Exergy Analysis in Hydrogen-Air Detonation

Abstract

The main goal of this paper is to analyze the exergy losses during the shock and rarefaction wave of hydrogen-air mixture. First, detonation parameters (pressure, temperature, density, and species mass fraction) are calculated for three cases where the hydrogen mass fraction in air is 1.5%, 2.5%, and 5%. Then, exergy efficiency is used as objective criteria of performance evaluation. A two-dimensional computational fluid dynamic code is developed using Finite volume discretization method coupled with implicit scheme for the time discretization (Euler system equations). A seven-species and five-step global reactions mechanism is used. Implicit total variation diminishing (TVD) algorithm, based on Riemann solver, is solved. The typical diagrams of exergy balances of hydrogen detonation in air are calculated for each case. The energy balance shows a successive conversion of kinetic energy, and total enthalpy, however, does not indicate consequent losses. On the other hand, exergy losses increase with the augment of hydrogen concentration in air. It obtained an exergetic efficiency of 77.2%, 73.4% and 69.7% for the hydrogen concentrations of 1.5%, 2.5%, and 5%, respectively.

1. Introduction

The Ramjet propulsion principle was invented at the beginning of the 20th century by Rene Lorin [1, 2], who published it in the technical review “Aérophile” in 1913. A Ramjet is an air-breathing jet engine which uses the engine’s forward motion to compress incoming air, without a rotary compressor. The development of this propulsion method was slow and was practically started only from the 1950s. Some years later appeared the pulsed detonation engine (PDE), which is a type of propulsion system that uses detonation waves to combust the fuel and oxidizer mixture [3, 4].

The excitement about pulse detonation engines (PDEs) stems from the fact that, theoretically at least, they can operate at thermal efficiencies about 30% higher than conventional combustion processes. This is due to the high detonation velocities between 1500 m·s⁻¹ and 2200 m·s⁻¹ in a hydrocarbon fuel/air mixture; for example, its thermodynamic cycle is effectively a constant volume or Humphrey cycle [5–7]. The Humphrey cycle can be considered a modification of the Brayton cycle in which the constant-pressure heat addition process of the Brayton cycle is replaced by a constant-volume heat addition process [8].

Due to this thermodynamic efficiency and mechanical simplicity, the PDE technology is more efficient when compared with current engine types. Additionally, the PDE can also provide static thrust for a ramjet or scramjet engine or operate in combination with turbofan systems showing an enormous potential to be applied in many sectors of the aerospace, aeronautic, and military industries. However, there are still engineering challenges that must be overcome, such as to improve methods for initiating the detonation process and develop materials able to withstand with extreme conditions of heat and pressure.

In the PDE technology the combustion takes place in an open-ended tube in which fuel is mixed with air and detonated. As the detonation wave travels down the tube at supersonic speed, a refraction wave propagates into the combustor, and exhaust products leave the chamber. Pressure within the chamber jumps down to charging conditions, which draws in fresh fuel and air, and the cycle is repeated. Each pulse lasts only milliseconds [9, 10]. Detonation is initiated by a predetonator, containing a more readily detonable mixture of fuel air, or by DDT (deflagration detonation transition), a process by which a flame, initiated at the closed end of a duct by a weak spark, accelerates to speeds on the order of 1000 m·s⁻¹ at which point a detonation wave is initiated within the flame-shock complex [11–13].

The existence of flames for problems with complex chemistry in the combustion approximation has been considered by many authors; see, for example, [14–18]. It has been pointed out in [19], that reaction mechanisms including chain branching can lead to significant qualitative differences in the initiation of detonations in comparison to single step reactions. The use of the exergy concept can be used with advantage to best understand the detonation problem [20]. The exergy gives a quantitative and illustrative description of the convertibility of different energy forms and is a function of the system and environment, whereas entropy is a function only of the system. Total exergy, thermomechanical and chemical, has been defined as the work which can be obtained by taking the system, by means of reversible processes, from its initial state to a state of total equilibrium (dead state), thermal, mechanical, and chemical (“unrestricted equilibrium”), with the environment. This definition entails that the system is in a state of internal equilibrium (uniformity of all intensive properties) and that the environment is in a state which is in internal equilibrium and does not change during time. The exergy of a fluid stream depends upon the choice of the reference environment. This choice, which seems obvious according to some authors [20–22], is, on the contrary, problematic, since every usual environment is a chemical state of nonequilibrium.

A simplified model of adiabatic detonation process in gases was considered by Petela [20] using the concept of exergy. The exergy loss for high pressure shock is larger than that for reactions of combustion. On the other hand, the total exergy loss for deflagration is also larger than that for detonation. Huntchins and Metghalchi [21] analyzing the thermal efficiency show that the Humphrey cycle reveals good advantage compared with Brayton cycle using the same fuel. They also compare the effectiveness of both cycles and conclude that the results using the Humphrey cycle have significant advantage comparing with a Brayton cycle based on the same fuel (methane). Also, the research work of Wintenberger and Shepherd [22] finds that this efficiency cannot be precisely translated into propulsive efficiency. Indeed, the results are only useful in comparing detonations with other combustion modes. They also find that the efficiency of cycles based on detonation and constant-volume combustion is very similar and superior to a constant-pressure combustion (Brayton) cycle when compared on the basis of pressure at the start of the combustion process.

This paper reports an exergy analysis on the detonation wave between the pulse shock wave to the end of the rarefaction wave for various concentrations of the hydrogen in air. Numerical simulation of a two-dimensional detonation of the hydrogen-air mixture is considered. The coupled hydrodynamical model and chemical model integrating the Euler system equations are combined with a detailed chemical reaction model.

2. Numerical Method

Supersonic combustion, called detonation, occurs coupled to a shock wave traveling at supersonic speed. If premixed gases inside a tube closed at both ends are ignited at one end, a laminar flame first develops, traveling pushed by the expanding hot products behind. The inverted small pressure jump across the flame generates local pressure pulses that wrinkle the flame, create turbulence, extend the burning area, and increase the burning rate, with a positive feedback that, if positively combined with pressure pulses reflected from the other end, might compress the fresh mixture to the autoignition temperature.

2.1. Chemical Model

Consider the reaction of hydrogen with oxygen to yield water that is usually written as H₂+12O₂⟶H₂O.(2.1)

The detonation of hydrogen in air must start by dissociation of a hydrogen molecule (it requires less energy than oxygen dissociation): H₂+M⟶H+H+M,(2.2) where M stands for an unspecified molecule (hydrogen, oxygen, a hot wire). The following basic reactions take place: H₂+M⟶H+H+M,H+O₂⟶OH+O,O+H₂⟶OH+H,OH+H₂⟶H₂O+H,H+OH+M⟶H₂O+M.(2.3)

The chemical kinetic scheme involves seven species, H₂, O₂, H₂O, OH, O, H, and N₂ and the above-described 5 elementary reversible reactions. The mechanism is the one proposed by Balakrishnan and Williams [23], and the empirical parameters from each one of the reactions [24] are depicted in Table 1. 𝑘(𝑇,𝑝) is the “rate coefficient,” which generally depends on temperature (𝑇), and activation energy and can be defined as follows: 𝑘(𝑇)=𝐴𝑇^𝑏𝑒^{−(𝐸𝑎/𝑅𝑇)},(2.4) where A is the preexponential factor as in Arrhenius law and 𝐸_𝑎 is the activation energy. The three parameters 𝐴, 𝑏, and 𝐸_𝑎 are determined in practice experimentally.

Table 1 

Empirical values for reactions in the combustion of hydrogen with air [24].

The chemical production rates are computed using the methodology suggested by Warnatz [25]. In general, for the reaction 𝐴+𝐵→ products not just to species 𝐴, it can be defined: 𝑑𝑐_𝐴𝑑𝑡=𝑑𝑐_𝐵𝑑𝑡=−𝑐_𝐴𝑐_𝐵𝐴𝑇^𝑏exp−𝐸_𝐴𝑅𝑇.(2.5)

2.2. Euler System Equations

The contributions of pressure-driven diffusion, bulk viscosity, and radiative heat transport can be neglected; then the balance equations of mass, momentum, and energy for the two-dimensional unsteady flow for a multicomponent chemically reacting gas mixture are written in the integral form as 𝜕𝜕𝑡_Ω𝜌𝑑𝑉+_𝜕Ω𝜌𝑉⃗‌⋅𝑛⃗𝑑𝑆=0,𝜕𝜕𝑡_Ω𝜌𝑉𝑑⃗‌𝑉+_𝜕Ω𝜌𝑉⃗‌𝑉⃗‌⋅𝑛⃗+𝑝𝑛⃗𝑑𝑆=0⃗‌,𝜕𝜕𝑡_Ω𝜌𝐸𝑑𝑉+_𝜕Ω(𝜌𝐸+𝑝)𝑉⃗‌⋅𝑛⃗𝑑𝑆=0.(2.6) The state equation can be defined as follows: 𝑝=(𝛾−1)⋅𝜌𝐸−12𝜌𝑉²⃗‌.(2.7) In these system equations the density 𝜌(𝑀,𝑡) is the total density of the whole gas in the position 𝑀, and at time 𝑡. 𝜌(𝑀,𝑡) is calculated as sum of mass fraction (𝑌_𝑖) and density (𝜌𝑖) of each species: 𝜌=^𝑘_𝑖=1𝑌_𝑖𝜌_𝑖.(2.8) The above equation system is rewritten in a control volume as follows: 𝜕𝑈⃗‌𝜕𝑡+𝜕𝐹𝑈⃗‌𝜕𝑥+𝜕𝐺𝑈⃗‌𝜕𝑦=𝑊,(2.9) where𝑈⃗‌≡⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜌𝜌𝑢𝜌𝑣𝜌𝐸𝜌𝑌₁𝜌𝑌₂⋮𝜌𝑌_𝑘−1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠;𝐹(𝑈)≡⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜌𝑢𝜌𝑢²+𝑃𝜌𝑢𝑣𝑢(𝜌𝐸+𝑃)𝜌𝑌₁𝑢𝜌𝑌₂𝑢⋮𝜌𝑌_𝑘−1𝑢⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠;𝐺(𝑈)≡⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜌𝑣𝜌𝑢𝑣𝜌𝑣²+𝑃𝑣(𝜌𝐸+𝑃)𝜌𝑌₁𝑣𝜌𝑌₂𝑣⋮𝜌𝑌_𝑘−1𝑣⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠;𝑊≡⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝0000𝑤̇‌₁𝑤̇‌₂⋮𝑤̇‌_𝑘−1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(2.10)

In these equations 𝑢 and 𝑣 are the velocity vector components in the directions 𝑥 and 𝑦, 𝑇 the mixture temperature, ρ the mixture density, 𝑝 the pressure, 𝑒 and 𝐸 are the internal and total energies, respectively. For each of the 𝑘 chemical species, 𝑌_𝑘 is the mass fraction, 𝑀_𝑘 is the molecular, and ℎ_𝑖 is the specific enthalpy. Then, the following equations of state can be also defined: 𝑌_𝑘=1−^𝑘−1_𝑖=1𝑌_𝑖,𝑃=𝜌𝑅𝑇^𝑘_𝑖=1𝑌_𝑖𝑀_𝑘,ℎ_𝑖=ℎ⁰+^𝑇_𝑇0𝑐_𝑝𝑖𝑑𝑇=𝑒_𝑖+𝑃𝜌,𝐸=𝑒+12𝑢²+𝑣²=^𝑘_𝑖=1𝑌_𝑖𝑒_𝑖+12𝑢²+𝑣².(2.11)

2.3. The Solution Procedure

The numerical methods that have been developed in this code for hydrogen and combustion are detailed in [26, 27]. These include lumped parameters, CFD-type codes, and also the CAST3M code [28] developed at the French Atomic Centre Energy (CEA). CAST3M is a general finite-element/finite-volume code for structural and fluid mechanics and heat transfer.

The CAST3M code is used to model both hydrogen dispersion and combustion phenomena over a very wide range of flow regimes, from nearly incompressible flow to compressible flow with shock waves. This could be done by developing a most suitable (i.e., accurate and efficient) numerical method for each type of flow or developing a single method suitable for all flow regimes. We are not aware of any single numerical method able to treat accurately and efficiently flow regimes which range from slowly evolving nearly incompressible buoyant flow to fast transient supersonic flow. However, in practice, for detonation simulation, the use of shock-capturing methods is preferred. In the hydrogen risk analysis tools CEA is developing; the choice was made to develop in the same computational platform CAST3M code for distribution calculations, an efficient pressure-based solver using a semi-implicit incremental projection algorithm which allows the use of “large” time steps, for combustion calculations, a robust and accurate density-based solver using a shock-capturing conservative method.

A brief description of the method we used here, typical for such hyperbolic system equations, follows. Consider the system of hyperbolic conservation laws, (2.10), with the initial conditions: 𝑢(𝑥,𝑦,0)=𝑢₀(𝑥,𝑦).(2.12)

Given a uniform grid with time step Δ𝑡 and spatial mesh size (Δ𝑥,Δ𝑦), we define an approximation of 𝑈 at the point 𝑥𝑖=𝑖Δ𝑥, 𝑦𝑗=𝑗Δ𝑦, 𝑡𝑛=𝑛Δ𝑡 by the finite volume formula: 𝑈^𝑛+1_𝑖,𝑗=𝑈^𝑛+1_𝑖,𝑗−Δ𝑡Δ𝑥𝐹^𝑛_{𝑖+1/2,𝑗}−𝐹^𝑛_{𝑖−1/2,𝑗}−Δ𝑡Δ𝑦𝐺^𝑛_{𝑖,𝑗+1/2}−𝐺^𝑛_{𝑖,𝑗−1/2}+𝑊^𝑛.(2.13)

Figure 1 shows the structured mesh used in this problem. Finding the exact solution to the Riemann problem for the equation system (2.9) is complex. Even if we would find it, it would be too complex to use it at each cell boundary, as it needed in (2.13). Therefore, we have to look for some approximate Riemann solver. Here, we adopt the approach described in [29], which we have already considered in case of conservation laws. Let us rewrite the homogeneous hyperbolic system (2.9) in primitive variables for one direction:𝜕𝑈𝜕𝑡+𝜕𝐹(𝑈)𝜕𝑥=𝜕𝑈𝜕𝑡+𝐴(𝑈)𝜕𝑈𝜕𝑥=0.(2.14) Consider the Riemann problem for (2.14) at each cell boundary 𝑗+1/2, that is, with the following initial data:𝑈(𝑥,0)=𝑈_𝑗,𝑥≤𝑥_𝑗+1/2𝑈_𝑗+1,𝑥≥𝑥_𝑗+1/2.(2.15) Following [10, 11], we calculate the Jacobian matrix 𝐴(𝑈) in the average state:𝑣_𝑗+1/2=𝑣_𝑗+𝑣_𝑗+12.(2.16) The intermediate state in the solution of the Riemann problem is 𝑣^∗_𝑗+1/2=𝑣_𝑗+_𝜆𝑖≺0𝑎_𝑖𝑟_𝑖,(2.17) where the eigenvalues 𝜆_𝑖 and the eigenvectors 𝑟_𝑖 of the Jacobian matrix 𝐴(𝑣_𝑗+1/2) and 𝑎_𝑖 are the coefficients of eigenvector decomposition of (𝑣_𝑗+1−𝑣_𝑗): 𝑣_𝑗+1/2−𝑣_𝑗=⁶_𝑖=0𝑎_𝑖𝑟_𝑖.(2.18) With the choice of the primitive variables, they are given by the following expressions:⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑎₀𝑎₁𝑎₂𝑎₃𝑎₄𝑎₅𝑎₆⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝Δ₁𝑟₀₁−Δ₃𝜌_𝑎𝑐_𝑎+Δ₄+𝑎₀𝑟₀₃𝜌_𝑎𝑐_𝑎−𝑟₀₄2𝜌_𝑎𝑐²_𝑎Δ₂−𝑎₀𝑟₀₂−𝜌_𝑎𝑎₁+𝑎₂Δ₃𝜌_𝑎𝑐_𝑎+Δ₄−𝑎₀𝑟₀₃𝜌_𝑎𝑐_𝑎+𝑟₀₄2𝜌_𝑎𝑐²_𝑎−Δ₆𝜌_𝑏𝑐_𝑏+Δ₇+𝑎₀𝑟₀₆𝜌_𝑏𝑐_𝑏−𝑟₀₇2𝜌_𝑏𝑐²_𝑏Δ₅−𝑎₀𝑟₀₅−𝜌_𝑏𝑎₄+𝑎₆Δ₆𝜌_𝑏𝑐_𝑏+Δ₇−𝑎₀𝑟₀₆𝜌_𝑏𝑐_𝑏+𝑟₀₇2𝜌_𝑏𝑐²_𝑏⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(2.19) where 𝑟_0𝑘 are the components of 𝑟₀ and Δ𝑘 is the k th component of (𝑣_𝑗+1−𝑣_𝑗). Recalculating 𝑣_𝑗+1/2 into the conservative vector 𝑢𝑗+1/2, we fully determine the Godunov-type scheme (2.17) for the homogeneous system (2.14).

Figure 1 

Structured mesh.

3. Exergy Calculation

A practical definition of exergy (E) for a closed system is the maximal work that can be obtained when a thermodynamic system is allowed to attain equilibrium with an environment defined as the “dead state,” typically defined as the system ambient. If that ambient is at temperature 𝑇₀ and pressure, 𝑝₀, the exergy is 𝐸=𝑈−𝑈₀+𝑝₀𝑉−𝑉₀−𝑇₀𝑆−𝑆₀,(3.1) where 𝑈,𝑉, and 𝑆 denote, respectively, the internal energy, volume, and entropy of the system, and 𝑈₀, 𝑉₀ and 𝑆₀ are the values of the same properties in the system which were called dead state.

The exergy expression can be generalized considering 𝑄 as a heat transfer across a system boundary where the temperature is constant at 𝑇_𝑏 and taking in account the mass flow across the boundary of a control. The exergy rate balance for a control can be derived using these described approaches, where the control volume forms of mass and energy rate balances are obtained by transforming the closed system forms. The exergy accompanying mass flow and heat transfer can be written as 𝑑𝐸𝑑𝑡=_𝑗1−𝑇₀𝑇⋅𝑄_𝑗+_exit𝑚̇_𝑒𝜀_𝑒−_inlet𝑚̇_𝑖𝜀_𝑖,(3.2) where 𝜀_𝑒𝑒 and 𝜀_𝑖 are the exit and inlet specific exergy and 𝑄_𝑗 is the heat transport rate. The sum of the thermomechanical and chemical exergies is the total exergy associated with a given system at a specified state, relative to a specified exergy reference environment. The chemical exergy is defined as 𝜀^𝑐ℎ=_𝑖𝑛_𝑟⋅ℎ_𝑟−_𝑖𝑛_𝑝⋅ℎ_𝑝_{(𝑇0,𝑃0)}−𝑇₀_𝑖𝑛_𝑟⋅𝑆_𝑟−_𝑖𝑛_𝑝⋅𝑆_𝑝_{(𝑇0,𝑃1)}(3.3) or in the following form, when the entropy at (𝑇₀,𝑃₁) is calculated: 𝜀^𝑐ℎ=_𝑖𝑛_𝑟⋅ℎ_𝑟−_𝑖𝑛_𝑝⋅ℎ_𝑝_{(𝑇0,𝑃0)}−𝑇₀_𝑖𝑛_𝑟⋅𝑆_𝑟−_𝑖𝑛_𝑝⋅𝑆_𝑝_{(𝑇0,𝑃0)}+𝑅𝑇₀ln⎛⎜⎝∏^𝑛𝑟_𝑖𝑌_𝑖∏^𝑛𝑝_𝑖𝑌_𝑖⎞⎟⎠,(3.4) where 𝑌_𝑟, 𝑌_𝑝, are the mass fraction of reactants and products of involved reactions and ℎ_𝑟, 𝑆_𝑟, ℎ_𝑝, and 𝑆_𝑝 are the specific enthalpy and specific entropy for the reactants and products, respectively. The specific enthalpy terms are determined using the enthalpy of formation for respective substances. The specific entropy appearing in the above equation is absolute entropy. Even if the logarithmic term normally contributes a few percent, it will not be neglected in the present paper. The Gibbs function for each one of the components could be expressed as follows: 𝜀^𝑐ℎ=_𝑖𝑛_𝑟⋅𝑔_𝑟−_𝑖𝑛_𝑝⋅𝑔_𝑝+𝑅𝑇₀ln⎛⎜⎝∏^𝑛𝑟_𝑖𝑌_𝑖∏^𝑛𝑝_𝑖𝑌_𝑖⎞⎟⎠.(3.5)

The energy associated with a specific state of a system is the sum of two contributions: the thermomechanical contribution and the chemical contribution. On unit mass, the total exergy is: 𝜀=(𝑈−𝑈₀)+𝑝₀(𝑉−𝑉₀)−𝑇₀(𝑆−𝑆₀)_{(1)}+𝜀^𝑐ℎ_(2),(3.6) where (2.1) is the thermo mechanical contribution and (2.2) is the chemical contribution.

The molar exergy of each reactant and product, at temperature 𝑇₁, is given by 𝜀₁=𝜀_O+Δℎ_O→1−𝑇_OΔ𝑠_O→1.(3.7) The standard chemical exergies of the reactants and products are given in Table 2.

Table 2 

Standard chemical exergies [30].

The specific chemical exergy of a material stream is that part of the exergy that results from the sum of the chemical potential differences between the pure reference substances and the pure material stream components at 𝑇₀ and 𝑃₀. The specific mixing exergy of a material stream is that part of the exergy resulting from the sum of two contributions. (i)The differences in potential between the exergies of the environmental components in environmental concentrations and pure environmental components at 𝑇₀ and 𝑃₀. (ii)The difference in potential of the exergy due to mixing of the pure components into a mixed material stream at 𝑇₀ and 𝑃₀.

The different exergy components are listed in Table 3.

Table 3 

Subdivision of the exergy.

Finally, it can be defined as a “performance factor” the ratio of outlet exergy to inlet exergy: 𝜙_eff=𝜀_out𝜀_inl.(3.8)

4. Results and Discussion

The structure of the pulse detonation and the shock tube is depicted in Figures 2 and 3, respectively.

Figure 2 

Structure of the pulse detonation.

Figure 3 

Shock tube.

The detonation field parameters in this work were computed by using the Euler multispecies equations of detonation of H₂-air mixture. They were solved using the first author’s solution procedure [26, 27]. Analyses were carried out for different hydrogen concentrations in air optimum pressure with corresponding temperatures; species concentrations were determined and given first case in figures as function of time. As an example, for first case where the mass fraction of H₂ is equal to 1.5%, peak detonation and end rarefaction density of the selected location were taken at 1.4 kg·m⁻³ and 1.1 kg·m⁻³, respectively. As a result, optimum shock and rarefaction temperatures were found to be 1930 K and 700 K, respectively. At the same time, the optimum pressures were found to be 18 bar and 6 bar, respectively for the same concentration of hydrogen in air.

As seen in Figure 4 representing the pressure distribution along the tube at 0.003 ms, the optimum pressure at the peak of detonation was found to be 18 bar, 22 bar, and 36 bar for 1.5%, 2.5%, and 5% of hydrogen mass fractions, respectively. Figure 5 shows the result of the temperature at 0.003 ms corresponding to the end of reactions for the case of hydrogen concentration equal to 1.5%. Optimum values vary from 1930 K for the peak of detonation to 700 K for the end of rarefaction. These results show that the structure of the detonation wave is dependent on hydrogen concentration.

Figure 4 

Pressure evolution along shock tube and as function of hydrogen mass fraction in air (case 1: 1.5%, case 2: 2.5%, case 3: 5%).

Figure 5 

Temperature distribution along the tube length for case 1.

Figure 6 shows the shock pressure variation with time for the three different concentrations of hydrogen in air.

Figure 6 

Pressure evolution as function of time with hydrogen mass fraction cases.

The first case considers a hydrogen-air mixture at an initial pressure 𝑃₀ = 1 bar and initial temperature 𝑇₀ = 298 K. On the left side of the tube (Figure 3) a pressure pulse (10 bars) is considered to initialize the shock wave. At 𝑡=0 s, the shock wave moves from the left to the right side of the tube and initiates the reactions. Shock waves generated due to the formation of a new reaction front are called “reaction shocks” [20]. The reaction shock overtakes the shock front at around 27 μs, strengthening it and sharply increasing its propagation speed; that is, a detonation wave is formed. Subsequently, the detonation wave speed decreases gradually and a high-frequency, low-amplitude propagation mode is established. In this study no reaction delay is considered, so when the shock wave is initialized, the reactions produced energy. Generally, chemical reactions behind the shock front start generating weak shock waves after approximately 20 μs [31]. The plots show the variation in shock front speed with time. Initially the shock wave travels at a constant speed (determined by the initial conditions) slightly greater than 1500 m·s⁻¹, 1800 m·s⁻¹, and 2200 m·s⁻¹ for respective hydrogen concentrations of 1.5%, 2.5%, and 5%.

Figures 6 and 7 show the variation with time of pressure and density as function of time and for mass fraction of hydrogen in air: 1.5%, 2.5%, and 5%. As can be seen from these figures, the peak of detonation is characterized by the high-pressure values followed by the rarefaction wave. During the rarefaction wave, the pressure, temperature, and density decrease; this decrease is stopped by the reflected wave from the end of the tube. The most detonation energy is absorbed by the shock when the wave is reflected. This is because by assuming that the detonation is instantaneous and all hydrogen is consumed in the first way of the shock wave, then the energy is completely kinetic, and, as a result, the density and pressure decrease.

Figure 7 

Density evolution for each hydrogen mass fraction in air.

Figure 8 shows the mass fraction of H₂O, calculated as function of time for the three studied cases: 1.5%, 2.5%, and 5% of hydrogen concentration. This figure gives us a good idea about mass fraction of H₂O during the detonation for each of hydrogen concentration in air. In fact, the contribution of H₂O in exergy calculation is not negligible since the standard chemical exergy of H₂O, O₂, and N₂ are, respectively, 8.63 kJ·mol⁻¹, 3.90 kJ·mol⁻¹, and 0.60 kJ·mol⁻¹.

Figure 8 

Mass fraction as function of temperature for each hydrogen concentration in air.

Mass fraction for the H₂, OH, and H₂O system is plotted in Figure 9. Here, the energy levels were chosen to be consistent with those realized in the detonation calculation. As seen in this figure the hydrogen, radical OH, and product H₂O as function of time for the referent element shown on Figure 3, optimum values at the end of reactions, were found to be 0.04%, 0.7%, and 1.2% carried out for species H₂, OH, and H₂O. These results are presented for the case when the initial mass fraction of hydrogen in air is assumed to be equal to 2.5%.

Figure 9 

Mass fraction profile of the H2, OH- and products H2O.

It is possible to consider the shock and rarefaction processes as the separate successive steps. Therefore, the exergy loss due to the irreversible detonation and the end of rarefaction can be calculated as the product of the initial temperature and the difference between entropies in the start of the shock wave and the end of rarefaction wave. The typical diagrams of exergy balances of hydrogen detonation in air are shown in Figure 10 as function of hydrogen concentration (1.5%, 2.5%, and 5%). Data were calculated for the point of steady detonation. It was shown that exergy losses increase with the augment of hydrogen concentration in air; consequently low hydrogen concentrations in air achieve slight higher efficiencies. It was also verified that the exergy efficiency decreases for all the hydrogen concentrations as a function of the time. It was obtained an exergetic efficiency of 77.2%, 73.4%, and 69.7% for the hydrogen concentrations of 1.5%, 2.5% and 5%, respectively.

Figure 10 

Exergy evolution as function of time with hydrogen mass fraction in air.

On the other hand, the energy balance shows a successive conversion of kinetic energy and total enthalpy, however, does not indicate consequent losses.

5. Conclusions

The CFD code was developed using finite volume discretization method coupled with implicit scheme for the time discretization. The code solves the Euler system equations including a chemistry model, with 5 reactions and 7 species, using an implicit total variation diminishing (TVD) algorithm based on Riemann solver.

There exists a large literature about the numerical discretization of the detonation of hydrogen-air mixture problems: Eulerian schemes, Lagrangian schemes, front tracking techniques, and level set methods, among others. Our main goal, in the present work, is to use detonation parameters, from the beginning to the end of detonation of hydrogen-air mixture to calculate and analyze the exergy loss and the efficiency. The main steps of H₂-air mixture detonation calculation were explained. The developed program is able to calculate the pressure, temperature, density, and mass fraction, for each species from the chemical model. The exergy efficiency was used as objective criteria of performance evaluation.

We conclude in this work that the exergy losses increase with the augment of hydrogen concentration in air, and the efficiency of hydrogen detonation, with low concentration in air, is slightly elevated than the detonation using high hydrogen concentration.

Acknowledgments

The authors would like to express our gratitude to the Portuguese Foundation for Science and Technology (FCT) for the given support to the grant SFRH/BPD/71686 and to the project PTDC/AAC-AMB/103119/2008 and also to The French Atomic Centre Energy (CEA) for all the provided help.