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ThrustOne Theory Manual

Purpose and scope

This book explains the physical models behind ThrustOne: the balances they solve, the assumptions that close those balances, and the meaning of their outputs. It is intended for a reader who wants to understand a calculation rather than obtain a number without its context. Familiarity with elementary thermodynamics, mechanics and calculus is helpful; the opening chapters establish the conventions needed to connect those subjects to propulsion calculations.

The first edition addresses 49 explicitly bounded calculation modes across nine engine families. Some modes describe complete idealized cycles. Others describe a component, a geometry construction, a chemical state or a segment of a flowpath. A family name does not imply that all of its physical processes are implemented. In particular, a combined-cycle submodel is not a vehicle-level transition analysis, and a nozzle reference contour is not a solved wall-pressure field.

The user guide explains where to enter values, choose a branch and export a result. This theory manual explains why those inputs are needed and what can be concluded from the result. The two documents serve different purposes. Field inventories and operating instructions do not replace the derivations here; theoretical background does not create an additional software capability.

How the book is organized

Chapters 1–4 establish control boundaries, static and total states, normalization and freestream conditions. These conventions recur throughout the book. Chapters 5–8 explain mixture properties, equilibrium and finite-rate chemistry. They distinguish a frozen composition from constant heat capacity, and an equilibrium state from a time-dependent chemical process.

Chapters 9–15 develop turbojet and turbofan cycles, components, mixing and reference-based off-design calculations. Chapters 16–22 cover rocket performance, chemical expansion, reference geometry, parameter studies and established-burning solid-motor models. Chapters 23–29 connect shocks and duct balances to ramjet and scramjet flowpaths, including variable properties and finite-rate species transport.

Chapters 30–32 explain the bounded combined-cycle submodels. Their differing connections matter: explicit independent branch allocation, momentum-based entrainment and turbine-to-fan shaft transfer are distinct closures. Chapters 33–34 explain numerical results and defensible parameter comparisons. The appendices provide notation, normalization, domain summaries and a mode-to-chapter directory.

Reading paths

For a first propulsion course, begin with Chapters 1–4, then select the relevant engine family. Read Chapter 33 before interpreting a failed solve or a state close to a sonic boundary. A reader who understands mass and energy conservation but is new to propulsion will usually benefit most from following one complete station chain before exploring many model options.

For turbine cycles, follow Chapters 9–10 and 13–14 before studying component maps and off-design closure in Chapters 11–12 and 15. A design-point pressure ratio and a pressure ratio solved from a reference engine are not interchangeable inputs. Mixed-flow calculations also require a more careful mass-flow basis than a single-stream cycle.

For rocket calculations, read Chapters 16–17 before sizing in Chapter 18 or contour construction in Chapter 19. Thermochemical state, performance scale and reference geometry answer different questions. Read Chapter 20 when comparing altitude, pressure, expansion ratio or mixture-ratio results. Solid-propellant readers should then use Chapters 21–22, which explicitly exclude ignition and tail-off modeling.

For reacting flows, read Chapters 5–8, then the reconstruction in Chapter 26 and the flowpath formulation in Chapter 29. This order is important: a species enthalpy reference affects energy closure, and a chemically changing mixture cannot be interpreted using one fixed gas constant. Chapters 23–24 supply the shock, friction, heat and mass-source background.

For combined cycles, use Chapters 30–32 together with the underlying turbine, ramjet and mixing chapters. Compare the control diagrams before comparing performance. An identical word such as “bypass” can denote a prescribed allocation in one model and a mass ratio determined by a balance in another.

Four questions to carry into every calculation

The first question is what crosses the control boundary. Captured air, onboard primary material, fuel, oxidizer, heat and shaft work enter different balances. A rocket and an airbreathing engine do not have the same inlet momentum debit.

The second is which state is specified. Static pressure acts on a surface; total pressure describes an isentropic reference state. Chamber static temperature and stagnation temperature coincide only under an appropriate velocity approximation. An apparently small label difference can change the calculation.

The third is what closes the problem. A prescribed exit pressure, a fixed area, a selected Mach branch, an energy constraint and a fitted inlet relation are different closures. One must not silently substitute one for another while comparing results.

The fourth is what the output is divided by. Specific thrust may use core air, total captured air or another stream. Specific impulse may use all injected mass rather than fuel alone. A ratio without its denominator is an incomplete physical statement.

Common conventions

Equations use SI units unless a local example explicitly identifies another unit. Temperatures in thermodynamic relations are absolute. Pressures are absolute unless stated otherwise; a gauge pressure is not a valid direct replacement in a pressure ratio. Internal energy and enthalpy may be negative under a formation-inclusive reference convention. Their signs alone do not determine physical admissibility.

A subscript t usually denotes a total state, a subscript 0 a freestream reference, and a superscript star a critical or sonic reference defined locally. Neither station numbering nor star notation is globally interchangeable across every model. The chapter’s definition controls its meaning.

The positive flow direction is downstream through the engine. Thrust is normally reported as the useful reaction-force magnitude opposite that flow direction. When a model permits negative net force, the sign indicates an unfavorable result under its accounting. Other modes explicitly reject nonpositive performance; the applicable chapter states the distinction.

Common symbols are listed in Appendix A. Some established symbols have more than one use: F may be thrust or a momentum function; s may be entropy or specific thrust; beta may denote a shock angle or a flow ratio. Local definitions and units resolve those collisions. The notation reference is a guide to reading, not permission to substitute symbols across equations without checking their definitions.

Worked examples and evidence

Examples use original inputs and explanations. A simple accounting example may isolate mass flow, energy or force without constructing a complete engine. Such an example is labeled accordingly. It demonstrates the balance being discussed and must not be mistaken for an independently confirmed App operating point.

Where a numerical value has a documented local reference, the internal editorial record preserves its actual location and qualifications. Independent source evidence, direct illustrative arithmetic and application-generated workflow results have different roles. The book does not turn a value calculated by the application into proof of that same application’s correctness.

Derivations use mathematical relations and original exposition. Diagrams establish control boundaries, station conventions or geometric constructions. They are not reproductions of source illustrations and do not imply that the application solves every feature shown as background context.

Model hierarchy and limits

An ideal constant-property calculation can reveal a useful relation with very few inputs. A variable-property calculation changes how the state is recovered. An equilibrium calculation changes the composition constraint. A finite-rate calculation adds chemical evolution. None of these changes automatically adds spatial transport, wall cooling, shocks, multiphase physics or a complete hardware design.

Greater mathematical detail is useful when it answers the intended question and its extra inputs are meaningful. It is not a universal ordering from an inferior model to a superior one. A complicated mechanism with unsuitable coefficients can provide a less defensible conclusion than a simple model used within its stated scope.

An interrupted flow history is presented as a valid prefix, not a completed nozzle or engine. A sampled maximum is identified as a sampled result, not a continuous optimum. A geometrically plausible contour is distinguished from an optimized flow solution. These distinctions are part of the result, rather than qualifications to remove when presenting a favorable number.

The equations in this book support interpretation of the current calculation modes. They do not constitute component qualification, operating approval, structural certification or a guarantee of a real engine’s performance. The productive use of the models is to formulate a clear question, preserve its assumptions and report the answer with the same boundaries.

1. Control volumes and engine performance

1.1 Begin with the physical boundary

An engine performance calculation begins by choosing a boundary, not by choosing a thrust equation. The boundary identifies which mass streams enter and leave, where pressures act, and which mechanical or thermal interactions are included. Two calculations may use individually correct equations and still disagree because they describe different boundaries.

For an airbreathing engine, a useful stationary control volume in the engine frame begins upstream of the inlet and ends at the exhaust plane. Captured air enters at the flight speed relative to the engine. Fuel enters through a separate supply. A turbofan has either separate core and bypass exit planes or one common mixed-flow exit. A rocket has no captured atmospheric-air stream: its propellant enters from onboard supplies. A standalone nozzle has an upstream state supplied by the user and does not, by itself, define the thrust of an entire vehicle.

Draw these boundaries before reducing the equations. Mark each inlet and exit area, the outward surface normal, static pressure, mass flow and axial velocity. Static pressure belongs in the surface-force balance. Total pressure is a useful state variable for losses and expansion, but it is not the pressure exerted on a plane in the momentum equation.

Throughout this chapter, the positive exhaust direction is downstream relative to the engine. We report the positive magnitude of the useful reaction force on the engine in the opposite direction. A signed force-vector formulation is equally valid, provided every normal and velocity uses the same convention.

1.2 Mass conservation and the choice of reference flow

The integral mass balance is

$$\frac{d}{dt}\int_{CV}\rho\,dV +\int_{CS}\rho\mathbf{V}\cdot\mathbf{n}\,dA=0.$$

For steady, one-dimensional streams this reduces to

$$\sum_{\mathrm{out}}\dot m_j=\sum_{\mathrm{in}}\dot m_i.$$

For a simple turbojet without bleed or auxiliary discharge,

$$\dot m_e=\dot m_a+\dot m_f=(1+f)\dot m_a, \qquad f=\frac{\dot m_f}{\dot m_a}.$$

The factor $1+f$ is a mass balance, not an efficiency correction. Neglecting it is an explicit ideal-cycle approximation. It must not disappear merely because fuel flow is small. Its effect also depends on exhaust speed, so a percentage error in mass flow need not produce the same percentage error in net thrust after inlet momentum is subtracted.

For a separate-flow turbofan, define core air flow $\dot m_c$, bypass air flow $\dot m_b$, and bypass ratio $\alpha=\dot m_b/\dot m_c$. Total captured air is

$$\dot m_a=(1+\alpha)\dot m_c.$$

Fuel-air ratio normally refers to core air, $f=\dot m_f/\dot m_c$. Therefore fuel per total captured air is $f/(1+\alpha)$. A comparison that divides total-air-based specific thrust into core-air-based fuel ratio overstates fuel consumption by a factor of $1+\alpha$.

For a rocket, total propellant flow includes oxidizer and fuel. With mixture ratio $MR=\dot m_{ox}/\dot m_f$, the fuel flow is $\dot m_p/(1+MR)$. Rocket specific impulse uses total propellant consumption, not fuel consumption alone. This distinction becomes especially important when comparing a rocket with an airbreathing engine whose oxidizer comes from the atmosphere.

1.3 Momentum flux is not thrust

The axial momentum balance for a steady control volume equates net external force on the fluid to the change in axial momentum flux. For uniform streams, each momentum-flux contribution is $\dot mV$. Pressure forces also act on each control surface. To compare an exhaust plane with its surrounding atmosphere, the relevant excess-pressure contribution is $(p_e-p_a)A_e$.

For one captured air stream and one exhaust, with negligible axial momentum supplied through the fuel line and with installation drag excluded, the customary uninstalled net thrust is

$$F_N=\dot m_eV_e-\dot m_aV_0+(p_e-p_a)A_e.$$

The assumptions matter. If an injected stream has appreciable axial momentum at the chosen boundary, its inlet momentum belongs in the balance. If an engine model includes multiple exits, the exhaust terms must be summed. If a propulsion installation includes spillage, external nacelle forces or other drag, an uninstalled cycle calculation does not automatically account for them.

Gross thrust is the exhaust contribution,

$$F_G=\dot m_eV_e+(p_e-p_a)A_e,$$

and inlet momentum drag for the simple captured-air boundary is $\dot m_aV_0$. Thus $F_N=F_G-\dot m_aV_0$. Calling gross thrust “net thrust” silently removes the cost of accelerating captured air from its incoming momentum state.

A different quantity appears in one-dimensional duct and mixer calculations:

$$J=pA+\dot mV.$$

This is a stream momentum function. It combines absolute-pressure force and momentum flux at one station. It is useful for connecting stations across a selected duct model. It is not, in general, an engine thrust. In particular,

$$F_G=J_e-p_aA_e,$$

and a net airbreathing calculation must still account for its inlet boundary. ThrustOne’s F/G state-reconstruction input uses a momentum function for F; it must not be populated with a whole-engine thrust simply because both quantities have units of newtons.

1.4 Pressure mismatch and the expansion condition

When $p_e=p_a$, the pressure-thrust term vanishes. This is the fully expanded condition, not a general property of nozzles. A design calculation may prescribe this condition and recover the exit area. A fixed-area nozzle in a changing atmosphere generally cannot remain fully expanded at every altitude.

For an underexpanded exit, $p_e>p_a$, the excess-pressure term is positive. For an attached overexpanded calculation, $p_e<p_a$, it is negative. That algebra alone does not establish whether an actual overexpanded nozzle remains attached. Boundary-layer separation, shock structure and asymmetric loads require additional physical information. A reference contour and an ideal exit state do not supply that information.

In the simplest fixed-internal-state altitude comparison,

$$\frac{\partial F_G}{\partial p_a}=-A_e.$$

This is a partial derivative holding the nozzle exit state and area fixed. It is not a universal engine-altitude law. It ceases to describe the full variation when the internal state, mass flow or flow topology responds to ambient pressure. When interpreting an App sweep, identify what the selected mode actually keeps fixed before assigning a trend to engine physics.

1.5 Worked example: why the reference flow matters

Consider an illustrative airbreathing stream with the following quantities. These are invented round-number inputs for understanding the accounting, not specifications of a real engine.

Mass conservation gives an exhaust flow of 82 kg/s. The three thrust terms are

$$\dot m_eV_e=82(750)=61{,}500\ \mathrm{N},$$ $$\dot m_aV_0=80(250)=20{,}000\ \mathrm{N},$$ $$(p_e-p_a)A_e=(100{,}000-95{,}000)(0.30)=1{,}500\ \mathrm{N}.$$

Gross thrust is therefore 63,000 N and net thrust is 43,000 N. Specific thrust based on captured air is

$$F_s=\frac{43{,}000}{80}=537.5\ \mathrm{N\,s/kg}.$$

Dividing by the exhaust mass flow instead gives approximately 524.39 N s/kg. That value is not an arithmetic error; it describes a different normalization. A result label that merely says “specific thrust” without identifying its reference mass flow is incomplete.

The fuel-air ratio is $2/80=0.025$, so thrust-specific fuel consumption can be calculated either directly or through consistently normalized quantities:

$$TSFC=\frac{2}{43{,}000} =\frac{0.025}{537.5} \approx4.65116\times10^{-5}\ \mathrm{kg/(N\,s)}.$$

If fuel mass addition were incorrectly omitted while all other supplied values were retained, the momentum term would fall by 1,500 N, giving 41,500 N net thrust. In this example that error happens to equal the pressure-thrust contribution. Omitting either one alone would happen to give the same 41,500 N result; omitting both would give 40,000 N. Agreement between two numbers can therefore conceal different accounting errors. Comparisons should inspect the terms, not only the final force.

As a second thought experiment, hold the exit state and geometry fixed but raise ambient pressure to 100 kPa. Pressure thrust becomes zero and net thrust becomes 41,500 N. This illustrates a controlled change in one boundary condition; it does not claim that a physical nozzle would retain all its internal states under an arbitrary ambient-pressure change.

1.6 Consumption, impulse and power describe different questions

Specific thrust measures force obtained per unit reference mass flow. Thrust-specific fuel consumption measures fuel flow required per unit force. Specific impulse measures thrust relative to a specified consumable mass flow and standard gravity:

$$I_{sp}=\frac{F}{\dot m_{\mathrm{consumable}}g_0}, \qquad g_0=9.80665\ \mathrm{m/s^2}.$$

The consumable is normally total propellant for a rocket and fuel for a fuel-based airbreathing impulse measure. For a mixed injected stream, the appropriate denominator depends on the model’s stated convention. A number in seconds is not comparable across engines until that convention is known.

Useful propulsive power is $F_NV_0$ for the uninstalled force convention used above. At zero flight speed it is zero even when static thrust is substantial. Thus zero propulsive power at a static condition is not evidence that a thrust calculation failed. Efficiency describes a rate of useful energy delivery; force describes momentum exchange. They answer different engineering questions.

Overall efficiency based on fuel chemical power requires a consistent heating-value convention and a compatible model boundary. Relations that split overall efficiency into thermal and propulsive factors often assume fully expanded streams. They should not be applied automatically to a pressure-mismatched nozzle without accounting for the associated pressure work. The App’s omission of an efficiency under an unsupported convention is more informative than a plausible-looking number with an undefined denominator.

1.7 Species and elements: two different conservation statements

Total mass conservation remains valid when chemical reactions change the composition. Conservation of each species does not. A mixture can lose molecular hydrogen and molecular oxygen while gaining water, with no loss of total mass. Therefore a balance that treats every inlet species as a conserved passive tracer describes frozen transport or nonreacting mixing, not chemical equilibrium.

Let $\dot n_i$ be the molar flow of species $i$, in kmol/s, and $\dot\xi_r$ the rate of progress of reaction $r$, also in kmol/s for its chosen stoichiometric equation. With signed stoichiometric coefficients $\nu_{ir}$, positive for products, a steady reactor balance can be written

$$\dot n_{i,out}-\dot n_{i,in}=\sum_r\nu_{ir}\dot\xi_r.$$

Changing the scale of a reaction equation changes its numerical rate-of-progress convention. Species production, rather than the numerical value of $\dot\xi_r$ alone, is the physical quantity that must remain consistent.

Let $a_{ki}$ count atoms of element $k$ in one molecule of species $i$. A chemically balanced reaction satisfies

$$\sum_i a_{ki}\nu_{ir}=0\qquad\text{for every element }k.$$

Multiplying the species balances by $a_{ki}$ and summing over species eliminates the reaction sources:

$$\sum_i a_{ki}\dot n_{i,out}=\sum_i a_{ki}\dot n_{i,in}.$$

Elements are conserved even though the individual species are not. If the molar masses $W_i$ are constructed consistently from the same elemental inventory, multiplying by $W_i$ also recovers total mass conservation. In an unsteady reactor, the corresponding accumulation terms must be retained; in a duct with injection, the injected element flows appear on the inlet side.

These equations explain the different questions asked by three App model classes. Frozen-property calculations evaluate a composition supplied by the user. Equilibrium calculations find a composition compatible with an elemental inventory and the selected thermodynamic constraints. Finite-rate calculations evolve species according to a specified reaction mechanism and finite elapsed time or flow distance. A conserved element balance alone cannot distinguish a chemically equilibrated mixture from a reacting or frozen mixture: all three may satisfy it.

An algebraic equilibrium composition is therefore not proof that a short combustor has enough residence time to reach equilibrium. Conversely, an incomplete finite-rate reaction does not violate thermodynamics merely because its composition differs from equilibrium. Chapters 5–8 develop these distinctions and the data conventions needed to use them.

1.8 Energy and entropy complete the control-volume description

The momentum balance defines force, but it does not determine an exhaust speed from temperature, fuel and component losses. For that purpose the energy balance must be joined to thermodynamic state relations. Define heat transfer $\dot Q$ as positive into the fluid and shaft power $\dot W_s$ as positive out of the fluid. For steady streams,

$$\dot Q-\dot W_s= \sum_{out}\dot m\left(h+\frac{V^2}{2}+gz\right) -\sum_{in}\dot m\left(h+\frac{V^2}{2}+gz\right).$$

Enthalpy $h=u+p/\rho$ contains the flow work required to carry material across the control surface. This is why adding a further pressure-work term to an enthalpy-based stream balance would count flow work twice. In this statement $h$ and $V^2/2$ have units of J/kg; multiplication by kg/s gives watts. Gravitational potential is usually negligible across an engine’s internal height, but neglecting it is a geometric approximation, not a property of all flow systems.

Write $h_t=h+V^2/2$ after neglecting height changes. An adiabatic stationary nozzle without shaft work then obeys $h_{t,e}=h_{t,in}$, so

$$V_e^2=V_{in}^2+2(h_{in}-h_e).$$

A decrease in static enthalpy provides the kinetic-energy increase. A compressor instead receives shaft work: $\dot W_s<0$, raising total enthalpy in an adiabatic single stream. A turbine extracts work, giving a positive $\dot W_s$ and a drop in total enthalpy. Connecting turbine and compressor balances requires the mechanical-efficiency convention and the correct mass flow through each component. It is not sufficient to equate their temperature drops when the fluids or flows differ.

For a combustor with air and fuel streams, the reference state of enthalpy is decisive. With thermochemical species enthalpies that include formation enthalpy, an idealized steady adiabatic balance is

$$\dot m_a h_{t,a}+\dot m_f h_{t,f} =(\dot m_a+\dot m_f)h_{t,p}.$$

Chemical conversion changes the species composition and hence the mixture enthalpy-temperature relation. It does not require an additional heating-value source in this particular equation. Adding the fuel heating value as well would count chemical energy twice. A simplified cycle formulation based on sensible enthalpies may instead represent released chemical energy through an explicit heating-value term. The two conventions can describe compatible physics, but their terms cannot be intermixed. Chapter 10 follows the actual conventions used by the variable-property cycle model.

Energy conservation also does not determine irreversibility. Entropy supplies a further constraint:

$$\sum_{out}\dot m s-\sum_{in}\dot m s =\sum_j\frac{\dot Q_j}{T_{b,j}}+\dot S_{gen}, \qquad \dot S_{gen}\ge0.$$

Here $T_{b,j}$ is the absolute boundary temperature associated with each heat interaction, and $\dot S_{gen}$ has units of W/K. An adiabatic single-stream device can preserve total enthalpy while generating entropy and losing total pressure. An inlet shock is an important example. Thus an unchanged total temperature, under an appropriate fixed-composition property model, does not imply an unchanged total pressure.

For multiple streams, do not replace the entropy-flow statement with a comparison of a single unweighted inlet and outlet entropy. Mixing changes both the flow weights and composition. The component chapters use these control-volume balances to distinguish an ideal expansion from a lossy inlet, a nonreacting mixer from a combustor, and a shaft-work interaction from heat addition.

1.9 What propulsive efficiency rewards

A simple fully expanded, single-stream model exposes an important design tradeoff. Neglect fuel mass addition and external installation losses, and let the inlet and exit mass flows both be $\dot m$. Then

$$F_N=\dot m(V_e-V_0),\qquad \dot E_{jet}=\frac{\dot m}{2}(V_e^2-V_0^2).$$

The useful propulsive power is $F_NV_0$. Dividing by the increase in stream kinetic-energy flux gives

$$\eta_p=\frac{F_NV_0}{\dot E_{jet}} =\frac{2V_0(V_e-V_0)}{(V_e-V_0)(V_e+V_0)} =\frac{2V_0}{V_e+V_0},$$

for a positive-thrust state with $V_e>V_0\ge0$. The cancelled factor is also the factor that determines thrust. At $V_e=V_0$, the original force and power expressions both vanish for fixed mass flow. The limiting value of one for $\eta_p$ is not a finite-thrust operating point. Maintaining a prescribed positive thrust while taking the velocity increment to zero requires an unbounded mass flow in this simplified model.

This result explains why accelerating more air through a smaller velocity increment can improve propulsive efficiency. It does not establish that an indefinitely large bypass ratio is best. Larger capture area, nacelle drag, component weight, fan limits and other installation effects are outside this elementary derivation. In ThrustOne, the ideal trend is useful for understanding the cycle, while the absence of a full installation model must remain explicit.

For an original illustrative comparison, take $V_0=200$ m/s and require 20 kN net thrust. An ideal stream with $V_e=600$ m/s requires 50 kg/s and adds 8 MW of jet kinetic power. A stream with $V_e=400$ m/s requires 100 kg/s and adds 6 MW. Both deliver 4 MW of useful propulsive power. Their propulsive efficiencies are therefore 0.50 and approximately 0.667. The second stream produces the same force with a smaller kinetic-energy increment but twice the air flow. No fuel consumption follows until a cycle establishes the fuel power needed to produce that jet power.

If a compatible thermal efficiency is defined as $\eta_{th}=\dot E_{jet}/(\dot m_f LHV)$, then multiplication gives

$$\eta_o=\eta_p\eta_{th}=\frac{F_NV_0}{\dot m_f LHV}.$$

This identity follows from consistent definitions, not from an independent physical law. An inappropriate intermediate jet-power definition cannot be repaired by multiplying two efficiency values. Pressure-mismatched exits, additional power exports and changing reference conventions require the energy accounting to be stated before using the decomposition.

1.10 Worked example: a two-exit engine

Consider a separate-flow engine with core air flow 20 kg/s, bypass ratio 4, fuel flow 0.6 kg/s and flight speed 200 m/s. Let the fully expanded core and bypass exit speeds be 700 and 300 m/s. There is no bleed, no separate power export and no installation drag in this example. We prescribe exit states to study accounting; these values are not claimed to be the solution of a specified thermodynamic cycle.

The bypass flow is 80 kg/s and the total captured air flow is 100 kg/s. The core exit includes fuel, so it carries 20.6 kg/s. The common whole-engine force is the sum of both exhaust contributions minus the captured-air momentum:

$$F_N=20.6(700)+80(300)-100(200)=18{,}420\ \mathrm{N}.$$

Equivalently, one may subtract each stream’s own inlet momentum. The core contribution is $20.6(700)-20(200)=10{,}420$ N; the bypass contribution is $80(300)-80(200)=8{,}000$ N. Their sum gives the same whole-engine result. Subtracting the full 100 kg/s inlet contribution from each stream would count inlet momentum drag twice.

On a total-air basis, $F_s=184.2$ N s/kg. On a core-air basis it is 921 N s/kg. Their ratio is five, exactly $1+\alpha$. The fuel-air ratio based on core air is $f=0.03$, whereas fuel per total air is 0.006. Consequently,

$$TSFC=\frac{0.6}{18{,}420} =\frac{0.006}{184.2} =\frac{0.03}{921} \approx3.25733\times10^{-5}\ \mathrm{kg/(N\,s)}.$$

Mixing the numerator from the core convention with the denominator from the total-air convention would give five times this consumption. The arithmetic would look ordinary; only an explicit flow definition exposes the mistake.

Now compare an independent rocket accounting example: 10 kg/s total propellant, exit speed 2,500 m/s, exit area 0.05 m², exit pressure 150 kPa and ambient pressure 100 kPa. Its thrust under the stated ideal nozzle boundary is 27,500 N, comprising 25,000 N momentum thrust and 2,500 N pressure thrust. The effective exhaust velocity is $F/\dot m_p=2,750$ m/s, which exceeds the actual exit speed because pressure also contributes to force. Specific impulse is approximately 280.42 s. Calling 2,750 m/s the actual flow velocity would transfer the pressure-force contribution into the kinetic-energy calculation incorrectly.

1.11 Closing an internal duct problem

Figure 1.1 distinguishes the whole-engine boundary from a boundary that encloses only a duct. For a steady single stream with no mass addition, gravity or other axial body force, let $F_{w,x}$ be the wall force on the fluid, positive downstream. The momentum equation is

$$p_1A_1-p_2A_2+F_{w,x}=\dot m(V_2-V_1),$$

or, after collecting terms,

$$J_2-J_1=F_{w,x}.$$

The reaction of the fluid on the wall has the opposite sign. In a frictionless constant-area duct with no axial wall-pressure resultant, this reduces to $J_2=J_1$. In a variable-area duct, the inclined-wall pressure contributes to $F_{w,x}$; friction contributes another force. A constant momentum function cannot therefore be assumed for every nozzle or source-term segment. Mass injection adds momentum terms associated with the incoming stream as well.

The relation illustrates why a pair of station states does not automatically define an entire propulsion system. A station momentum function does not contain the external ambient subtraction appropriate to a complete nozzle, nor the captured-air momentum of a complete engine. The wall reaction of an internal component also need not equal the final vehicle propulsive force.

A solvable component problem must specify enough independent information to determine its outlet. For example, continuity, momentum and energy alone do not supply an equation of state, a wall-friction law, a chemical composition or a nozzle branch. Conversely, prescribing geometry, all inlet and outlet states, heat addition and wall force independently may overconstrain the problem. Chapters 2, 24 and 26 show how particular closures turn the same balances into different computable models.

1.12 Applying the chapter in ThrustOne

Before comparing two outputs, identify the selected mode, reference mass flow, fuel or propellant denominator, exit-pressure closure and included boundary forces. The same engine label can contain a complete cycle model and a standalone component model with different boundaries.

Use exported SI inputs to recover the physical state behind a result. A changed display unit does not change the control volume. A changed model, nozzle closure or normalization does. The purpose of the following chapters is to make those differences explicit before deriving the individual model equations.

The following distinctions apply to the current implementation. They are physical interpretation rules; they do not require the reader to memorize an export schema.

Two efficiency conventions require particular care. The constant-property turbojet and turbofan calculations retain a thrust-power overall efficiency when an exit is not fully expanded, but suppress the thermal/propulsive kinetic-energy split in that condition. The variable-property cycle implementation reports no such efficiency outputs even with fully expanded exits: it uses formation-enthalpy combustion and does not supply the compatible heating-value denominator used by the other model. A missing metric is therefore not equivalent to a zero metric or proof of failed calculation.

The balances in this chapter determine relationships among states and streams. Subsequent chapters supply the properties, losses, geometry and root choices needed to determine those states. That separation makes a meaningful comparison possible: first establish that both calculations describe the same boundary, then compare how their model assumptions predict the flow through it.

2. Static, stagnation and sonic states

2.1 A flowing state contains more than temperature and pressure

At a station in an engine, static temperature and static pressure describe the local thermodynamic state of a specified gas. Velocity supplies the additional information needed to describe its motion relative to the engine. A gas can therefore have the same static temperature and pressure at two stations yet carry different kinetic energy and different mass flux through equal areas.

The term static does not mean that the gas is stationary. It distinguishes local thermodynamic properties from properties obtained by imagining a particular process that brings the moving gas to rest. A fluid element moving at high speed still has a static temperature, density and pressure. In the notation of this book, an unsubscripted T or p is static unless the selected model states otherwise; the subscript t denotes a stagnation or total state.

Mass flow requires geometry as well as state. For a uniform one-dimensional stream normal to its section,

$$\dot m=\rho VA.$$

Neither T/p/M nor T/p/V alone fixes an absolute mass flow without an area. Conversely, prescribed mass flow can be used to recover area after the thermodynamic state and speed have been determined. This difference separates a state-reconstruction task from a nozzle-sizing task even when both display the same station quantities.

The following derivations assume a steady stream, negligible gravitational-height change and an ideal gas. Additional restrictions are stated as they are introduced. The distinction between a frozen gas and an equilibrium reacting mixture is deferred until Section 2.10; it cannot be hidden in the choice of a numerical gamma.

2.2 Thermally and calorically perfect descriptions

An ideal gas with fixed composition obeys

$$p=\rho RT,\qquad dh=c_p(T)\,dT,\qquad c_p(T)-c_v(T)=R.$$

Here R is the mixture-specific gas constant in J/(kg K). A thermally perfect description permits temperature-dependent heat capacity while retaining ideal-gas behavior. A calorically perfect description additionally holds $c_p$, $c_v$ and $\gamma=c_p/c_v$ constant. It follows that

$$c_p=\frac{\gamma R}{\gamma-1},\qquad c_v=\frac{R}{\gamma-1}.$$

These are connected properties, not three arbitrary independent constants. Supplying mutually inconsistent values of R, cp and gamma would cause the energy equation and equation of state to describe different gases. The App’s constant-property models may derive one property from the others; a comparison with an external calculation must follow that same convention.

For a fixed composition the entropy differential is

$$ds=\frac{c_p(T)}{T}\,dT-R\frac{dp}{p}.$$

It separates thermal and pressure contributions. With constant cp, integration between two states gives

$$s_2-s_1=c_p\ln\frac{T_2}{T_1}-R\ln\frac{p_2}{p_1}.$$

Setting this change to zero produces an isentropic relation. It does not make an arbitrary real flow isentropic: a shock, friction or mixing can generate entropy while conserving total enthalpy. The state relation must be used with the physical process, not as a replacement for it.

2.3 Stagnation enthalpy comes from energy conservation

For an adiabatic stream with no shaft work, Chapter 1 gives

$$h_t=h+\frac{V^2}{2}=\text{constant along the modeled stream}.$$

The associated stagnation temperature is defined by $h(T_t)=h_t$ for the same frozen composition. Under the constant-cp approximation,

$$h(T_t)-h(T)=c_p(T_t-T)=\frac{V^2}{2},$$

and therefore

$$T_t=T+\frac{V^2}{2c_p}.$$

This temperature relation follows from energy; it does not require the actual deceleration to be reversible. A stationary adiabatic normal shock preserves total enthalpy even though it generates entropy. For the same calorically perfect gas this means total temperature is unchanged while total pressure falls.

A hot gas can possess considerable static enthalpy but modest kinetic energy. A colder gas moving rapidly can have the same total enthalpy. A nozzle uses this exchange to accelerate the stream; a diffuser converts kinetic energy into static enthalpy. Neither component creates energy simply by changing velocity.

For a variable-cp gas the correct relation is instead

$$\int_T^{T_t}c_p(\theta)\,d\theta=\frac{V^2}{2}.$$

The temperature on the upper limit is an unknown. Replacing the integral with $c_p(T)(T_t-T)$ is a further approximation. Formation-enthalpy constants cancel between states with unchanged composition, but they cannot generally be discarded when species compositions differ. Chapters 5 and 10 make the reacting and frozen reference conventions explicit.

Stagnation properties also depend on the velocity reference frame. A freestream that has one total enthalpy relative to the ground has another relative to a fast-moving engine. The cycle calculations use the engine frame. Rotating blade relative-total properties introduce a different frame and should not be substituted into an absolute-frame nozzle state.

2.4 Stagnation pressure requires an isentropic reference process

Stagnation pressure is the pressure reached by bringing the local flow to rest isentropically, with the selected composition constraint. It is a constructed state property; it is not generally the pressure that an actual lossy stopping process delivers.

For a calorically perfect gas, set $s_t=s$ in the entropy relation:

$$0=c_p\ln\frac{T_t}{T}-R\ln\frac{p_t}{p}.$$

Rearrangement gives

$$\frac{p_t}{p}=\left(\frac{T_t}{T}\right)^{c_p/R} =\left(\frac{T_t}{T}\right)^{\gamma/(\gamma-1)}.$$

At zero velocity, Tt=T and pt=p. At nonzero velocity, the same gas has Tt>T and pt>p. Substituting total pressure into $\rho=p/(RT)$ while leaving the temperature static mixes two states and overestimates the local density. The static mass-flow equation must use a mutually compatible static density and speed.

If a fixed-composition adiabatic device preserves Tt but generates entropy, its inlet and outlet total states satisfy

$$s_2-s_1=-R\ln\frac{p_{t,2}}{p_{t,1}}.$$

A positive entropy change then corresponds to a total-pressure ratio below one. Total-pressure recovery can describe a loss even when total temperature is constant. Inlet recovery and combustor pressure-loss parameters should be interpreted in this way, rather than as direct changes in temperature or energy.

A pressure probe that creates a shock in a supersonic stream measures a post-shock stopping pressure. That measured value is not the upstream isentropic total pressure unless the shock loss is accounted for. This chapter defines the state quantities; it does not provide a probe-correction model or claim that the App reconstructs an arbitrary measurement installation.

2.5 Mach number and sonic reference ratios

For a frozen ideal-gas composition, the local acoustic speed follows from an isentropic infinitesimal disturbance:

$$a_f^2=\left(\frac{\partial p}{\partial\rho}\right)_{s,Y} =\gamma(T)RT,\qquad M_f=\frac{V}{a_f}.$$

The subscript Y indicates that composition is held fixed. Under constant properties, writing V²=M²γRT in the stagnation-temperature relation gives

$$\tau(M)\equiv\frac{T_t}{T} =1+\frac{\gamma-1}{2}M^2.$$

Temperature, pressure and density ratios then become

$$\frac{T}{T_t}=\tau^{-1},\qquad \frac{p}{p_t}=\tau^{-\gamma/(\gamma-1)},\qquad \frac{\rho}{\rho_t}=\tau^{-1/(\gamma-1)}.$$

The starred state is the sonic state on the same constant-property isentrope, with M=1:

$$\frac{T^*}{T_t}=\frac{2}{\gamma+1},\qquad \frac{p^*}{p_t}=\left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)}.$$

For gamma=1.4 these are approximately 0.833333 and 0.528282. They are ratios of static sonic properties to total properties. They are not ratios to a general inlet static state, and they change with gamma. In particular, 0.833333 follows directly from 2/2.4; a corrupted table or OCR value near it must not override the defining equation.

A star does not always identify a physical throat present in the geometry. A subsonic duct can be assigned a hypothetical sonic reference area for its existing mass flow and total state. That reference becomes the actual minimum-area section only when the assumptions and boundary conditions establish a choked smooth-nozzle solution.

2.6 Deriving the maximum mass flux

Define axial mass flux $j_m=\dot m/A=\rho V$. This notation avoids confusion with the G input used elsewhere for total specific enthalpy. Substituting the ideal-gas law, acoustic speed and static-to-total ratios gives

$$j_m=\frac{p_t}{\sqrt{T_t}}\sqrt{\frac{\gamma}{R}}\, M\left(1+\frac{\gamma-1}{2}M^2\right)^{-\frac{\gamma+1}{2(\gamma-1)}}.$$

For fixed gas, pt and Tt, only the last two factors vary with Mach number. Put $b=(\gamma-1)/2$ and differentiate the logarithm with respect to M:

$$\frac{d\ln j_m}{dM} =\frac{1}{M}-\frac{\gamma+1}{2(\gamma-1)}\frac{2bM}{1+bM^2} =\frac{1-M^2}{M(1+bM^2)}.$$

The derivative is positive for 0<M<1, zero at M=1 and negative for M>1. Thus the sonic state maximizes mass flux at fixed total conditions. The critical value is

$$j_m^*=\frac{p_t}{\sqrt{T_t}}\sqrt{\frac{\gamma}{R}} \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}.$$

An increase in velocity on the supersonic branch is accompanied by a sufficiently large density decrease that mass flux falls. Faster flow does not automatically mean more mass through the same area when total conditions are held fixed.

The formula also exposes its scale dependence: doubling pt doubles the critical mass flux, while multiplying Tt by four halves it if gas properties remain unchanged. In a reacting chamber, temperature and properties can change together, so the latter relation is a controlled constant-property comparison, not a complete combustion trend.

Choking means that a selected throat has reached this limiting mass flux for its upstream total state. Further changes downstream cannot simply demand a larger steady mass flow through that same section without changing the upstream state or the physical flow pattern. It does not mean that every station is sonic or that the entire engine stops responding to operating conditions.

2.7 Area–Mach inversion and the two roots

For the same stream and isentropic total conditions,

$$\dot m=A j_m=A^*j_m^*.$$

Taking the ratio gives

$$\frac{A}{A^*}=\frac{1}{M} \left[\frac{2}{\gamma+1} \left(1+\frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}.$$

This function reaches its minimum value of one at M=1. For a ratio greater than one, it has a subsonic root and a supersonic root. The specified area ratio by itself cannot choose between them. A ratio below one has no root under these assumptions. An algorithm that silently returns one branch has added a physical assumption without informing the reader.

The differential form explains the geometry. Continuity gives $d\rho/\rho+dV/V+dA/A=0$. For an inviscid isentropic stream without added mass or heat, momentum gives $dp=-\rho V\,dV$, and the acoustic relation gives $dp=a^2d\rho$. Combining them yields

$$\frac{d\rho}{\rho}=-M^2\frac{dV}{V},\qquad \frac{dA}{A}=(M^2-1)\frac{dV}{V}.$$

A subsonic flow accelerates through a contraction, whereas a supersonic flow accelerates through an expansion. Smooth passage through M=1 requires an area extremum in this restricted model; a converging-diverging nozzle supplies the familiar minimum. The relation does not guarantee that a supplied back pressure will establish the supersonic solution downstream. Shock-containing and entirely subsonic solutions require the appropriate boundary treatment.

Heat transfer, friction and injection alter this differential equation. Consequently the common statement “Mach one occurs at the minimum area” is not a general theorem for the source-term ducts used in other App modes. A sonic event in such a model is a compatibility condition, not merely a geometric feature.

2.8 Worked example: the same area supports different states

Use gamma=1.4, R=287 J/(kg K), pt=300 kPa and Tt=600 K. Let a choked reference throat have area 0.004 m². The constant-property model gives

$$T^*=600\frac{2}{2.4}=500\ \mathrm{K}, \qquad p^*\approx158.485\ \mathrm{kPa}.$$

The critical mass flux is approximately 495.023 kg/(m² s), so the mass flow is

$$\dot m=0.004(495.023)\approx1.98009\ \mathrm{kg/s}.$$

Select a downstream area ratio of 1.6875, giving A=0.00675 m². The supersonic area root is M=2. At that state,

$$T=\frac{600}{1+0.2(2)^2}=333.333\ \mathrm{K},$$ $$p=300\left(\frac{333.333}{600}\right)^{3.5} \approx38.3414\ \mathrm{kPa},$$ $$V=2\sqrt{1.4(287)(333.333)}\approx731.938\ \mathrm{m/s}.$$

Using static density gives approximately 0.400781 kg/m³. Multiplying density, speed and exit area recovers 1.98009 kg/s.

The subsonic root for the same area ratio is approximately 0.372244. It has a much higher pressure and density, and a much lower speed:

The two states share total conditions, mass flux and area ratio, but their pressures are very different. They are alternative algebraic branches, not two simultaneous exit states. Connecting a selected root to an actual throat and prescribed ambient pressure requires a complete nozzle-flow solution. In particular, this table is not a model of a normal shock between the two states: a physical shock would change total pressure and would not connect them on the same isentrope.

2.9 Near-sonic inversion is sensitive

Let $M=1+\delta$. Expanding the area relation around its minimum gives

$$\frac{A}{A^*}=1+\frac{2}{\gamma+1}\delta^2+O(\delta^3).$$

There is no linear term. The two branches approach each other as the square root of the area excess:

$$M\approx1\pm\sqrt{\frac{\gamma+1}{2}\left(\frac{A}{A^*}-1\right)}.$$

For gamma=1.4, an area-ratio excess of 0.0001 corresponds to a leading-order Mach deviation of about 0.0109545. An area displayed as 1.000 to only three decimal places can therefore conceal a physically relevant difference between slightly subsonic and slightly supersonic states.

Direct substitution provides another illustration: M=0.99 gives A/A*=1.000083802, while M=1.01 gives 1.000082876. The small asymmetry comes from terms beyond the quadratic approximation. Rounding both ratios to one would erase both the deviation and the branch.

This is conditioning, not automatically a defective root solver. The slope of the forward area function vanishes at the minimum, so its inverse becomes sensitive. A solver should retain the selected branch and sufficient internal precision; a report should not imply more information than the input area ratio supports. Chapters 26 and 33 discuss the related distinction between a valid limiting state, an unbracketed root and a terminated integration.

2.10 Frozen variable-property expansion

For a frozen ideal-gas mixture, R remains constant with temperature but cp and gamma may change. Specify the total state and composition. A trial static temperature supplies enthalpy and entropy, from which the isentropic pressure and speed follow:

$$\ln\frac{p_t}{p}=\frac{1}{R}\int_T^{T_t}\frac{c_p(\theta)}{\theta}\,d\theta, \qquad V(T)=\sqrt{2[h(T_t)-h(T)]}.$$

Then $\rho(T)=p(T)/(RT)$, $j_m(T)=\rho(T)V(T)$, and the local frozen Mach number follows from the local heat-capacity ratio. This sequence replaces a single global gamma power law. The property data must support every trial state, not only the inlet and final accepted exit.

An original analytic example makes the distinction visible without relying on a species database. Let a hypothetical frozen ideal gas have R=287 J/(kg K) and

$$c_p(T)=1000+0.1T\quad\mathrm{J/(kg\,K)}$$

over the chosen illustrative interval. The coefficient 0.1 has units J/(kg K²). Between T=600 K and Tt=900 K,

$$h_t-h=1000(900-600)+0.05(900^2-600^2) =322{,}500\ \mathrm{J/kg}.$$

Therefore V≈803.119 m/s. At the static state cp=1060 J/(kg K), gamma≈1.37128 and the local frozen acoustic speed is approximately 485.937 m/s, giving M≈1.65272. The exact pressure ratio for this hypothetical property law is

$$\frac{p_t}{p}=\exp\left[ \frac{1000\ln(900/600)+0.1(900-600)}{287}\right] \approx4.55990.$$

If cp were frozen at its 600 K value for the entire deceleration, the same kinetic energy would instead give Tt≈904.245 K and a pressure ratio of approximately 4.54904. The point is not that this particular difference is always large. The approximation has changed both the energy integral and entropy integral, and no single local gamma guarantees that both are represented consistently. The App’s NASA-property models use their actual property functions, not this illustrative linear law.

For fixed composition, maximizing the isentropic mass flux still leads to the local frozen sonic condition. Along the isentrope, energy gives $V\,dV=-dh=-dp/\rho$, while $dp=a_f^2d\rho$. It follows that

$$\frac{d(\rho V)}{d\rho}=V-\frac{a_f^2}{V},$$

which vanishes at V=af. This is a local property result; it does not restore the constant-gamma area formula. The critical temperature and pressure must still be found using the variable-property enthalpy and entropy relations.

In an equilibrium expansion, composition is allowed to change along the path. The path derivative used to identify maximum mass flux is then different from the frozen acoustic derivative. A reported frozen Mach number need not equal one at the equilibrium critical state. ThrustOne distinguishes the equilibrium critical mass-flux closure from its local frozen_mach diagnostic; Chapter 17 develops the chemical and sonic conventions in detail.

2.11 What choking does and does not determine

A convergent nozzle connected to a reservoir provides a simple boundary example. If downstream pressure is sufficiently high, the nozzle can remain subsonic, with exit static pressure matching the downstream condition in the ideal model. As back pressure falls to the critical ratio, the minimum-area exit becomes sonic. Further lowering ambient pressure does not lower the internal exit pressure in the ideal choked convergent model; instead the jet is underexpanded outside the exit.

This distinction prevents an invalid combination of assumptions: setting the exit of a convergent nozzle to M=1 while also assigning it an arbitrary pressure below the critical static pressure at unchanged total conditions overdetermines the state. A calculation that fully expands to a specified low pressure may be sizing an exit area for a converging-diverging nozzle rather than predicting a fixed convergent nozzle.

A supersonic area-root calculation has a similarly limited meaning. It determines a candidate isentropic exit state from area ratio and total conditions. It does not automatically determine whether the physical nozzle has an internal shock, separated flow or a start transient. Those features require further equations and information. The selected App mode must tell the reader which boundary problem is actually closed.

The constant-property cycle, frozen-property nozzle and equilibrium rocket nozzle share conservation laws but impose different closures and property assumptions. Their results should be compared by matching total state, composition, area and ambient convention, not simply by matching a label such as “nozzle exit Mach.”

2.12 Relating the derivation to the selected App mode

The same physical relation appears under different boundary conditions in the App. The distinctions below are essential to choosing an interpretable calculation.

Temperature-domain checks in variable-property calculations are physical-data constraints. A requested expansion that would fall below the mixture’s common data range is rejected instead of extrapolating the species fits. The existence of a constant-gamma answer outside that interval does not authorize the variable-property model to continue there.

When reading a result, first identify whether its boundary was specified by exit pressure, area ratio, an inlet state and increasing area, or reference-matched convergent geometry. Only then interpret its Mach number, mass flow and pressure-thrust contribution. The algebra of an isentrope is common to these problems; their boundary conditions decide which particular state it describes.

3. Performance normalization and similarity

3.1 Why one physical engine needs several coordinate systems

A compressor running at the same shaft speed does not necessarily operate at the same aerodynamic condition when its inlet temperature changes. The blade speed can remain fixed while the acoustic speed changes. Similarly, the same air mass flow passing through a component at lower inlet pressure occupies a larger volume and may require a different flow velocity. Raw rpm and kg/s therefore do not isolate the similarity of the internal flow.

Normalization removes selected scales so that meaningful comparisons become easier. It does not remove the underlying physics. A corrected coordinate can compensate for inlet temperature and pressure while leaving Reynolds-number effects, tip clearance, gas composition and geometry changes unresolved. Two points close together on a corrected map are not automatically interchangeable engines.

Three different operations must be kept separate. A dimensionless quantity has no physical units, such as a pressure ratio. A corrected quantity retains units while referring a measured value to a chosen temperature and pressure convention. A reference-normalized quantity is a ratio to one selected operating point. A corrected mass flow in kg/s, a mass-flow parameter containing dimensional gas constants, and a flow ratio relative to a design point are related but are not the same object.

The distinction matters in ThrustOne because component-map coordinates, standalone component states and reference-based off-design calculations coexist. External maps may use rpm, a fraction of design corrected speed, or a percentage. ThrustOne expects its map coordinates and dependent tables to be dimensionless and design-point normalized; it does not perform the physical temperature/pressure correction on import. An external map must first be expressed in that convention.

3.2 Obtaining flow and speed similarity groups

Consider geometrically similar rotating components with characteristic diameter D. Let inlet total pressure and temperature be pt and Tt, and let the working gas have fixed R and gamma. The acoustic velocity scale is proportional to $\sqrt{RT_t}$. A density scale is proportional to $p_t/(RT_t)$. Their product with an area scale gives a mass-flow scale:

$$\rho_{scale}V_{scale}D^2 =\frac{p_t}{RT_t}\sqrt{RT_t}D^2 =\frac{p_tD^2}{\sqrt{RT_t}}.$$

A corresponding dimensionless flow group is

$$\Pi_m=\frac{\dot m\sqrt{RT_t}}{p_tD^2}.$$

Let N denote shaft revolutions per second. Blade speed is proportional to ND, so a dimensionless speed group is

$$\Pi_N=\frac{ND}{\sqrt{RT_t}}.$$

Using angular velocity instead of revolutions per second introduces the constant 2π; using rpm introduces a factor of 60. Such constants are harmless only when they are carried consistently. An rpm coordinate cannot be inserted into a formula calibrated with revolutions per second without conversion.

For a fixed component diameter and fixed gas, holding these groups constant gives the familiar proportionalities

$$\dot m\frac{\sqrt{T_t}}{p_t}=\text{constant}, \qquad \frac{N}{\sqrt{T_t}}=\text{constant}.$$

The first result also follows from the mass-flux relation in Chapter 2 for fixed Mach number and area. A compressor does not have one uniform Mach number everywhere, but the scaling motivates useful correlating coordinates for similar internal flow patterns.

Additional dimensionless parameters remain. Reynolds number involves viscosity, speed, density and length. Relative clearance involves a clearance-to-diameter ratio. Gas-property changes affect gamma and the thermodynamic conversion from work to pressure ratio. Consequently a family of maps at matched corrected flow and speed may still show changes in efficiency or stability boundaries. In this book, similarity is a stated approximation or a data correlation, never a claim that two dimensional scales determine every feature of a real compressor.

3.3 Corrected quantities retain a reference convention

Choose reference temperature Tref and pressure pref. At the inlet station relevant to the component, define

$$\theta=\frac{T_t}{T_{ref}},\qquad \delta=\frac{p_t}{p_{ref}}.$$

The usual fixed-gas corrected mass flow and corrected speed are

$$\dot m_c=\dot m\frac{\sqrt{\theta}}{\delta}, \qquad N_c=\frac{N}{\sqrt{\theta}}.$$

Because theta and delta are dimensionless, corrected mass flow has the same units as actual mass flow and corrected speed has the same units as actual speed. These definitions can be inverted:

$$\dot m=\dot m_c\frac{\delta}{\sqrt{\theta}}, \qquad N=N_c\sqrt{\theta}.$$

The pressure and temperature belong to the chosen component inlet, not automatically to the atmosphere. In a turbojet, the compressor inlet total state follows the inlet’s ram rise and pressure recovery. In a turbofan, the high-pressure compressor inlet follows the fan or booster path. Correcting every spool with freestream static conditions would mix incompatible reference stations.

A turbine and compressor on one shaft share actual mechanical speed, but their corrected speeds usually differ because their inlet total temperatures differ. If the same Tref is used for both,

$$\frac{N_{c,t}}{N_{c,c}}=\sqrt{\frac{T_{t,c}}{T_{t,t}}}.$$

There is no mechanical contradiction: the two coordinates describe the same shaft motion relative to different local acoustic scales.

The selected reference conditions are part of a data definition. They should accompany a map or case documentation. A corrected value without its inlet station and reference temperature/pressure is an incomplete specification, even if the number is finite and lies inside a tabulated axis range.

3.4 Worked example: preserving an aerodynamic coordinate

Use the illustrative convention Tref=288 K and pref=100 kPa. Suppose a reference component point has inlet totals equal to these values, actual mass flow 20 kg/s and actual speed 10000 rpm. Its corrected values are then 20 kg/s and 10000 rpm.

Now change the inlet totals to 360 K and80 kPa while asking for the same corrected coordinates. The ratios are theta=1.25 and delta=0.8. Therefore,

$$\dot m=20\frac{0.8}{\sqrt{1.25}} \approx 14.3108\ \mathrm{kg/s},$$ $$N=10000\sqrt{1.25}\approx 11180.34\ \mathrm{rpm}.$$

The hotter stream requires a higher actual speed to maintain the same corrected speed. The reduced pressure and higher temperature reduce the actual mass flow corresponding to the selected corrected flow. Feeding the old actual values into the new state would move the operating point on the map.

For example, retaining 10000 rpm at 360 K gives a corrected speed of approximately8944.27 rpm under this convention. A controller holding actual speed fixed and a calculation holding corrected speed fixed are imposing different conditions. Neither is intrinsically wrong; they answer different questions.

These calculations do not establish that the same point is physically attainable at both conditions. They presume the same gas, geometry and usable map correlation. Temperature limits, Reynolds-number changes and shaft constraints can invalidate the supposed correspondence. The example isolates the coordinate transformation so those additional decisions remain visible.

3.5 Changing the correction convention does not change the engine

Suppose the same map data are re-expressed using Tref,2 and pref,2 rather than Tref,1 and pref,1. Applying the definitions to the same actual state gives

$$\frac{\dot m_{c,2}}{\dot m_{c,1}} =\sqrt{\frac{T_{ref,1}}{T_{ref,2}}} \frac{p_{ref,2}}{p_{ref,1}}, \qquad \frac{N_{c,2}}{N_{c,1}} =\sqrt{\frac{T_{ref,2}}{T_{ref,1}}}.$$

For the preceding 20 kg/s and 10000 rpm corrected point, changing the convention to300 K and101.325 kPa gives approximately 19.8556 kg/s and 10206.21 rpm. The physical engine has not changed. Every point and every query must be transformed consistently before the new coordinates can be used.

A design-normalized map may instead use $\widehat N_c=N_c/N_{c,R}$. A value of 0.95 then means 95% of the reference corrected speed. It does not mean 0.95 rpm, and it does not necessarily mean 95% of reference mechanical speed. With $T_t/T_{t,R}=1.21$, a mechanical speed ratio of 0.95 corresponds to

$$\frac{N_c}{N_{c,R}} =\frac{N}{N_R}\sqrt{\frac{T_{t,R}}{T_t}} =\frac{0.95}{1.1}\approx 0.863636.$$

The corrected speed ratio has changed more because the inlet is hotter. Conversely, holding a corrected speed ratio at 0.95 under these temperature conditions would imply a mechanical speed ratio of1.045. Confusing the two could conceal a shaft-speed exceedance in a real application, even though a simple map interpolation produces a valid number.

3.6 Pressure, temperature and work ratios describe different effects

A component total-pressure ratio is $\pi=p_{t,out}/p_{t,in}$; its total-temperature ratio is $\tau=T_{t,out}/T_{t,in}$. These definitions do not include an efficiency by themselves. A compressor model supplies the relation between them.

For a calorically perfect gas and an isentropic compressor, $\tau_s=\pi^{(\gamma-1)/\gamma}$. With an isentropic efficiency defined as ideal work divided by actual work,

$$\eta_c=\frac{\tau_s-1}{\tau-1}, \qquad \tau=1+\frac{\pi^{(\gamma-1)/\gamma}-1}{\eta_c}.$$

The actual specific work is $\Delta h_t=c_pT_{t,in}(\tau-1)$. Thus the same pressure ratio and efficiency require more work per kilogram when the inlet total temperature is higher. A pressure-ratio match alone is not a shaft-power match.

A polytropic efficiency uses a different differential process convention and produces a different finite pressure-to-temperature relation. One must not replace isentropic efficiency with a polytropic number in the equation above. Chapters11 and15 derive the component conventions; here the important distinction is between a coordinate definition and the physical relation that connects coordinates.

For a fixed-geometry compressor family, a simplified work-coefficient assumption may give

$$\Delta h_t=K N^2,$$

where K contains geometry, units and the assumed work coefficient. This is an operating-region approximation, not a universal compressor law. Combining it with the constant-property isentropic-efficiency relation and dividing by a reference point gives

$$\left(\frac{N}{N_R}\right)^2 =\frac{T_{t,in}}{T_{t,in,R}} \frac{\pi^{(\gamma-1)/\gamma}-1} {\pi_R^{(\gamma-1)/\gamma}-1} \frac{\eta_{c,R}}{\eta_c}.$$

If efficiency is unchanged, the last factor is one. The inlet-temperature ratio disappears only after converting this mechanical-speed ratio into a corrected-speed ratio. This derivation explains why a speed estimate in a reference-normalized cycle can resemble a map relation without actually using a measured map.

Changing gas composition or cp changes the derivation as well. A variable-property compressor requires an enthalpy calculation, and the assumed constant K may no longer represent the same blade loading. Similarity relations should therefore be restricted to the gas and component approximation under which they were obtained.

3.7 A reference point replaces constants, not missing equations

Referencing is useful when a constant is difficult to identify separately but an operating point is known. For a choked section with fixed area, gas and discharge convention, Chapter2 gives

$$\dot m=C\frac{p_t}{\sqrt{T_t}}.$$

The constant C includes the area and critical-flow factor. Comparing with a reference state eliminates it:

$$\frac{\dot m}{\dot m_R} =\frac{p_t}{p_{t,R}}\sqrt{\frac{T_{t,R}}{T_t}}.$$

If the section area changes, an area ratio must be restored. If gamma or R changes, the critical-flow factors must also be restored. If the section becomes unchoked, the Mach-dependent mass-flow function replaces the fixed sonic factor. A reference value does not authorize a model to ignore those changes.

As a simple original example, a fixed choked section with total-pressure ratio0.7 and total-temperature ratio1.2 has a mass-flow ratio of approximately 0.639010. Multiplying a reference flow of 25 kg/s gives about 15.9752 kg/s. The reduction is stronger than the pressure ratio alone because the gas is hotter. This applies to the mass passing through the selected section. Converting hot-gas flow to captured air requires the fuel-addition and bleed assumptions, rather than silently equating the two flows.

A full off-design engine must satisfy multiple constraints simultaneously: compatible component mass flows, shaft power balances, nozzle capacity, specified geometry and an operating control such as fuel flow or turbine temperature. A reference point can supply constants in these equations. It cannot replace the equations themselves. If a model assumes two sections remain choked, its result is conditional on that assumption even when every reference-normalized residual is small.

3.8 Map coordinates and physical operating lines

A component map relates performance quantities over an operating domain. In the App’s representation, a dimensionless corrected-speed axis and a beta axis locate a point, while normalized corrected flow, pressure-ratio multiplier and efficiency multiplier are dependent tables. Beta is an ordering coordinate along a tabulated line; it is not inherently a blade angle, a bypass ratio or a physical control setting.

This representation permits interpolation without requiring every speed line to have the same corrected-flow values. It also means that a requested corrected flow is not simply the second map coordinate. In general an inverse query would search for beta locations whose interpolated output matches a target quantity, and a nonmonotone map could have multiple inverse roots. The current App implements a narrower inverse: it requires consistently monotone pressure-ratio rows and solves for beta at a prescribed normalized speed and pressure-ratio multiplier. It rejects a nonmonotone map for this inverse task rather than providing a multiple-root selector. It does not invert corrected flow.

A component’s operating line is generated by the rest of the engine and its controls. Interpolating a map at a prescribed speed and beta does not solve for that operating line. Neither does drawing a smooth curve through available points establish a surge boundary from first principles. A surge-pressure-ratio array, where supplied, is additional normalized information. The App reports the pressure-ratio indicator pi_surge/pi_operating − 1. That is neither a mass-flow-distance margin nor a dynamic stability prediction.

Out-of-range coordinates require new physical information, not just a more convenient interpolation formula. The App rejects extrapolation from the supplied map domain. That refusal is materially different from predicting that the real component cannot operate anywhere beyond the table: the table may be incomplete, while the App has no justified model for the unprovided region.

3.9 Using normalized results without overstating similarity

Before comparing two runs, identify the correction inlet station, reference temperature and pressure, gas model, geometry, reference operating point, and whether a displayed speed is actual, corrected or reference-normalized. Chapter1’s flow-basis rules still apply: a corrected core flow is not automatically corrected total inlet air, and a core-based specific work is not a whole-engine power.

Data provenance must be carried by the case or accompanying map documentation. A file that preserves the numeric map axes but omits their engineering meaning can round-trip perfectly while remaining ambiguous. A successful import proves that a data structure was read, not that its corrected coordinates match the intended reference convention.

Reference-normalized turbine-engine modes are useful because they expose changes around a specified design state without demanding a complete measured component set. Their limits should be stated positively and precisely: fixed-area or choked assumptions, supplied reference states, chosen efficiency relations and the matching equations actually solved. They should not be described as arbitrary engine simulation, a validated stability envelope or a general substitute for measured maps.

The following chapters use these distinctions to connect atmospheric inputs, component states and engine cycles. A normalization is successful when it makes the model’s assumptions easier to inspect, rather than hiding them behind apparently universal curves.

3.10 Interpreting the App’s map multipliers

A small original example shows why a normalized map output is not itself a physical pressure ratio. Suppose the standalone compressor calculation supplies a baseline total-pressure ratio of10 and a baseline isentropic efficiency of0.85. At the selected dimensionless corrected speed and beta, let the map return flow multiplier0.90, pressure-ratio multiplier1.10 and efficiency multiplier0.96. The component uses

$$\pi_c=10(1.10)=11.0,\qquad \eta_c=0.85(0.96)=0.816.$$

The flow ordinate0.90 remains a dimensionless corrected-flow ratio. It is not0.90 kg/s and cannot be turned into an absolute flow without the missing design-flow scale and correction convention. If an inverse query is intended to find this point, its target map pressure-ratio multiplier is1.10, not the component’s physical pressure ratio11.0.

If the normalized surge pressure ratio at this speed is1.20, the displayed pressure-ratio margin is

$$\frac{1.20}{1.10}-1\approx0.090909.$$

The common normalization cancels in this ratio. A margin of about9.09% under this convention says nothing by itself about transient surge onset or the distance in mass flow to a physical stability boundary. It is a useful interpretation of supplied map data, not a replacement for such data.

The analytic demonstration-map generator selects a design point and normalizes its dependent tables there. It illustrates the workflow; it is not a measured manufacturer’s map. The generic beta axis is only required to be finite and strictly increasing. Although the demonstration generator uses beta from0 to1, that interval does not establish a universal physical meaning for beta.

The cycle-level reference models use another set of quantities. Their mechanical spool-speed ratios include the appropriate inlet total-temperature factors in the work relation. They are inferred speed ratios under a closed-form reference model, not measured shaft speeds and not the normalized corrected-speed coordinates of the standalone component map. The single-spool turbojet also reports a corrected-flow ratio obtained from its own reference closure; that output is not automatically the ordinate of an independently supplied map.

The distinction prevents a tempting but invalid shortcut: taking a cycle’s mechanical speed ratio, inserting it directly as map corrected speed, and treating the returned flow multiplier as a completed whole-engine match. A coupled match would require compatible reference stations and scales, component flow continuity, shaft-power constraints and nozzle conditions to be solved together. The present map lookup and reference-cycle modes are separate bounded calculations.

4. Freestream and inlet bookkeeping

An engine calculation begins before air crosses its physical inlet. The flight condition establishes a static thermodynamic state, the speed of the incoming stream, and the energy and momentum carried toward the engine. The inlet then changes that state. Keeping these operations distinct prevents an atmospheric pressure from being mistaken for compressor entry pressure, or an inlet pressure loss from being applied twice.

This chapter develops the route from a flight specification to inlet boundary conditions. The atmosphere supplies an optional relation between altitude, pressure and temperature. Gas properties turn Mach number into velocity. A stagnation calculation accounts for ram heating. An inlet model supplies losses or a downstream state. Finally, a chosen capture area converts incoming mass flux into mass flow. Each operation requires its own assumptions.

The reader should be able to reconstruct a resolved flight state, explain which quantities are independently specified, and identify when two superficially similar input routes describe different physical conditions. Chapter 2 supplies the detailed stagnation derivation; Chapter 3 supplies corrected-flow conventions. The calibrated variable-property inlet receives its full treatment in Chapter 27.

4.1 Stations before components

Use station 0 for the undisturbed incoming stream and station 2 for a representative inlet exit or compressor entry. At station 0, distinguish static pressure $p_0$ from total pressure $p_{t0}$. The subscript 0 identifies a location; it does not itself mean stagnation. Likewise, $T_0$ is the local static temperature and $T_{t0}$ is the temperature associated with bringing that stream to rest adiabatically without shaft work.

For a calorically perfect ideal gas,

$$\rho_0=\frac{p_0}{RT_0},\qquad a_0=\sqrt{\gamma RT_0},\qquad V_0=M_0a_0.$$

These relations require composition through $R$, and the heat-capacity model through $\gamma$. Mach number alone does not specify speed. Two streams at the same Mach number and different temperatures have different velocities even when their composition is identical.

An adiabatic inlet without shaft work preserves total enthalpy while dissipative mechanisms increase entropy. Its static pressure may increase as velocity decreases, while total pressure decreases. These statements are compatible: static pressure measures one part of the local state, while total pressure also reflects the stream’s recoverable mechanical potential.

A cycle inlet can be represented by total-pressure recovery without resolving its internal geometry. A flowpath inlet instead needs a downstream static state, velocity and area relationship. The amount of information returned by these two models differs. A recovery factor alone cannot locate shocks, predict boundary-layer separation or determine an exit Mach number.

4.2 Standard atmosphere as a closure

An altitude is a geometric coordinate. It becomes a thermodynamic state only after an atmospheric model has been selected. A standard atmosphere provides a reproducible reference profile; it does not measure the weather encountered by a vehicle.

ThrustOne’s standard-atmosphere helper uses a constant-molecular-weight approximation over geometric altitudes from 0 to 86 km. Its sea-level values are

$$T_{\mathrm{SL}}=288.15\ \mathrm{K},\quad p_{\mathrm{SL}}=101325\ \mathrm{Pa},\quad R=287.05287\ \mathrm{J\,kg^{-1}K^{-1}}.$$

The profile uses geopotential altitude $H$ internally. With geometric altitude $z$ and reference Earth radius $R_E=6356766\ \mathrm{m}$,

$$H=\frac{R_Ez}{R_E+z},\qquad z=\frac{R_EH}{R_E-H}.$$

This transformation allows a constant reference gravitational acceleration $g_0=9.80665\ \mathrm{m\,s^{-2}}$ in the hydrostatic equations. Under the corresponding inverse-square gravity approximation,

$$\frac{dH}{dz}=\frac{R_E^2}{(R_E+z)^2},\qquad g(z)\,dz=g_0\,dH.$$

Geometric and geopotential altitude are close near sea level, but their difference grows with height. Layer boundaries below are geopotential values; entering a listed boundary as geometric altitude does not place the calculation exactly on that boundary.

The lapse rate is $dT/dH$, with its sign included. The final layer is used only up to the supported upper boundary. This is a piecewise atmospheric reference, not a promise that constant gas properties or an engine model remain physically adequate throughout the full altitude range.

Hydrostatic pressure in a layer

A stationary atmosphere obeys

$$\frac{dp}{dH}=-\rho g_0=-\frac{pg_0}{RT}.$$

Within a layer beginning at $H_b$, write

$$T=T_b+L(H-H_b).$$

For nonzero lapse rate $L$, integration gives

$$\ln\frac{p}{p_b}=-\frac{g_0}{RL}\ln\frac{T}{T_b},\qquad p=p_b\left(\frac{T}{T_b}\right)^{-g_0/(RL)}.$$

For an isothermal layer, integrate with $T=T_b$:

$$p=p_b\exp\left[-\frac{g_0(H-H_b)}{RT_b}\right].$$

The exponential relation is also the limit of the lapse relation as $L$ approaches zero. Substituting a zero lapse rate directly into the power-law expression creates a division by zero; the separate expression is part of the model’s definition.

Pressure at a new layer base is inherited from the preceding layer. It is not reset to sea-level pressure. This propagation preserves pressure continuity; the specified base temperatures preserve temperature continuity. The slope of temperature can change abruptly even though temperature itself remains continuous. Consequently, derivatives of downstream quantities may change at layer boundaries without indicating a solver defect.

4.3 Dynamic pressure is not stagnation-pressure rise

Dynamic pressure is defined by

$$q=\frac{1}{2}\rho_0V_0^2.$$

For the constant-property ideal gas above,

$$q=\frac{\gamma}{2}p_0M_0^2.$$

Temperature cancels because density varies inversely with temperature while squared speed at fixed Mach varies directly with temperature. This cancellation is conditional on the selected gas model. With temperature-dependent heat capacities, $\gamma$ itself changes with temperature.

The corresponding inversions are

$$p_0=\frac{2q}{\gamma M_0^2},\qquad M_0=\sqrt{\frac{2q}{\gamma p_0}}.$$

The first inversion requires nonzero Mach number. At $M_0=0$, every finite positive static pressure gives $q=0$; the pair $M_0=0,q=0$ cannot determine pressure. Positive $q$ paired with zero Mach is inconsistent. The second inversion is well defined at zero dynamic pressure when static pressure is already known.

Total pressure follows the compressible stagnation relation,

$$\frac{p_{t0}}{p_0}= \left(1+\frac{\gamma-1}{2}M_0^2\right)^{\gamma/(\gamma-1)}.$$

Expanding this relation for small Mach number gives $p_{t0}-p_0=q+O(M_0^4)$. Thus dynamic pressure approximates the stagnation-pressure rise at low Mach number. It is not their exact difference at general Mach number. Using $p_{t0}=p_0+q$ at supersonic speed discards the compressible pressure relation even if every other part of the cycle uses it correctly.

Dynamic pressure is useful as a loading coordinate. It does not independently establish a vehicle’s structural limit, thermal limit or engine operability boundary. Those limits require additional models or supplied constraints.

4.4 What each flight input route determines

The generic turbojet flight adapter resolves the following routes before the applicable cycle calculation. This adapter is not a universal input resolver for every engine family; component and dual-spool modes have separate handling.

The generic dynamic-pressure routes use $\gamma=1.4$ to resolve the flight condition. This choice is independent of an editable cold-gas ratio used later in a cycle. Changing a cycle property does not automatically redefine the flight adapter’s dynamic-pressure convention. For a comparison involving a different gas-property model, reconstruct the physical ambient state and velocity consistently rather than assuming that matching labels guarantees identical conditions.

The Mach-and-dynamic-pressure route imposes more than the algebraic pressure relation. It also imposes the standard-atmosphere temperature associated with that pressure. A manually entered ambient temperature is therefore not an independent degree of freedom on this route.

Pressure decreases monotonically through the supported standard atmosphere. The App uses that monotonicity to invert pressure to geometric altitude within its supported interval. A pressure outside that interval cannot be assigned a supported standard altitude. Such rejection does not mean the pressure is thermodynamically impossible; it means the chosen atmospheric closure does not represent it.

The altitude-and-dynamic-pressure route permits zero dynamic pressure and resolves zero Mach. The Mach-and-dynamic-pressure route requires both quantities to be positive because pressure inversion would otherwise be underdetermined or inconsistent. Different boundary behavior follows from the different unknowns.

4.5 Ram heating and inlet recovery

Define the ram temperature ratio

$$\tau_r=1+\frac{\gamma-1}{2}M_0^2.$$

Then

$$T_{t0}=T_0\tau_r,\qquad p_{t0}=p_0\tau_r^{\gamma/(\gamma-1)}.$$

These are freestream total conditions before an inlet loss is applied. For an adiabatic, no-work constant-property inlet,

$$T_{t2}=T_{t0},\qquad p_{t2}=\pi_d p_{t0}.$$

Here $\pi_d$ is total-pressure recovery. Multiplying static pressure by recovery instead produces the wrong station quantity. Reducing total temperature by the same factor would incorrectly remove energy from an adiabatic inlet.

The current constant-property turbojet cycle forms its nonideal inlet ratio from the supplied recovery factor and an empirical Mach-dependent ram-recovery factor:

$$\pi_d=\pi_{d,\mathrm{supplied}}\eta_r(M_0),$$

$$\begin{gathered} \eta_r(M)= \begin{cases} 1,&M\leq1,\\ 1-0.075(M-1)^{1.35},&1<M<5,\\ 800/(M^4+935),&M\geq5. \end{cases} \end{gathered}$$

The ideal cycle setting uses an inlet ratio of one. The empirical correlation summarizes a loss allowance for a cycle calculation. It does not compute a particular shock arrangement, spill flow or inlet unstart. Nor should its piecewise join be interpreted as a newly predicted physical discontinuity.

For a fixed-composition ideal gas at unchanged total temperature, the entropy change is

$$\Delta s=-R\ln\pi_d.$$

Recovery below one therefore corresponds to positive entropy production. This provides a physical interpretation of the loss: the total enthalpy remains available in the energy balance, but less pressure potential remains for subsequent components. It also explains why a compressor pressure ratio must multiply $p_{t2}$, not the loss-free $p_{t0}$.

4.6 Capture area and incoming mass flow

For a uniform freestream normal to a capture plane,

$$\dot m_a=\rho_0V_0A_c.$$

The area $A_c$ belongs to the upstream streamtube entering the engine. It need not equal a geometric inlet lip area under conditions with spillage, external compression or a nonuniform incoming flow. Assigning an area to a calculation is not equivalent to predicting the streamtube that a physical installation captures.

For a steady inlet with no bleed or added mass,

$$\rho_0V_0A_c=\rho_2V_2A_2.$$

Defining contraction ratio as $CR=A_c/A_2$ gives

$$CR=\frac{\rho_2V_2}{\rho_0V_0},\qquad A_2=\frac{A_c}{CR}.$$

This is a continuity statement. It does not independently determine the exit state. Energy, entropy or a loss closure, and an appropriate branch selection are still required.

If an inlet slows the flow, density usually rises through compression; whether the area decreases depends on their combined effect on mass flux. Interpreting a velocity decrease alone as evidence for a larger exit area ignores density. The same warning applies when comparing cold ambient air with heated downstream gas.

A specific-thrust result also does not determine absolute thrust until its normalization and mass flow are established. Use the air-flow basis of the selected engine model. In particular, fuel addition changes downstream mass flow, while bypass engines may distinguish core air from total captured air, as developed in Chapter 1.

4.7 Worked flight-to-inlet calculation

Consider an illustrative constant-property stream at geometric altitude 10,000 m and Mach 2. Use standard air, $\gamma=1.4$, supplied inlet recovery 0.95, and an independently assigned capture area of $0.20\ \mathrm{m^2}$. This example demonstrates bookkeeping; it does not establish the design of a physical inlet.

First convert altitude:

$$H=\frac{6356766(10000)}{6356766+10000}=9984.2934\ \mathrm{m}.$$

The state lies in the first atmospheric layer. Consequently,

$$T_0=288.15-0.0065H=223.2521\ \mathrm{K},$$

$$p_0=101325\left(\frac{223.2521}{288.15}\right)^{9.80665/(0.0065\,287.05287)} \approx26499.87\ \mathrm{Pa}.$$

Using unrounded intermediate values gives the following chain:

At this Mach number, $p_{t0}-p_0\approx180.847\ \mathrm{kPa}$, much greater than $q\approx74.200\ \mathrm{kPa}$. Their difference is a compressibility effect, not an inlet loss. The inlet loss subsequently reduces total pressure by approximately 25.141 kPa.

The same standard-atmosphere condition can be specified by Mach 2 and the computed dynamic pressure: the algebra gives the original static pressure, and atmospheric inversion returns the original altitude and temperature. Altitude 10,000 m with that dynamic pressure likewise recovers Mach 2. Small differences from entering rounded values are expected.

If corrected mass flow is needed at compressor entry, Chapter 3’s correction must use $T_{t2}$ and $p_{t2}$. Using loss-free freestream total pressure would reduce the reported corrected flow and conceal the inlet loss. A reference station is part of the definition of a corrected quantity.

4.8 A nonstandard atmosphere is a different case

Now prescribe $T_0=250\ \mathrm{K}$, $p_0=20\ \mathrm{kPa}$, and Mach 2 directly, with the same constant gas properties and capture area. The dynamic pressure is

$$q=\frac{1.4}{2}(20000)(2^2)=56000\ \mathrm{Pa}.$$

The speed is approximately 633.935 m/s and the captured mass flow is 35.3348 kg/s. This state is fully specified without assigning a standard altitude.

Entering only Mach 2 and 56 kPa dynamic pressure into the generic atmospheric route recovers 20 kPa static pressure, but it obtains temperature from the standard profile. It therefore does not preserve the independently supplied 250 K state. Equal Mach and dynamic pressure are insufficient to guarantee equal velocity or mass flow when atmospheric temperature is not the same.

At fixed $p_0$, $M_0$, composition and constant $\gamma$,

$$V_0\propto\sqrt{T_0},\qquad \rho_0\propto T_0^{-1},\qquad \dot m_a\propto T_0^{-1/2}$$

for fixed capture area. Warming the stream raises speed but lowers captured flow; dynamic pressure remains unchanged. A study at fixed dynamic pressure therefore does not hold every physically important inlet quantity fixed. State explicitly whether altitude, static pressure, temperature, Mach number or capture area is constrained when interpreting a trend.

4.9 The separate calibrated inlet convention

The calibrated inlet uses frozen variable-property air with nitrogen and oxygen mole fractions 0.79 and 0.21. Its current implementation requires the NASA9 property representation. It computes sound speed and enthalpy from that gas model, rather than treating the generic flight adapter’s constant $\gamma=1.4$ as a universal convention.

Its altitude routes are bounded to geometric altitudes from 0 to 70 km. In the Mach-and-dynamic-pressure route, it searches altitude using dynamic pressure evaluated with its own gas properties. In the altitude-and-dynamic-pressure route, it derives Mach from the dynamic pressure of a unit-Mach state at that altitude. Both procedures preserve the calibrated model’s property convention.

The inlet solution requires supersonic freestream flow. Its reference condition sets an entropy-loss relation, and that relation is then used with total-enthalpy conservation and continuity to obtain downstream states. A constant loss coefficient calibrated from one reference condition is a modeling assumption, not a measured inlet map spanning every flight condition.

The full calibrated scramjet flowpath additionally requires a supersonic inlet exit. A mathematically available subsonic exit cannot silently replace that required branch. The separate inlet model and the complete engine model therefore have different admissibility conditions.

In that flowpath, the captured air flow is computed from freestream density, Mach, local frozen sound speed and the supplied capture area. Inlet exit area follows from contraction ratio. This arrangement connects absolute geometry to mass flow, but it still does not predict external spillage or aircraft installation effects.

Do not multiply the calibrated inlet’s pressure recovery by the generic turbojet ram correlation merely because both are called inlet recovery. Each belongs to a separate loss closure. Combining them without redefining the model counts losses twice and changes the meaning of the calibration.

4.10 Reading a resolved result

A useful flight-state report identifies the input route, the resolved static state, the property convention, and the inlet closure. An altitude label alone omits the atmospheric assumption. A Mach label alone omits the speed of sound. A pressure label without its static or total qualification omits the energy-state definition.

Before comparing two runs, reconstruct those definitions. The most informative discrepancies are often structural: a standard-temperature substitution, a recovery applied to the wrong pressure, a capture area changed between cases, or a different sound-speed convention. These can alter a comparison even when both numerical solutions have converged accurately.

The chapter’s equations provide a short route from independent inputs to interpretable inlet conditions. They also identify what remains outside the result: weather prediction, installation drag, spillage, internal shock layout and general inlet operability. Later engine chapters build on the resolved state and add only the component closures supported by their particular models.

5. Frozen mixtures and property polynomials

A fixed composition does not imply a fixed heat capacity. A stream can retain the same species proportions while its molecular energy storage changes strongly with temperature. Frozen thermodynamics describes that situation: the composition is prescribed, while enthalpy, heat capacities, entropy, density and sound speed are evaluated at the requested state.

The distinction matters in propulsion because temperature changes can be large. A single heat-capacity ratio may offer a useful simplified cycle, but a more detailed temperature calculation requires integrated property relations. Replacing a constant gamma with a local value in every familiar power law does not perform those integrations.

This chapter develops the property model used by the frozen-gas calculation and shared by several flowpath models. It explains composition normalization, polynomial evaluation, reference energies, entropy of mixing and temperature inversion. Chemical equilibrium changes the composition and is treated in Chapter 6. Finite-rate chemistry changes composition over a specified time or distance and is treated in Chapters 7–8 and 29.

5.1 Specifying composition without specifying batch size

Let $n_i$ be an amount of species $i$, $W_i$ its molecular weight in kg/kmol, and $n=\sum_i n_i$. The mole fraction is

$$x_i=\frac{n_i}{n},\qquad \sum_i x_i=1.$$

All amounts must use a common basis. Their absolute scale is irrelevant to an intensive property calculation: amounts 79 and 21 describe the same mixture as 0.79 and 0.21. The App accepts nonnegative relative molar amounts, merges repeated entries for the same recognized species, and normalizes their sum. At least one amount must be positive. Negative amounts and unknown species do not define an admissible mixture.

The mixture molecular weight follows from total mass divided by total molar amount:

$$\overline W=\frac{\sum_i n_iW_i}{\sum_i n_i}=\sum_i x_iW_i.$$

The mass fraction is therefore

$$Y_i=\frac{n_iW_i}{\sum_jn_jW_j}=\frac{x_iW_i}{\overline W}.$$

Conversely, supplied mass fractions imply

$$x_i=\frac{Y_i/W_i}{\sum_jY_j/W_j},\qquad \overline W=\left(\sum_i\frac{Y_i}{W_i}\right)^{-1}.$$

These transformations explain why mass percentages cannot be entered unchanged in a molar-composition field. The error becomes especially large when light hydrogen is mixed with heavier species. Normalization only makes the entries sum to one; it cannot determine that the wrong physical basis was supplied.

A species with exactly zero amount makes no thermodynamic contribution. In the current property calculation, recognized zero-amount entries are omitted after normalization, including from the active temperature-range intersection. An unknown species name is still rejected even if its entered amount is zero. This separates a valid absent component from an unrecognized component definition.

5.2 Molar and mass-specific quantities

Use an overbar for a molar quantity, such as $\overline h_i$ in J/kmol. Unbarred $h$ denotes J/kg. The universal gas constant in the App is

$$R_u=8314.46261815324\ \mathrm{J\,kmol^{-1}K^{-1}}.$$

The mixture mass-specific gas constant is

$$R=\frac{R_u}{\overline W}.$$

For an ideal mixture with frozen composition,

$$\overline c_p=\sum_i x_i\overline c_{p,i},\qquad \overline h=\sum_i x_i\overline h_i,$$

$$c_p=\frac{\overline c_p}{\overline W}=\sum_iY_ic_{p,i},\qquad h=\frac{\overline h}{\overline W}=\sum_iY_ih_i.$$

Either molar or mass-weighted expressions are valid when the weights and property units match. A mole-weighted sum of species values already expressed in J/kg generally gives the wrong mixture value.

The ideal-gas relation is $p=\rho RT$. Since $h=u+RT$ and composition is fixed,

$$c_v=c_p-R,\qquad \gamma=\frac{c_p}{c_v},\qquad \rho=\frac{p}{RT},\qquad a_f=\sqrt{\gamma RT}.$$

The subscript on $a_f$ identifies frozen sound speed. The thermal response used in this derivative holds species proportions fixed. An equilibrium acoustic response can include composition changes and need not equal the frozen response. Likewise, the heat capacity of an equilibrium path is not obtained merely by evaluating frozen heat capacities at its current composition: the changing species amounts also contribute to the energy derivative.

Within this ideal frozen model, pressure changes density and entropy but does not change enthalpy or heat capacity at a fixed temperature. Real-gas pressure dependence, liquid phases and condensation are outside the property calculation.

5.3 NASA9: one heat-capacity fit and two integration constants

A property record consists of a temperature interval, a reference pressure and polynomial coefficients. The NASA9 representation can be written with coefficients $a_1,\ldots,a_9$:

$$\frac{\overline c_p^{\circ}(T)}{R_u}= a_1T^{-2}+a_2T^{-1}+a_3+a_4T+a_5T^2+a_6T^3+a_7T^4.$$

The superscript circle denotes the species standard-state function. Temperature is in kelvin; the coefficient units are those required by this expression. Changing the input temperature unit without changing the coefficients destroys the fit.

Enthalpy follows from $d\overline h/dT=\overline c_p$. Integrating each term gives

$$\frac{\overline h^{\circ}(T)}{R_u}= -\frac{a_1}{T}+a_2\ln T+a_3T+\frac{a_4T^2}{2} +\frac{a_5T^3}{3}+\frac{a_6T^4}{4}+\frac{a_7T^5}{5}+a_8.$$

Equivalently,

$$\frac{\overline h^{\circ}(T)}{R_uT}= -\frac{a_1}{T^2}+\frac{a_2\ln T}{T}+a_3+\frac{a_4T}{2} +\frac{a_5T^2}{3}+\frac{a_6T^3}{4}+\frac{a_7T^4}{5}+\frac{a_8}{T}.$$

At fixed reference pressure, $d\overline s^{\circ}/dT=\overline c_p^{\circ}/T$. Integrating once more yields

$$\frac{\overline s^{\circ}(T)}{R_u}= -\frac{a_1}{2T^2}-\frac{a_2}{T}+a_3\ln T+a_4T +\frac{a_5T^2}{2}+\frac{a_6T^3}{3}+\frac{a_7T^4}{4}+a_9.$$

The logarithms use the temperature convention of the fitted coefficients. The integration constants carry the selected enthalpy and entropy references. They are not optional numerical decorations. Deleting $a_8$ preserves the derivative $d\overline h/dT$ while changing chemical energy differences; deleting $a_9$ preserves the entropy derivative while changing chemical potentials.

The representation contains nine coefficients because seven terms describe heat capacity and two constants establish the integrated reference states. It does not mean that heat capacity is a ninth-degree polynomial. The inverse powers and logarithms are essential parts of the representation.

Each interval has its own complete coefficient set. The current NASA9 records use a lower interval from 200 to 1000 K and an upper interval from 1000 to 6000 K; the upper set is selected at exactly 1000 K. Coefficient changes approximate the same species functions across a broad range. Small fit inconsistencies at a join should not be interpreted as a phase transition, nor should a polynomial be extended past its recorded upper limit simply because it still returns a number.

5.4 NASA7 and database consistency

The hydrogen–oxygen mechanism uses NASA7 records. With coefficients $b_1,\ldots,b_7$, its heat capacity is

$$\frac{\overline c_p^{\circ}}{R_u}=b_1+b_2T+b_3T^2+b_4T^3+b_5T^4.$$

The integrated functions are

$$\frac{\overline h^{\circ}}{R_uT}=b_1+\frac{b_2T}{2}+\frac{b_3T^2}{3} +\frac{b_4T^3}{4}+\frac{b_5T^4}{5}+\frac{b_6}{T},$$

$$\frac{\overline s^{\circ}}{R_u}=b_1\ln T+b_2T+\frac{b_3T^2}{2} +\frac{b_4T^3}{3}+\frac{b_5T^4}{4}+b_7.$$

The App evaluates these through the same algebraic representation by placing two zero coefficients before the seven NASA7 coefficients. This is an exact embedding of the NASA7 functional form, not a refit and not a conversion to a different physical dataset.

A database choice changes more than the number of coefficients. It changes the available species, fitted functions, reference conventions and valid intervals. The standalone frozen-property mode uses the general NASA9 set. The NASA7 set is associated with the supported hydrogen–oxygen mechanism where selected; it is not an arbitrary replacement for every model’s property data. In particular, calibrated inlet/engine calculations require their NASA9 convention.

No single common range covers every imaginable mixture. For an actual set of active species, the usable interval is

$$T_{\min}=\max_iT_{\min,i},\qquad T_{\max}=\min_iT_{\max,i}.$$

For example, an active NASA7 mixture of H2, O2 and N2 is restricted to 300–3500 K. Pure H2 in that database has a 200 K lower boundary; adding positive N2 narrows it to 300 K. A later reacting calculation may need species that were absent initially, so its admissible domain must account for its own species requirements. An initial-mixture property range alone does not guarantee that an entire reactive trajectory is supported.

5.5 Ideal mixing entropy and the pressure term

For ideal gases, each species contributes entropy at its partial pressure $p_i=x_ip$:

$$\overline s(T,p,\boldsymbol{x})= \sum_i x_i\left[\overline s_i^{\circ}(T) -R_u\ln\left(\frac{x_ip}{p_i^{\circ}}\right)\right].$$

The reference pressure $p_i^{\circ}$ must match the species record. When every record shares one reference pressure $p^{\circ}$, this separates into

$$\overline s=\sum_i x_i\overline s_i^{\circ}(T) -R_u\sum_i x_i\ln x_i-R_u\ln(p/p^{\circ}).$$

The middle term is the ideal entropy of mixing. It is nonnegative because every nonzero mole fraction is at most one. The last term represents the pressure effect: isothermal compression reduces entropy. The mass-specific value is $s=\overline s/\overline W$.

For fixed composition between two states,

$$s_2-s_1=\int_{T_1}^{T_2}\frac{c_p(T)}{T}\,dT -R\ln\frac{p_2}{p_1}.$$

The mixing term cancels because composition is unchanged. It must remain when comparing different compositions, such as in a mixing or equilibrium calculation. Removing it because it cancels along one frozen path would corrupt those other problems.

A zero mole fraction contributes the limiting value $x\ln x\to0$. The implementation omits zero components and splits the logarithm into $\ln x_i+\ln(p/p_i^{\circ})$ for active species. This avoids forming a tiny partial-pressure product that could underflow before taking a logarithm. It does not remove the physical mixing term or turn a trace species into a finite concentration floor.

Changing the reference pressure also requires changing the standard-state entropy convention if the physical entropy is to stay unchanged. Retaining the old entropy coefficients while substituting a new pressure reference shifts the result. This is why the difference between 1 bar and 1 atmosphere belongs in the property definition, even when its numerical effect seems modest.

5.6 Formation enthalpy and sensible heating

A species enthalpy can be expressed schematically as

$$\overline h_i(T)=\Delta_f\overline h_i^{\circ}(T_r) +\int_{T_r}^{T}\overline c_{p,i}(\vartheta)\,d\vartheta,$$

with the reference temperature and elemental conventions consistent with the dataset. The polynomial integration constant carries that reference information. Species with negative formation enthalpy can have negative enthalpy over part of the supported range. Negative enthalpy is not negative temperature and does not itself make the state invalid.

For a frozen stream of unchanged composition, the reference contribution cancels from $h_2-h_1$. This is why a constant-cp approximation can sometimes reproduce a temperature rise even when it uses a different zero of enthalpy. In a reacting stream, however, species amounts change. The difference between product and reactant formation terms is then part of the energy balance.

If the energy equation already uses formation-inclusive species enthalpies, adding a fuel heating value as a second chemical-energy source counts reaction energy twice. A heating-value cycle and a formation-enthalpy chemistry calculation are different accounting formulations. Each can be meaningful, but their terms cannot be combined casually.

Similarly, a supplied total enthalpy

$$h_t=h(T,\boldsymbol{x})+\frac{V^2}{2}$$

uses the same enthalpy reference as the property model. A numerical value previously formed as $c_pT+V^2/2$ is not automatically transferable. The reference offset affects an absolute target even when it has no effect on a frozen temperature difference. This point is particularly important for the state-reconstruction modes developed in Chapter 26.

5.7 Inverting an enthalpy or entropy target

The direct property problem supplies temperature and pressure. The enthalpy inversion instead supplies composition, pressure and a mass-specific enthalpy target, and seeks temperature from

$$f(T)=h(T,\boldsymbol{x})-h_{\mathrm{target}}=0.$$

For an admissible frozen ideal gas, $dh/dT=c_p>0$. Enthalpy is therefore increasing within the supported fit intervals. The App brackets the calculation by the common temperature range, evaluates its endpoint enthalpies, and rejects a target outside them. It then uses bounded bisection to locate the temperature. Its finite iteration count is a numerical procedure, not a statement that the fitted thermodynamic data have that many digits of physical accuracy.

Pressure remains a required positive input for the reported state, even though ideal-gas enthalpy at fixed composition does not depend on it. A pressure change should affect the returned density and entropy, not the temperature recovered from a fixed enthalpy target.

The shared property layer also supports entropy inversion at a supplied pressure. The derivative $(\partial s/\partial T)_p=c_p/T$ is positive over an admissible positive-temperature range, so the same bracketing reasoning applies. Unlike enthalpy inversion, changing the specified pressure generally changes the temperature required to recover the same entropy.

A target outside the bracket does not authorize extrapolation. It may indicate an incorrect unit, an incompatible enthalpy reference, or a real physical state outside the available data. These cases have different engineering meanings, even if all produce a refusal to return an extrapolated temperature.

5.8 Worked composition and entropy example

Consider 79 relative molar units of N2 and 21 of O2 in the general database. Their molecular weights are 28.0134 and 31.9988 kg/kmol. Normalization gives $x_{N2}=0.79$ and $x_{O2}=0.21$, followed by

$$\overline W=0.79(28.0134)+0.21(31.9988)=28.850334\ \mathrm{kg/kmol}.$$

The mass fractions and gas constant are

$$\begin{gathered} Y_{N2}=0.76708249,\\ Y_{O2}=0.23291751,\\ R=288.19294\ \mathrm{J\,kg^{-1}K^{-1}}. \end{gathered}$$

Thus 21% oxygen by moles corresponds to approximately 23.29% by mass. The gas constant differs from the standard-atmosphere value in Chapter 4 because this explicit binary mixture has a different molecular-weight definition. The difference should be retained when moving between these models.

The ideal mixing contribution is

$$s_{\mathrm{mix}}=-R[0.79\ln0.79+0.21\ln0.21] =148.1187\ \mathrm{J\,kg^{-1}K^{-1}}.$$

This is the mixing contribution, not the complete entropy of the gas. The species standard-state entropies and the pressure term are still required for the complete value.

If pressure is doubled at unchanged temperature and composition,

$$\Delta s=-R\ln2=-199.7601\ \mathrm{J\,kg^{-1}K^{-1}}.$$

Density doubles, while enthalpy, heat capacities and frozen sound speed remain unchanged. This contrasting case separates the ideal pressure effect from temperature and composition effects without requiring a second polynomial evaluation.

5.9 Worked property and inverse example

Argon provides a transparent example in the lower NASA9 interval. Its recorded coefficients reduce to $\overline c_p/R_u=2.5$, and the enthalpy integration constant gives

$$R=208.1321372\ \mathrm{J\,kg^{-1}K^{-1}},\quad c_p=520.3303431\ \mathrm{J\,kg^{-1}K^{-1}},$$

$$h=c_p(T-298.15\ \mathrm{K}),\qquad \gamma=5/3.$$

At 600 K and 150 kPa the direct calculation yields

For the inverse calculation, supply $h=157061.714\ \mathrm{J/kg}$, the same pure composition and a positive pressure. The analytical lower-interval inverse is

$$T=298.15+\frac{h}{520.3303431}\approx600\ \mathrm{K}.$$

The App uses its general bracketed property inversion rather than an argon-specific shortcut. The simple analytical result makes the unit and reference convention visible: entering 157.061714 as J/kg would describe a state close to the enthalpy reference temperature, not 600 K.

Heating this gas from 600 to 900 K requires an enthalpy rise of approximately 156.099103 kJ/kg. Isentropically doubling its pressure from the 600 K state instead gives

$$c_p\ln(T_2/600)=R\ln2,\qquad T_2=600\,2^{0.4}=791.704746\ \mathrm{K}.$$

Both calculations remain within the lower interval, where the constant-cp reduction is exact for the recorded fit. That reduction must not be generalized to every species or temperature interval. A variable-cp mixture requires the integrated functions developed above.

5.10 From a property result to a flow model

The standalone Frozen Composition · Variable-Specific-Heat Properties mode evaluates the prescribed composition directly or recovers temperature from its enthalpy target. Its returned molecular weight, gas constant, heat capacities, gamma, enthalpy, entropy, density and sound speed describe one thermodynamic state. It does not predict whether the mixture would react, how fast it would react, or which composition would be in equilibrium.

A frozen nozzle adds mass continuity, total-enthalpy conservation and an entropy or loss condition. A frozen component adds a particular efficiency or pressure closure. A reacting flowpath instead updates species amounts while maintaining its energy and momentum conventions. The property model supplies constitutive values to those equations; it does not supply the missing component closure.

When reading a result, retain the composition basis, database, reference pressure, enthalpy convention and supported temperature interval alongside the displayed temperature and pressure. Those definitions allow the number to be reused correctly. The later thermochemistry and flowpath chapters rely on this same distinction between a local property evaluation and the physical process that connects two states.

6. Gas-phase equilibrium

A chemical-equilibrium calculation determines a composition compatible with a specified thermodynamic constraint and a conserved elemental inventory. It does not merely balance one reaction equation. It compares all admissible compositions within the selected species set and finds the state with no remaining thermodynamic tendency to change along an allowed reaction direction.

ThrustOne provides two related problems. The temperature–pressure mode fixes temperature and pressure, then determines composition. The constant-pressure energy mode fixes pressure and the incoming enthalpy plus supplied heat, then determines both temperature and composition. Their common chemical model is an ideal gas with a bounded set of 13 species. A converged result is equilibrium within that set; it does not establish that the set contains every important real species or phase.

Chapter 5 developed the formation-inclusive enthalpy and ideal mixing entropy used here. This chapter adds elemental constraints, chemical potentials and the energy closure. Reaction rates and residence times enter later chapters, not this equilibrium calculation.

6.1 The inventory that reactions cannot change

Let $n_i$ be the amount of species $i$, and let $A_{ki}$ count atoms of element $k$ in one molecule of that species. The elemental inventory is

$$b_k=\sum_iA_{ki}n_i,\qquad \boldsymbol A\boldsymbol n=\boldsymbol b.$$

For example, using columns H2, O2, H2O, H, O and OH gives

$$\begin{gathered} \boldsymbol A= \begin{bmatrix} 2&0&2&1&0&1\\ 0&2&1&0&1&1 \end{bmatrix}, \end{gathered}$$

where the rows are hydrogen and oxygen. An initial amount of 1.2 H2 and 0.7 O2 contains 2.4 hydrogen atom units and 1.4 oxygen atom units. Any admissible equilibrium composition must retain both quantities, regardless of how much water, atomic gas or hydroxyl it contains.

Molecular amount need not be conserved. Dissociating one molecule into two changes total molecular count without changing the elemental inventory or total mass. Consequently, equilibrium mole fractions must be formed from the final total amount $N=\sum_i n_i$, not from the initial total.

The App derives inventory from the declared initial species. It then admits database species whose constituent elements all have positive inventories. Supplying hydrogen and oxygen does not authorize production of a nitrogen species; nitrogen is absent from that problem. Conversely, a species need not be present initially to appear among the products when its elements are available.

At fixed temperature and pressure, two initial mixtures with the same elemental inventory define the same equilibrium problem. Their chemical preparation history is not an additional constraint. In the energy-closure mode, their incoming enthalpies may differ, so identical inventories alone do not guarantee identical final states.

6.2 What is minimized at fixed temperature and pressure

For a homogeneous ideal gas mixture, the total Gibbs energy is

$$G=\sum_i n_i\mu_i,$$

where the chemical potential $\mu_i$ is the partial molar Gibbs energy. With a common standard pressure $p^{\circ}$,

$$\mu_i(T,p,\boldsymbol x)=\overline g_i^{\circ}(T) +R_uT\ln\left(x_i\frac{p}{p^{\circ}}\right), \qquad \overline g_i^{\circ}=\overline h_i^{\circ}-T\overline s_i^{\circ}.$$

The logarithmic term includes both mixing and pressure. It follows directly from the ideal entropy expression in Chapter 5. Omitting mixing entropy removes the $\ln x_i$ contribution and changes the composition selected by the calculation.

At fixed $T$, $p$ and elemental inventory, equilibrium minimizes $G$ over nonnegative species amounts. The imposed temperature implies that heat can be exchanged as needed to maintain that temperature. A TP equilibrium result therefore does not, by itself, predict the temperature of an adiabatic combustor.

For the ideal mixture, the composition-dependent Hessian has quadratic form

$$\delta\boldsymbol n^T\nabla^2G\,\delta\boldsymbol n =R_uT\left[\sum_i\frac{(\delta n_i)^2}{n_i} -\frac{(\sum_i\delta n_i)^2}{N}\right]\geq0.$$

The inequality follows from the weighted Cauchy–Schwarz inequality. It expresses the convex character of the ideal-mixture problem within its positive domain. The scale direction must still be constrained by inventory; a freely scalable amount is not an independent physical equilibrium branch. This thermodynamic structure is useful, but it does not make finite-precision iteration infallible or make a restricted species set complete.

6.3 Reaction directions and dimensionless equilibrium constants

A reaction direction $\boldsymbol\nu$ changes composition through an extent $\xi$:

$$\boldsymbol n=\boldsymbol n^{(0)}+\boldsymbol\nu\xi.$$

Take products as positive and reactants as negative. Element conservation requires $\boldsymbol A\boldsymbol\nu=0$. For an interior equilibrium, differentiation at fixed temperature and pressure gives

$$\frac{dG}{d\xi}=\sum_i\nu_i\mu_i=0.$$

Dividing by $R_uT$ and inserting the chemical potentials yields

$$\sum_i\nu_i\frac{\overline g_i^{\circ}}{R_uT} +\ln\prod_i\left(x_i\frac{p}{p^{\circ}}\right)^{\nu_i}=0.$$