.97)

For a gas molecular weight of 20.1 kg/kmol, the ratio of specific heats γ = 1.26, and specific heat of the solid (aluminum) particles = 903 J/kg ·K, we calculate and graph Equation 12.97 in Figure 12.37.

Figure 12.37 shows a drop in rocket performance with increasing fraction of the solid flow. The impact of cycle pressure ratio is small, as expected. The presence of solid particles in the propellant is, however, to enhance the burning rate and combustion temperature T_c, which was assumed constant in this calculation. The effect of higher burning rate and combustion temperature due to solids should, therefore, outweigh the penalty of the two-phase flow in the nozzle, as depicted in Figure 12.37.

The second scenario for the particle momentum/acceleration is to assume the solid particles are not accelerated by the gas, that is, dV_s = 0 while their temperature has reached that of the gas. The temperature–pressure relation remains the same as Equation 12.93b. However, the energy equation gives the variation of the gas speed with temperature drop of the mixture, that is,

(12.98a)

The solution of the above equation is

(12.98b)

The thrust of the two streams of gas and solid, one moving at speed V_g2 and the second moving at speed V_s, is

(12.99)

Therefore, the stream-thrust-averaged speed at the nozzle exit (for a perfectly expanded nozzle) is

(12.100a)

Since the second term on the RHS of Equation 12.100a is the product of two small numbers (remember that the solid was assumed to be unaccelerated and X is the small solid fraction), therefore, the thrust-averaged exit velocity is

(12.100b)

and the specific thrust is expressed in terms of the thrust-averaged exit velocity as

(12.101)

The ratio of specific impulse with/without solid particles that are in thermal equilibrium with the gas, but are not accelerated by the gas, is

(12.102)

The graph of Equation 12.102 superimposed on the rocket performance chart of Figure 12.37 is shown in Figure 12.38. Note that the specific impulse is further degraded when the solid particles are not accelerated by the gas (i.e., the case of small particle drag).

Case 2

The solid and gas are not in thermal equilibrium, that is, when the solid temperature remains constant, dT_s = 0 as a result of negligible heat transfer with the gas. In this case, Equation 12.91 reduces to

(12.103)

In addition to T_s = constant, either for full acceleration where V_s = V_g or when the solid particle remains unaccelerated, that is, when dV_s = 0, Equation 12.103 gives

(12.104)

The energy Equation 12.89 reduces to the following form for V_s = V_g, that is, full acceleration case,

(12.105)

Therefore, the specific impulse ratio is

(12.106)

The energy Equation 12.89 for the case of constant V_s, that is, unaccelerated solid particles in the two-phase flow reduces to

(12.107)

which has a solution following

(12.108)

Although the gas phase seems to be unaffected by the presence of the solid particles, the thrust is affected since the two phases of the flow have different momentums at the nozzle exit. The thrust averaged exit velocity is

(12.109)

which again for negligible solid velocity V_s in the exhaust plane, the ratio of the specific impulse will be

(12.110)

Now, let us graph Equations 12.106 and 12.110 in Figure 12.39. The specific impulse diminishes with solid flow fraction, as in the previous cases. The nozzle (or cycle) pressure ratio has no effect on specific impulse ratio if the solid particles are not heated to the gas temperature, that is, when they are not in thermal equilibrium with the gas. The unaccelerated solid particles cause a larger reduction in specific impulse than the fully accelerated ones. The reason is that the fully accelerated solid particles contribute to momentum, whereas the unaccelerated particles with negligible exit momentum cause a reduction in thrust. Therefore, the solid particle drag causes the particles to accelerate and contribute to thrust. The case studies in this section are useful only in establishing a range for the impact that multiphase flow makes on rocket performance. In reality, multiphase flow is more complex and accurate simulation needs computational approach.

12.14.2 Flow Expansion in Rocket Nozzles

The products of combustion do not in reality stop reacting once they enter the nozzle. Actually, the conversion of thermal to kinetic energy in the nozzle causes a cooling down effect of the gases as well as the pressure drop, which impact the reaction rates of some of the products of combustion. Therefore, chemical reaction takes place continually from the combustor through the nozzle. Sometimes the nozzle reverses some of the reactions that took place in the combustor. For example, dissociation in the combustor, which is a direct result of high combustion temperature, may reverse, through the recombination process, to some extent in the nozzle when the temperature of the mixture drops. In addition, chemical reactions between the constituents are often considered to be in equilibrium. However, that too is an assumption that leads to major simplifications in the analysis and may be questioned. Whether chemical equilibrium is reached in a mixture depends on the chemical kinetics and the reaction rates in the mixture. In a rocket expansion process where the timescales are very short (in milliseconds) due to high (convection) speeds, chemical equilibrium may be questioned legitimately in such environments. Computational approaches to complex chemical reactions are fortunately available, for example, Gordon and McBride (1996), which may be used for an accurate analysis of chemically reacting mixtures of gases in different thermal environments.

However, the choice of frozen (concentration of) species of gas mixture in the nozzle (known as the frozen chemistry model) is often made for preliminary rocket design calculations. As evidenced from the computational results of Olson (1962), shown in Figure 12.40 (or Figure 12.22 from Sutton and Biblarz, 2001), the corrected specific impulses for hydrogen–oxygen and JP4–oxygen are both indicated to be higher with chemical reactions in the nozzle. Also, the corrected experimental results in Olson show that the equilibrium model gives an overestimation of the actual performance. This important result points to the frozen chemistry model as a conservative estimation of the rocket performance. A conservative estimation is precisely what is needed in a preliminary design stage.

12.14.3 Thrust Vectoring Nozzles

Conventional gimbals with hydraulic or electromechanical actuators may be used to swivel the entire rocket thrust chamber. The range of control authority using this approach is ∼ ±7°. Injection of a fluid, that is, either a propellant or an inert gas, in the nozzle, known as the secondary injection, causes an oblique shock formation and thus secondary injection thrust vector control (SITVC). The range of control authority using secondary injection is ∼ ±5°. The use of rocket clusters in the vehicle design opens the opportunity to differential operation of the opposite rockets in the cluster to achieve control. The use of jet deflectors, as in jet tabs or jet vanes, offers a limited-duration thrust vector control (in short burn-time rockets) since they are exposed to the hot exhaust gases. The control authority offered in this scheme is ∼ ±10°. Humble et al. (1995) present a practical engineering approach to rocket thrust vector control and is recommended for further reading.

A modern approach to (an old) nozzle design has produced linear aerospike configuration that uses many (e.g., 20) rocket engine cells.Figure 12.41 and 12.42 show aerospike nozzle thrust coefficient variation with nozzle pressure ratio; RS-2200 is a linear aerospike rocket engine on a test stand (from Stennis Space Center; courtesy of NASA). The linear aerospike nozzles have superior off-design performance over a bell nozzle as well as having a vector control capability.

12.15.1 Supersonic Combustion Ramjet

Conventional ramjets cannot produce static thrust. They require ram compression, that is, some forward flight speed, before these simple airbreathing engines (or “ducts with burners”) can produce thrust. It is seemingly ironic that the same engines (i.e., ramjets) cease to produce thrust at very high ram compressions! There are two aspects of ram compression that, at high speeds, are detrimental to thrust production. We studied them in Example 12.13. These are

1. the inlet total pressure recovery that exponentially deteriorates with flight Mach number, and
2. the rising gas temperature in the inlet that cuts back (ΔT_t)_burner to eventually zero.

The worst offender in total pressure recovery of supersonic/hypersonic inlets is the normal shock, which incidentally is also responsible for the rising gas temperatures in the burner. Therefore, if we could only do away with the normal shock in the inlet, we should be getting a super efficient ramjet. But the flow in a supersonic inlet without a (terminal) normal shock would still be supersonic! How do we burn fuel in an airstream that is moving supersonically? The short answer is not very efficiently! Despite the obvious challenges, supersonic combustion ramjets, or scramjets, are born out of necessity. Scramjets hold out the promise of being the most efficient airbreathing engines, that is, with the highest (fuel)-specific impulse, at flight Mach numbers above ∼6 to suborbital Mach numbers; although the upper limit in scramjet operational Mach number is still unknown.

Scramjet engines, more than any other airbreathing engine, need to be integrated with the aircraft. The need for integration stems from a long forebody that is needed at hypersonic Mach numbers to efficiently compress the air. In addition, a long forebody offers the largest capture area possible for the engine, A₀. Aft integration with the aircraft allows for a large area ratio nozzle suitable for high altitude hypersonic vehicles. Figure 12.45 shows a generic scramjet engine that is integrated in a hypersonic aircraft. The inlet achieves compression both externally and internally through oblique shocks. The series of oblique shock reflections inside the inlet create a shock train, known as the isolator. The flow that emerges from the isolator is supersonic, say Mach 3.0, as it enters the combustor. Achieving efficient combustion at supersonic speeds is a challenge.

Fuel injection, atomization, vaporization, mixing, and chemical reaction timescales should, by necessity, be short. If air is moving at Mach 3, where the speed of sound is ∼333 m/s, it traverses 1 m in ∼1 ms. This is just an indication of the convective or residence timescale in the combustor. Hydrogen offers these qualities. For example, hydrogen offers ∼1 /10 of the chemical reaction timescale of hydrocarbon fuels, as discussed in Chapter 7. Additionally, hydrogen is in a cryogenic state as a liquid; therefore, it offers aircraft and engine structure regenerative cooling opportunities.

By accepting high Mach numbers inside the scramjet engine, the static temperature of the gas will be lower throughout the engine and thus the burner is allowed to release heat in the combustor that is needed for propulsion. The h–s diagram of the scramjet cycle is shown in Figure 12.46.

12.15.1.1 Inlet Analysis

Scramjet inlets are integrated with the aircraft forebody, involving several external and internal oblique shocks. In addition, the vehicle’s angle of attack, α, impacts the inlet recovery through oblique shock waves’ angles. For a given forebody shape, we may use shock-expansion theory (discussed in Chapter 2) to calculate the waves’ orientations and the associated total pressure recovery. In the absence of the detailed aircraft forebody geometry, or in the preliminary phase, we may assume an inlet recovery or resort to standards such as AIA or MIL-E-5008B. We listed these in Chapter 6; they are repeated here for convenience.

(6.34)

(6.35)

(6.36)

12.15.1.2 Scramjet Combustor

Although the detailed design and analysis of supersonic combustion is beyond the scope of this book, we can still present global models and approaches that are amenable to closed-form solution. Supersonic branch of Rayleigh flow, that is, a frictionless, constant-area duct with heating is a possible (and the simplest) model for a scramjet combustor. The fact that the analysis was derived for a constant-area duct is not a major limitation, since we can divide a combustor with area change into a series of combustors with constant areas. Rayleigh flow has been detailed in Chapter 2and we will not review it here. However, we may offer to analyze a variable-area combustor that maintains a constant pressure. From a cycle analysis viewpoint, constant-pressure combustion is advantageous to cycle efficiency and thus we consider it here further. Figure 12.47 shows a frictionless duct with heat exchange, but with constant static pressure. We assume the gas is calorically perfect and the flow is steady and one-dimensional. The contribution of fuel mass flow rate to the gas in the duct is small; therefore, combustion is treated as heat transfer through the wall. The problem statement specifies the inlet condition and heat transfer rate (or fuel flow rate or fuel-to-air ratio) and assumes a constant static pressure in a duct with a variable area. The purpose of the analysis is to calculate the duct exit flow condition as well as the area A₄.

We apply conservation principles to the slab of fluid, as shown. The continuity requires

(12.111)

The balance of x-momentum gives

(12.112)

Equation 12.112 signifies a constant-velocity flow, that is,

(12.113)

Therefore, the continuity Equation 12.111 demands ρA = constant, or area is inversely proportional to density.

The energy balance written for the duct gives

(12.114a)

Since with constant velocity, the kinetic energy (per unit mass) remains constant, then

(12.114b)

(12.115)

The RHS of Equation 12.115 is known (as an input to the problem), which then establishes the exit static temperature T₄. Since static pressure is constant, we can calculate the exit density ρ₄. Also, the area ratio follows the inverse of the density ratio, that is,

(12.116)

which establishes the exit flow area. The exit static temperature calculates the exit speed of sound and Mach number (since V = constant).

We may calculate a critical heat flux q* (or critical fuel-to-air ratio f^*) that will choke the duct at its exit. Equation 12.115 is written as

(12.117)

Also, Mach number ratio M₄/M₂ is related to the ratio of speeds of sound following

(12.118)

For a choked exit, M₄ = 1; therefore, the critical heat flux q* in terms of duct inlet flow conditions is expressed as

(12.118a)

In terms of fuel-to-air ratio, we may express q^* as f^* QR η_b to get

(12.118b)

With calculated exit Mach number, we may arrive at the total pressure at the exit, following

An input to this analysis is the burner efficiency η_b. However, combustion efficiency in supersonic streams is not as well established or understood as the conventional low-speed burners. Mixing efficiency in supersonic shear layers, without chemical reaction, forms the foundation of scramjet combustion efficiency. A supersonic stream mixes with a lower speed stream (in our case the fuel) along a shear layer by vortex formations and supersonic wave interactions. These interactions require space along the shear layer or flow direction. Consequently, burner efficiency is a function of the combustor length and continually grows with distance along the scramjet combustor. Burrows and Kurkov (1973) may be consulted for some data and analysis related to supersonic combustion of hydrogen in vitiated air. Heiser et al. (1994) should be consulted for detailed discussion of hypersonic airbreathing propulsion.