DEVICE AND METHOD FOR FILLING A PRESSURIZED-GAS TANK

Abstract

The invention relates to a method for filling a pressurized-gas tank by means of a filling device comprising a gas source, a filling pipe connecting the source to the tank, a flow-rate and/or pressure control valve, an electronic control member configured to bring filling to a stop when an estimated temperature of the gas present in the tank reaches a temperature limit value, the method comprising, prior to the tank being filled, a step of determining the ambient temperature at the filling device, a step of determining the pressure of the gas present in the tank and a preliminary step of estimating the initial temperature of the gas present in the tank, the initial temperature of the gas present in the tank being a value estimated on the basis of the ambient temperature and on the basis of the pressure of the gas present in the tank prior to the tank being filled, the initial temperature of the gas present in the tank being higher than or equal to or lower than or equal to the ambient temperature.

Claims

1-16. (canceled)

17. A method for filling a pressurized-gas tank, the method comprising the steps of: providing a filling device comprising: a gas source, a filling pipe connecting the source to the tank, a flow rate and/or pressure control valve in the filling pipe, and an electronic control component configured to stop filling when an estimated temperature or density of the gas present in the tank reaches a temperature or density limit value, determining an ambient temperature on the filling device, determining a pressure of the gas present in the tank and estimating an initial temperature of the gas present in the tank, wherein the initial temperature of the gas present in the tank is a value that is estimated as a function of the ambient temperature and as a function of the pressure of the gas present in the tank before filling, with said initial temperature of the gas present in the tank being greater than or equal to or less than or equal to the ambient temperature; and filling the pressurized-gas tank.

18. The method as claimed in claim 17, wherein the initial temperature of the gas present in the tank is divided into a first computed initial temperature corresponding to a state of the tank that is considered to be recently filled to a first initial density, and a second computed initial temperature corresponding to a state of the tank that is considered to be recently drawn off to a second initial density.

19. The method as claimed in claim 18, wherein the first computed initial temperature of the gas present in the tank is within a high temperature range with a lower limit that is determined to be greater than or equal to the ambient temperature and an upper limit corresponding to a determined maximum temperature limit, for example, equal to 85 C.

20. The method as claimed in claim 18, wherein the first computed initial temperature is determined from a first predetermined predictive curve provided by a predetermined physical model simulating a reference filling of the tank over the high temperature range.

21. The method as claimed in claim 18, wherein the second computed initial temperature of the gas present in the tank is within a low temperature range with an upper limit that is determined to be less than or equal to the ambient temperature and a lower limit corresponding to a determined minimum temperature limit, for example, ranging between zero and 5 C.

22. The method as claimed in claim 18, wherein the second computed initial temperature is determined from a second predetermined predictive curve provided by a predetermined physical model simulating a reference draining of the tank over the low temperature range.

23. The method as claimed in claim 18, further comprising: modeling, during filling, a first temperature variation curve or a first density variation curve of the gas present in the tank, wherein when the first temperature variation curve is modeled, having the first computed initial temperature of the gas present in the tank as the starting condition, wherein when the first density variation curve is modeled, having the first initial density, wherein when the second temperature variation curve is modeled, having the second computed initial temperature of the gas present in the tank as the starting condition, wherein when the second density variation curve is modeled, having the second initial density of the gas present in the tank as the starting condition.

24. The method as claimed in claim 23, further comprising a step of estimating, during filling: a temperature variation curve of the gas present in the tank, with said temperature variation curve ranging between the first temperature variation curve and the second temperature variation curve, or a density variation curve of the gas present in the tank, with said density variation curve ranging between the first density variation curve and the second density variation curve.

25. The method as claimed in claim 18, wherein the first computed initial temperature and/or the second computed initial temperature is recomputed during filling as a function of the flow rate and as a function of the temperature of the gas present in the filling pipe, with said flow rate and said temperature being determined by computation and/or by sensors on the filling device.

26. The method as claimed in claim 17, wherein the temperature limit value is a determined fixed value, or a value provided by a reference temperature curve, with said reference temperature curve being provided by a predetermined physical model that simulates the thermodynamic conditions of the gas during a reference filling of the tank, wherein the density limit value is a determined fixed value or a value provided by a reference density curve, with said reference density curve being provided by a predetermined physical model that simulates the thermodynamic conditions of the gas during a reference filling of the tank.

27. The method as claimed in claim 20, wherein the physical model is based on a system of equations comprising at least one from among: an internal energy balance equation applied to the gas present in the tank; a mass balance equation applied to the gas present in the tank; an energy conservation equation in a tank wall; a heat flow continuity equation between the gas present in the tank and the tank wall; a heat flow continuity equation between the tank wall and the ambient air; and a flow rate equation connecting a mass flow rate of the filling device to a pressure difference between the filling device and the tank.

28. The method as claimed in claim 27, wherein the first or second computed initial temperature and the initial pressure of the gas present in the tank are obtained by solving said system of equations.

29. The method as claimed in claim 17, wherein the ambient temperature of the filling device and the initial pressure of the gas present in the tank are determined by computation and/or are measured by sensors on the filling device.

30. The method as claimed in claim 17, wherein the electronic control component is configured to control the flow rate and/or pressure control valve in order to generate a predetermined pressure curve or ramp during filling.

31. The method as claimed in claim 23, wherein the electronic control component is configured to simulate and estimate the temperature variation curve and/or the density variation curve of the gas present in the tank in a dynamic manner when filling the tank and/or in an anticipated manner before filling.

32. A device for filling a pressurized-gas tank, the device comprising: a gas source, a filling pipe connecting the source to the tank, a flow rate and/or pressure control valve in the filling pipe, a set of one or more sensors configured to measure the pressure in the tank and/or the ambient temperature on the filling device, and an electronic control component configured to perform the steps of: stop filling when an estimated temperature or density of the gas present in the tank reaches a temperature limit value or density limit value; and estimate, before filling the tank, an initial temperature of the gas present in the tank, wherein the initial temperature of the gas present in the tank is a value that is estimated as a function of the ambient temperature and as a function of the pressure of the gas present in the tank before filling, with the initial temperature of the gas present in the tank being greater than or equal to or less than or equal to the ambient temperature.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0033] These and other features, aspects, and advantages of the present invention will become better understood with regard to the following description, claims, and accompanying drawings. It is to be noted, however, that the drawings illustrate only several embodiments of the invention and are therefore not to be considered limiting of the invention's scope as it can admit to other equally effective embodiments.

[0034] Further particular features and advantages will become apparent upon reading the following description, which is provided with reference to the figures, in which:

[0035] FIG. 1 schematically and partially illustrates a filling device according to the invention;

[0036] FIG. 2 schematically illustrates steps of a filling method by means of the filling device;

[0037] FIG. 3 illustrates a reference temperature curve providing, for a given ambient temperature, a change in the temperature limit of the gas as a function of its pressure;

[0038] FIG. 4 illustrates a change in relation to the ambient temperature of a lower limit (respectively, of an upper limit) associated with a high temperature range (respectively, a low temperature range) in which a reference filling occurs;

[0039] FIG. 5 illustrates curves providing, for a simulated reference filling at a given ambient temperature, the variation in the temperature of the tank that is considered to be recently filled (respectively, that is considered to have been recently drawn off), the variation in the temperature, the density and the pressure of the gas present in said tank, the variation in the temperature and the pressure of the gas at the outlet of the filling device;

[0040] FIG. 6 illustrates a surface curve corresponding to several reference temperature curves established for various ambient temperatures;

[0041] FIG. 7 illustrates the reference temperature curve of FIG. 1 with a reduced temperature variable between 0 (zero) and 1 (one);

[0042] FIG. 8 illustrates reference temperature curves and predictive curves of the first initial temperature of the tank or of the gas present in said tank;

[0043] FIG. 9 illustrates curves representing a change in the temperature of the tank and in the temperature of the gas present in said tank, with said change being linked to a thermal diffusion after a reference filling;

[0044] FIG. 10 illustrates the curves of FIG. 8 after reducing the temperature variable;

[0045] FIG. 11 illustrates predictive curves of the second initial temperature of the tank or of the gas present in said tank;

[0046] FIG. 12 illustrates the curves of FIG. 11 after reducing the temperature variable and reducing the pressure variable between zero (0) and one (1);

[0047] FIG. 13 illustrates an embodiment of a temperature variation curve of the gas in order to control filling of the tank and to prevent it from overheating;

[0048] FIG. 14 illustrates an embodiment of a temperature variation curve of the gas in combination with the reference temperature curve in order to control filling of the tank and to prevent it from overheating;

[0049] FIG. 15 shows an example of radial discretization of a tank wall;

[0050] FIG. 16 shows details of a mesh obtained after the discretization of FIG. 15.

DETAILED DESCRIPTION OF THE INVENTION

[0051] The device 10 for filling pressurized-gas tanks 1 is, for example, a station for filling pressurized-hydrogen tanks.

[0052] The device 10 comprises a gas source 2, a filling pipe 3 connecting the source 2 to the tank 1, a flow rate and/or pressure control valve 4 in the filling pipe 3, a set of one or more sensors 6a, 6bconfigured to measure the pressure in the tank 1 and/or the ambient temperature Tamb on the filling device 10.

[0053] Furthermore, the device 10 comprises an electronic control component 5 configured to control filling and notably to stop filling when an estimated temperature (respectively, a density) of the gas present in the tank 1 reaches a temperature limit value (respectively, a density limit value). The control component 5 is also configured to estimate an initial temperature of the gas present in the tank 1 before filling the tank 1.

[0054] The electronic control component 5 comprises, for example, a microprocessor, a computer or any suitable electronic controller.

[0055] According to the invention, the initial temperature Tini of the gas present in the tank 1 is a value that is estimated as a function of the ambient temperature Tamb and as a function of the pressure Pini of the gas present in the tank 1 before filling, with said initial temperature Tini being greater than or equal to or less than or equal to the ambient temperature Tamb.

[0056] Thus, for a given ambient temperature Tamb, the initial temperature Tini of the gas present in the tank is no longer a fixed quantity as provided by the prior art, but rather a quantity that depends on the initial pressure Pini of the gas present in the tank 1.

[0057] Advantageously, the initial temperature Tini of the gas present in the tank 1 is divided into a first computed initial temperature Tini_1 corresponding to a state of the tank 1 that is considered to be recently filled to a first initial density ini_1, and a second computed initial temperature Tini_2 corresponding to a state of the tank 1 that is considered to be recently drawn off to a second initial density ini_2.

[0058] The ambient temperature Tamb and the initial pressure Pini are respectively determined during a step S1 and a step S2 of a filling method implementing the filling device and schematically illustrated in FIG. 2. Steps S1 and S2 may or may not be carried out simultaneously. As for the initial temperature Tini, which is divided into a first initial temperature Tini_1 and a second initial temperature Tini_2, it is determined during a step S3 that is also illustrated in FIG. 2.

[0059] Hereafter, the tank 1 that is considered to have been recently filled will also be likened to a tank recently exposed to direct sunlight or in a relatively hot environment or even to a tank being hot-filled. In order to refill such a tank, the fluid that is already present is considered to have been compressed and therefore to have increased in temperature well beyond the ambient temperature.

[0060] The tank that is considered to have been recently drawn off will also be likened to a tank recently exposed to a relatively cold environment or to a tank being cold-filled. In order to fill such a tank, the residual fluid is considered to have previously experienced significant expansion and therefore significant cooling.

[0061] The tank that is considered to have been recently filled and the tank that is considered to have been recently drawn off are modeled virtual tanks that may or may not be identical. In the second case, the tank that is considered to have been recently drawn off notably can have a surface area: volume ratio that is greater than that of the tank that is considered to have been recently filled, which makes it more able to discharge the heat and to cool.

[0062] Advantageously, the first initial temperature Tini_1 of the gas present in the tank is within a high temperature range with a determined lower limit TLgas_min_1 that is greater than or equal to the ambient temperature and an upper limit TLgas_max_1 corresponding to a determined maximum temperature limit. The change in the lower limit TLgas_min_1 as a function of the ambient temperature of the filling device is provided by a curve Thotsoak (see FIG. 4) derived, for example, from recommendations of the SAE relating to filling a hot tank. The upper limit TLgasmax_1 is set, for example, to 85 C. This value takes into account the thermophysical properties of the tank.

[0063] Advantageously, the second initial temperature Tini_2 of the tank is within a low temperature range having a determined upper limit TLgas_max_2 that is less than or equal to the ambient temperature Tamb and a lower limit TLgas_min_2 corresponding to a determined minimum temperature limit, for example, ranging between zero and5 C. In particular, the upper limit TLgas_max_2 is a function of the ambient temperature and follows a curve Tcoldsoak (see FIG. 4) derived, for example, from recommendations of the SAE relating to filling a cold tank.

[0064] Advantageously, the temperature limit value (respectively, the density limit value) is a determined fixed value or a value provided by a reference temperature curve Tpgasmax (respectively, a reference density curve). In particular, the reference temperature curve Tpgasmax (respectively, the reference density curve) allows, for a given pressure of the gas in the tank, the temperature limit value (respectively, the density limit value) to be estimated.

[0065] An example of a reference temperature curve Tpgas_max is illustrated in FIG. 3. For a maximum pressure of the gas present in the tank 1 the temperature limit value is 85 C. and coincides with the upper limit TLgasmax_1 of the high temperature range. For a minimum pressure of the gas in the tank, the temperature limit value is approximately 20 C. and coincides with the lower limit TLgasmin_1 of the high temperature range.

[0066] The density limit value depends on the service pressure of the tank. For example, for a service pressure of 350 barg, the density limit value is set to 24.1 kg/m3, for example. For a service pressure of 700 barg, the limit density value is set to 40.2 kg/m3, for example. In general, the density limit value can be defined by a state of filling SOC (State of Charge) percentage that defines a recommended maximum amount of hydrogen in a tank.

Determining the Reference Temperature Curve Tpgasmax and the Reference Density Curve

[0067] Advantageously, the reference temperature curve Tpgasmax (respectively, the reference density curve) is provided by a first physical model that simulates a reference filling of the tank over the high temperature range (respectively, over the low temperature range) with pre-cooling at the source. The reference filling is simulated from a minimum pressure, which in this case is set to 5 barg, up to a maximum pressure corresponding to the service pressure of the tank.

[0068] In particular, the reference temperature curve Tpgasmax (respectively, the reference density curve) is obtained from a curve Pgashot providing the change in the pressure of the gas present in the tank as a function of time and from a curve Tgasmax (respectively, a curve RHO_cold) providing the change in the temperature as a function of time (respectively, the change in the density) of the gas contained in the tank. The curves Pgashot, Tgasmax (respectively RHO_cold) are also provided by the first predetermined physical model. They are illustrated in FIG. 5.

[0069] For various ambient temperatures, the first predetermined physical model provides as many reference temperature curves Tpgasmax (respectively, as many reference density curves) that form a reference temperature surface curve (respectively, a reference density surface curve). FIG. 6 illustrates an example of a reference temperature surface curve.

[0070] In order to facilitate the use of the reference temperature curve Tpgasmax (respectively, the reference density curve), the temperature variable is reduced between the value of zero (0) corresponding to the lower limit TLgasmin_1 and the value of one (1) corresponding to the upper limit TLgasmax_1. FIG. 7 illustrates a curve Reduc_TPgasmax obtained after reducing the temperature variable on the reference temperature curve TPgasmax.

Determining the First Initial Temperature Tini_1 of the Gas

[0071] The first initial temperature Tini_1 of the gas present in the tank is provided by a second predetermined predictive curve TPgashot as a function of the initial pressure Pini of the gas (see FIG. 8). In other words, for a given initial pressure Pini of the gas present in the tank, the predictive curve Tpgashot allows the first initial temperature Tini_1 to be determined.

[0072] The predictive curve Tpgashot is drawn from a second physical model that simulates a reference filling of the tank over the high temperature range, optionally with pre-cooling of the gas supplied by the source. Preferably, the second physical model also simulates a thermal diffusion on the tank and/or on the gas present in the tank. Thus, the second physical model can differ from the first physical model by taking into account the thermal diffusion.

[0073] The thermal diffusion occurs after the reference filling of the tank and the pre-cooling of the source. Advantageously, the thermal diffusion is simulated over a period of 2 minutes, corresponding to the minimum time that is required between two successive fillings of the tank on the filling station. Still advantageously, during the thermal diffusion period, the pressure of the gas in the tank is assumed to be constant.

[0074] For a given ambient temperature, the thermal diffusion induces a temperature drop for the gas present in the tank. Such a drop is shown by a curve Stgasmax illustrated in FIG. 9.

[0075] It should be noted that the curve Tpgashot is obtained from the curve Pgashot and from the curve Tgashot that are both established for the same ambient temperature. In particular, the curve Tgashot results from a subtraction performed between the curve Tgasmax and the curve STgasmax.

[0076] Advantageously, in order to facilitate the use of the curve TPgashot, the temperature variable can be reduced between the value of zero (0) corresponding to the lower limit TLgasmin_1 and the value of one (1) corresponding to the upper limit TLgasmax_1. The curve Reduc_TPgashot thus obtained is illustrated in FIG. 10.

Determining the Second Initial Temperature Tini_2 of the Gas

[0077] The second initial temperature Tini_2 of the gas is obtained from a third curve TPgascold as a function of the initial pressure Pini of the gas (see FIG. 11). In other words, based on the initial pressure Pini of the gas present in the tank, the curve Tpgascold allows the second initial temperature Tini_2 of the gas present in the tank to be determined.

[0078] The curve TPgascold can be provided by a third predetermined physical model (described below) that simulates a reference draining of the tank over the low temperature range, i.e., between the upper limit TLgasmax_2 and the lower limit TLgasmin_2.

[0079] The upper limit TLgasmax_2 corresponds to the start of the reference draining where the tank is in an at least partly filled state, and preferably is 100% filled. At this upper limit TLgasmax_2 there is a corresponding maximum pressure Pgasmax_2, called service pressure (Nominal Working Pressure), and a maximum density gasmax_2. It should be noted that the maximum pressure Pgasmax_2 can be written as a function of the upper limit TLgasmax_2 through the following mathematical expression taken from literature:

[00001]$\mathrm{Pgasmax\_}2\left[\mathrm{barg}\right]=A*\mathrm{TLgasmax\_}2\left[C.\right]+B$

with A ranging between 1 and 2, and B ranging between 300 and 400.

[0080] The lower limit TLgasmin_2 corresponds to the end of reference draining where the tank has a minimum pressure Pgasmin_2, in this case set to 5 barg, and a minimum density gasmin_2, called second initial density ini_2. Furthermore, the lower limit TLgasmin_2 can be obtained from the following mathematical expression taken from the third physical model:

[00002]$\mathrm{Tgasmin\_}2=A*\mathrm{Tamb}+B$

where

[0081] Tamb is the ambient temperature on the filling device;

[0082] A ranges between 0.5 and 1.5;

[0083] B ranges between 30 and 15.

[0084] For various ambient temperatures, the third physical model provides as many curves Tpgascold that form a surface curve (not illustrated). In order to facilitate the use of these curves TPgascold, the temperature variable on each of them is reduced between the value of zero (0) corresponding to the lower limit TLgasmin_2 and the value of one (1) corresponding to the upper limit TLgasmax_2. Similarly, the pressure variable can be reduced between the value of zero (0) corresponding to the minimum pressure Pgasmin_2 and the value of one (1) corresponding to the maximum pressure Pgasmax_2 before draining the tank. The curve Reduc_TPgascold illustrated in FIG. 12 is the result of such a reduction performed on the curve TPgascold.

[0085] Ultimately, for an initial pressure Pini measured in the tank and an ambient temperature Tamb taken from the filling device, at least one predetermined physical model considers that this initial pressure Pini is the final pressure of a previous filling (hot case) or of a previous draw-off (cold case). These two hypotheses respectively define a value for the first initial temperature Tini_1 (corresponding to the hot case) and a value for the second initial temperature Tini_2 (corresponding to the cold case). The value of the first initial temperature Tini_1 and the value of the second initial temperature Tini_2 thus defined can differ from the initial temperature value determined according to the prior art, for example, deviating by 10 to 20 C. from the ambient temperature.

Estimating the Temperature Variation Curve of the Gas Between a First Temperature Variation Curve of the Gas (Hot Case) and a Second Temperature Variation Curve of the Gas (Cold Case)

[0086] Advantageously, the control component 5 can be configured to estimate, during filling, a temperature variation curve of the gas present in the tank as a function of the initial temperature of the gas present in the tank, i.e., with the initial temperature of the gas present in the tank as the starting point (or starting condition).

[0087] More specifically, the control component 5 can be configured to estimate such a curve between a first temperature variation curve of the gas (hot case) and a second temperature variation curve of the gas (cold case). Thus, for a given pressure of the gas present in the tank, the corresponding temperature is estimated between a first temperature located on the first temperature variation curve of the gas (hot case) and a second temperature located on the second temperature variation curve of the gas (cold case).

[0088] The first temperature variation curve (respectively, the second curve) of the gas has the first initial temperature Tini_1 (respectively, the second initial temperature Tini_2) of the gas as the starting point (or starting condition). Furthermore, the first temperature variation curve (respectively, the second curve) of the gas is supplied by a predetermined physical model that simulates reference filling of the tank from the first initial temperature Tini_1 (respectively, from the second initial temperature Tini_2) of the gas, with said filling being accompanied by pre-cooling at the source and followed by a thermal diffusion.

[0089] It should be noted that the physical model that is the source of the predictive curve Tpgashot of the first initial temperature Tini_1 of the gas is preferably the same as that which is the source of the first temperature variation curve of the gas. The two curves are distinguished by their starting points, which are respectively the lower limit TLgasmin_1 (for the predictive curve TPgashot of the first initial temperature Tini_1) and the first initial temperature Tini_1 (for the first temperature variation curve of the gas).

Estimating the Density of the Gas Present in the Tank Between a First Density Variation Curve (Hot Case) and a Second Density Variation Curve (Cold Case)

[0090] Advantageously, the control component 5 can be configured to estimate (for example, compute), during filling, a density variation curve of the gas present in the tank as a function of the initial density of the gas present in the tank. This means that the density variation curve has the initial density of the gas present in the tank as the starting point. It should be noted that the initial density can be computed by the ideal or real gas equation while knowing the volume of the tank (known or estimated, for example, via the known technique based on a pressure pulse before filling), the pressure and the temperature of the gas.

[0091] More specifically, the control component 5 can be configured to estimate the density variation curve of the gas between a first density variation curve of the gas (hot case) and a second density variation curve of the gas (cold case). Thus, for a given pressure of the gas present in the tank, the corresponding density is estimated between a first density located on the first density variation curve (hot case) and a second density located on the second density variation curve (cold case).

[0092] The first density variation curve (respectively, the second curve) has the first initial density (respectively, the second initial density) of the gas present in the tank as the starting point.

[0093] Furthermore, the first density variation curve (respectively, the second curve) of the gas is provided by a physical model that simulates a reference filling of the tank from the first initial density (respectively, from the second initial density) of the gas with pre-cooling at the source.

Determining the Reference Temperature Curve of the Tank and the Temperature Variation Curves of the Tank

[0094] Advantageously, the condition for stopping filling of the tank relating to a limit value of the temperature of the gas present in the tank can be replaced by a stop condition relating to a limit value of the temperature of the tank.

[0095] The limit value of the temperature of the tank can be provided by a reference temperature curve of the tank, with said curve being provided by the first physical model. In particular, for a tank formed by a plurality of layers, for example, an inner layer, called liner layer, in contact with the gas and an outer layer, called composite layer, in contact with the ambient air, the first physical model provides three reference temperature curves, illustrated in FIG. 8, namely, a first curve TPgasliner_max for the inner wall, a second curve TPlinercompo_max for the inner layer/outer layer interface, and a third curve TPexternal_max for the outer wall.

[0096] The curves TPgasliner_max, TPexternal_max, TPlinercompo_max can be obtained from the curve Pgashot, respectively in combination with the curves Tgasliner_hot, Texternal_hot, Tlinercompo_hot providing the change in the temperature as a function of time for the inner wall, the inner layer/outer layer interface, and the outer wall. The curves Tgasliner_hot, Texternal_hot are illustrated in FIG. 5.

[0097] Furthermore, in order to be easily used, the curves TPgasliner_max, TPexternal_max, TPlinercompo_max can be respectively converted into a curve Reduc_TPgasliner_max, a curve Reduc_TPexternal_max, and a curve Reduc_TPlinercompo_max, each defined between a value of zero (0) corresponding to the lower temperature limit TLgasmin_1 and a value of one (1) corresponding to the upper temperature limit TLgasmax_1.

Estimating the Initial Temperature of the Tank Between a First Initial Temperature (Hot Case) and a Second Initial Temperature (Cold Case) of the Tank

[0098] The control component 5 can be configured to estimate the initial temperature of the tank between a first initial temperature of the tank corresponding to a state of the tank that is considered to be recently filled to the first initial density (hot case), and a second initial temperature of the tank corresponding to a state of the tank that is considered to be recently drawn off at a second initial density (cold case).

[0099] The first initial temperature Tini_1 of the tank can be provided by a predictive curve TPgasliner_hot, TPlinercompo_hot, Tpexternal_hot (see FIG. 8) as a function of the initial pressure Pini of the gas. In other words, for a given initial pressure Pini of the gas present in the tank, the predictive curve TPgasliner_hot, TPlinercompo_hot, TPexternal_hot allows the first initial temperature Tini_1 of the tank to be determined.

[0100] The curves TPgasliner_hot, TPexternal_hot, Tplinercompo_hot can be provided by the second physical model. More specifically, these curves are obtained from the curve Pgashot, respectively in combination with the curves Tgasliner_hot_bis, Texternal_hot_bis, Tlinercompo_hot_bis (not illustrated) providing the change in the temperature as a function of time, respectively for the inner wall, the outer wall and the inner layer/outer layer interface of the tank.

[0101] The curve Tgasliner_hot_bis is obtained from the curve Tgasliner_hot modulated by a curve STgasliner. The curve Texternal_hot_bis is obtained from the curve Texternal_hot modulated by a curve STexternal. The curve Tlinercompo_hot_bis is obtained from the curve Tlinercompo_hot modulated by a curve STlinercompo. The curves STgasliner, STexternal, STlinercompo are illustrated in FIG. 9. They represent a thermal diffusion after the reference filling and that occurs in the vicinity of the inner wall, the outer wall and the inner layer/outer layer interface of the tank.

[0102] In the example illustrated in FIG. 9, the thermal diffusion corresponds to a temperature drop over time for the inner wall of the tank. However, for the outer wall and the inner layer/outer layer interface of the tank, the thermal diffusion corresponds to an increase in temperature.

[0103] It should be noted that the curves TPgasliner_hot, TPexternal_hot, TPlinercompo_hot can be respectively reduced into a curve Reduc_TPgasliner_hot, a curve Reduc_TPexternal_hot, and a curve Reduc_TPlinercompo_hot, each defined between a value of zero (0) corresponding to the lower temperature limit Tlgas_min_1 and a value of one (1) corresponding to the upper temperature limit Tlgas_max_1 (see FIG. 10).

[0104] The second initial temperature Tini_2 of the tank is provided by a predictive curve TPgasliner_cold, TPexternal_cold, TPlinercompo_cold as a function of the initial pressure Pini of the gas (see FIG. 11). In other words, for a given initial pressure Pini of the gas, the predictive curve TPgasliner_cold, TPexternal_cold, TPlinercompo_cold allows the second initial temperature Tini_2 of the tank to be determined.

[0105] It should be noted that the predictive curve TPgasliner_cold, TPexternal_cold, TPlinercompo_cold can be provided by the third physical model that simulates a reference draining of the tank. On completion of this reference draining, the temperature of the tank reaches a minimum value that varies across the thickness of the tank. For a tank made up of two layers, the minimum temperature Tmin in the vicinity of the inner wall, the outer wall or the inner layer/outer layer interface can be provided, for example, by a linear law of the Tmin=A*Tamb+B type,

where

[0106] Tamb is the ambient temperature;

[0107] A ranges between 0.5 and 1.5;

[0108] B ranges between 20 and 10.

[0109] The predictive curves TPgasliner_cold, TPexternal_cold, TPlinercompo_cold can be obtained from the curve Pgascold (providing the change in the pressure of the gas as a function of time), respectively in combination with the curves Tgasliner_cold, Texternal_cold, Tlinercompo_cold providing the change in the temperature of the tank as a function of time (cold case). The curves Tgasliner_cold, Texternal_cold and Pgas_cold are illustrated in FIG. 5.

Estimating the Temperature Variation Curve of the Tank Between a First Temperature Variation Curve of the Tank (Hot Case) and a Second Temperature Variation Curve of the Tank (Cold Case)

[0110] The control component 5 can be configured to estimate, during filling, a temperature variation curve of the tank (see step S5 of the method illustrated in FIG. 2). This estimate is made between a first temperature variation curve of the tank (hot case) and a second temperature variation curve of the tank (cold case). Said first curve and said second curve are determined during a step S4 of the method illustrated in FIG. 5.

[0111] The first variation curve has the first initial temperature Tini_1 of the tank as the starting point. The second variation curve has the second initial temperature Tini_2 of the tank as the starting point.

[0112] The first temperature variation curve (respectively, the second curve) of the tank can be obtained from a model that simulates a reference filling of the tank from the first initial temperature Tini_1 (respectively, from the second initial temperature Tini_2) of the tank, with pre-cooling of the gas at the source, and thermal diffusion after the reference filling.

Effect of the Thermal Diffusion on the Predictive Curves of the First Initial Temperature Tini_1 of the Tank or of the Gas Present in the Tank

[0113] An analysis of FIG. 8, providing the reference temperature curves of the tank, or of the gas present in the tank, (i.e., the curves TPgasliner_max, TPlinercompo_max, TPexternal_max, TPgas_max), and the predictive curves of the first initial temperature Tini_1 of the tank, or of the gas in the tank, (i.e., the curves TPgasliner_hot, TPlinercompo_hot, TPexternal_hot, TPgas_hot), shows that the gas and the inner wall of the tank experience a reduction in their temperatures following the thermal diffusion, while the outer wall and the inner layer/outer layer interface of the tank experience an increase in their temperatures following the thermal diffusion.

[0114] Taking into account this singular change in the temperature of the tank following the thermal diffusion allows the accuracy of the reference temperature curve of the tank and/or the reference temperature curve of the gas present in the tank to be improved. Thus, the invention allows safety to be improved during filling by limiting the risk of exceeding the temperature limit value.

Embodiments of the Reference Temperature Curve Tgas_max and of the First Temperature Variation Curve Tgas_hot_bis of the Gas

[0115] When filling the tank, in order to prevent any overheating, the first temperature variation curve Tgas_hot_bis of the gas present in the tank can be used alone (see the example of FIG. 13) or in combination with the reference temperature curve Tgasmax of the gas present in the tank (see the example of FIG. 14).

[0116] As shown in each of these examples, after connecting the vehicle to the end of the filling pipe (step 1) and initializing the filling by means of the filling device (step 2), initial overheating is detected in the gas present in the tank (step 3). In the example of FIG. 13, the overheating is detected when the curve Tgas_hot_bis reaches the temperature limit value that in this case is represented by a horizontal line at 85 C. In the example of FIG. 14, the initial overheating is detected below the temperature limit value (85 C.). This overheating corresponds to the moment when the curve Tgas_hot_bis catches the curve Tgas_max.

[0117] Following this initial overheating, the control component stops the filling while the tank is not filled to its maximum. The stop is extended (step 4) for a certain period of time that is required before a new filling cycle. Steps 5, 6, 7 correspond to this new filling cycle, for which the initial temperature of the gas present in the tank is determined as a function of the temperature reached by the same gas at the end of the preceding cycle, but also as a function of the pressure of this same gas at the beginning of the new cycle.

[0118] By taking into account the temperature reached during a previous filling cycle, then during a subsequent filling cycle the invention limits the risk of overheating, and opens the possibility of reaching a maximum load in the tank.

[0119] Of course, the curves Tgas_max and Tgas_hot_bis of the above examples can be replaced by the curves TPgasmax and TPgas_hot in order to control filling of the tank. When the stop condition relates to a limit value of the temperature of the tank, the filling can be controlled using one of the curves (hot case) providing the variation in the temperature of the tank (for example, the curve TPgasliner_hot). This variation curve can be used alone or in combination with the associated reference temperature curve, (for example, the curve TPgasliner_max).

Presentation of the Physical Models

[0120] The aforementioned physical models that are the source of the various predictive curves presented in this description are based on a system of energy balance equations applied to the tank and to the gas present in the tank. These equations are presented below. [0121] a. The internal energy balance equation of the gas in the tanks is written as follows:

[00003]$\begin{array}{cc}\frac{d\left(m_g{u}_g\right)}{\mathrm{dt}}=k_i{S}_i\left(T_{\mathrm{wi}}-T_g\right)+{\mathrm{Qh}}_{g\mathrm{inlet}}& \left(1\right)\end{array}$

[0122] This balance uses the mass of the gas in the tanks m.sub.g, the mass internal energy of the gas in this tank u.sub.g, the heat exchange between the gas in the tanks and the inner wall thereof with the exchange coefficient k.sub.i and the inner surface of these tanks S.sub.i, the total hydrogen

[00004]$Q=\frac{{\mathrm{dm}}_g}{\mathrm{dt}},$

as well as the incoming mass enthalpy of the gas h.sub.g inlet in the tanks. T.sub.g and T.sub.wi are respectively the spatial average temperatures of the gas in the volume of these tanks and of the inner wall thereof.

[0123] One of the two safety criteria not to be exceeded involves the density in the model. As the PLC can know the total volume of the tanks of the vehicle V.sub.CHSS (Compressed Hydrogen Storage System), equation (1) can be written using m.sub.g=V.sub.CHSS.sub.g:

[00005]$\begin{array}{cc}{V}_{\mathrm{CHSS}}\frac{d\left({}_g{u}_g\right)}{\mathrm{dt}}=k_i{S}_i\left(T_{\mathrm{wi}}-T_g\right)+Q{h}_{g\mathrm{inlet}}& \left(2\right)\end{array}$

[0124] In order to solve this equation (2), it needs to be discretized by replacing the time derivatives with finite differences. The equation is thus obtained, by dividing by V.sub.CHSS:

[00006]$\begin{array}{cc}\frac{{}_g\left(t+t\right)u_g\left(t+t\right)-{}_g\left(t\right)u_g\left(t\right)}{t}=k_i\frac{S_i}{V_{\mathrm{CHSS}}}\left(T_{\mathrm{wi}}\left(t+t\right)-T_g\left(t+t\right)\right)+\frac{Q}{V_{\mathrm{CHSS}}}{h}_{g\mathrm{inlet}}& \left(3\right)\end{array}$

where t is the value of a time step (approximately one second).

[0125] The material balance equation in the tanks is used to write:

[00007]$\begin{array}{cc}{V}_{\mathrm{CHSS}}{}_g\left(t+t\right)=V_{\mathrm{CHSS}}{}_g\left(t\right)+Q& \left(4\right)\end{array}$

that is, by dividing by V.sub.CHSS:

[00008]$\begin{array}{cc}{}_g\left(t+t\right)={}_g\left(t\right)+\frac{Q}{V_{\mathrm{CHSS}}}& \left(5\right)\end{array}$

[0126] Equation (5) allows the value of .sub.g(t+t) to be obtained. Equation (3) therefore allows the value of the internal energy u.sub.g(t+t) to be obtained:

[00009]$\begin{array}{cc}{u}_g\left(t+t\right)=\frac{k_i\frac{S_i}{V_{\mathrm{CHSS}}}\left(T_{\mathrm{wi}}\left(t+t\right)-T_g\left(t\right)\right)t+\frac{Q}{V_{\mathrm{CHSS}}}{h}_{g\mathrm{inlet}}t+{}_g\left(t\right)u_g\left(t\right)}{{}_g\left(t+t\right)}& \left(6\right)\end{array}$ [0127] b. The energy conservation equation in the wall is written as follows:

[00010]$\begin{array}{cc}{c}_p\frac{T}{t}=\frac{}{r}\frac{}{r}\left(\frac{T}{r}\right)& \left(7\right)\end{array}$

[0128] This balance involves the density of the wall , its heat capacity v and its thermal conductivity , as well as the radius r from the center of the tank.

[0129] At the interface between the gas and the liner, the flow continuity equation is written as follows:

[00011]$\begin{array}{cc}{k}_i{S}_i\left(T_g-T_{\mathrm{wi}}\right)=-{}_{S_i}{}_{\mathrm{liner}}\frac{T}{r}\mathrm{dS}& \left(8\right)\end{array}$

[0130] At the interface between the composite and the ambient environment, the flow continuity equation is written as follows:

[00012]$\begin{array}{cc}-{}_{S_e}{}_{\mathrm{comp}}\frac{T}{r}\mathrm{dS}=k_e{S}_e\left(T_{\mathrm{we}}-T_{\mathrm{amb}}\right)& \left(9\right)\end{array}$

[0131] where k.sub.e is the heat exchange coefficient between the ambient environment and the outer wall of the tank, S.sub.e is the outer surface of these tanks and T.sub.amb is the ambient temperature.

[0132] Finally, at the interface between the two liner and composite materials inside the wall, the flow continuity equation is written as follows:

[00013]$\begin{array}{cc}-{{}_{\mathrm{liner}}\left(\frac{T}{r}\right)}_{\mathrm{liner}}=-{{}_{\mathrm{comp}}\left(\frac{T}{r}\right)}_{\mathrm{comp}}s& \left(10\right)\end{array}$ [0133] c. Discretization of the equations in the wall.

[0134] In order to solve equations (7), (8), (9) and (10), the wall needs to be radially discretized into various meshes. The number of meshes in the liner n_liner and in the composite n_comp can vary. The wall temperatures are computed on each node, and then n_mesh=n_liner+n_comp+1 nodes. Therefore, the size of the meshes is (r).sub.liner=e.sub.liner/n.sub.liner in the liner and (r).sub.comp=e.sub.comp/n.sub.comp in the composite, as shown in FIG. 13. The thickness of the meshes at the interfaces is (r)/2. In the illustrated example, 3 meshes are provided in the liner and 3 other meshes are provided in the composite.

[0135] Equation (7) is discretized and applied for each of the various meshes forming the wall so that it can be solved. FIG. 14 shows the radial discretization used to solve the equation. In this figure, a mesh, centered on the point P, is surrounded by a point W to the west and by a point E to the east, with the inside of the tank being located to the west. A fictitious face w is defined between the points W and P, and a second fictitious face e is defined between the points P and E. The distances between points W and P, on the one hand, and P and E, on the other hand, are respectively denoted (r).sup.w and (r).sup.e. The faces w and e are separated by r.sub.p. Points W, P and E are respectively located at r.sub.W, r.sub.P and r.sub.E from the center of the tank.

[0136] Equation (7) is integrated over a time step and is spatially integrated from west to east, in order to obtain:

[00014]$\begin{array}{cc}A={}_w^e{}_t^{t+t}\left(c_p\frac{T}{t}\right)r\mathrm{dr}\mathrm{dt}={}_t^{t+t}{}_w^e\left[\frac{}{r}\left(r\frac{T}{r}\right)\right]\mathrm{dr}\mathrm{dt}=B& \left(11\right)\end{array}$

[0137] The left term is called A, while the right term is called B.

[0138] Assuming that the density and the thermal capacity c.sub.p hardly change between the instants t and t+t, the left term is written as follows:

[00015]$\begin{array}{cc}A={}_w^e{}_t^{t+t}\left(c_p\frac{T}{t}\right)r\mathrm{dr}\mathrm{dt}{}_w^e{c}_p\left({}_t^{t+t}\frac{T}{t}\mathrm{dt}\right)r\mathrm{dr}={}_w^e{c}_p\left[T\left(r,t+t\right)-T\left(r,t\right)\right]r\mathrm{dr}& \left(12\right)\end{array}$

[0139] It is assumed that the temperature difference T(r, t+t)T(r,t) remains constant between points w and e and is equal to T(r.sub.p, t+t)T(r.sub.p,t). The term A is written as follows:

[00016]$\begin{array}{cc}A{c}_p\frac{T\left(r_P,t+t\right)-T\left(r_P,t\right)}{t}{}_w^e\mathrm{rdr}& \left(13\right)\end{array}$

[0140] By writing:

[00017]${}_w^e\mathrm{rdr}=\left(\frac{r_E^2-r_W^2}{2}\right)=\left(r_E-r_W\right)\left(\frac{r_E+r_W}{2}\right)=r_p\left(\frac{r_E+r_W}{2}\right),$

the term A is ultimately written as follows:

[00018]$\begin{array}{cc}A{c}_p\frac{T\left(r_P,t+t\right)-T\left(r_P,t\right)}{t}{r}_P\left(\frac{r_E+r_W}{2}\right)& \left(14\right)\end{array}$

[0141] The term B, namely, the right term of equation (11), can be written as follows:

[00019]$\begin{array}{cc}B={}_t^{t+t}\left[\left(r_E,t\right)r_E\frac{T}{r}\left(r_E,t\right)-\left(r_W,t\right)r_W\frac{T}{r}\left(r_W,t\right)\right]\mathrm{dt}& \left(15\right)\end{array}$

[0142] By assuming that

[00020]$\frac{T}{r}\left(r_E,t\right)=\frac{T\left(r_E,t\right)-T\left(r_P,t\right)}{{\left(r\right)}_e}\mathrm{and}\frac{T}{r}\left(r_W,t\right)=\frac{T\left(r_P,t\right)-T\left(r_W,t\right)}{{\left(r\right)}_w},$

and by assuming that the thermal conductivity hardly changes between the instants t and t+t, the term B is written as follows:

[00021]$\begin{array}{cc}B=\left(r_E,t\right)r_E{}_t^{t+t}\frac{T\left(r_E,t\right)-T\left(r_P,t\right)}{{\left(r\right)}_e}\mathrm{dt}-\left(r_W,t\right)r_W{}_t^{t+t}\frac{T\left(r_P,t\right)-T\left(r_W,t\right)}{{\left(r\right)}_w}\mathrm{dt}& \left(16\right)\end{array}$

[0143] The values taken at the instant t are noted with an exponent .sup.0 , and the values taken at the instant t +t are noted with an exponent .sup.1. The values taken at points W, P and E, respectively, are noted with an index .sub.W, P and .sub.E. For example, the value T(r.sub.p, t+t) is noted

[00022]$T_p^1.$

[0144] An implicit scheme is used, and therefore equation (11) becomes:

[00023]$\begin{array}{cc}{}^0{c}_p^0\frac{T_P^1-T_P^0}{t}{r}_P\left(\frac{r_E+r_W}{2}\right)={}_E^0{r}_E\frac{T_E^1-T_P^1}{{\left(r\right)}_e}-{}_W^0{r}_W\frac{T_P^1-T_W^1}{{\left(r\right)}_w}& \left(17\right)\end{array}$

[0145] At the interface between the gas and the liner, the flow continuity equation (8) is discretized as follows:

[00024]$\begin{array}{cc}{k}_i^0{S}_i\left(T_g^0-T_1^1\right)=-{}_{liner}^0{S}_i\frac{T_2^1-T_1^1}{{\left(r\right)}_{liner}}& \left(18\right)\end{array}$

[0146] At the interface between the composite and the ambient environment, the flow continuity equation (9) is discretized as follows:

[00025]$\begin{array}{cc}-{}_{comp}^0{S}_e\frac{T_{n_{mesh}}^1-T_{n_{mes{h}^{-1}}}^1}{{\left(r\right)}_{comp}}=k_e^0{S}_e\left(T_{n_{mesh}}^1-T_{amb}^1\right)& \left(19\right)\end{array}$

[0147] Finally, at the interface between the two liner and composite materials inside the wall, the flow continuity equation (10) is discretized as follows:

[00026]$\begin{array}{cc}-{}_{liner}^0\frac{T_{n_{line{r}^{+1}}}^1-T_{n_{liner}}^1}{{\left(r\right)}_{liner}}=-{}_{comp}^0\frac{T_{n_{line{r}^{+2}}}^1-T_{n_{line{r}^{+1}}}^1}{{\left(r\right)}_{comp}}& \left(20\right)\end{array}$

[0148] The system of equations (17) for each point P that is not located at an interface and (18), (19) and (20) can be solved by the Thomas algorithm, also called TDMA (Tri-Diagonal Matrix Algorithm).

Physical Properties and Coefficients

[0149] a. Tg and Pg from and u

[0150] For each time step, the model computes the temperature and the pressure of the gas based on the density and the internal energy u of the gas:

[00027]$\begin{array}{cc}{T}_g=f_1\left(,u\right)& \left(21\right)\end{array}$$\begin{array}{cc}{P}_g=f_2\left(,u\right)& \left(22\right)\end{array}$

[0151] These functions f.sub.1 and f.sub.2 are adjusted in order to best correspond to the thermophysical properties of the gas. [0152] b. Heat transfer coefficients k.sub.i and k.sub.e

[0153] The model also needs values for the heat transfer coefficients k.sub.i and k.sub.e. For the internal transfer coefficient k.sub.i, a function is used that is calibrated based on the correlation presented by Bourgeois, T. et al., (see Bourgeois T., Ammouri F., Weber M., Knapik C., Evaluating the temperature inside a tank during a filling with highly-pressurized gas. International Journal of Hydrogen Energy 2015; 40: 11748-55).

[00028]$\begin{array}{cc}N{u}_{\mathrm{Dint}}=\frac{D_{int}{k}_i}{{}_g}=a.Math.{\mathrm{Ra}}_{\mathrm{Dint}}^b+c.Math.{\mathrm{Re}}_{\mathrm{dinj}}^d& \left(23\right)\end{array}$

[0154] For the external transfer coefficient k.sub.e, a function is used that is calibrated based on the correlation presented by Massard, F. (see heat engineer's checklist. ELSEVIER; 1997):

[00029]$\begin{array}{cc}{k}_e=0.675{}_{air}\frac{R{a}_{Dext}^{0.058}}{\mathrm{Dext}},R{a}_{\mathrm{Dext}}<10^{-2}& \left(24.1\right)\end{array}$$\begin{array}{cc}{k}_e=1.02{}_{air}\frac{R{a}_{Dext}^{0.148}}{\mathrm{Dext}},10^{-2}<R{a}_{\mathrm{Dext}}<10^2& \left(24.2\right)\end{array}$$\begin{array}{cc}{k}_e=0.85{}_{air}\frac{R{a}_{Dext}^{0.188}}{\mathrm{Dext}},10^2<R{a}_{\mathrm{Dext}}<10^4& \left(24.3\right)\end{array}$$\begin{array}{cc}{k}_e=0.48{}_{air}\frac{R{a}_{Dext}^{0.25}}{\mathrm{Dext}},10^4<R{a}_{\mathrm{Dext}}<10^8& \left(24.4\right)\end{array}$$\begin{array}{cc}{k}_e=0.125{}_{air}\frac{R{a}_{Dext}^{0333}}{\mathrm{Dext}},10^8<R{a}_{\mathrm{Dext}}& \left(24.5\right)\end{array}$

[0155] Advantageously, the physical models described above can be associated with a heat transfer model between the filling device and the one or more tanks. This transfer model is described below.

[0156] The term of incoming mass enthalpy of the gas h.sub.g inlet of equation (1) is obtained based on the mass enthalpy of the gas on the dispenser by virtue of a heat transfer model. Its aim is to estimate the heat exchanges of the gas between the dispenser of the filling device and the inlet into the tanks of the vehicle, along the length of the hose and the other elements connecting the dispenser to the tanks.

[0157] The energy conservation equation applied to the gas inside the piping is written in equation (25), for each elementary length dx along the piping. In this case, it is assumed that the piping has a homogeneous temperature over its entire length:

[00030]$\begin{array}{cc}\stackrel{}{m}{c}_{p,g}\frac{d{T}_{\mathrm{gas},\mathrm{pipe}}}{dx}=k_{i,\mathrm{pipe}}\left(t\right)d_{\mathrm{int},\mathrm{pipe}}\left(T_{\mathrm{pipe}}\left(t\right)-T_{\mathrm{gas},\mathrm{pipe}}\left(x,t\right)\right)& \left(25\right)\end{array}$

where m is the gas density, c.sub.p,g is the mass thermal capacity of the gas, x is the abscissa along the piping, T.sub.gas,pipe is the temperature of the gas in the piping depending on x and t, k.sub.i,pipe is the internal exchange coefficient in the piping, d.sub.int,pipe is the average internal diameter of the piping and T.sub.pipe is the temperature of the piping solely depending on t.

[0158] Equation (25) can be integrated between 0 (the dispenser) and the position x along the piping in order to obtain the expression of T.sub.gas,pipe(x, t). For the sake of greater clarity, L.sub.C(t)={dot over (m)}c.sub.p,g/(k.sub.i,pipe(t)d.sub.int,pipe) is noted as a characteristic length:

[00031]$\begin{array}{cc}d{T}_{\mathrm{gas},\mathrm{pipe}}/\left(T_{\mathrm{pipe}}\left(t\right)-T_{\mathrm{gas},\mathrm{pipe}}\left(x,t\right)\right)=1/L_C\left(t\right)\mathrm{dx}& \left(26.1\right)\end{array}$$\begin{array}{cc}{\left[-\ln \left(T_{\mathrm{pipe}}\left(t\right)-T_{\mathrm{gas},\mathrm{pipe}}\left(x,t\right)\right)\right]}_0^x=1/{L_C\left(t\right)\left[x\right]}_0^x& \left(26.2\right)\end{array}$$\begin{array}{cc}\left(T_{\mathrm{pipe}}\left(t\right)-T_{\mathrm{gas},\mathrm{pipe}}\left(x,t\right)\right)/\left(T_{\mathrm{pipe}}\left(t\right)-T_{\mathrm{disp}}\right)=\exp \left(-x/L_C\left(t\right)\right)& \left(26.3\right)\end{array}$

[0159] The energy conservation equation applied to the piping is written in equation (27):

[00032]$\begin{array}{cc}{\left(m{c}_p\right)}_{\mathrm{pipe}}\frac{d{T}_{\mathrm{pipe}}}{\mathrm{dt}}=\stackrel{}{m}{h}_{\mathrm{pipe}}\left(t\right)-k_{e,\mathrm{pipe}}\left(t\right)S_{e,\mathrm{pipe}}\left(T_{\mathrm{pipe}}\left(t\right)-T_{amb}\left(t\right)\right)& \left(27\right)\end{array}$

where (mc.sub.p) pipe is the heat capacity of the piping, h.sub.pipe is the energy transferred from the gas to the piping, k.sub.e,pipe is the external exchange coefficient of the piping and S.sub.e,pipe is the external surface of the piping.

[0160] The term {dot over (m)}h.sub.pipe(t), by virtue of equation (26.3), can be written as follows:

[00033]$\begin{array}{cc}\stackrel{}{m}{h}_{\mathrm{pipe}}\left(t\right)=-{}_0^L{k}_{i,\mathrm{pipe}}\left(t\right)d_{\mathrm{int},\mathrm{pipe}}\left(T_{\mathrm{pipe}}\left(t\right)-T_{\mathrm{gas},\mathrm{pipe}}\left(x,t\right)\right)\mathrm{dx}& \left(28.1\right)\end{array}$$\begin{array}{cc}\stackrel{}{m}{h}_{\mathrm{pipe}}\left(t\right)=-k_{i,\mathrm{pipe}}\left(t\right)d_{\mathrm{int},\mathrm{pipe}}{}_0^L\left(T_{\mathrm{pipe}}\left(t\right)-T_{\mathrm{disp}}\right)\exp \left(-x/L_C\left(t\right)\right)\mathrm{dx}& \left(28.2\right)\end{array}$$\begin{array}{cc}\stackrel{}{m}{h}_{\mathrm{pipe}}\left(t\right)=\stackrel{}{m}{c}_{p,g}\left(T_{\mathrm{pipe}}\left(t\right)-T_{\mathrm{disp}}\right)\left[\exp \left(-L_{\mathrm{pipe}}/L_C\left(t\right)\right)-1\right]& \left(28.3\right)\end{array}$

where L.sub.pipe is the length of the piping.

[0161] Equation (27) is discretized:

[00034]$\begin{array}{cc}{T}_{\mathrm{pipe}}\left(t+t\right)=T_{\mathrm{pipe}}\left(t\right)+\frac{t}{{\left(m{c}_p\right)}_{\mathrm{pipe}}}\left[\stackrel{}{m}{h}_{\mathrm{pipe}}\left(t\right)-k_{e,\mathrm{pipe}}\left(t\right)S_{e,\mathrm{pipe}}\left(T_{\mathrm{pipe}}\left(t\right)-T_{\mathrm{amb}}\right)\right]& \left(29\right)\end{array}$

[0162] While the invention has been described in conjunction with specific embodiments thereof, it is evident that many alternatives, modifications, and variations will be apparent to those skilled in the art in light of the foregoing description. Accordingly, it is intended to embrace all such alternatives, modifications, and variations as fall within the spirit and broad scope of the appended claims. The present invention may suitably comprise, consist or consist essentially of the elements disclosed and may be practiced in the absence of an element not disclosed. Furthermore, if there is language referring to order, such as first and second, it should be understood in an exemplary sense and not in a limiting sense. For example, it can be recognized by those skilled in the art that certain steps can be combined into a single step.

[0163] The singular forms a, an and the include plural referents, unless the context clearly dictates otherwise.

[0164] Comprising in a claim is an open transitional term which means the subsequently identified claim elements are a nonexclusive listing (i.e., anything else may be additionally included and remain within the scope of comprising). Comprising as used herein may be replaced by the more limited transitional terms consisting essentially of and consisting of unless otherwise indicated herein.

[0165] Providing in a claim is defined to mean furnishing, supplying, making available, or preparing something. The step may be performed by any actor in the absence of express language in the claim to the contrary.

[0166] Optional or optionally means that the subsequently described event or circumstances may or may not occur. The description includes instances where the event or circumstance occurs and instances where it does not occur.

[0167] Ranges may be expressed herein as from about one particular value, and/or to about another particular value. When such a range is expressed, it is to be understood that another embodiment is from the one particular value and/or to the other particular value, along with all combinations within said range.

[0168] All references identified herein are each hereby incorporated by reference into this application in their entireties, as well as for the specific information for which each is cited.