Experimental Thermofluidic Characterization of Different Metallic Regenerators Crossed by Alternating Air Flow

Abstract

The regenerator is one of the most important elements of a thermal machine operating in an alternating flow regime. The present study constituted an experiment on the dynamic characterization of different metallic regenerators crossed by alternating air flow. During an experiment, gas temperature, velocity, and pressure were measured at both ends of the regenerator. The acquisition frequency was set at 1 kHz for each temperature, pressure, and velocity signal. This enabled us to fully characterize an oscillating flow cycle. Experiments were performed for different regenerators (structures and porosities), frequencies, temperature gradients, and displaced air volumes. The experimental results showed that the density ratio is significant at high frequencies for all structures. The friction coefficient is determined based on a classical correlation at the time of maximum velocity. The friction factor$f$seems to decrease with the kinetic Reynolds number$R{e}_{\omega }$for a 30% porosity regenerator. For the other tested regenerators (35% and 40% porosity), we observed that it is almost constant with a little dispersion. To minimize the dispersion effects, another definition was proposed to calculate the friction factor at the time of maximum pressure drop. The results showed that for all regenerators a single clear trend is the function of the kinetic Reynolds number. A significant phase shift was observed between the velocity and pressure drop. It increases with the increasing kinetic Reynolds number. It was found that the phase shift only depends on frequency. Finally, a correlation equation was proposed to predict the phase shift of different regenerator structures. It was found that the effect of the kinetic Reynolds number$R{e}_{\omega }$on the phase shift is more dominant than that of the hydraulic diameter to length ratio $D_h/L$.

1. Introduction

The alternating or oscillating flows within a regenerator are of scientific and industrial interest. They are frequently found in many industrial applications, such as Stirling machines, magneto-caloric coolers with active regeneration, heat exchangers, compressors, and internal combustion engines. These machines operate according to thermodynamic cycles requiring the use of an alternating fluid flow. The characterization of regenerators operating in an alternating regime constitutes an essential need for the thermal or thermo-mechanical modeling of these systems. Experimental and theoretical works are being devoted to the thermo-fluidics of alternating flows within a regenerator. The first regenerator was developed in 1816 by Robert Stirling for the improvement of his hot air engine. The regenerator or economizer of a Stirling machine increases the efficiency of the thermodynamic cycle and it is located between the hot and cold heat exchangers. It often consists of either a porous material or an assembly of small grids of fine wire cloth, plates, or balls, through which the working gas flows in an alternating fashion. Various authors, such as Tong and London [1], Blass [2], and Miyabe et al. [3], have proposed correlations for the pressure drop (or the Darcy friction factor) in a regenerator based on unidirectional steady flow experiments.

In previous works, the Darcy friction factor$f$has been calculated using the following equation (Tanaka et al. [4]):

$$f=\frac{∆P{D}_h}{\frac{1}{2}\rho u^{2}L},$$

(1)

where$∆P$is the pressure drop, $D_h$ is the hydraulic diameter, $\rho$is the fluid density, $L$ is the length of the regenerator matrix, and $u$ is the velocity of the fluid flow.

Previous experimental and analytical investigations showed that the steady flow condition is not exactly the same as that of the oscillating flow. Several authors (Taylor and Aghili [5], Tanaka et al. [4], Gedeon and Wood [6], and Pamuk and Özdemir [7]) conducted experiments that reveal that pressure drops across the regenerator are different under oscillating flow conditions than under steady flow at the same Reynolds number.

Zhao and Cheng [8] investigated the oscillatory pressure drop characteristics in packed columns (composed of three different sizes of woven screen) subjected to a periodically reversing flow of air. They found that the cycle-averaged pressure drop of the oscillatory flow in a packed column is four to six times higher than that of a steady flow.

In an experimental work, Xia et al. [9] presented two test rigs that could generate steady and oscillating flows through a woven screen regenerator under similar conditions. They measured velocity at both ends of the regenerator. This is different from theoretical velocity based on piston velocity, which should begin at a 0° crank angle of the piston (bottom dead center) and end at 180° (top dead center). It was also shown that the effect of gas compression cannot be ignored for steady and oscillating flows through a regenerator when mass flow increases. They stated that the theoretical velocity could be used only when the effect of gas compression is negligible. By studying the pressure drop for a smooth tube with tri-orifice baffle inserts under steady and oscillatory flow conditions, Muñoz-Cámara et al. [10] showed that the pressure drop coefficient is identical to the steady flow one at a very low kinetic Reynolds number$\left(R{e}_{\omega }<30\right)$. The kinetic Reynolds number$R{e}_{\omega }$is a dimensionless expression for the oscillating inertial forces in relation to viscous forces (Simon and Seume [11]):

$$R{e}_{\omega }=\frac{\rho \omega D_h^2}{\mu },$$

(2)

where$\rho$is the density, $\omega$is the angular frequency, and$\mu$is the dynamic viscosity. The kinetic Reynolds number is also called the oscillating Reynolds number.

They also observed that the pressure drop signal under oscillatory flow is not a purely sinusoidal wave.

Richardson and Tyler [12] were among the first to experimentally investigate oscillatory flow in a pipe and discovered the so-called “annular effect”, i.e., that the maximum velocity in an oscillatory flow occurs near the wall rather than at the center of the pipe, as is the case under steady flow. Experiments such as those of Hino et al. [13], Ohmi et al. [14], and Zhao and Cheng [15] showed that the appearance of turbulence (perturbation) in an alternating flow is different from that in a quasi-stationary, unidirectional flow. The onset of turbulence happens only in the early deceleration phase, whereas in the acceleration phase the flow regime is quasi-laminar.

Other works, performed by Hino et al. [16], Akhavan et al. [17] and Vittori and Verzicco [18], also show that the oscillating flow regime can be either laminar, disturbed laminar (small perturbations appearing during the acceleration phase of the cycle), highly turbulent for the entire cycle, or intermittently turbulent (at the beginning of the deceleration part of the cycle).

Not only is there a fundamental difference between steady and oscillatory flow, but many experimental and analytical investigations have shown that under conditions of oscillating flow, the pressure drop is not always in phase with the velocity. Thus, predicting phase shift in alternating flow is crucially important for designing alternating flow-based devices. This is why the phase shift has received particular attention.

For example, Chen et al. [19] showed that the influence of the regenerator length on the phase shift between pressure and velocity is negligible compared to that of frequency. The authors prove that the pressure drop coefficient is difficult to determine due to the phase shift variation with velocity (up to 50°). Choi et al. [20] used modified phase shift as a dimensionless parameter to predict the phase shift${\varphi }_{\mathsf{\Delta }P}$between the pressure drop and the mass flow rate at the warm end of oscillating flow through a twill screen regenerator. They found that all their experimental cases can be expressed by a single relationship:

$${\varphi }_{\mathsf{\Delta }P}=1.32\times {10}^{-8}{\left(D_h/L\right)}^{-2.62} Va,$$

(3)

where$Va=R{e}_{\omega }/4$is the Valensi number (same as kinetic Reynolds number, ratio of the time scale of viscous penetration to the oscillation period [11]).

Another analytical correlation was obtained by Khodadadi [21] to predict the phase shift between the velocity and the pressure of oscillating flow through a porous medium channel bounded by two impermeable parallel plates. He found a significant phase shift of 90° between the velocity and the pressure. His results showed that the phase shift can be predicted using the Stokes number, external diameter, and porous medium shape parameter. The annular effect reported by Khodadadi [21] could be responsible for this phase shift. Ni et al. [22] observed significant phase differences between the movements of gas inside porous media and the movements of the piston. This phase shift increases with the increasing pressure drop of porous media.

A combined theoretical and experimental study on fluid motion in the wire mesh regenerator of a Stirling engine in an oscillatory flow was conducted by Isshiki et al. [23]. They found a small phase shift between the pressure drop and velocity. It was stated that the turbulent intensity in the regenerator exit flow field is stronger during the decelerating period than during the accelerating one. Friction factors defined by adjusting the phase shift between the pressure drops and velocity variations show that it appears bigger in the accelerating period than in the decelerating period. Zhao and Cheng [8] found that the phase shift between the pressure drop and the velocity strongly depends on the kinetic Reynolds number $R{e}_{\omega }$and less so on the dimensionless displacement of the fluid.

Guo et al. [24] experimentally investigated the flow characteristics of a split cycle Stirling cryocooler regenerator. They stated that the phase shift between the pressure drop and the mass flow depends on the porosity and the regenerator outlet conditions. Zhao and Cheng [25], Leong and Jin [26], Jin and Leong [27], Bağcı et al. [28], Rogdakis et al. [29], and Dellali et al. [30] experimentally investigated the characteristics of oscillating flow. Their results showed that the phase shift between the velocity and the pressure drop is small under high porosity.

In this study, the characteristics of alternating air flow through three types of regenerator (Str_30%, Str_35% and Str_40%) with different geometries, lengths, and porosities were experimentally investigated. We present new experimental results for velocity, pressure, and temperature under alternating air flow. The friction factors through the regenerator were also calculated and are discussed herein. Based on all the experimental data, a new correlation equation for the phase shift between instantaneous pressure drop and velocity was obtained.

2. Description of Experimental Setup

In this section, the experimental setup designed and built to perform experiments of alternating flow interaction with porous media is described. The device (Figure 1) is similar to an alpha Stirling engine configuration. Its main components are an electrical motor (1), a cylinder and crank shaft (2), a double-acting pneumatic cylinder (3), two connecting tubes (4 and 5), two air/water heat exchangers (6 and 7), and a regenerator (8). These parts are mounted on an aluminum support of sufficient mechanical rigidity to withstand the substantial vibration caused by the alternating motion of the system. The shaft is pierced so that the position of the crank and hence its travel, and thus the volume swept by the piston, can be adjusted. The piston stroke can vary from 64.4 mm to 128.4 mm in 3.20 mm increments.

The experimental device allowed for the insertion of different regenerators to be tested. The working fluid is air, which is displaced as an alternating in-line flow by the motion of the piston (cylinder). The regenerator is connected between the two connecting tubes at the extremities.

2.1. Regenerators

The studied regenerators are all made of 316 L stainless steel and are manufactured using the direct metal laser sintering (DMLS) technique. Using this technique, their dimensions and shapes can be precisely defined. They all have an outer diameter $D_{reg}$of 9.5 mm and a length$L$of 60 or 70 mm. The differences between them lie in their porosities (ε), which result from their structure (Figure 2). We examined a 30% porosity regenerator presenting a pyramidal structure [31], a 35% porosity regenerator with a straight channel structure, and a 40% porosity regenerator with heterogenous straight channels. The hydraulic diameter$D_h$of each regenerator structure (Str) is obtained using the defining Equation (4) of that parameter:

$$D_h=\frac{V_{fluid}}{S_{fluid}},$$

(4)

where$V_{fluid}$is the fluid volume contained within the regenerator and $S_{fluid }$ is the regenerator surface in contact with the contained fluid. These parameters’ values are given by the CAD software used to design the regenerator.

2.2. Instrumentation and Measurement Techniques

In order to characterize the fluid regenerator interaction from thermal and dynamic perspectives, temperatures, pressures, and velocities were measured as indicated in Figure 3 and Figure 4.

For temperature measurements, homemade 12.7 μm diameter K type (Chromel-Alumel) microthermocouples were used (Label 3 on Figure 4). They were placed at both ends of the regenerator, and at five spots that were manufactured on the regenerator (Figure 2). In this paper, we only present the temperature evolution measured at both ends of the regenerator. Thermocouples were characterized under static and dynamic states (Lanzetta and Gavignet [32]). Their accuracy is ±0.1 °C and their cut-off frequency $F{r}_c$ is 30 Hz for forced convection. The alternate flows have a frequency range between 0.65 to 5.88 Hz. Therefore, no temperature correction is required as the cut-off frequency is higher than that of the flow$(Fr<F{r}_c)$. Local instantaneous pressures were measured using XTL-140M-5BARA Kulite sensors (Label 2 in Figure 4). The sensors were calibrated and placed at both ends of the regenerator. Two hot-wire sensors (TSI, Model: 1201) were used for measurements of the instantaneous axial velocity at each end of the regenerator (Label 1 in Figure 4). The hot-wire sensor was calibrated using a TSI IFA-300 system. The velocity signal was digitized using an A/D converter card and the data were processed via Thermal-Pro software. Sensors were placed on each regenerator side in the same section as pressure and temperature sensors. The hot-wire sensor only measures the velocity of the flow, i.e., an absolute velocity value. As the flow oscillates, a sign should be added to that value in order to give information regarding flow direction.

Finally, the angular position of the piston$\theta$was obtained using a magnetic proximity sensor (FESTO SME-8-K-LED-24), with a response time of less than 5 ms. The proximity sensor was positioned on the cylinder (Label 4 on Figure 4) in such way that it delivered a signal of maximum value for the initial position of the piston A (Figure 5).

The calculation of the temporal angular position of the piston$\theta$ was conducted theoretically. It consists of calculating the complete cycle time that corresponds to the two successive positions of port A and B (Figure 5). The compression phase $\left(A\to B\right)$corresponds to the first half of the cycle and the expansion phase $\left(B\to A\right)$corresponds to the other half.

The formula used to obtain the transient angular position of the piston$\theta$is given by:

$$\theta =360\frac{t\left(i\right)-t\left(max\right)}{∆t},$$

(5)

where t(i) is the variation of time during a cycle, t(max) is the time corresponding to the second position of the maximum dead center of the piston, and Δt is the time between the two successive positions of the maximum dead center of the piston.

Since the flow alternates, it is necessary to define a temporal reference in order to describe the parameters’ temporal evolutions. Based on sensor position detection, the zero-time reference for a single cycle matches the detection of a piston’s lower spatial position. For the study of successive cycles, zero time matches the lower piston position of the first cycle of the series. Conveniently, we used rotation angle$\theta$instead of time to plot the thermo-fluidic parameters’ evolution.

In this paper, we consider that the piston motion pushes fluid from the hot side to cold side of the regenerator during the compression phase. During the expansion phase, fluid passes from the cold to the hot side of the regenerator (Figure 6).

The hot side represents the volume between the regenerator and the hot exchanger, and the cold side is between the regenerator and the cold exchanger. The time corresponding to the minimum value of the velocity (u) is taken as a reference for the starting point of the compression phase or the expansion phase (reverse phase) of the fluid flow.

All the signals were synchronized using a National Instrument USB-6211 I/O 250 KHZ multifunction board that was connected to both the external trigger system of the IFA 300 board and the NI SCXI-1600 trigger module of the NI SCXI-1000 chassis.

2.3. Uncertainty Analysis

The uncertainties$q$were analyzed by the method described in the Guide to the expression of Uncertainty (GUM) [33]. It can be classified into two groups. In Type “A”, standard uncertainty is evaluated using statistical methods. In Type “B”, standard uncertainty is evaluated by means other than statistical methods [33].

The equation to estimate the experimental uncertainty (type “A” uncertainty) of a quantity $X,$estimated from $n$independent observations$X_i,$is represented by:

$$q_A=\frac{1}{\sqrt{n}}{\left[\frac{1}{n-1}{\displaystyle \sum }{\left(X_i-\overline{X}\right)}^2\right]}^{1/2}.$$

(6)

When performing the cycle averaging of the data obtained over 45 successive cycles, the Type “A” uncertainty of the measured velocity is ±0.02 ms⁻¹. The dynamic uncertainty measurement is ±1.85 × 10⁻³ bar.

For the Type “B” evaluation of standard uncertainty, we used data provided in the calibration apparatus for the velocity (±1%) and the pressure (±5 × 10⁻³ bar). Considering that these intervals represent a rectangular distribution between the values$\left[-y,+y\right]$, the Type “B” evaluation of standard uncertainty was calculated according the following equation:

$$q_B=\frac{y}{\sqrt{3}}$$

(7)

The combined standard uncertainty is:

$$q_C=\sqrt{q_A^2+q_B^2}.$$

(8)

The expanded uncertainty$q$is obtained by multiplying the combined standard uncertainty$q_c$by a coverage factor$k=2$corresponding to a 95% confidence interval:

$$q=k{q}_C.$$

(9)

The total relative percent uncertainty is computed using the following formula:

$$\mathrm{Total}\mathrm{relative}\mathrm{percent}\mathrm{uncertainty}=\frac{\mathrm{expended}\mathrm{uncertainty}\left(q\right)}{\mathrm{value}\mathrm{of}\mathrm{measurment}}100[\%].$$

(10)

The uncertainties of frequency$Fr$and porosity$\epsilon$were estimated to be 0.64% and 1%, respectively. The thermocouples were calibrated under static and dynamic states to ensure the accuracy uncertainties of ±0.1 °C (Lanzetta and Gavignet [32]). The calculated total relative percent uncertainties for pressure and velocity are given in

Table 1. Total relative percent uncertainty of experiment.

Structure of Regenerator	Total Relative Percent Uncertainty
	Pressure	Velocity
Str_30%	±0.63 to ±0.70%	±1.16 to ±2.33%
Str_35%	±0.60 to ±1.6%	±2.30%
Str_40%	±0.61 to ±0.95%	±2.30%

for different structures of the regenerator with all experimental data.

Considering that the errors are uncorrelated variables, the standard uncertainties can be calculated using Equations (11) and (12) according to the principle of uncertainty propagation.

$$Z\left(r,\dots ,v,\dots w\right),$$

(11)

$$q_Z=\pm \sqrt{{\left(\frac{\partial Z}{\partial r}\right)}^2{q}_r^2+\dots {\left(\frac{\partial Z}{\partial v}\right)}^2{q}_v^2+\dots {\left(\frac{\partial Z}{\partial w}\right)}^2{q}_w^2},$$

(12)

where $r,\dots ,v,\dots ,w$are measured quantities with uncertainties$q_r,\mathrm{\dots }{q}_v,\mathrm{\dots }{q}_w$, and the measured values are used the compute the function$Z\left(r,\dots ,v,\dots w\right)$.

The maximum uncertainties of the kinetic Reynolds number$R{e}_{\omega }$and the friction factor$f$are 1.55% and 8.9%, respectively, with a 95% confidence level.

The relative uncertainties depend on the frequency $Fr$and vary from:

• 0.72% to 1.55% for the kinetic Reynolds number $R{e}_{\omega }$;
• 7.6% to 2.5% for the pressure drop$∆P$;
• 8.9% to 3.6% for the friction factor$f$.

2.4. Experimental Protocol

The tests were performed for different structures of metallic regenerators (Str_30%, Str_35% and Str_40%) and two lengths (L₁ = 60 mm and L₂ = 70 mm). These tests were operated with ten frequencies from 0.65 Hz to 5.88 Hz, two strokes of the piston (C₁ = 64.4 mm and C₂ = 128.4 mm), and two thermal axial temperature gradients (∆T₁ = 70 °C/10 °C; ∆T₂ = 40 °C/10 °C).

Table 2. Test conditions for alternating flow.

Structure of Regenerator	Porosity ε	Length L (mm)	Stroke C (mm)	Frequency Fr (Hz)	$\mathit{R}{\mathit{e}}_{\mathit{\omega } }=\frac{\mathit{w}{\mathit{D}}_{\mathit{h}}^2}{\mathit{\nu }}$	Thermal Condition THot–TCold (°C)
Str_30%	0.3	60 or 70	64.4 or 128.4	0.65–5.88 (10 values)	0.0083–0.0732
Str_35%	0.35	60 or 70			0.0145–0.125	70 °C–10 °C or 40 °C–10 °C
Str_40%	0.4	60 or 70			0.0164–0.139

includes all the tested parameters in our study. Experiments were conducted for each combination of parameters for a given regenerator. The results are discussed in the next section.

3. Results and Discussion

In the present section, the exploitation and analysis of the measurements data are presented in order to investigate the overall thermo-fluidic behavior of the alternating flow through the regenerator. In the presented experiments, the hot heat exchanger (HHEX) temperature was set at 70 °C, while the cold heat exchanger (CHEX) temperature was set at 10 °C. The piston stroke was set at C₁ = 64.4 mm. We start by analyzing the temperature before moving on to the velocities and then the pressures.

3.1. Evolution of Side Temperatures

In this paper, we only present the temperature evolution measured at each side of the regenerator. The effect of frequency on the temperature is plotted in Figure 7 for three types of regenerators, Str_30%, Str_35%, and Str_40%, at a length of L₁ = 60 mm.

At low frequency, the fluid temperature in the hot side increases in amplitude due to the heating effect as the piston motion pushes fluid from the hot side to the cold side of the regenerator. The fluid temperature in the cold side also increases due to the heating effect from the hot side. The temperature fluid drops when the piston motion pushes fluid from the cold side to the hot side. It should also be noted that fluid temperatures reach their maximum value earlier for Str_35% and Str_40% regenerators than for the 30% one at low frequencies. Moreover, maximum temperatures are reached before the mid-period.

When increasing the rotation velocity of the motor, the maxima are reached for a rotation angle closest to 180°. The fluid maximum temperature reaches 80 °C on the hot side and 35 °C on the cold side (Figure 7). These temperatures are 10 and 25 °C higher than those of the hot and cold exchangers, respectively. This is due to compressibility effects (pressure and, therefore, density) implied by alternations, as can be seen in Section 3.3 and Section 3.4. The evolutions of the maximum temperatures of the hot side and cold side for the three types of regenerator are out of phase by about 180°. Neither geometry nor porosity affects this value. This was also observed by Bonnet [34] and Gheith [35] in studies on heat engines operating on alternating flow.

3.2. Evolution of Side Velocities

Figure 8 shows the velocity evolution measured by the hot-wire sensor. One can clearly see the outward and return phase. However, one can also describe the cycle as a combination of compression and expansion phases. Those phases occur between the two successive minimum velocity values. The compression phases match the hot to cold flow displacement where velocity should be counted positively, whereas the expansion phase is associated with the cold to hot displacement where velocity should be counted negatively. Typical velocity evolutions on each regenerator’s sides are shown on Figure 8a–c for porosities of 30%, 35%, and 40%, respectively. Experiments were conducted for the same frequency (close to Fr ≈ 0.7 Hz) that corresponds to the lowest tested frequency. The relative uncertainties depend on the frequency Fr and vary from ±1.16% to ±2.33% for the Str_30% regenerator, whereas it is almost constant at ±2.30% for the Str_35% and Str_40% regenerators.

The first observation is that the amplitude of the velocity of the fluid in the hot side is less when the piston pushes the fluid from the hot side to the cold side of the regenerator during first half cycle. This is because the volume of fluid during this phase is subjected to compression generated by the movement of the piston. As a result, the density increases under the effect of the increase in pressure. In addition, the regenerator plays the role of an obstacle, slowing the flow of the fluid as a result of head losses. At the same time, the fluid in the cold side of the regenerator is decompressed. Its velocity increases when fluid exits the regenerator. We can also observe that exiting velocities are more disturbed, implying turbulence generation in the exiting flow. Measured axial velocities are similar to those obtained by Hino et al. [16], Simon and Seume [11], Zhao and Cheng [15], and Xiao et al. [9]. During the second half cycle, fluid passes from the cold to the hot side of the regenerator, and the reverse situation occurs.

The second observation is that the minimum velocity, obtained with the structures Str_35% and Str_40%, do not correspond to the moment when the piston is in the top dead center (TDC) extreme position. The absolute speed reaches its minimum value at a rotation angle of about θ = 15° and 190° (instead of 0° and 180°). This divergence also exists with the Str_30% structure and is more significant. The explanation may lie in the geometry of the Str_30% structure, which generates a large variation in density (the effect of gas compression) as a result of its more compact and disturbing structure.

3.3. Evolution of Side Pressures

Figure 9 shows the pressures curves at each regenerator side. The pressure amplitude is weakly sensitive to frequency for the Str_30% structure (Figure 9a). For the Str_35% and Str_40% structures, the pressure amplitude increases by approximately 0.2 bar between the lowest and highest frequency (Figure 9b,c). This corresponds to 20% of the average pressure. This phenomenon is probably due to the fact that the compressibility of the fluid in the presence of the Str_30% structure is greater than with the other two structures.

In Figure 9b,c the pressure maximum amplitude on either side of the regenerators (Str_35% and Str_40%) does not match the predicted theoretical temporal forecasts. The pressure reaches its maximum value for θ = 120° at a low frequency. For higher frequencies, the maximum is obtained just before the neutral point (aka TDC), i.e., θ = 180° instead of θ = 90°. This shift also occurs for Str_30% (Figure 9a), regardless of the frequency, and always for θ = 180°. This phenomenon is probably due to compressibility, which is greater for the Str_30% regenerator. This could explain the most disturbed structure and the lowest porosity. These results are confirmed by those obtained at different frequencies. This difference between the theoretical and experimental results was also experimentally observed by Zhao and Cheng [8].

3.4. Effect of Air Compressibility

In this section, we take interest in the gas compression effect of alternating fluid flow. We carried out isothermal flow tests (no initial temperature gradient) in order to solely investigate the fluid gas compression effect due to the regenerator’s presence.

The ideal gas model is used to approximate gas density. The resulting equation for density estimation is:

$$\rho =\frac{P}{rT} ,$$

(13)

where $r=287{\mathrm{J}\mathrm{kg}}^{-1}{\mathrm{K}}^{-1}$for the air.

The effect of frequency on the gas density $\rho$ at both ends of the regenerator for a porosity of ε = 40% is plotted in Figure 10a (similar results are obtained for other porosities; Kahaleras et al. [34]). We observe an increase in the gas density at high frequencies. We also observe a significant variation in the density (1.08 kg m⁻³ < ρ < 1.50 kg m⁻³), even if the measurements were carried out for isothermal flow.

The density variation ratio for the three types of tested regenerators is calculated with respect to the average density${\rho }_{avg}$over a complete cycle, which is defined by Kahaleras et al. [34]:

$$\frac{∆\rho }{{\rho }_{avg}}(\%)=\frac{{\rho }_{max}-{\rho }_{min}}{{\rho }_{avg}}100,$$

(14)

where ${\rho }_{min}$and ${\rho }_{max}$are the minimum and maximum densities over a complete cycle, respectively.

We can observe that the density variation ratio is significant at a high frequency for any structure (Figure 10b) as it reaches 30%. For the most disturbed structure (Str_30%), this compressibility is already obtained at low frequencies. For Str_35% and Str_40% structures, we observe that this ratio increases with frequency until it stabilizes at a maximum value (28%), beyond which this ratio remains almost constant.

3.5. Friction Factor Evaluation

As we have seen, in alternating flow, all the flow parameters (pressure, velocity, and temperature) vary over time. One can therefore determine the friction factor$f$at a particular time $\left(\theta \left(0\right), \theta \left(u_{max}\right),\dots \mathrm{\dots }\right)$. The literature study showed that many authors, such as Tanaka et al. [4], Hsu et al. [35], and Jin and Leong [27], determine the friction factor at the time of maximum velocity as:

$$f\left(u_{max}\right)=\frac{\mathsf{\Delta }P\left(u_{max}\right)D_h}{\frac{1}{2}\rho \left(u_{max}\right)u{\left(u_{max}\right)}^{2}L},$$

(15)

where the velocity $u$is obtained by dividing the frontal velocity by porosity$\epsilon$.

We first determine the friction factor$f$on the basis of this relationship. In a second step, we discuss the suitability of this relationship for an alternating flow.

In the formula for calculating$f$, the flow parameters, such as density$\rho$and pressure drop$\mathsf{\Delta }P$, are determined at the position of maximum velocity$\theta \left(u_{max}\right)$ and denoted$\rho \left(u_{max}\right)$and $\mathsf{\Delta }P\left(u_{max}\right),$ respectively. These quantities are calculated during the compression phase (when the hot gas enters the regenerator) at the time when the fluid speed reaches its maximum value.

Figure 11a–c show the evolution of the friction factor $f\left(u_{max}\right)$ as a function of the kinetic Reynolds number$Re\left(u_{max}\right)$for different regenerator structures. Since the tested friction factor is a function of the velocity and pressure drop, the uncertainties of the tested friction factor, denoted by the error bars in Figure 11, are estimated by the law of propagation (Equation (11)). The largest uncertainty in the friction factor was computed to be approximately 5.50% for Str_30%, 7.60% for Str_35%, and 8.9% for Str_40%. We note that the uncertainty for the Str_30% regenerator is lower than that of Str_35% and Str_40% structures.

In addition, the friction factor seems to decrease with the frequency of the Reynolds number for the regenerator of porosity 30%, despite some dispersion of values. This result is consistent with the works of Ibrahim et al. [36] on complex geometry regenerators. Nothing like this is clearly visible for other regenerators. Indeed, we observe that it is almost constant (less than 0.05% dispersion).

These dispersions are due to the velocity patterns being relatively flat around the maximum value (Figure 8). This leads to uncertainty in the determination of $\theta \left(u_{max}\right),$ which can have a significant impact. Indeed, the pressure drop$\mathsf{\Delta }P\left(u_{max}\right)$measured for maximum fluid velocity time can vary considerably over a small range of rotation angles $\theta$(Figure 12).

This implies potential substantial dispersion in the friction factor $f$values. Although it is difficult to determine clear trends, there is a real quantitative difference in the friction factor depending on the type of regenerator. Indeed, if it seems logical that the porosity impacts the friction factor (ε decreases, $f$increases), it can be seen that for identical porosity variations of 5% between regenerators, the friction factor changes significantly in the first case (passage from 30% to 35%) and very little in the second (passage from 35% to 40%). This is due to the difference in structure geometry. The Str_30% regenerator is the only one with a pyramid structure that is much more disruptive than the channel structures of the Str_35% and Str_40% regenerators. Thus, the 30% to 35% evolution is more relevant than the 35% to 40% one.

We have seen that the determination of$u_{max}$in our installation is a source of dispersion. Conversely, the maximum pressure drop is easily detectable (Figure 12).

We therefore propose to calculate the pressure drop coefficient related to the maximum pressure drop:

$$f\left(∆P_{max}\right)=\frac{\mathsf{\Delta }P\left(\mathsf{\Delta }{P}_{max}\right)D_h}{\frac{1}{2}\rho \left(\mathsf{\Delta }{P}_{max}\right)u{\left(\mathsf{\Delta }{P}_{max}\right)}^{2}L}.$$

(16)

The results obtained and presented in Figure 13a–c show a clearer trend for each of the regenerators. We observe that the effects of dispersion have therefore been attenuated by the choice of the maximum pressure drop$\mathsf{\Delta }{P}_{max}$as a study parameter. The evolution of the friction factor seems to consist of two distinct phases. At low kinetic Reynolds numbers$\left(R{e}_{\omega }<0.04\right),$the friction factor seems almost constant. When the kinetic Reynolds number is sufficiently large$\left(R{e}_{\omega }>0.04\right),$the friction factor increases with the kinetic Reynolds number.

3.6. Pressure-Drop and Velocity Phase Shift

In this subsection, the evolution of pressure drops $\mathsf{\Delta }P$ and velocities are compared. The pressure drop is calculated using the following formula:

$$∆P=\left|P_{hotside}\left(\theta \right)-P_{coldside}\left(\theta \right)\right|,$$

(17)

where $P_{hot side}\left(\theta \right)$and $P_{cold side}\left(\theta \right)$are pressures at the hot and cold inlet sides of the regenerator, respectively.

In Figure 14, different evolutions are plotted together as a function of the angle of rotation$\theta$for the three regenerator structures (Str_30%, Str_35%, and Str_40%). As fluid velocity shows an almost constant plateau around the maximum velocity$u_{max }$ over a significant rotation angle interval, measuring phase shift between the pressure drop and velocity maximum appears hazardous.

Substantial uncertainties in the calculation of the angular phase difference between the head loss and velocity are probable. Therefore, it appears to be more justified to determine this phase shift between the minimum values. Furthermore, as experiments were conducted for different frequencies and regenerators, we gathered results on a single phase shift versus kinetic Reynolds number $R{e}_{\omega }$ (defined in the Introduction) pattern.

Those evolutions in phase shift $\varphi \left(\mathsf{\Delta }P, u\right)$are presented for all regenerators regarding:

• Strokes of the piston (C₁ = 64.4 mm and C₂ = 128.4 mm) in (Figure 15a).
• Temperature gradient (ΔT₁ = 70 °C/10 °C and ∆T₂ = 40 °C/10 °C) in (Figure 15b).
• Regenerator length (L₁ = 60 mm and L₂ = 70 mm) in (Figure 15c).

Figure 15 shows that the phase shift between the pressure drop and velocity is significant and can reach 71° at high Reynolds frequency numbers for any structures. We can also observe that the phase shift increases linearly with the kinetic Reynolds number, regardless of the configuration. This means that there are negligible effects of displacement, regenerator length, and temperature gradient on the phase shift in our investigated parameter area. This confirms the theoretical and experimental results of Simon and Seume [11], Khodadadi [21], Zhao and Cheng [8], and Isshiki et al. [23]. However, the above results show that the phase shift depends on the structure of the regenerator.

Thus, all the results have been plotted for each regenerator for all the corresponding experiments independently of other parameters (length, stroke’s motion, and temperature gradient). The obtained graphs are presented in Figure 16.

In each graph in Figure 16, the phase difference appears as a growing linear function of the oscillating Reynolds number $R{e}_{\omega }$for each regenerator. Seeking a law confirms this observation, as the regression curves added to the figure proves it. Regressions are given in

Table 3. Phase shift correlations between the pressure drop and speed for different regenerator structure.

Structure of Regenerator	Dh (mm)	Correlation Model	α	R-Square
Str_30%	0.178	φ(ΔP,u) =αReω	732.93	0.98
Str_35%	0.237		432.91	0.99
Str_40%	0.250		397.63	0.98

, assuming that at zero frequency a phase shift should be null.

If we relate the phase shift ${\varphi }_i$of a regenerator of any structure (i) to the phase shift of the 30% porosity regenerator${\varphi }_{30\%}$_, the following relationship may be written:

$$\frac{{\varphi }_i}{{\varphi }_{30\%}}=\frac{{\alpha }_{i}R{e}_{\omega ,i}}{{\alpha }_{30\%}R{e}_{\omega ,30\%}}=\frac{{\alpha }_i}{{\alpha }_{30\%}}\frac{{\rho }_i{\omega }_i{D}_{hi}^2}{{\rho }_{30\%}{\omega }_{30\%}{D}_{h30\%}^2}=\frac{{\alpha }_i}{{\alpha }_{30\%}}\frac{{\rho }_i{\mu }_{30\%}}{{\rho }_{30\%}{\mu }_i}\frac{D_{hi}^2}{D_{h30\%}^2}.$$

(18)

Furthermore, the ratio $\rho /\mu$is quite constant; it is then possible to write:

$$\frac{{\varphi }_i}{{\varphi }_{30\%}}=\frac{{\alpha }_i}{{\alpha }_{30\%}}\frac{D_{hi}^2}{D_{h30\%}^2}.$$

(19)

The calculations of those ratios for regenerators Str_35% and Str_40% are given in

Table 4. Slope ratio and hydraulic diameter inversed ratio.

Structure of Regenerator	αi/α30	(Dh30%/Dhi)2
Str_35%	0.59	0.56
Str_40%	0.54	0.51

. It appears that they are the inverse of each other.

$$\frac{{\alpha }_i}{{\alpha }_{30\%}}\cong \frac{D_{h30\%}^2}{D_{hi}^2}.$$

(20)

Thus, Equation (20) leads to:

$$\frac{{\varphi }_i}{{\varphi }_{30\%}}=\frac{{\alpha }_i}{{\alpha }_{30\%}}\frac{D_{hi}^2}{D_{h30\%}^2}=1,$$

(21)

meaning:

$${\varphi }_i\cong {\varphi }_{30\%}.$$

(22)

This result proves that the phase shift is only frequency-dependent in our experiments. When plotting the experimental phase shift measurements versus frequency altogether, Figure 17 is obtained.

The arrangement of the experimental measurement points thus obtained seems to confirm the previous result.

In order to obtain correlations depending on characteristics of the regenerators, a modified phase shift${\varphi }^{*}$as a new dimensionless parameter (as defined by Choi et al. [20]) is introduced to obtain the proper following:

$${\varphi }^{*}\equiv \varphi {\left(D_h/L\right)}^{2.62}$$

(23)

Thus, all our results are plotted for each regenerator for all the corresponding experiments independently of other parameters (length L, stroke of the piston C, and temperature gradient ∆T). The proposed equation is also plotted on the graph for comparison in Figure 18.

Therefore, the experimental modified phase shift ${\varphi }^{*}$is well fitted by the following correlated equation, as shown in Figure 18. The linear regression plot definitively validates it, as the R-square of the fitting curves is 0.97 with an error of ±2.39%.

Thus:

$${\varphi }^{*}=2.12\times {10}^{-4}R{e}_{\omega }=\varphi {\left(D_h/L\right)}^{2.62}.$$

(24)

Therefore, in our case, a correlation equation for phase shift$\varphi$between the velocity and pressure drop is proposed for our experimental data:

$$\varphi =2.12\times {10}^{-4}R{e}_{\omega }{\left(D_h/L\right)}^{-2.62},$$

(25)

which is valid in the range of$\left(0.0025\le D_h/L\le 0.0041;0.0057\le R{e}_{\omega }\le 0.177\right)$.

Equation (25) shows that the phase shift between the velocity and the pressure drop is governed by the hydraulic diameter to length ratio $D_h/L$ and the kinetic Reynolds number$R{e}_{\omega }$. It indicates that the effect of the kinetic Reynolds number $R{e}_{\omega }$on the phase shift is more dominant than that of the hydraulic diameter to length ratio $D_h/L$.

4. Conclusions

This study was focused on our understanding of the physical parameters characterizing alternating flows at different frequencies submitted to a temperature gradient and passing through different metallic regenerator structures that can be found in Stirling cycle thermal machines, and in so-called alternative machines. These investigations were experimentally performed using a specific test bench, allowing for the measurement of instantaneous pressures, velocities, and temperatures at both sides of the regenerator over several cycles.

From the results presented above, the following main conclusions may be drawn:

• The dynamic characteristics of the regenerator are finally determined from the instantaneous measurements of the pressures, velocities, and temperatures on each side of our regenerators. Temperature measurements required the development of microthermocouples of 12.7 µm (type K). They were manufactured in our laboratory and characterized both statically and dynamically.
• The effect of the compressibility of the working fluid (variation of the density) in our experimental device is not negligible, which brings us closer to the conditions actually generated in thermal machines operating an alternating regime.
• The friction coefficient is calculated using a classical correlation at the time of maximum velocity. The friction factor$f$seems to decrease with the kinetic Reynolds number $R{e}_{\omega }$for a 30% porosity regenerator. For the other tested regenerators, we observe that it is almost constant with a little dispersion. To minimize the dispersion effects, another definition was proposed to calculate the friction factor at the time of maximum pressure drop. The results then show, for the three regenerators, a single trend that is clearer as a function of the frequency Reynolds number.
• A new correlation for the phase shift between the velocity and pressure drop of different regenerator structures was obtained. It was found that the effect of $R{e}_{\omega }$ on the phase shift is more dominant than that of the hydraulic diameter to length ratio $D_h/L$. This correlation equation is useful in the design of regenerators in a Stirling machine or alternating flow-based devices.

In future, this experiment will be carried out for other regenerators, especially those with different porosities and shapes (mesh, grid, and channel). In order to do so, the test bench will be enhanced. We also aim to determine the energy balance on the regenerator and on each heat exchanger. The last improvement will be the temperature measurement inside and at the wall of the regenerators.