## Abstract

Experiments were conducted to determine the pressure rise that results from either the combustion of a localized gas volume or the expansion of a pressurized gas volume adjacent to an inert gas in a closed vessel. The experiments consisted of either pressurized air or the combustion of stoichiometric and fuel-lean hydrogen–air mixtures compressing an inert gas. The pressure rise in the inert gas was measured as a function of either the volume fraction or the initial pressure of the expanding gas. Helium, nitrogen, air and carbon dioxide were tested to explore the effect of inert gas heat capacity on the pressure rise. The final pressure of the inert gas increased with the volume fraction and initial pressure of the expanding gas, and was influenced to a lesser extent by the heat capacity of the inert gas. A model was assessed using the experimental data, and the theoretical results were consistent with the observed trends. This model and other published models were assessed and compared using prior data for localized gas combustion surrounded by an inert gas and the partial combustion of homogeneous methane–air mixtures.

## 1. Introduction

A number of instances arise when a localized gas volume expands in a closed vessel and the subsequent pressure rise is desired. One example is when a portion of a homogeneous fuel–oxidizer mixture burns and compresses the unburned combustible gases, a process that will be referred to as partial combustion. Another example is when a local volume of a combustible gas is adjacent to or surrounded by an inert gas. The process of the combustion of the local gas pocket compressing the inert gas will be referred to as localized gas combustion. The last example occurs when a non-reacting pressurized gas volume is allowed to expand against the surrounding gas. All three examples share the common characteristic that the expansion of a local gas volume generates a pressure rise in the surrounding gas.

In a seminal paper, Flamm & Mache (1917) described the development of a model to predict the pressure rise during partial combustion. The motivation for the model was to develop an expression that related the pressure rise to the fraction of gas burned so that the normal burning velocity of a flame in a spherical bomb could be determined from an experimental pressure-history measurement. The partial combustion process was modelled as infinitesimal layers of gas burning at a constant pressure, although each subsequent layer of gas burned at an increasingly higher pressure. The unburned gas was compressed isentropically. Using their model, Lewis & von Elbe (1961) developed an approximate equation for the pressure rise as a function of the fraction of gas burned by assuming that the temperature of the unburned compressed gas was equal to the initial gas temperature. The expression is exact when the ratios of the specific heats of the burned and unburned gases are equal.

Babkin & Kononenko (1967) developed a model to predict the pressure rise during partial combustion, which was also used to determine normal burning velocity in a spherical bomb from recorded pressure histories. Their model assumed that a homogeneous fuel–air mixture was centrally ignited within a spherical geometry and that differential layers burned at a constant but increasingly higher pressure. Their model also assumed that the unburned gas was compressed isentropically. However, their model did not assume that the temperature of the unburned compressed gas was close to the initial gas temperature, and it modelled the burned and unburned gases with different ratios of specific heats. The approximate relationship described by Lewis & von Elbe (1961) was recovered when the ratios of the specific heats were equal.

Babkin & Babushok (1977) used the expression developed by Lewis & von Elbe (1961) to study the time for pressure rise in explosion development from partial combustion in vessels of arbitrary shape. Because Babkin and Babushok incorporated Lewis and von Elbe’s model in their model for combustion time, the model was limited to burned and unburned gases with equal ratios of specific heats. It was also limited to initial stages of gas combustion because of the restriction of spherical geometry. Babkin and Babushok’s model was valid until the flame made contact with a wall in a vessel of arbitrary shape.

Babkin *et al*. (1979) used the expression developed by Babkin & Kononenko (1967) to develop expressions for rate of pressure rise for partial combustion in closed spherically symmetric vessels with central ignition for three cases: burned and unburned gases having ratios of specific heats that were unequal, equal and both equal to one. Using these special cases, they were able to recover the form of various approximate equations from other researchers that had previously investigated accidental explosions from industrial operations.

Metghalchi & Keck (1980) developed a laminar burning velocity model and used it for propane–air mixtures. The model divided the mixture into regions of burned and unburned gases. The unburned gas was assumed to be compressed isentropically. The conservation of volume and energy was approximated using the first term of a Taylor series expansion and solved numerically because temperature-dependent specific heats and equilibrium gas compositions were used in the analysis. Metghalchi & Keck (1982) used a similar approach to predict the laminar burning velocity of three hydrocarbon–air mixtures, but studied a number of refinements to the idealized assumptions of the earlier model. The approach proposed by Metghalchi and Keck was extended by Elia *et al*. (2001) and Saeed & Stone (2004) to divide the burned gas into a number of regions. This approach permitted the calculation of the temperature gradient within the burned gas.

In an attempt to develop an analytical expression for partial combustion, Luijten *et al*. (2009) used the same approach as Metghalchi & Keck (1980), but assumed constant specific heats and constant burned and unburned gas compositions. Analytical models were developed for the cases when the burned gas was modelled as a single region and multiple regions. Using the approach of Lewis & von Elbe (1961), they also extended their work to include the effect of different specific heats for the burned and unburned gases. The expression for this extended model was identical to the expression developed by Luijten *et al*. for the case of single regions for the burned and unburned gases, even though different development approaches were used.

An early investigation of the pressure rise from local gas combustion was by Sibulkin (1980), who examined the problem of combustion of a gas pocket in a nuclear reactor containment vessel. Sibulkin used an equilibrium thermodynamic approach to develop an expression for the pressure rise in the inert gas as a function of the volume of the combustible gas. The first law of thermodynamics was applied to the combustible gas volume between the unburned and burned states, along with the assumption that the gases behaved as ideal gases and that the inert gas was compressed isentropically, resulting in three coupled equations that were solved simultaneously. Because the relationship between the pressure and volume in the combustible gas during the combustion process was unknown, Sibulkin assumed that the pressure in the work term of the first law was constant and equal to the initial pressure to obtain an analytical expression for the work. However, this approximation forced a restriction to the model that the pressure rise is small. Sibulkin limited his calculations to a pressure rise of 50 per cent of the initial pressure, which corresponded to the local gas volumes that occupied approximately less than 5 per cent of the total volume.

Another model for the pressure rise owing to local combustion in a closed vessel was developed by Babkin *et al*. (1985) for applications to industrial accidents. A purely thermodynamic approach was taken to develop the model. The problem with developing an appropriate expression for the work term in the first law encountered by Sibulkin (1980) was overcome by taking the contents of the entire vessel (burned, unburned and inert gases) as the system. Since the system contained burned and unburned gases, the enthalpy across the flame front was assumed to be constant to provide expressions for corresponding terms in the first law. By using expressions for conservation of mass and volume and the assumption of isentropic compression of the unburned gas, an expression was obtained for the pressure developed during local combustion as a function of the volume of the combustible gas, the expansion coefficient of the combustion products at constant pressure (or, alternately, the amount of heat liberated in the combustion of a unit mass of mixture) and gas properties.

Boyack *et al*. (1993) developed a model to examine the scenario when a portion of a combustible gas mixture deflagrates in a nuclear reactor containment vessel prior to a detonation. For example, if the vessel is filled with a combustible hydrogen–air–steam mixture and ignition occurs in the uncluttered dome region of the upper containment, the gases can burn as a deflagration until the flame enters the lower containment. If a detonation then occurs in the compartmentalized lower containment, the unreacted combustible gases have been pre-compressed by the partial combustion in the upper containment and the peak detonation pressures are higher than the scenario that the entire mixture detonated. Even though the application involved partial combustion of a homogeneous mixture, the system modelled included regions with two different gases. This allowed the model to be applicable to both partial and local combustion.

The model was developed based on thermodynamic principles only. Because the first law depends only on the initial and final states of the mixtures, the entire local combustible gas volume was assumed to deflagrate completely before the burned region was allowed to expand and compress the unburned gases isentropically. The problem with the work term in the first law encountered by Sibulkin (1980) was overcome by taking a system containing both gases. By assuming ideal gases along with the conservation of mass for both gases (burned and unburned) and the constitutive relation that the sum of the different gas mixture volumes must always be a constant, a set of five coupled equations was solved simultaneously for the final pressure.

Although a number of analytical models have been developed, data to assess the models are limited. Vykhristyuk *et al*. (1988) performed local combustion experiments and summarized both partial combustion and local combustion data from earlier reports with limited distribution. The local combustion experiments performed by Vykhristyuk *et al*. involved igniting different sized rubber balloons filled with either a stoichiometric or fuel-lean (equivalence ratio equal to 0.83) methane–air mixture contained within a sealed test chamber filled with air. Data were also presented from previous local combustion experiments using polyethylene bags filled with various fuel–air mixtures (acetone, natural gas and city gas) and enclosed in sealed test vessels filled with air. The data presented from partial combustion experiments involved homogeneous stoichiometric mixtures of methane or natural gas with air in a sealed rigid vessel.

Unfortunately, the ability to use previously reported data to assess the models’ parameters is limited. The main parameters that influence the final pressure in a local combustion event are the volume fraction of gas that burns and the pressure ratio of the gas that expands to the gas that is compressed (Boyack *et al*. 1993). The ratio of specific heats of the compressed gas influences the final pressure to a lesser extent. The local combustion data reported by Vykhristyuk *et al*. (1988) are limited to a combustible gas volume fraction of less than 10 per cent. That is, the volume of the combustible mixture tested is less than 10 per cent of the total volume. The pressure ratio is limited to less than the adiabatic isochoric complete combustion (AICC) value, typically a maximum value of approximately 8 for stoichiometric mixtures and less for off-stoichiometric mixtures. Likewise, since air always surrounded the combustible gas in previous experiments, the ratio of specific heats of the inert gas was not varied.

The purpose of the present study is to determine the pressure rise in a gas compressed by partial or local combustion or from the expansion of a pressurized gas. Experiments were performed to vary key parameters that influence the final pressure of the gas, including the volume fraction of the burned gas, the pressure ratio of the gases and the ratios of specific heats of the compressed gases. Parameters were varied over sufficiently large ranges to fully assess models for partial and local combustion.

## 2. Experimental method

Experiments were conducted in the test facility shown in figure 1, comprising a horizontal instrumented pressure vessel, a vacuum pump (VP) and a series of compressed gas cylinders. The pressure vessel was composed of three cylindrical flanged pipe segments separated by two flanged ball valves and closed at the ends by blind flanges. The volume of either the combustible gas or pressurized gas could be varied by use of the ball valves. The pressure vessel had a total length of 1.985 m and an internal diameter of 97.3 mm. The compressed gases consisted of the inert gases, fuel, oxidizer and a fuel–oxidizer mixture tank. All cylinders included a two-stage regulator and a needle valve for fine pressure adjustment, except for the mixture tank, which consisted of a flame arrester and a by-pass line containing a needle valve for filling. The compressed gas cylinders, a vacuum pump and a plumbing branch were connected through a manifold. The plumbing branch contained a pressure relief valve, a ball valve for manual blow down, a pressure gauge for quick visual inspection of manifold pressure and a pressure transducer. A strain-gauge type pressure transducer (Kobold model KPK-30/60 2221) was used to establish initial conditions in the pressure vessel for experimental tests and to prepare fuel–air mixtures in the mixture tank. The pressure transducer range was 0–5 atm absolute with a 0.25 per cent accuracy full-scale output. Ball valves on two inlet lines were used to isolate the manifold from the pressure vessel.

The blind flanges and a series of bosses along the top of the pressure vessel were tapped to install high-frequency-response pressure transducers and an igniter. Two pressure transducers were used to measure the pressure history of the gases, one each for the driver and driven gases. The piezo-resistive pressure transducers (Endevco model 8530C-100) had a range up to 690 kPa absolute, a natural frequency of 500 kHz and an accuracy of 0.2 per cent full-scale output. The igniter operated by a capacitive discharge through a gapped spark plug. A 250 μF capacitor at 90 V discharged within 40 μs through a silicon-controlled rectifier when a manual trigger completed the circuit.

Prior to the tests, the hydrogen–air mixture was prepared by means of the method of partial pressures. The mixture tank was initially evacuated and then dry air was added followed by hydrogen. Prior to testing, the gases were allowed to mix in the tank for a minimum of 3 days and typically much longer. Three equivalence ratios, Ø, were tested: Ø=0.997±0.009; Ø=0.707±0.008; and Ø=0.371±0.007.

Three parameters were varied in the combustion tests: the volume fraction of combustible gas (driver gas) to inert gas (driven gas), the pressure of the driver gas as manifested by the equivalence ratio of the combustible gas and the type of gas that was compressed (driven gas). Depending on the volume fraction tested, part of the pressure vessel may have been isolated by closing one of the large ball valves. Furthermore, a piston was used to separate the driver and driven gases during the tests. The piston was a 96.3 mm diameter hard plastic hollow cylinder 50.8 mm long with two truncated hemispherical caps, each 31.8 mm long for a total length of 114.4 mm. Prior to the test, the piston was placed in the driven gas section against the ball valve separating the driver and driven gases.

The general testing procedure was to evacuate the pressure vessel, add the driver gas and then the driven gas. The evacuation of the pressure vessel and addition of the driver gas were performed with the large ball valves open. One atmosphere of combustible gas was added to the test chamber, and the ball valve was closed, trapping the combustible gas. The driven gas section was then evacuated and filled with 1 atm of the inert gas. The total uncertainty of the initial pressure was estimated to be ±3.4 kPa at worst and typically ±2.1 kPa. Just before ignition, the ball valve was opened and the igniter closest to the piston was activated. As the combustible gas deflagrated, the piston compressed the inert gas until the pressures equilibrated. The total uncertainty of the final pressure was estimated to be ±6.3 per cent.

Another series of tests was conducted using a pressurized inert gas instead of the combustible gas as the driver gas. The same variables were tested and the same testing procedure was used with some minor exceptions. Air at nominally 5 atm of pressure (absolute) was used for the driver gas, and the pressure of the driven gas was varied to get pressure ratios that varied from approximately 2–20. The test began when the large ball valve separating the driver and driven gases was manually opened, thereby compressing the driven gas. The total uncertainty of the final pressure was estimated to be ±1.1 per cent.

Pressure histories were recorded using automatic data acquisition software with a conditional software trigger based on the pressure rise in the driven gas. The sampling rate for the combustion tests was 200 000 samples s^{−1}, except for the leanest hydrogen–air mixture, which was sampled at 10 000 samples s^{−1}. The sampling rate for the pressure tests was 1000 samples s^{−1}.

## 3. Model

The experimental data were used to assess an analytical model developed to calculate the final pressure resulting from a local or partial combustion in a closed vessel (Boyack *et al*. 1993). The model assumes that two ideal gases, identified as A and B, are contained in a rigid insulated vessel and are separated by a frictionless insulated piston. Gas A is assumed to have a higher pressure than gas B and, as the piston is released, gas A expands (driver gas) and gas B is compressed (driven gas) until the final pressures equilibrate. When modelling an event involving a deflagration, the two gases initially may be the same gas, such as a fuel–air mixture, for the case of partial combustion or different gases, such as a combustible mixture and an inert gas, for the case of local combustion. Under either of these circumstances for modelling purposes, gas A is assumed to be the post-deflagration products of the combustible mixture and modelled to exist at the AICC state, whereas gas B is either the initial combustible gas mixture or the inert gas, depending on the case of either partial or local combustion. Because of the generality of the model, the case of the rupture of a pressure vessel contained within a rigid closed vessel can also be modelled. Gases A and B may be the same or different inert gases, and gas A is assumed to have the higher pressure.

The initial thermodynamic state, defined as state 1, of both gases is known: gas A exists at AICC conditions and gas B at the specified initial conditions. The initial volume fraction of the combustible gas will be specified. For the case of local combustion, it will be determined by the relative amounts of combustible and inert gases. The volume fraction will also be specified for partial combustion. For example, in the analysis of a severe nuclear accident scenario, it may be specified that gases in the dome of the containment deflagrate prior to transitioning to a detonation in the lower compartments. Likewise, the flame speed can be calculated at different radii of the spherical vessel using models that show gas pressure as a function of volume fraction burned. No variables are known at the final state, defined as state 2, except that the final pressure of the gases are equal. Gas A will cool as it expands and its volume will increase, while gas B will be compressed and heat up. Together, there are five unknowns: the final temperatures and volumes of the two gases and the final pressure.

The analytical model was developed using the conservation equations for energy and mass, the assumption of isentropic compression of the driven gas and a constitutive relation for the volume. The conservation of energy applied to the entire volume yields an expression for the final temperature of gas A, *T*_{A2}, assuming that the change in internal energy is represented as the product of the specific heat at constant volume and the temperature difference, Δ*u*=*C*_{v}Δ*T*,
3.1
where *T*,*P*,*V* and *γ* are the temperature, pressure, volume and ratio of specific heats, respectively, of the gases. The conservation of mass for gases A and B yields the following equations for the final volumes:
3.2
and
3.3
The final temperature of gas B, *T*_{B2}, is obtained from the equation for isentropic compression of the gas,
3.4
Finally, because the gases are contained in a rigid volume,
3.5
Equations (3.1)–(3.5) represent five coupled independent equations that must be solved simultaneously for the five unknowns at the final state, resulting in an implicit non-dimensional equation for the final pressure,
3.6
This is the same as eqn (13) from Boyack *et al*. (1993), but rearranged to eliminate the temperatures from the original equation.

The AICC solution is recovered when the gas in the entire volume burns, that is, when *V*_{A1}/*V* =1 and *V*_{B1}/*V* =0. Inserting these values into equation (3.6) yields the solution that *P*_{2}=*P*_{A1}, which is the AICC pressure. Once the final pressure, *P*_{2}, is determined from equation (3.6), the final temperature of gas B, *T*_{B2}, can be determined from equation (3.4), the final temperature of gas A, *T*_{A2}, from equation (3.1) and the final volumes, *V*_{A2} and *V*_{B2}, from equations (3.2) and (3.3), respectively.

## 4. Results

A series of local combustion tests were performed, and the results were used to assess the model’s prediction of the final gas pressure. The results of these tests, which are summarized in the first part of table 1, show the final gas pressure as a function of the volume fraction of combustible gas with variations in the ratio of specific heats of the inert gas at a fixed ratio of driver gas to driven gas pressure. The combustible gas for this test series was a stoichiometric hydrogen–air mixture with initial conditions nominally at 1 atm of pressure and room temperature, which was typically 25 ^{°}C. The inert gases included carbon dioxide (*γ*=1.289), nitrogen (*γ*=1.40) and helium (*γ*=1.667), which cover the practical range of ratio of specific heats, *γ*, for common gases.

For the combustion tests, the driver gas was the deflagration products and the driven gas was the inert gas. The driver gas pressure was required only by the model to calculate the final gas pressure and was not required to determine the experimentally measured final gas pressure. According to the model, the driver gas pressure is assumed to be the AICC pressure. Theoretical values of the AICC pressure can be computed with chemical equilibrium codes. However, given the size of the pressure vessel and the reduction of theoretical pressure because of radiative heat losses, an experimentally measured deflagration pressure was used in the model. In a companion set of tests conducted in the test facility shown in figure 1, the peak pressure was measured for all three equivalence ratios when the entire vessel was filled with hydrogen–air mixtures. The ratio of the peak pressure to the initial mixture pressure was measured and used to calculate the driver gas pressure shown in table 1. The driven gas pressure and the final gas pressure were measured directly using the piezo-resistive pressure transducers. The final gas pressure was non-dimensionalized using the initial pressure of the driven gas to facilitate comparison between tests with slightly different initial gas pressures. The last two columns of table 1 show the theoretical results for the final gas pressure for each test conducted. Using values for *V*_{A1}/*V*,*P*_{A1}/*P*_{B1} and *γ*_{B}, the final gas pressure, *P*_{2}, was calculated using equation (3.6) for each test shown in table 1.

The model was also used to calculate the final gas pressure for local combustion over the entire range of combustible gas volume fractions from 0–1, as shown in figure 2. The calculations were made at a fixed ratio of driver gas to driven gas pressure nominally equal to the experimental values, and were made for the three different driven gases to show the effect of the ratio of specific heats. The model shows that the non-dimensionalized final gas pressure increases with the volume fraction of combustible gas. This is an intuitive result since there is an increasing fraction of combustible gas compressing a diminishing fraction of inert gas. The model also shows higher final pressures for gases that have higher ratios of specific heats, which corresponds to gases with simpler molecular structure. Monatomic gases, such as helium, have fewer modes of storing internal energy from the work of compression, which manifests itself as higher temperatures and, subsequently, higher final pressures when compared with a more structurally complex gas, such as nitrogen or carbon dioxide. The experimental data are also cross-plotted against the model predictions shown in figure 2. The predicted trend that the final gas pressure increases with increasing volume fraction of combustible gas is supported by the data, as is the predicted trend that gases with structurally simpler inert gas molecules have higher final pressures.

The measured pressures in figure 2 generally fall below the predicted values. This may be a result of radiative heat losses, reducing the driving pressure during the experiment. For initially equal starting lengths, the time required to burn a given amount of combustible mixture will be longer when the volume expands than for a fixed volume. The peak pressure from a constant volume combustion was used as the driving pressure in the model. However, the additional time during expansion would permit more radiative heat losses and a lower experimental driving pressure, resulting in a measured final pressure lower than the predicted value.

A series of tests were performed to explore the effect of the volume fraction of driver gas on the final gas pressure. However, this series used pressurized air as the driver gas, instead of the products of the combustible gas mixture, to eliminate any issues associated with radiative heat losses. This configuration would simulate the rupture of a pressure vessel in a rigid enclosure. The driver gas consisted of air at approximately 5 atm of pressure (absolute), and the pressure of the driven gas was adjusted to yield a pressure ratio similar to that used in the combustion tests. The driver gas, the driven gas and the final gas pressures were measured experimentally and recorded in table 2. Using values for *V*_{A1}/*V*,*P*_{A1}/*P*_{B1} and *γ*_{B}, the final gas pressure, *P*_{2}, was calculated using equation (3.6) for each test shown in table 2. In addition to inert gases composed of helium, nitrogen and carbon dioxide, this series also included air as an inert gas. A ratio of specific heat value of 1.40 was used for air in the model calculations.

The final gas pressure was calculated using the model with the same driver gas to driven gas pressure ratio used in the local combustion test series over the complete range of combustible gas volume fractions. The calculations were cross-plotted against the data and are shown in figure 3. The data agree with the model predictions and show that the final gas pressure increases with increasing driver gas volume fraction. Likewise, the data generally support the predicted trend that the final gas pressure increases for gases with larger ratios of specific heats, although the effect on the final gas pressure is not as significant as variations in volume fraction.

A series of local combustion tests were performed to observe the effect of variations in the driver gas pressure on the final gas pressure when the tests were conducted at a fixed driver gas volume fraction. Tests were conducted with three different hydrogen–air mixture stoichiometries (Ø=0.371, 0.707 and 0.997) to yield a range of driver gas to driven gas pressure ratios between 2.61 and 5.74. The experimental results are shown at the bottom of table 1, along with predicted values for each test. The ability to assess the model with data is somewhat limited since the maximum driver gas pressure of the combustion tests cannot exceed the AICC value. Another series of tests was conducted using pressurized air to create a wider range of driver gas to driven gas pressure ratios. The driver gas consisted of air at approximately 5 atm of pressure (absolute), and the pressure of the driven gas was adjusted to yield pressure ratios between approximately 2 and 20. The experimental results are shown in table 3, along with predicted values for each test.

The final gas pressure was calculated for air compressing different inert gases using the model with the same experimental driver gas volume ratio over the range of driver gas to driven gas pressure ratios tested. The calculations were cross-plotted against both the combustion data and the pressurized gas data and are shown in figure 4. The data agree with the model predictions and show that the final gas pressure increases with increasing driver gas to driven gas pressure ratio. The model does not show a large variation in the final gas pressure with variations in the driven gas ratio of specific heats. However, this is consistent with the results shown in figure 3, which shows that the influence of the ratio of specific heats is not large for large volume fractions. The results for variations in driver gas pressure shown in tables 1 and 3 were conducted at a driver gas volume fraction of nearly 0.8. The data do not exhibit any significant variation in the final pressure with the different inert gases tested and agree with the model predictions.

## 5. Discussion

Vykhristyuk *et al*. (1988) reported local combustion data for methane–air systems, which may be used to assess the present model and other local combustion models reported in the literature. Tests were conducted by igniting either a stoichiometric or fuel-lean (Ø=0.828) methane–air mixture contained within a thin rubber balloon, which was placed at the centre of a large rigid 3.2 m diameter cylindrical vessel (33.8 m^{3}) filled with air. The data are shown in figure 5 and are limited to small combustible gas volume fractions less than 10 per cent.

The data reported by Vykhristyuk *et al*. (1988) was used to assess various local combustion models. Equation (3.6) from the present model was used to predict the final gas pressure for both the stoichiometric and fuel-lean methane–air mixtures. The driver gas was assumed to be the products of combustion of the methane–air mixture, which existed at the AICC state, whereas the driven gas was air at the initial conditions. The theoretical results were calculated over the range of combustible gas volume fractions from 0 to 0.1 and plotted in figure 5. Vykhristyuk *et al*. (1988) also presented various forms of a model developed by Babkin *et al*. (1985). By neglecting the terms representing the effects of unequal specific heats in the original equation of Babkin *et al*. (1985), Vykhristyuk *et al*. (1988) presented an approximate form (eqn (2) in) and an alternate approximate form for small combustible gas fractions (eqn (3) in), both of which are shown in figure 5 for stoichiometric methane–air mixtures. Finally, the results of a model developed by Sibulkin (1980) for stoichiometric methane–air mixtures are shown in figure 5 using Sibulkin’s equation (eqn (12) in Sibulkin 1980) and methane values for equation parameters reported in the original paper. The results for Sibulkin’s model begin to deviate from the data at larger pressure ratios. This is an artefact of an approximation made by Sibulkin to develop an expression for the term representing the work of compression of the inert gas. Sibulkin noted this approximation and suggested that the model cannot be used for a non-dimensionalized final pressure (*P*_{2}/*P*_{B1}) greater than 1.5. However, this limits the use of Sibulkin’s model to very small combustible gas volume fractions (*V*_{A1}/*V* <0.06).

Vykhristyuk *et al*. (1988) also present data that can be used to assess partial combustion models. Models developed to predict peak pressures from local combustion may also be used for partial combustion by assuming that both the driver and driven gases are initially the same combustible gas mixture.

The experiments were performed in a 2.15 l spherical vessel with central ignition of stoichiometric methane–air mixtures at approximately 293 K and initial pressures ranging from 0.06 to 0.225 MPa. Vykhristyuk *et al*. (1988) were able to determine the pressure of the unburned gas as a function of the volume of combustible gas consumed based on the pressure history of the combustion event, and the results are plotted in figure 6. As expected, the non-dimensionalized final pressure is similar for all initial pressures, and the maximum experimental pressure does not exceed the AICC pressure ratio, which is 8.878 for stoichiometric methane–air mixtures.

Models were assessed using the partial combustion data presented by Vykhristyuk *et al*. (1988), and the results are shown in figure 6. Theoretical predictions of the compressed gas pressure from the present model were calculated using equation (3.6) over the entire range of combustible gas volume fractions. The driver gas was assumed to be the products of combustion of the stoichiometric methane–air mixture existing at the AICC state, whereas the driven gas was the methane–air mixture at the initial mixture conditions. Results from the models presented by Vykhristyuk *et al*. (1988) are also shown in figure 6. These include results from the original model of (eqn (1) in Vykhristyuk *et al*. 1988) and the approximation proposed by Vykhristyuk *et al*. (eqn (2) in Vykhristyuk *et al*. 1988), which neglects the effects of unequal specific heats between the driver and driven gases. The second approximation given by Vykhristyuk *et al*. (eqn (3) in Vykhristyuk *et al*. 1988), which is shown in figure 5, is not included in figure 6 since that form of the equation is limited to small combustible gas volume fractions. Both forms of the model generally agree with the data, except at combustible gas volume fractions above approximately 82 per cent, in which the predicted final pressure exceeds the AICC pressure. The results of the model proposed by Lewis & von Elbe (1961) are also shown in figure 6. As will be shown later, this model is identical to the present model when the ratios of specific heats of the driver and driven gases are equal. However, since the driver gas (methane–air combustion products) and the driven gas (methane–air mixture) are different, the model of Lewis and von Elbe is approximate and there is a small difference between the predictions of the two models. The results from the Lewis and von Elbe model are linear, and the results from the present model are curved slightly upwards, yet the curvature in the data is more pronounced. The discrepancy between the model and data would be an interesting area of future work.

The present model can be reduced to a partial combustion expression developed by Lewis & von Elbe (1961) under certain conditions. Lewis & von Elbe (1961) outline a theory put forth by Flamm & Mache (1917) to predict the pressure rise when a fraction of the gas burns,
5.1
where *n* is the fraction of total gas burned, *T*_{i} and *P*_{i} are the temperature and pressure of the gas before ignition, *T*_{u} and *P* are the temperature and pressure of the compressed unburned gas after an amount *n* of gas burns, *P*_{e} is the pressure after all of the gas burns and is equal to the AICC pressure, *γ*_{ b} and *γ*_{u} are the ratios of specific heats of the burned and unburned gases and *K* is a constant equal to *γ*_{u}(*m*_{e}/*m*_{i}*C*_{vb}*T*_{bi}−*C*_{vu}*T*_{i}), where *m*_{e}/*m*_{i} is the ratio of moles after and before combustion, *C*_{vb} and *C*_{vu} are the constant specific heats of the burned and unburned gases and *T*_{bi} is the temperature of the burned gas when burned at the initial pressure and temperature of the mixture. Lewis and von Elbe proposed an approximate form of equation (5.1) by assuming that the compressed unburned gas temperature is close to the initial gas temperature, *T*_{u}≅*T*_{i}, and, after using other relationships developed in the paper, yields the approximate form
5.2
However, the approximation that *T*_{u}≅*T*_{i} imposed by Lewis and von Elbe to develop equation (5.2) was not necessary since it was already implicitly assumed that *γ*_{ b}=*γ*_{u} in the development of equation (5.1). With this assumption, equation (5.1) becomes
5.3
An expression for *K* can be obtained when none of the mixtures has burned, that is, when *n*=0. At this condition, *P*=*P*_{i} and (*γ*_{ b}−1)*K*=*R**T*_{i}(*P*_{e}−*P*_{i})/*P*_{i}. When this expression for *K* is substituted into equation (5.3), equation (5.2) is obtained. The significance of this alternate approach is that it removes one of the restrictions imposed on equation (5.2) by Lewis and von Elbe. Equation (5.2) is an exact expression when *γ*_{ b}=*γ*_{u} regardless of the temperature of the compressed unburned gas relative to the initial gas temperature. Equation (5.2) is an approximate relationship only when *γ*_{ b}≠*γ*_{u}.

More importantly, the fact that *γ*_{ b}=*γ*_{u} is the only restriction imposed on the development of equation (5.2) provides the basis to simplify the present model to get the same form as the equation of Lewis and von Elbe under these conditions. In terms of the present paper’s nomenclature, the ratios of specific heats of the burned and unburned gases are represented as *γ*_{A} and *γ*_{ B}. When it is assumed that *γ*_{A}=*γ*_{ B}, equation (3.6) becomes
5.4
which can be rearranged into the form
5.5
Equation (5.5) is identical to equation (5.2) by noting that *V*_{A1}/*V* is the fraction of gas that has burned, *n*,*P*_{2} is the final pressure, *P*, that results when *n* gas fraction has burned, *P*_{B1} is at the initial pressure of the mixture, *P*_{i}, and *P*_{A1} is the AICC pressure, which is the same as *P*_{e}.

Luijten *et al*. (2009) developed an expression that extended the Lewis and von Elbe approximation, equation (5.2), for the case when *γ*_{ b}≠*γ*_{u},
5.6
Equation (5.6) is based on a combination of the original eqns (19) and (20) of Luijten *et al*. (2009). By rearranging terms, equation (5.6) can be placed into the form
5.7
Noting that (*γ*_{u}−*γ*_{ b})/(*γ*_{u}−1)=−((*γ*_{ b}−1)/(*γ*_{u}−1)−1), equation (5.7) is the same as equation (3.6) of the present model. Furthermore, by adding and subtracting a *P*_{e}/*P*_{i} term to equation (5.7) and doing some algebraic manipulation, the following equation is derived:
5.8
which is identical to eqn (13) of Babkin & Kononenko (1967), recognizing that (ρ/ρ_{i})^{γu}=*P*/*P*_{i}, the form used in Babkin and Kononenko’s paper.

Equations (3.6), (5.6) and (5.8) are different forms of the same equation, yet are derived using different approaches. Since the mass fraction is shown to be a function of thermodynamic properties, the process for reaching the final state apparently is not important. That is, the integrative effect of a series of infinitesimal layers of mixture burning at increasingly higher but constant pressure yields the same result as the cumulative mixture burning at constant volume and then expanding against the remaining unburned mixture. Since equations (3.6), (5.6) and (5.8) were developed using models that treated the burned gas as either a single region or an infinite number of differential layers, this would suggest that any model that used a finite number of regions for the burned gas should also yield the same results.

## 6. Conclusions

Experiments were performed to measure the pressure that develops when a gas is compressed by an adjacent gas that either burns or expands. This scenario occurs during: (i) local combustion, that is, when a local gas volume burns and compresses an adjacent inert gas, (ii) partial combustion, that is, when part of a uniform combustible mixture burns and compresses the remaining portion of the mixture, or (iii) from the expansion of a locally pressurized gas compressing a surrounding gas at a lower pressure, as in the rupture of a pressure vessel inside a rigid container. The pressure of the compressed gas increases when either the volume fraction or driving pressure of the combustible gas or pressurized gas increases. The pressure of the compressed gas is influenced to a lesser extent by the molecular structure of the compressed gas. Simple gases with high ratios of specific heats, such as monatomic gases, generally exhibit higher pressures than more structurally complex molecules, such as triatomic gases, when compressed under equal driving conditions of the expanding gas.

Local combustion models were assessed with available data. Owing to an approximation in its development, the model of Sibulkin (1980) is limited to combustible gas volume fractions less than approximately 6 per cent. The model of Babkin *et al*. (1985) and approximations to this model by Vykhristyuk *et al*. (1988) generally agree with the data, but predict compressed gas pressures in excess of the AICC pressures, a non-physical result, for combustible gas volume fractions that exceed approximately 82 per cent. The present model generally agrees with the data over the full range of combustible gas volume fractions and with the trends shown in the data. It has been assessed with different types of fuels and inert gases, as well as different geometric shapes of the test vessels. The present model is applicable to local combustion, partial combustion and expansion of locally pressurized gases.

## Acknowledgements

Funding for support of this project through the University of Evansville’s UExplore, ARSAF and BEAC grants is gratefully acknowledged.

## Footnotes

- Received March 17, 2009.
- Accepted June 29, 2009.

- © 2009 The Royal Society