Linde-Hampson Anti-Machine: Self-heated Compressed Van-Der-Waals Gas as an Energy Carrier for Pneumatic Vehicles

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In Figure 1b, the diagram of internal energy calculated by finite difference method for one kg of compressed air in PT plane is shown. The U(P,T) diagram ranges pressures up to 800 bar and temperatures up to 1500 K. The map of the internal energy dependence is described by 16 isoenergetic curves numbered beginning with zero specific energy up to 1.05 MJ/kg with a difference in approximately 0.05 MJ/kg: U_s= -0.05 + 0.05·s, s = 1,2..22. The parameter K in the modified van der Waals equation of state (1) to be 3.42. A motivation to choose the working point A, B, B’, C follows from Figure 1b, where the system of isoenergetic curves covers the actual range of parameters: A represents the final point of the adiabatic process (U_A = 0.217 MJ/kg), B is the initial point of the adiabatic process (U_B = 1.011 MJ/kg). Below, the role of special point B’ as an intermediate state of preliminary prepared compressed air fuel (U_B = 0.989 MJ/kg) will be explained. At last, the thermodynamic state of compressed air in point C is chosen as an initial state of fuel before the reinforced process is started (U_C = 0.148 MJ/kg). The diagram of internal energy is convenient to determine the possible mechanical work, which may be picked out the compressed air fuel in an adiabatic process. The difference between points А and В(1400 K, 1.011 MJ/kg) equals to 0.794 MJ/kg, what is of interest to consider air as a fuel. The jump of the internal energy between points A and C is about 0.069 MJ/kg whereas the needed minimal isothermic work to produce compressed air in point C is about 0.692 MJ/kg. The calculated amount of extracted heat in the isothermic process Aà C is about 0.383 MJ/kg.

Figure 2. (Color online) Calculated air enthalpy P-T diagram. Isenthalpic curves: H = 0.95·s MJ/kg, s = 1,2…16; A, B, B’, C, chosen special points (Figure 1); a, Joule-Thomson inversion curve at K = 3.42, b, approximate Joule-Thomson inversion curve built by (Figure 1, curves 2, 3); lines c and d note the Joule-Thomson process possible pressure limits 300bar and 700 bar

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In Figure 2, the calculated by finite differences diagram of enthalpy for one kg of compressed air in PT plane is shown where isenthalpic curves s = 1,2…16 correspond to formula H = 0.95·s MJ/kg. The arc a is plotted by the Joule-Thomson inversion curve (3) taken at K = 3.42 whereas the arc b is the approximate Joule-Thomson inversion curve built by data ^[7] (also Figure 1, curve 3). Both these curves confine the area of positive inclination of the enthalpy contour lines (positive Joule-Thomson effect) within intervals of pressures P<9P_cr ≈ 329 bar and temperatures 77≥Т≤697 K for the modified Van der Waals gas at K = 3.42. The throttling process taken inside these arcs leads to decreasing of temperature. The outside area may be used to increase the temperature in the process of throttling. The polynomial approximation to the experimental Joule-Thomson inversion curve was found in ^[7] for gaseous N₂, Ar, and other gases (Figure 2, curve b, Figure 1a, curve 3). The lower and upper lines c and d depict possible pressure limits 700 bar and 300 bar in a throttling process. The chosen special points A, B, B’, C, were discussed above.

Note that two ways to produce a kg of compressed air fuel i.e., to reach the position C on the pressure-temperature plane, exist. The first way is represented by an isochoric transfer of one kg of liquid air taken at 1 bar into the state C shown in Figure 1a by the bold line. This process needs an amount of heat about 0.15 MJ that should be added to energy expended during the liquefying process. To avoid wasting of energy one more demand should be fulfilled-the initial gas-liquid mixture density must exceed critical density 331 kg/m³ of our van der Waals model with accepted Kopp-Neumann rule. As follows from Figure 1a, the shown isochoric line corresponds to density 415 kg/m³.

Figure 3. (Color online) Calculated specific energy and density of the van-der-Waals model of air at K = 3.42. 1, isothermal compression work (P_A = 1bar) vs final pressure P_C (left axis). 2, fuel density measured in g/cm³(right axis)

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Further we will consider the second way which is based on direct compressing beginning with rarefied state of air at atmospheric pressure and room temperature. Since states A and C are taken at a coinciding temperature, the minimal work at the path Aà C may be reached in the isothermal process. In Figure 3, shown are the dependences of energy expense and density of final state of compressed air on the final state pressure P_C by an isothermal process in interval ranged from initial pressure 1 bar to the final 1000 bar at 300 К (Figure 1, Figure 2, point A). The calculation gives for the chosen initial thermodynamic parameters of fuel P_C = 700 bar and T_C = 300 K the energy of production W = 0.692(0.5215 at k = 8/3) MJ/kg and fuel density ρ = 413 kg/m³. It should be emphasized, that relative increasing of density at P_C <300 bar (curve 2) is not significant whereas the CAF energy capacity (curve 1) seriously hoicks at small pressures and then transfers into a plateau: W(300) = 0.621 MJ/kg, W(1000) = 0.718 MJ/kg.

In this work, a way to reinforce the real gas energy capacity using the negative Joule-Thomson effect is considered. The energy capacity i.e., work that a gas may perform in an adiabatic expansion process, depends on the location of point B on the PT plane. By technical conditions, the pressure should become atmospheric at the end of the process whereas the final temperature in general case not coincides with Т_C. Moreover, the less is temperature T_A, the more is useful mechanical work obtained in the process. The bigger magnitudes of P_B and P_B correspond to bigger energy capacity of the compressed air. Due to the negative Joule-Thomson process performed in area limited by lines c and d in Figure 2 the compressed gas transfers from the point С to point B’ at the PT plane and then to point B. From this point of view, the required magnitudes P_C begin with pressures of the order of 400 bar and higher in correspondence with curves a and b in Figure 2. It worth to note, that a simple isenthalpic process leads to a weak growth of temperature at these conditions. In Sections 3 and 4, nevertheless, we will show how the use of a heat exchanging procedure helps to extend the range of final temperatures T_B’.

The adiabatic work W(T_A, T_B) done by one kg of a dense gas may be found both immediately from the P-T diagram for internal energy of compressed air shown in Figure 1b and as a process function

(4)

The surface W(T_A, T_B) calculated in axes T_A, T_B is plotted in Figure 4. For the chosen interval of temperatures, the surface W(T_A, T_B) is practically plain. The maximal work 0.77734 MJ/kg corresponds to point W₂(120, 1300), whereas the minimal zeroth one corresponds to nearest right point (300К, 300К). The most interesting for technical aims interval of pressures from 30 bar to 90 bar is highlighted by light gray. A conclusion can be made from the behavior of the highlighted band that to obtain specific work more than 0.6 MJ/kg one should use the regimes with outlet temperatures T_A more than 300 K. The pressure of shown points W(T_A, T_B) in the highlighted band increases along the direction of gradient, whereas magnitudes of the done work decrease along the band from left to right. Therefore, the most acceptable point of area T_B≤1300 K should be acknowledged a point with maximal work in the highlighted band. Our calculation gives for this region the point W(300,1079.4) with pressure P_B = 90 bar and potential work W = 0.559 MJ/kg. The nearest left point of the surface W(300,1300) = 0.7165 MJ/kg has P_B = 172.5 bar. The most remote left point of the surface W₂corresponds to P_B = 4.086 Kbar that exhibits relatively strong pressure increasing along the surface gradient.

The adiabatic work surface shown in Figure 4 being close to a plane may be approximated by the equation

(5)

where T₀ = 300K, T₁ = 1300K , T₂ = 120K, W(T₁, T₀) = W₁ = 0.7165 MJ/kg, W(T₁, T₂) = W₂ = 0.84 MJ/kg. The expression (5) gives a very good approximation for nearest low temperature part of the adiabatic work surface at chosen parameters, in the middle of the shown surface in point W(240, 650) the accuracy is about 0.6%, whereas the most remote high temperature part of the surface gives value W₂ = 0.84 MJ/kg instead of the exact one 0.77734 MJ/kg (15%) and for the low temperature point W₃ = 0.12351 MJ/kg instead of the exact magnitude of potential work 0.12595 MJ/kg (2%). As the matter of fact, the considered way gives minimal energy to produce one kg of compressed air fuel in point C(700, 300) about 0,693 MJ, at the same time the difference U_A-U_C ~ 0,069 MJ may be transformed into heat and returned. The same surface taken at K = 3.42 is more attractive in applied sense: though the both reference points W₁ and W₂ are close to previous ones, the technically important band of pressures 30<P<90 bar now begins from the very left corner of the surface, and it includes point W₁ = 0.721 MJ/kg. Therefore an opportunity arises to operate by work magnitudes more than 0.5 MJ/kg at temperatures T_A ranged from 200 K to 300 K.

Figure 4. (Color online) Calculated surface of specific work W depended on the adiabatic process limit temperatures T_A, T_B. The transverse dotted lines s = 1, 2…7 correspond to energies W_s = 0.1·s MJ/kg; K = 8/3

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The relatively high pressure of the considered carrier of energy in state С allow the compressed gas to be transmitted into a state with higher internal energy which in turn may be converted into mechanical work.

3. Linde-Hampson’s Anti-Machine: to Reinforce the Compressed Air Fuel

Here we consider a way to transform the compressed gas from a state most convenient for storing in a tank to a state suitable to produce maximal work at maximal power using the throttling process. During the Joule-Thomson process, the working body at point C(300, 700) may be transmitted into an arbitrary state B’ lying at the corresponding isenthalpic curve (see Figure 2). Motion along the isenthalpic curve beginning with point C at 700 bar to an appropriate point B’ at 300 bar causes the temperature shift about 23.4 К at K = 8/3 and 17.1 К at K = 3.42. Due to the negative work the system does under the van der Waals gas, the internal energy increases to about 0.046 MJ/kg at K = 8/3 and 0.042 К at K = 3.42. With the help of a heat exchanger which redistributes the heat between the input and output of the throttle the working body state may be transmitted into any point B’ of the Р-Т plane inside the pressure interval P<P_C. In Figure 5a, the Linde-Hampson anti-machine design is schematically shown. It includes the heat exchanger and throttling device connected so that output of the throttling device creates the hot circle of exchanger whereas the cold exchanger branch is the input of the throttling device. The axis X shows chosen direction of increasing for gas coordinate x (L≥ x ≥0) both in the input circle before the throttle 2 and output circle after throttling. Just when the heat exchanger begins to act, the leg of isenthalpic curve СB’ begins to shift in side of a higher temperature. Therefore the energy gain may reach 1 MJ/kg and more. The time interval of this regime establishing will be evaluated below, it may reach seconds or dozens of seconds depending on the quality of the heat exchanger. It is worth noting that the energy capacity of the considered working body may exceed the minimal work spent initially to produce 1 kg of the compressed air fuel (point С). On the one side, the exceedence arises due to increasing of internal energy of a real gas in considered Joule-Thomson process and, on the other side, one more way exists of energy capacity rising - to chose the outlet temperature Т_А lower than fuel temperature Т_С. Doing so, one can formally reach the fuel performance factor bigger than 100%. One more circumstance is of interest: if the temperature Т_B’ is high enough, what is the nature of the additional energy reached in a throttle procedure accompanied by the heat exchange? This and other questions like principal restrictions of gain and maximal magnitudes at given initial parameters depend on taken model and needs more detailed investigation.

4. An Introduction to the Theory of Joule-Thomson Heat Exchanger

In contrary to the well known Joule-Thomson cooler or Linde-Hampson machine ^{[10, 11, 12, 13, 14, 15]}, the proposed anti-machine acts like a heater outside the inversion region which is marked by arcs in Figure 1a and Figure 2. One more difference is manifested in different pressure regimes for heater and refrigerator. The cooling effect arises at relatively low decompressing pressures whereas the Linde-Hampson anti-machine needs a jump of several hundred bar between the input and output chambers of throttling device

As it follows from Figure 2, the heat exchanger increases temperature of compressed gas decreasing simultaneously its pressure to point B’. And the final stage of reinforcing procedure is one more throttling process decreasing pressure to the working pressure magnitude Р_B, which is also accompanied by a little increasing of temperature when the phase point moves from the position B’ to position B.

In Figure 5a, the Linde-Hampson anti-machine design is schematically shown. It includes the two-loop heat exchanger united with a throttling device. The intake compressed air fuel enters the heat exchanger in point C(P_C, T_C) and left the system in point B’(P_B’, T_B’). The first, high pressure, loop is kept at pressure P_C and the second one is at pressure P_B’. Due to the difference P_C– P_B’ the throttling process performed in area between lines c and d (Figure 2) leads to heating of the flowing gas depending on the heat exchanger parameters L and α. The throttle device 2 divides the high pressure input chamber P_C = 700 bar with compressed gas fuel and the low pressure output chamber 1 at P_B’ = 300 bar.

The heat exchange processes in the proposed simplified system may be described in time by two equations for temperature distributions T(x) and Ѳ(x) inside the input and output circuits CP_C’ and P_B’B’ shown in Figure 5a:

(6)

where L≥ x ≥0, the parameters of the equation (6):

(7)

(8)

Here κ(x) is thermal conductivity coefficient, α(x) is the heat exchange coefficient measured in W/m²K, is fuel consumption, is the perimeter of heat contact between cold and hot tubes so that gives the corresponding elementary surface of heat contact; S, are sections of cold and hot tubes inside the heat exchanger. The inhomogeneous gas density distributions are indicated as and . The equations (6) take into consideration the gas movement, thermal diffusion in both branches and heat exchange between hot and cold branches. The exchange of heat between the hot and cold branches of the exchanger is described by the right parts of the system (6). In general the heat exchange processes are essentially depended both on the system design and the dense gas properties ^{[12, 13]} therefore we undertake the simplified approach (6) to analyze the principal parameters of the Linde-Hampson anti-machine.

Our estimation show that for a 10 kW power vehicle the average density magnitudes are ~0.4 g/cm³, ~0.2 g/cm³ and the corresponding mean velocities are of the order of 40 cm/s and 20 cm/s for sections S, which are taken about 1 cm².

The needed partial solution of equation (6) depends on the initial and boundary conditions.

(9)

where δT_H(t) has been determining at every moment of time as the temperature jump caused by the Joule-Thomson process. To evaluate in general the parameters of the Linde’s anti-machine, further we develop an analytical approach neglecting the nonhomogeneity of coefficients in both equations (6), then after the transfer from temperatures.

The general solution of (6) may be obtained using the discrete Fourier transform in linear space of dimension N = 2n + 1. The direct space is determined by coordinates

(10)

whereas points of the reciprocal space are chosen as

(11)

It is easy to find the set of basic functions of the approach

(12)

obeying the demands of completeness, orthogonality and normality. As a matter of fact, the parameters in equations (6) like a, b, ω have a nonhomogeneous nature due to the coordinate dependence of density in both branches of the exchanger. To evaluate in general the parameters of the Linde’s anti-machine, further we develop an analytical approach neglecting the nonhomogeneity of coefficients in both equations (6), then after the transfer from temperatures T(x, t), Ѳ(x, t) to Fourier images , the system (6) takes the form

(13)

The general solution of (13) found for the constant coefficients κ, , ω and gives the time behavior of the Fourier components and

(14)

where the roots ν₁, ν₂ of the characteristic equation have the following view:

(15)

Further we will take into account that the roots ν_j have real and imaginary parts ξ_j and ζ_j , correspondingly (j = 1,2). Then the general solutions for T(x, t) and Ѳ(x, t) may be written in a real view

(16)

where the real coefficients A_k , B_k which are present in the expansions (16) should be found from the first two boundary equations in (6). The coefficients of expansion , have been transformed here in T(x, t) and Ѳ(x, t) taken at t→∞ when the expressions in brackets go to zero. Approximate time of transfer to infinity may be taken as

(17)

Figure 5. (Color online) (a) Approximate design of the Linde-Hampson anti-machine:1, heat exchanger: inletting cold high pressure gas, 2, throttling device and ambient hot low pressure gas. The fuel intake point, C(PC, TC), heated fuel point, B’(PB’, TB’), PC, pressure of the input loop, PB’ , pressure of outlet loop. (b) Calculated by (6) final (t>>τ) temperature distribution curves T(t, x), Ѳ(t, x): initial, 1, 2 and final, 1’, 2’(cut at T = 1500 K). PC = 700 bar, TC = 300 K, PB’ = 300 bar, TB’ = 1134 K, Tmax(x = 0) = 1937 K, α = 150 W/m2·K, L = 1.6 m, τ = 22.4 s, δTH = TC’- TB’ = 16.13 K’

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In Figure 5b, the initial (t = 0) and final (t>>τ) curves of temperature distribution Ѳ(x, t) and T(x, t) calculated by the equations (6) and solution (16) are plotted. The initial temperature jump δT_H (0) = T_C - T_B’ in (9) was found as 17.1 K, whereas the evaluation gives δT_H = T_C’- T_B’ = 16.13 K for final times t>>τ. The coefficient of heat transfer ‘α’ depends on several factors including the nature of fluids at both sides of the wall, wall surface geometry, fluid velocity and thermal conditions. Different sources give ‘α’ for dense gases at pressures of hundreds bar in a wide interval from hundreds to thousands of W/m²K. Following ^[12] we have chosen for our estimating model α = 150 W/m²·K. The calculation shows that at taken heat exchanger parameters and L = 1.6 m the temperature gradient in the high pressure and low pressure circuits is established near 502 K/m through the interval of time τ = 22.4 s. There exists a strong dependence of the final temperature distribution and the inlet temperature Ѳ(∞, L) on the heat exchanger parameters α, S and L. The corresponding combination of these parameters allows one to find a wide spectrum of work regimes. One more circumstance is important that though the negative Joule-Thomson effect at room temperatures has the lowest pressure limit 300 bar (or 400 bar by data ^[7]), the process of fuel reinforcing keeps its efficiency up to 100 bar. Therefore, the upper pressure limit, shown by line d in Figure 2, may also be lowered.

In Figure 6, the intake temperature Ѳ_L = Ѳ(L, ∞) dependence in P_C ,P_B_’axes is shown for the model K = 3.42. The most attractive technically region of 800K<Ѳ_L<1200K is highlighted by light gray. The chosen temperature interval corresponds to possible mechanical work inside an interval from 0.5 to 0.8 MJ/kg at pressures not higher than 90 bar according to obtained data of the fuel energy capacitance.

Figure 6. (Color online) Calculated by (13) the intake temperature Ѳ_L = Ѳ(L, ∞) dependence in the P_C-P_B’ plane. K = 3.42, α = 150 W/m²·K, L = 1.6 m, surface is cut at Ѳ_L = 2500 K; light gray (green online) band corresponds to 800 K< Ѳ_L<1200 K.

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To explain the nature of energy reinforcing during the Joule-Thomson process one should take into account the inevitable decreasing of initial pressure Р_С in tank with time. As the matter of fact, the initially forced gain turns by corresponding loss at final stages of the process when the fuel pressure in a tank is sufficiently decreased. Nevertheless if knowingly stop the process remaining some amount of ballast fuel in the tank, then the proposed way то allows significantly increase the energy capacity of compressed air.

Above the main idea of reinforcing the fuel parameters for a pneumatic thermodynamic engine was considered. Like a steam engine, this thermodynamic engine produces mechanical work due to the energy of compressed gas without using oxidation or any other chemical process.

5. General Scheme of CAF Circulation

Note that mentioned above way to prepare the initial state C using an isochoric transfer of one kg of liquid air beginning with 1 bar pressure at density 0.415 g/cm³ (Figure 1a, bold line) needs an amount of heat about 0.15 MJ. It may be taken from the heat exchanger. As a result the working point C due to the correspondingly modified heat exchanger will shift to lower temperatures for approximately 200 K. For example, B(96.3, 1400) à B(61.0, 1200) and specific adiabatic work decreases from 0.793 MJ/kg to 0.649 MJ/kg. Here we remain outside the scope of our study technical questions like pressure keeping systems or ways to transport liquid air to state C in an isochoric process. In Figure 7, shown are six types of pressure regimes in a pneumo-vehicle based on liquid air fuel. The process of fuel reinforcing inside the engine begins from liquid air in a tank (cycle 1), which intakes (cycle 6) into the circuit of high pressure 700 bar in the isochoric process with heating due to the energy transfer from the high temperature cycle 3 to cycle 4. The increasing of temperature during the process 6 needs about 0.15 MJ/kg. Therefore the output temperatures T_B and T_B’ of cycle 4 should be decreased to approximately 200 К from 1200 K to 1000 K. The latter in turn is the temperature of the fuel intake in the engine cylinder to produce mechanical work. As was mentioned above the intake pressure should be chosen in an approximate interval from 30 to 90 bar. The heat exchanger output 300 bar pressure cycle 3 (curve 2 in Figure 5a) is characterized by the temperature fall from about 2100 K to 1200 K due to heat flows out of the cycle 3. The pre-injection pressure cycle 5 corresponds to state B shown in Figures 1 and Figure 2. In general the parameters of state B should be chosen in technically valuable band of pressures 30-90 bar corresponding the internal combustion and diesel engines. In time when mechanical work is performed the working body state transforms from point B to point A. The cycle 3 to cycle 5 temperature drop 1200à 1000 is caused by a need to supply the isochoric liquid-gas transfer of the cycle 6. Also, cycle 2 is shown in Figure 7 servicing the pneumo-system of the vehicle: doors, windows, windshield viper blade and conditioning.

Figure 7. (Color online) Six cycles of pressure in a pneumo-vehicle. 1, liquid air in tank; 2, servicing pneumo-system; 3, heat exchanger’s output pressure cycle; 4, heat exchanger’s input pressure cycle; 5, pre-injection pressure cycle; 6, fuel heating isochoric cycle

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An important point of our study is technical perspective of air as the energy carrier and energy accumulator called to replace gasoline filling stations and oil bases. In a more wide treatment other gases or fluids also may be considered as carriers of mobile energy. Of course, additional amount of energy is needed to prepare compressed gas or steam fuel and new kinds of sources should replace in future the niche of traditional sources. Therefore, the considered conception implies an idea of restructuring in power engineering so that electric energy partially is used in compressors to produce compressed air and partially returns to the power system through a new kind of electric power stations transforming heat which accompanies the compressed air fuel production into electrical energy. This new kind of power stations may be inscribed in existing systems of accumulating power stations. Many unsolved problems exist including the investigation of unknown phase diagrams of gaseous mixtures at high pressures.

Figure 8. (Color online) A concept of oxygen depleted air circulation: Industry-Fuel Station-Vehicle-Atmosphere.

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