Theoretical evaluation of stack-based thermoacoustic refrigerators

Standing wave thermoacoustic refrigerator uses stack, is the heart of the thermoacoustic cooling system. The porous stack in the resonator tube develops temperature difference across the stack for heat pumping upon loudspeaker sound interaction of oscillating gas. In this paper, the optimization of stack-heat exchangers system and resonator is discussed using linear thermoacoustic theory for better COP and cooling power of refrigerator. The loudspeaker is assumed to provide the required acoustic power with the back volume gas spring system. Helium and air are chosen because of their better thermophysical properties and cost, compared to other competent gases. The 200 mm diameter stack is optimized for the temperature difference of 28 K. The theoretical results of the optimized refrigerator models are compared with the DeltaEC simulation results for deriving conclusions. DeltaEC predicts the cooling power and COP of 349 W at 0.998 for helium, and 139 W at 1.133 for air, respectively.


Introduction
In the recent years, the research on thermoacoustic cooling has given impetus on optimizing the devices theoretically as well as building practically. Thermoacoustic cooling is considered as an alternative technology to replace the present dominant vapor compression units in the commercial world. It is a promising cooling technology compared to the other unconventional technologies because of its simplicity of construction, benign working substances and absence of moving parts, can be built with the indigenous materials, etc. Thermoacoustic effect is the interconversion of sound and heat energies happens in thermoacoustic devices. Thermoacoustic devices are of either thermoacoustic engines or refrigerators. Thermoacoustic engine convert heat input into acoustic work output, whereas the thermoacoustic refrigerator convert acoustic work input into refrigeration effect. Thermoacoustic refrigerator pumps heat from cold space to surrounding environment to produce refrigeration effect upon acoustic energy input. Thermoacoustic refrigerator is either powered by thermoacoustic engine called thermo-acoustically driven thermoacoustic refrigerator, or by loudspeaker called acoustically driven thermoacoustic refrigerator. The loudspeaker-driven thermoacoustic refrigerators (LDTAR) are quite promising and popular compared to the enginedriven (thermo-acoustically driven) thermoacoustic refrigerators (EDTAR) because of simplicity in construction and operations [1]. The schematic showing the components, oscillating gas, and energy conversion process in thermoacoustic refrigerator is shown in Fig. 1. Electro-acoustic energy conversion happens in the moving coil loudspeaker and supplies sound energy to the resonator contains a stack-heat exchangers system and pressurized gas (helium, air, or any other inert gas) at 10 bar [2]. The acoustic standing wave generated by loudspeaker causes gas to oscillate front and back through the small pores of the stack-heat exchangers system. The ambient and cold heat exchangers are placed on either side of the stack as shown in Fig. 1. The position of the stack-heat exchangers system in a resonator depends on the maximum compression and rarefaction regions.
The stack-heat exchangers system is placed such that the ambient heat exchanger (AHX) is at the maximum compression region and the cold heat exchanger (CHX) is at the maximum rarefaction region [3,4]. The successive gas oscillation pumps heat from CHX to AHX through stack produces temperature gradient across the stack (Fig. 1). The heat pumping mechanism in the stack-heat exchangers system is depicted with Pv, Ts, and Ph diagrams (Fig. 2), which is similar to Brayton cycle. It has two isentropic processes and two isobaric processes. The process 1-2 compresses gas isentropically and the compressed gas reject heat at constant pressure (process 2-3) to the AHX plates maintained at ambient temperature by water cooling system. The above two processes happen in the forward movement of the gas. In the reverse movement of the gas, it expands isentropically (process [3][4], and the cold gas absorbs heat from the CHX plates connected to cooling space at constant pressure (process 4-1). The cycle repeats and the variation of the size of the oscillating gas at the state points 1-2-3-4 are also shown (Fig. 2).
In this paper, an attempt is made to explain the method of optimizing stack-heat exchangers system in terms of the normalized stack length and center positions with helium and air considering the performance of the stack, cooling power and power density of the refrigerator compared to the published literature [5][6][7][8]. The spiral geometry is chosen for the stack-heat exchangers system because of its simplicity and ease of construction during fabrication compared to other geometries (parallel plate, circular, pin array, etc.) [4,5]. Also, the performance of the spiral geometry expressed in terms of half stack spacing distance y and thermal penetration depth δ k is on par with parallel plate geometry at y = 2δ k for helium and y = δ k for air [6]. The cooling capacity of the refrigerator increases with increase in size of the refrigerator and at the same time performance of the stack and refrigerator decreases, and vice versa [7]. The stack center position X is the distance between the loudspeaker position and the mid-point of the stack length l 3 . The performance of the stack increases with decrease in stack center position   and length. Because the decrease in stack center position and length decreases the acoustic wave disturbance of the oscillating gas in the stack region of the cooler. Hence, the presence of the small length of the stack will offer minimum resistance for the oscillating gas, leading to improvement in the performance of the stack as found in the published literature [4,8]. In the published literature [3,4,7,8], the stack-heat exchangers system is optimized by choosing the stack length and center position merely based on the highest performance of the stack. This makes the device compact and improves power density with poor cooling capacity of the refrigerator. The highest performance of the stack is chosen at the lowest possible stack length and center position such that which ensures the space for accommodating AHX and instrumentation between the hot end of the stack and loudspeaker [9,10]. The scope of the present work is limited to the design, analysis, optimization, and theoretical evaluation of the standing wave LDTAR systems [11]. The objectives of the current research work are to optimize the spiral stackheat exchangers system with helium and air, not merely at highest stack performance, but also at better cooling power and power density of the refrigerator compared to the published literature [8]. This is achieved by wisely choosing the next higher value for the normalized stack length and center position such that small drop in stack performance returns with drastic improvement in cooling capacity. Because the small increase in stack length and center position dramatically relives acoustic resistance in the stack-heat exchangers system. The small drop in stack performance is because of viscous losses of the oscillating gas in the stack-heat exchangers system. The optimization techniques for the stack-heat exchangers system are discussed in Sect. 2. The small drop in power density of the refrigerator model at the higher chosen value of the normalized stack length and center position is compensated while optimizing the resonator system discussed in Sect. 3. The loudspeaker optimization to improve electro-acoustic efficiency using back volume gas spring system discussed in Sect. 4. The theoretically optimized standing wave refrigerator models with helium and air as oscillating gases are tested in DeltaEC software [12] for validation discussed in Sect. 5 and the conclusions are drawn in Sect. 6.

Design and optimization of stack-heat exchangers system
In this section the design and optimization of the stack, cold and ambient heat exchangers having spiral geometry with 85% porosity to minimize acoustic field disturbances to improve cooler performance compared to 75% porosity is discussed [7,8]. The geometry and cross section of the stack and heat exchangers is same except the length. Hence the design and optimization techniques are same for stack, CHX and AHX except the material selection. The stack materials should not conduct heat in the direction opposite to heat pumping process (ambient end to cold end) ( Fig. 1) which decreases stack performance. Hence, the locally available Mylar sheets have reasonable cost and low thermal conductivity (0.15 Wm −1 K −1 ) for the stack, and copper sheets for the cold and ambient heat exchangers (401 Wm −1 K −1 ), is chosen for better performance [5,13]. The cross section of the spirally wound Mylar stack and copper heat exchangers is shown in Fig. 3. The thin Mylar sheet is wound over  The design requirements, operating parameters of the refrigerator and thermophysical properties of the oscillating helium and air is given in Table 1. The derived parameters used in the design process and their values obtained using the data given in Table 1 are given in Table 2. In standing wave thermoacoustic engine or cooler, the term which decides performance is the critical temperature difference between ambient and cold heat exchangers, θ c . At critical temperature, the stack does not pump heat because of zero temperature difference θ between ambient and cold heat exchangers. Critical temperature difference between ambient and cold heat exchangers is measured per unit length of the stack which is given by The normalization of design parameters is employed to simplify the design process by reducing the number of parameters involved in optimizing the stackheat exchangers system [14]. While optimizing stack-heat exchangers system, the basic seventeen design parameters (Table 1) and the seven derived parameters ( Table 2) are reduced to eleven as shown in Table 3. They are the drive ratio D (ratio of acoustic pressure amplitude to average pressure), normalized temperature gradient θ n (ratio of temperature gradient to mean temperature of gas between heat exchangers) and normalized critical temperature gradient Γ (ratio of temperature gradient to critical temperature gradient between ambient and cold heat exchangers). The ambient heat exchanger length l 2 , stack length l 3 , cold heat exchanger length l 4 , and stack center position X are normalized with the subscript n by multiplying with wave number of gas, c. The thermal and viscous penetration depths (δ k and δ v ) are normalized with the subscript n by dividing with half-spiral stack-heat exchangers sheets spacing, y. Similarly, the cooling power (Q) and acoustic power (W) are normalized with the subscript n by dividing with the product of (PuA). In the design process, the thermal conductivity of stack materials is neglected, and for helium and air the product of T mg is set equal to one. Using the normalized design parameters (Table 3), the normalized heat output (Q ns ) and acoustic work input (W ns ) equations obtained using Rott's thermoacoustic theory [13,15] is given by  Table 2 Derived design parameters and their values as a function of the oscillating gas  Table 3 Normalized design parameters of thermoacoustic refrigerator Normalized parameters The terms Λ 1 and Λ in Eqs. (2) and (3) are the dimensionless heat conduction correction factor and the thermal conductivity factor of the oscillating gas, respectively, are given by Using the normalized equations from (2) to (5), the COP of the spiral stack is given by the ratio of the normalized cooling power output to the normalized acoustic power input is given by Substituting the data given in Tables 1 and 2 in Eqs. (2) and (5) and the performance of the stack as a function of the normalized stack length l 3n and center position X n for helium and air are given in Tables 4 and 5. It is found that for every normalized stack length there is a maximum performance of the stack at a particular normalized stack center position for both helium and air. Hence, it is found that each stack length will have the optimum stack center position at the highest stack performance as found in the published literature [16]. But the designer chooses the optimum stack length l 3 and center position X based on ensuring the sufficient space to accommodate the loudspeaker, pressure and temperature sensors between loudspeaker and ambient heat exchanger during fabrication. Addition to these criteria the attempts are made in this section to choose the normalized stack length l 3n and center position X n based on better cooling power Q at the moderate performance of the stack COP S for both helium and air. From Tables 1 and 2 it is found that as the normalized stack length and center position increases, the performance of the stack decreases but the cooling power increases. By balancing between the stack performance and cooling power the four, stack length and center combinations (SLCC) are chosen (Table 6) for helium and air to study its effects on the performance of the cooler. The length of the oscillating gas (displacement amplitude) in the cold heat exchanger is given by In Eq. (7), x is the location of the cold heat exchanger away from the driver (the cold end of the stack). Using the data given in Tables 1 and 2, the length of the cold heat exchanger l 4 is equal to the length of the oscillating gas (2x 1 ) is determined. It is found that the ambient heat exchanger rejects heat nearly twice the heat supplied by the cold heat exchanger [4,5,15]. Hence, (7) 2x 1 = 2P a sin(cx) ρuω the length of the ambient heat exchanger l 2 is equal to 2l 4 . The normalized critical temperature gradient Γ calculated for both helium and air for the four SLCC are found to be less than one (Table 6), confirms the spiral stack behaves as refrigerator [4,5,13,[15][16][17]. The stack center position X, the position of the ambient heat exchanger l 1 , and the length of ambient heat exchanger l 2 , the length of stack l 3 , and the length of cold heat exchanger l 4 are calculated using wave number of gas, c as a function of SLCC is given in Table 6. Similar to the stack, the normalized acoustic power input to the ambient and cold heat exchangers (W nax and W ncx ) for both helium and air is calculated by setting Γ = 0 (for small lengths) in Eqs. (3) and (5) as a function of  Table 6 Normalized critical temperature gradient, center position, and geometric lengths of stack-heat exchangers system as a function of the normalized stack length and center combination and working gas SLCC is given in Table 6. The acoustic power dissipated into the stack W s , ambient heat exchanger W ax , and cold heat exchanger W cx is determined by using the date given in Table 1 in the equations W s = W ns (PuA), W ax = W nax (PuA), and W cx = W ncx (PuA), respectively for helium and air as a function of SLCC is given in Table 7.

Resonator optimization
Resonator is a solid material containing stack-heat exchangers system in the large diameter section. It should be light weight, made with poor thermal conductivity material to avoid heat leak losses and selfsufficient to withstand 10 bar in dynamic pressure. The portion of the resonator right side to the cold heat exchanger is optimized for the minimum heat dissipation losses by decreasing volume, which improves performance and power density. The quarter wavelength TDH (taper and divergent tube with hemispherical end) resonator design is efficient compared with the quarter wavelength taper and small diameter tube resonator design as found in the published literature [4,5,10,18,19]. Hence, the TDH resonator is optimized for the spiral stack with four SLCC values (Table 7) using helium and air as oscillating gas. The length of the resonator is proportional to the velocity of oscillating gas. The total length of the resonator expressed in terms of wavelength of the gas is given by For full-wavelength resonator n = 1, for half-wavelength n = 2, for one-third-wavelength n = 3, and for quarter wavelength n = 4, etc. In this section the attempts are made to optimize the length, surface area and volume of the TDH resonator ( Fig. 4) with helium and air as a function of the four SLCC (Table 7). For the current four SLCC resonator designs the small taper length is 20 mm, the diameter ratio at the throat is 0.15 and the radius of the hemispherical end is 100 mm, are chosen based on the minimum resonator losses [4,5,13,20]. For the fixed stack diameter of 200 mm, the length of the stack diameter varies as a function of the four SLCC. The length of the divergent section L 3 decides the diverging angle θ 2 in the divergent section of the resonator and the total length of resonator L t . The length of the divergent section is wisely chosen such that it should have small diverging angle for both helium and air (Table 8) to minimize gas turbulence during expansion. These angles are smaller compared to the published designs found elsewhere [8]. Further increase in the length of the divergent section increases the resonator length and hence the resonator losses. This decreases the performance of the cooler and power density. The total length of the resonator L t is expressed in terms of wavelength of gas, the total surface area of resonator (A t ), volume left to taper (VLT), volume right to taper (VRT), volume ratio (V r , the ratio of VRT and VLT) and total volume of the resonator (V t ) is determined (Table 8). Using the data given in Tables 1 and 2 in Eq. (9), the resonator components heat power dissipation losses per unit surface area is determined [4].
The first term in the right-hand side of Eq. (9) is the viscous loss and second term is the thermal loss in the resonator system.
The pressure p 1 = P a cos(cx) and velocity u 1 = P a sin(cx)/ερu in Eq. (9) are determined at the mean center position of the individual resonator component measured from the loudspeaker position. The total heat power dissipation losses for the whole resonator, W r is determined by adding the heat power dissipation losses at the individual resonator components as given in Table 8. In the design process it is assumed that the loudspeaker provides the necessary acoustic power required for heat pumping process in the stackheat exchangers system. The acoustic refrigeration capacity of the cooler strongly depends on the resonator power from loudspeaker. For the stack diameter of d 1 = 200 mm for both helium and air (Table 1), the cooling power of the stack Q is found to be 601 W for helium (Table 4) and 225 W for air (Table 5) at the optimized conditions of the normalized stack length and center combinations. Hence, the commercial loudspeaker with the power range of 500-1000 W is sufficient to power the resonator with a possibility of achieving the electroacoustic efficiency up to 90% with a back volume gas spring system (Fig. 4) [8,21,22]. In the real world, the designer can select the suitable commercial loudspeaker from the suppliers viz. Philips, Sony corporations, Pioneer corporation, Harman international, and many other local suppliers. The total acoustic power dissipated into the stack-heat exchangers and resonator W t , and COP, COPC, and COPR are calculated using the values given in Tables 4, 5, and 7 in Eqs. (10)-(13) as given in Table 8. For helium, at SLCC of l 3n = 0.2 and X n = 0.4 the cooling power of the refrigerator is 601 W at the stack performance of 2.54 (Table 4). Further increase in the SLCC decreases the performance of the stack drastically (1.86) for the small increase of cooling power (716 W). Hence the resonator with SLCC of l 3n = 0.2 and X n = 0.4 with the COP of 1.8 is optimized compared to the other three designs (Table 8) by considering the cooler performance,   cooling power, and the total volume of the resonator for its compact size. Similarly, the air resonator is optimized at the SLCC of l 3n = 0.2 and X n = 0.2 with the stack performance of 2.16 at 225 W (Table 5) where the COP of the cooler is found to be 1.66 (Table 8). Fig. 4 Layout of the TDH-resonator with back volume gas spring system uses helium and air as oscillating gas Table 8 Theoretical performance results of the optimized TDH resonator as a function of the normalized stack length and center combination and working gas

Acoustic driver
Acoustic driver is a commercially available loudspeaker modified with back volume gas spring system. It matches the loudspeaker operating frequency with the resonator frequency, which increases the electroacoustic efficiency of the loudspeaker from 3% to 50-90% possible [4,21,22]. The electromechanical parameters of the commercial 1000 W loudspeaker approximately are given in Table 9.
The spring constant for the acoustic driver s is defined by where f d is the driver frequency, and m is the moving mass of the moving coil loudspeaker. Using the data given in Table 9 in Eq. (14), the spring constant for the loudspeaker with helium and air is determined. The back volume of the gas spring system for the commercial loudspeaker is given by where γ is the specific heats ratio, P is the average gas pressure, A b is the cross-section of the back volume system, which is equal to the cross-section of the vibrating diaphragm of the loudspeaker, A d . Using the data given in Tables 1 and 9, the back volume V b and the length of the back volume l b (Fig. 4) with helium and air resonators are found to be 3479 cm 3 and 111 mm for helium, and 11,667 cm 3 and 371 mm for air, respectively.

DeltaEC
As a check on the assumptions of the boundary layer and short stack approximations [4,13,23] shows that the stack length l 3 for helium and air for the chosen SLCC are smaller (Table 6) than the wavelength of the gas λ ( Table 2). The temperature difference (28 K) chosen in the current research work is about a factor 10.3 s smaller than the mean temperature of helium and air (288 K). The thermal and viscous penetration depths are also smaller than the half gas spacing in the stack-heat exchangers system except the thermal penetration depth of air (Tables 1 and 2). Hence the results of the calculations done in Sects. 2 and 3 are the good estimates for the design optimization of thermoacoustic cooler with helium and air. In this section, the DeltaEC computer program [12] is used to predict the performance of the theoretically optimized refrigerator models with the four SLCC operating with helium and air is discussed. DeltaEC solves the same thermoacoustic equations in a geometry designed by the user, with the boundary conditions of the different geometric component variables. The thermoacoustic segments, boundary conditions and the use of DeltaEC software for validating thermoacoustic models are discussed elsewhere [24][25][26]. The specifications of the loudspeaker given in Table 9 with back volume gas spring system for helium and air are used in DeltaEC calculations. In DeltaEC program, the cold and ambient heat exchangers are targeted at 273 K and 301 K, respectively and hence the mean temperature of the heat exchangers T mx is 287 K. The cooler is operated with 10 bar helium and air with 3% drive ratio. In the theoretical design calculations, the mean temperature of gas T mg is taken to be 288 K, 1 K higher than the mean temperature of the heat exchangers 287 K as found in the published literature [4,7,8]. Similarly, the DeltaEC predicts the mean temperature of gas T mg is about 0.28 to 0.74 K higher than the mean temperature of the heat exchangers T mx with helium and air as given in Table 10. It increases slightly as the SLCC increases. DeltaEC assumes perfect insulation for the whole resonator including the loudspeaker with back volume gas spring system [13,26]. Hence, the AHX rejects the heat equal to the sum of heat absorbed at CHX and the electrical energy input to the loudspeaker with helium and air for the four SLCC as given in Table 10. The electrical power input to the driver W e and the heat rejection rate at the AHX Q r increases with increase in the SLCC for both helium and air. This is because of the increase in volume of the stack demands higher electrical power input for heat pumping and hence the heat rejection rate at the AHX increases. The electro-acoustic efficiency of the loudspeaker with back volume gas spring system increases with increase in the SLCC (29.5-87.9%) for both helium and air (Table 10). This is because of the decrease in acoustic resistance due Table 9 Electromechanical parameters of the loudspeaker with gas spring system for helium and air as working substance to gas turbulence in the stack-heat exchangers system. DeltaEC predicts that the cooling power of the refrigerator increases with increase in the SLCC, whereas the COP and COPR decreases (Table 10) similar to the theoretical results (Table 8). For the optimized 0.224 λ TDHhelium, and the 0.222λ TDH-air resonators (Table 8), the variation of the gas area Gas A, acoustic peak pressure amplitude P a , and solid material temperature T s , along the resonator length from the driver position are shown in Figs. 5, 6, 7, 8, 9, and 10. The gas area between the driver and AHX is 100% which is equal to the cross-section area of the stack 0.03142 m 2 . There is a 15% reduction in gas area in the stack-heat exchangers system, which drastically decreases in the taper section, and gradually increase in the divergent section terminated with the hemispherical end. DeltaEC shows the same nature of variation of gas area in the resonator for both helium and air as shown in Figs. 5 and 6. The only difference is the position and the length of the stackheat exchangers system. The acoustic pressure amplitude of gas at the driver position is 0.3 bar, remains same up to the length of AHX position, and gradually decrease in the stack-heat exchangers system. Also decreases drastically in the taper section and further goes negative in the divergent section for both helium and air as observed in Figs. 7 and 8. The variation of solid material temperature T s (stack-heat exchangers and resonator) from the driver position to the resonator tip for helium and air is shown in Figs. 9 and 10. There is a small drop of temperature of about 2 K in AHX, and decrease drastically in the stack region, reaches to 273 K in the CHX region. The temperature of the resonator remains the same for the rest of the portion for both helium and air since the DeltaEC assumes perfect insulation.

Conclusions
The design and optimization of the thermoacoustic refrigerators having 200 mm diameter spiral stack-heat exchangers system with 85% porosity operating with helium and air at 10 bar pressure and 3% drive ratio for  the temperature difference of 28 °C is discussed. In the design process, the AHX is assumed to be kept at 28 °C by water cooling system. The stack-heat exchangers system is optimized by considering both performance of the stack and cooling power. This improves performance as well as power density of the refrigerator as discussed in Sects. 1, 2, and 3. The current optimized 0.224 λ TDH-helium resonator has 2% improvement in COP, 79% improvement in cooling power and 3.1 times increase in power density compared to the published design [8]. But the optimized 0.222λ TDH-air resonator has 83% improvement in COP leading to 27% decrease in cooling power and 2.2 times increase in power density compared to the published design [8]. It is found that the air shows little poor performance compared to helium because of its low sound velocity and thermal conductivity (Table 1) and high Prandtl number (Table 2). But the cost of air is much cheaper and abundantly available in nature compared to helium, and hence the use of air as working gas is justifiable. The theoretical and DeltaEC results (Tables 8  and 10) show agreement with each other but the theoretical results show better performance compared to Del-taEC results. Because the theoretical design is optimized by neglecting loudspeaker parameters, and neglecting the effect of damping in the right portion of the resonator after CHX (Fig. 4). This affects the velocity of oscillating gas and frequency and hence the performance of the cooler [4,5]. Similar to helium and air, the design and optimization techniques discussed in this paper to improve performance, cooling power and power density of the thermoacoustic coolers are also holds good for the other working substances used elsewhere [4,12,27].

Nomenclature
A Cross-sectional area of stack, m 2 AHX Ambient heat exchanger A b Cross-sectional area of the back volume system A d Cross-sectional area of vibrating diaphragm of loudspeaker, cm 2 A t Total resonator surface area, cm 2 Bl Force factor, NA −1 or T-m CHX Cold heat exchanger COP Coefficient of performance of cooler COPC Carnot's Coefficient of performance COPR Coefficient of performance relative to Carnot's COP S Coefficient of performance of stack c Wave number of gas, cym. −1 C p Specific heat of gas at constant pressure, Jkg −1 K −1 C v Specific heat of gas at constant volume, Jkg −1 K −1 D Drive ratio DeltaEC Design environment for low-amplitude thermoacoustic Energy Conversion d 1 Stack diameter, mm d 2 Throat diameter, mm f Resonator frequency, Hz f d Driver frequency, Hz Gas A Gas area in resonator system, m 2 k g Thermal conductivity of gas, Wm −1 K −1 LDTAR Loudspeaker-driven thermoacoustic refrigerator L 1 Length of large diameter tube, m L 2 Length of taper tube, m L 3 Length of divergent tube, m L e Electrical inductance of loudspeaker, H L t Total resonator length, mm l Half spiral stack sheet thickness, mm l 1 Distance between loudspeaker and hot heat exchanger, mm l 2 Length of ambient heat exchanger, mm l 3 Length of stack, mm l 4 Length of cold heat exchanger, mm l b Length of back volume system, mm m Moving mass of driver, g P Average gas pressure, bar PVC Poly vinyl chloride Pv Pressure and volume Ph Pressure and enthalpy P a Acoustic or dynamic pressure amplitude, bar p 1 Pressure of gas at the mean center position of the component, bar