Regression Model of Dynamic Pulse Instabilities during Condensation of Zeotropic and Azeotropic Refrigerant Mixtures R404A, R448A and R507A in Minichannels

Abstract

This paper presents experimental research and mathematical modeling data concerning the impact of unit dynamic instabilities on the phase-transition condensation processes of the zeotropic mixtures R404A and R448A and azeotropic R507A refrigerants in pipe minichannels. The R507 refrigerant is currently used as a temporary substitute for R404A, whereas R448A is a sustainable prospective substitute for R404A. The study presents experimental testing data for the condensation processes of these refrigerants in pipe minichannels and a proposal for the use of dimensional analysis, including the Π-Buckingham theorem, to determine the regression relationship explaining the propagation of unit dynamic instabilities. Based on the experimental studies performed, regression computational models were developed and showed satisfactory agreement in the range of 20% to 25%. They give the possibility to identify, in a utilitarian, way the speed of propagation of temperature and pressure instabilities during the liquefaction of refrigerants. The study was carried out on pipe minichannels with an internal diameter of di = 3.3, 2.3, 1.92, 1.44 and 1.40 mm.

1. Introduction

In the broadly understood field of application of energy machines and devices, various thermodynamic factors are used to carry out transitions and cycles. Refrigerants occupy a special place among these factors and are used in various applications, including refrigeration, air-conditioning, heat pumps and distributed energy systems, including the ORC (Organic Rankine Cycle). The correct selection of these factors has posed a problem for many years, not only in technical terms but also in ecological and economic terms. The issues related to the safety of the refrigerants used are also significant. The lack of clear global legal regulations is not conducive to obtaining solutions in the foreseeable future. It is now known that the use of some factors will be prohibited or substantially limited in upcoming years. This restriction applies, in particular, to the so-called ecologically harmful F-gases.

From a thermodynamic point of view, the “perfect” refrigerant to be used in a circuit should have the following parameters [1,2,3,4]:

• Low normal boiling point (t_sat),
• High critical temperature (t_cr) and pressure (p_cr),
• Low freezing point,
• Stable properties throughout the operation range,
• Low specific volume,
• Low viscosity,
• Chemically inert regarding the structural materials of the system,
• Low water solubility,
• Nontoxic and nonexplosive,
• High heat-transfer coefficient in phase transitions,
• Low unit cost,
• Sustainable impact on the environment.

Unfortunately, none of the currently proposed substances meet all of these requirements. It should be emphasized that, at present, the priorities for the refrigerant selection criteria have changed and are used in the following order [5,6,7,8,9,10]:

• Environmental impact criteria,
• System operational safety criteria,
• Thermodynamic criteria,
• Technical criteria,
• Economic criteria.

In previous decades, the basic criteria used for refrigerant selection primarily involved its thermodynamic properties.

Under the legal regulations set out in the Montreal Protocol and Regulation of the European Parliament and Council no. 517/2014 of 16 April 2014 [11], fluorinated greenhouse gases (known as F-gases), particularly hydrofluorocarbons, perfluorocarbons, sulfur hexafluorides and other refrigerants containing fluorine, shall be withdrawn from use. This group of gases also includes a group of commonly used refrigerants, such as R401A, R404A and R407C.

At present, the measures of the destructiveness of refrigerants toward the environment are two basic indices, i.e., the GWP (global-warming potential) and ODP (ozone-depletion potential). These indices specify the impact of a given chemical on the destruction of the stratospheric ozone layer and an increased greenhouse effect.

The current state of knowledge on the thermodynamic properties of the proposed F-gas substitutes is limited in certain cases only to the information provided in commercial studies that have been made public (sometimes to a limited degree) by their manufacturers. However, there are no studies or widely documented publications describing the behavior of F-gas substitutes, e.g., during the boiling and condensation phase transitions occurring in conventional channels and in minichannels under steady and unstable conditions.

The current state of knowledge enables predicting the effects of the influence of some dynamic instabilities (generated, e.g., individually or periodically) on the condensation phase-transition processes of fluorinated refrigerants in pipe minichannels. Substantial changes in the regulatory framework and requirements for the use of new chemicals are the reason for analyzing the current trends regarding the discussed topic [12,13,14,15,16].

Therefore, expansion of the state of knowledge (especially considering the influence of unit instabilities) in the indicated area into substitutes for fluorinated refrigerants is now highly recommended.

Research on the impact of periodic dynamic instabilities in systems with phase transitions on two-phase condensing flow in minichannels has already been carried out and described for currently withdrawn refrigerants [12,14,15,16,17,18,19,20,21,22,23].

It is known that, under conditions of dynamic instability, the transfer of wave-like disturbances occurs during refrigerants’ flow condensation used thus far. Two-phase media are particularly susceptible to this phenomenon. A special role is assigned to the transmission of disturbances in the form of acoustic waves or wave changes of other sizes, including mass and heat flux density. All of these instabilities are distinguished by specific properties, including a substantially different propagation velocity in the flow [3,24]. The irreversible phenomena of dissipative and dispersive interactions occur in both of these mechanisms. Dissipative interactions involve problems with the irreversibility of the process (leading to increased entropy) and suppression of the propagation of disturbances in two-phase media. The dispersion properties of a two-phase medium affect the dependence of the propagation of disturbances on the frequency of their generation. The dynamic instabilities observed in single- and multiple-phase media occurring during condensation in the minichannel flow may assume different forms. This result stems from the nature of the disturbances that produce the instabilities, which depend on the following characteristics:

• The conditions of the initiation of a phase transition,
• Interactions related to dynamic changes in the mass flux density,
• Flow rate,
• Pressure change,
• Temperature change,
• Phase composition change,
• Conditions for the creation of the so-called shock wave,
• Interference interactions of surface waves (the so-called Kelvin–Helmholtz instabilities).

The occurrence of wave phenomena is characterized by the interdependence of all parameters specifying the system status. For example, static or dynamic growth (or decreases) in the mass flow in a system produces changes in basically all other parameters, such as the density, pressure, temperature, and saturation. The occurrence of disturbances caused by a change in one parameter typically leads to the formation of further disturbances and their overlap. An increase in the (particularly wave-like) propagation of disturbances may also occur. This study proposes a qualitative and quantitative description of the phenomena accompanying condensation under dynamic instability conditions for the sustainable substitutes of currently used refrigerants in pipe minichannels.

The conclusions derived from the theoretical and experimental analysis of the instability phenomena occurring in phase transitions should foster the development of the existing methods of their description and expand the knowledge in this field.

Many problems related to the generation of dynamic instabilities exist, including those occurring during the condensation of refrigerants in conventional and small section channels that need to be augmented.

In general, two issues inspired the undertaking of the present study. The first issue is the dynamically increasing number of centers that conduct research on the instabilities in minichannels. The second issue is related to the introduction of new refrigerants.

The confusion linked to the need for replacing refrigerants (including F-gases) resulted in a sudden increase in interest in testing their substitutes. This fact produced numerous new problems. One such problem includes the need to adapt the construction of heat exchangers whose structure is based on low-diameter channels. This basic requirement is needed so that such devices would work under stable conditions, and the determination of the range of stable operation generates the need to examine the impact of instability on their functioning.

In the analysis of the literature discussed in this study, regarding the condensation phase transition of the R404A refrigerant in minichannels, attention was drawn to the lack of sufficient information on the use of its proposed substitutes, i.e., R507 and R448A. This deficiency is due to the wide application of these working agents in areas related to heat pumps, refrigeration and air-conditioning, which also occur in stationary and mobile facilities.

2. Impact of Instabilities on Operating Medium-Phase Transitions

Analysis and the experimental studies of the authors of this study has shown that unstable interactions negatively affect the efficiency of boiling and the condensation phase transitions of refrigerants [25,26,27,28,29,30].

One of the phenomena causing negative effects is the so-called feedback manifested, for example, by the interaction of impulsive changes in the flow of the thermodynamic medium on the mechanical elements of the system, causing their vibrations. The critical heat flux (CHF) phenomenon may appear as a result of these interactions and is responsible for inducing impulsive changes in the power of a given system. As a consequence, disturbances in the control systems occur, which may lead to the failure or destruction of a given device. The mentioned phenomena occur in the clockwise cycles of thermodynamic heat engines and in the cycles of anticlockwise thermal working machines (cooling devices, heat pumps, etc.).

Considering the various mechanisms that cause instabilities, some characteristic types can be identified, which are most frequent in the transformation of machines and devices. Differences may exist in the way the instability propagates, the impulsive mode, the extent of its occurrence, and the methods for predicting them. Moreover, significance is attributed to the effect (influence) of the geometry of the flow channels and other elements of the installation.

Introducing the correct classification of the various categories of instability, which will allow a better understanding of their formation mechanism and control possibilities, poses an important problem.

2.1. Static Instabilities

As with dynamic instabilities, complex static instabilities can arise (in phase transitions) from oscillatory or impulsive nucleation interactions. The main difference between these instabilities is the ability to terminate the increase in static-induced oscillations by reversing the direction of flow.

The oscillation increase in the case of static instabilities has a finite value that is the outcome of reaching the so-called boundary cycle amplitude, which, by assumption, cannot be higher than half of the channel hydraulic diameter [31,32]. It is worth noting where bidirectional oscillations of the change in operating medium flow direction were observed, taking from 20 s to 4 h [28]. Another observed phenomenon consisted of chaotic impulsive interactions deriving from the reversal of the flow direction in a closed circuit, in which heat exchangers were placed in the vertical position relative to the channel axis. These solutions are used in nuclear reactor cooling systems. A characteristic feature of these structures is the use of the so-called primary reactor cooling system with an intermediate system, where the heating energy is transferred to an external heating system. It was observed that the intermediate exchanger was the site where static instabilities occurred, resulting from, for example, a sudden increase in vapor bubbles or an unexpected appearance of condensate blocking the flow [26,33]. This phenomenon resulted in a temporary flow stop and directed the medium in opposite directions, separated by the so-called fluid or vapor bridges. Gravitational interactions dynamizing the flow in accordance with their direction of action play a significant role in this phenomenon. The propagation of “unidirectional” instabilities was observed in a system where heat exchangers were located horizontally relative to the channel axis [34,35].

These phenomena were observed in true systems, such as cooling systems in nuclear power plants, and were experimentally studied and described considering geometric parameters, i.e., the evaporator and condenser length [27,36]. These studies allowed the identification of the limit values of an impulsive water hammer as the occurring amplitudes of instability, which depend on the existing heating powers in relation to the conditions of heat reception. The results were used for the mathematical analysis and development of numerical computational codes for calculating the increase in the oscillation amplitude of the occurring instabilities [33].

The mathematical models developed in this case were used to control the operation of nuclear reactors. Subsequently, they were implemented in other systems in which similar unstable phenomena were observed in diminutive spaces, e.g., between the fuel and control rods of a nuclear reactor, through which the working medium flows. Experimental verification of the aforementioned calculation methods was carried out in relation to the determination of the stability range of the cooling process of wet toroidal transformers. The conducted research has shown the usefulness of the developed calculation methods for their descriptions of water hammer type instability [28,33,37]. There are similar results in the literature for flows in rectangular loops, the design of which in the case of compact heat exchangers is equivalent to that of cooling circuits of electronic devices [21,26,36,38,39]. Conducted numerical simulations that showed the hydraulic impulsive conditions for the implementation of phase transitions occurring in these systems [40]. The analysis of the simulations revealed the possibility of identifying the lower and upper limits of impulsive, as well as oscillating, instabilities. The lower limit is associated with unidirectional propagation of instabilities, whereas, with the increase in the system power, bidirectional (“reversed”) flows are observed. Exceeding the upper limit of the amplitude of instabilities results in these interactions having a chaotic, purely probabilistic nature. The described phenomena have their source in the nucleation interactions initiating the phase transitions of condensation and boiling of a thermodynamic factor [36].

2.2. Dynamic Instabilities

The dynamic instabilities in single- or multiple-phase systems are associated with density wave instabilities, a sudden oscillation or impulsive pressure and temperature change in a boiling or condensation phase transition. The characteristic feature is the often very low frequencies of these instabilities in single-phase systems, attaining 0.0015–0.005 Hz, while reaching 1–10 Hz in two-phase flows. This result stems from the lower velocity of propagation of disturbances in single-phase systems than that of disturbances in two- or multiple-phase systems [41].

The relationship between the dynamic instability propagation velocity and the frequency of their incidence stems from the so-called dispersive properties of two- or multiple-phase systems. The realization of transitions in real conditions is always linked to the possibility of “unsettling” their thermodynamic stability status and entering the area of unstable states. Under these conditions, internal processes appear to strive for restoration of the system balance. Their duration is referred to as the relaxation time and is comparable to the time in which changes in parameter values occur in a medium during the flow in the conditions of unstable interactions.

The term “relaxation time” refers to the balanced state of a system, that is, without interactions with the environment (isolated), yet the phenomenon can be referred approximately to transients. Relaxation processes consequently lead to medium dispersity, which is reflected by the propagation of disturbances at the phase velocity calculated from the relationship [3,24,42,43,44]:

$$v_f\left(\omega \right)=\frac{\omega }{k\left(\omega \right)}$$

(1)

where ω is the circular frequency (pulsation) of a disturbance: ω = 2πf. The value f deter-mines the number of cycles of a periodic phenomenon occurring in a unit of time, where $f=\frac{1}{T}$, where T is the period between the occurrence of the same phase in a vibrating movement. The physical term, T, is the wave period equal to the period of propagating vibrations, whereas k is the wavenumber defined by the following relationship [3,24]:

$$k=\frac{2\pi }{\lambda }$$

(2)

where λ is the wavelength.

The value of the phase velocity, v_f, is strictly linked to the circular frequency, ω, which is identified as a pulsation. In systems where the instabilities are caused by nucleation interactions, this pulsation refers to the pulsation growth of the vapor bubbles during the boiling process or the condensate droplets during condensation. In the case of instabilities resulting from dynamic interactions, the circular frequency refers to the waves of the condensate layer at the interface or the oscillating changes in the flow structure during the boiling process. There are two types of phase velocities, the so-called equilibrium velocity and frozen velocity (Figure 1) [24,45].

If the frequency (pulsation) of the disturbances is low (ω → 0), then we are considering the equilibrium velocity v_r = v_f(ω → 0). The state of the system in the conditions of instabilities traveling at this velocity is characterized by a time(of changes in the flow) that is much greater than the relaxation time. It also means that the system can reach a new state of thermodynamic equilibrium: θ(t >> θ), where θ is the relaxation time.

If disturbances with high frequencies (pulsations) occur, that is, ω → ∞, then we are dealing with the frozen velocity: v_zr = v_f(ω → ∞). Then the system status is characterized by a very short relaxation time (t << θ) approaching zero, meaning that it cannot “follow” the created changes.

In practice, instabilities with frequencies reaching infinitely large amplitudes cannot be generated. Increasing the frequency of the generated instabilities reduces the wavelength. This result necessitates the identification of the limit values of phase velocities, which particularly concerns the frozen speed v_zr. The literature analysis showed that, for conventional channels and minichannels, v_zr is determined by using the wavelength of instabilities propagating in a two-phase medium [24,46,47].

Based on the theoretical and experimental studies, it was established that, with pulsations greater than the limit frequency ${\omega }_g=\frac{2\pi }{\theta }$, a two-phase system can be treated as “frozen” for a relaxation time of θ = (1 − 0.01) s. The limit pulsation, ω_g, corresponds to the limit length of a disturbance wave, i.e., λ_g = (1 − 0.1) m, at a phase velocity of v_f ≈ 10 m/s. In two-phase flows, the frozen velocity, v_zr, is referred (in certain cases) to the speed of sound or critical equilibrium velocity [3,24,47].

Figure 1 shows the typical range of the equilibrium velocity and frozen velocity and their dependence on the frequency of their generation. For conventional channels, the following values of the instability propagation velocities related to the frozen velocities and the equilibrium velocities are recorded. According to the investigations, the propagation velocity of pressure turbulences remains in the range of v_p = 40–340 m/s, whereas the propagation velocities of temperature turbulences of front condensation are v_T = 0.2–6.2 m/s [48,49,50,51,52].

The study for the condensation process of the R134a and R404A refrigerants in pipe minichannels under the conditions of dynamic instabilities demonstrated that the velocity of the movement of the pressure change signal v_p, remains in the range of 40–205 m/s, whereas the velocity of the front condensation movement v_T, is 0.30–4.5 m/s [53].

Importantly, apart from dispersive interactions, the irreversible phenomena of dissipative interactions occur during condensation in channels under conditions of dynamic instability. Interactions of this nature involve problems with the irreversibility of processes, leading to an increase in entropy and a suppression of disturbance propagation in two-phase media. The dynamic nature of the condensation phase transition determines the state of the system in which physical wave phenomena or oscillations of parameter values are the source of energy losses due to friction or turbulence. The wave phenomena them-selves refer in this situation to the occurrence of correlations between instabilities of varying nature [3].

The number of studies describing the phenomena of instability in mini- and microchannels during the condensation phase transitions of the currently used refrigerants is relatively extensive, but there are no significant publications available on the subject of the impact of this type of interaction on the phase transitions of new sustainable refrigerants proposed as replacements for the ones currently withdrawn from use, especially F-gas substitutes [3,12].

This issue is the subject of this study, which considers the phenomena accompanying condensation in pipe minichannels under unstable conditions and the substitutes of refrigerants currently used and planned for withdrawal. A qualitative and quantitative description was carried out for alternative agents, and the test results were verified with regard to the refrigerants currently used in pipe minichannels under dynamic instability conditions.

The qualitative approach provided answers to questions concerning the determination of the influence of instability phenomena on parameters such as heat transfer and flow resistance. On the other hand, the quantitative approach allowed the velocity of pressure and temperature instability propagation to be determined.

This study presents the conclusions drawn from a theoretical and experimental analysis of these phenomena, which allowed the gaps in this area of knowledge to be filled.

The issues that have been considered are of a fundamental nature and relate to phenomena occurring in conditions of dynamic instabilities. Thus, the study included such issues as the influence of the following:

• Type of refrigerant on the condensation mechanism under instability conditions,
• Impulsive wave propagation velocity of pressure instabilities moving in the two-phase condensation process,
• Impulsive temperature instability propagation velocity moving in the form of front condensation,
• Implementation of the assumptions concerning the solution of the aforementioned questions required appropriate experimental testing under the conditions in which they were realized for the R404A refrigerant.

In the field of application, the developed models can be helpful for designers of heat exchangers especially compact ones built on minichannels in their design, taking into account impacts of hydraulic nature.

3. Subjects of Experimental Tests

The refrigerants R404A, R507 and R448A constituted the experimental test items. Presently, R404A is intended for withdrawal from use. R507 is already being used in its place, with the prospect of a complete replacement of these two agents by the new and sustainable R448A.

The R404A refrigerant, which was the reference in the experimental testing, is a mix-ture of 1,1,1,2-trifluoroethane, pentafluoroethane, and 1,1,1,2-tetrafluoroethane in the following composition: 52% R143a, 44% R125 and 4% R134a. According to The Bitner Company Report [10], R404A is widely used in industrial refrigeration for cold stores and freezers; in commercial refrigeration, such as for, refrigeration installations in shops; in air-conditioning units; and in refrigerated transport. Considering the laws on climate protection, its application will be restricted in the near future in favor of other refrigerants with a low GWP, e.g., R407F, R448A (Solstice N40) and R744. The R404A refrigerant has GWP = 3922 and ODP = 0, and it is currently used as a replacement for R502 and R22. On the other hand, alternative refrigerants for R404A are R507, R404F, R448A (Solstice N40) and R449A (Opteon XP40).

The currently used replacement for R404A is R507. The chemical composition of this compound is an azeotropic mixture of 1,1,1,2-trifluoroethane and pentafluoroethane in the ratio 50% R125 and 50% R143a. It is used as a liquefied pressurized gas. Currently, it is commonly used in synthetic refrigerants, and it is popular in industrial refrigeration for cold stores and freezers and in commercial refrigeration, including in refrigeration installations in stores. Other areas of its application include air-conditioning units and refrigerated transport. Due to its composition, it can be used for centrifugal compressors and in pump systems utilizing flooded evaporators. Considering the laws on climate protection, its application will be restricted in the near future in favor of other refrigerants with a low GWP, e.g., R407F, R448A (Solstice N40), R744, etc. This restriction stems from the value of GWP = 3985 at ODP = 0 for R507. Due to its parameters, R507 may be used as a substitute for the withdrawn refrigerants, e.g., R502 or R22 (or their substitutes), and as an alternative to R404A, R404F, R448A (Solstice N40) and R449A (Opteon XP40).

As stated by Schnotale [4], R448A is a zeotropic mixture developed for use as a substitute for R22 and R404A (R507) in existing systems or as a target agent in newly designed systems. Its scope of application includes industrial refrigeration, that is, cold storages, freezers and commercial refrigeration, including catering refrigeration systems. This refrigerant is considered a replacement for the R22, R404A and R507 refrigerants (or their substitutes), which are being withdrawn from the market, but its use requires an oil change and the replacement and adjustment of some components in the installation according to appropriate procedures. The R448 indices are as follows: ODP = 0 and GWP = 1386.

Due to its impact on the environment, R448A is treated as a sustainable chemical compound.

The studies of Kuczyński et al. [14,15,16] Teng et al. [26] and Zhang et al. [30] revealed the results concerning the determination of the impact of dynamic instabilities on the condensation of the R404A, R507 and R448A refrigerants. The present study provides the results of research leading to a regression model describing the propagation velocity of pressure and temperature instabilities in pipe minichannels resulting from dynamic impulsive interactions.

4. Experimental Test Methodology

The experimental testing took place under the conditionsfor the identification of the impulsive instabilities of condensation of R404A and R134a refrigerants in pipe minichannels [14,15,16]. In the case of R507 and R448A, the detailed parameters of the study were as follows:

The scope of the preliminary research covered the following:

• Refrigerants—R404A, R507 and R448A,
• Three internal diameters, d = 3.3, 2.3, 1.9, 1.44 and 1.40 mm,
• A single-channel configuration,
• Refrigerant mass flux G = 60–316 kg/(m²∙s),
• Refrigerant inlet pressure pin = 1.09–7.5 MPa (saturation temperature t_sat = 42.6–45.5 (°C)).

Figure 2 shows a view of the measurement section, a chart of which is shown in Figure 3.

Impulsive instabilities were generated with the use of electromagnetic cutoff valves (Figure 1). The experiment was performed with the use of a methodology enabling identification of the development and disappearance of the condensation process of the tested refrigerants.

To intensify the condensation process, the mass flux was increased by opening the supply valve with adjustable time, t. On the other hand, disappearance of the condensation process was obtained by suddenly closing the cutoff valve. It was determined that the minimum opening time in which the system reaction to the produced impulsive disturbance was observed amounted to t_o = 0.3 s [14,15]. The experiment was carried out in the valve opening and closing time of t_o = 0.3–3.5 s, increasing or decreasing this time by 0.05 s in subsequent measurements. Figure 4, below, presents the results of the testing concerning the discussed issue.

As a result of the sudden influx of refrigerant vapors to the minichannel, the pressure change signal is moved. In Figure 4, the range of temperature and pressure changes in the time intervals in which they were observed are shown by the lines.

The movement of pressure instabilities, v_p, is accompanied by a propagation of the temperature instability velocity, v_T. These motions are referred to as the front condensation, v_FC. It has been observed that, in a condensation process, the front condensation velocity, v_FC, is reduced and moves in the same direction as the flow of vapor in the minichannel. The decrease in velocity is caused by an increase in the frequency of generated disturbances. On the other hand, if the condensation process ceases, this instability moves in the opposite direction of the incoming vapor, which results in a reduction of the most effective true condensation area in terms of heat transfer.

Section 5 presents the methodology for constructing models for individual instabilities and refrigerants.

The values obtained for condensation front velocity, v_T, were in the range of 0.3 and 4.5 m/s, while the velocity of pressure-change signal, v_p, ranged from 40 to 205 m/s. Data presented by Bohdal et al. [17] showed that, in large diameter (conventional) channels, the condensation front velocity, v_T, was 0.2 and 1 m/s, while the pressure-change signal propagation velocity, v_p, was between 40 and 240 m/s. It should be noted that, in the case of minichannels, v_p is highly influenced by the internal diameter of the channel. The velocity of pressure-change signal propagation decreases with the diameter. The velocity of the condensation front, v_T, rises due to a smaller ∆T between the vapor phase and the mixture inside the minichannel.

5. Evaluation of Measurement Errors

On the example of experimental determination of velocity of movement of the signal of change (oscillation) of pressure, v_P, and velocity of movement of the signal of temperature change (condensation front), v_T, in the condenser (pipe mini channel), the procedure of determination of measurement uncertainty is presented. The values of v_Pi and v_Ti were characterized by a noticeable “scattering” from the mean values of ${\overline{v}}_P$ and ${\overline{v}}_T$, i.e., standard deviation.

The determination of the velocity of movement of the temperature and pressure change signal was linked with the determination of the time, t, where the signal temperature and pressure changes move along the length of the tubeminichannel. According to the point estimation, the measurements data were determined as follows:

• Average value estimator:

  $$\overline{x}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{x}_i}$$

  (3)
• Standard deviation estimator:

  $${\overline{S}}_{\overline{x}}=\sqrt{\frac{1}{\left(n-1\right)}{\displaystyle \sum _{i=1}^n{\left(x_i-\overline{x}\right)}^2}}$$

  (4)

  which is expressed as y_A—type A uncertainty of measurement.

It was assumed that the accuracy, q_α, for the confidence interval α = 0.9, defined as the absolute mean deviation for n = 19 measurements, and the values ∆t_T and ∆t_p will not be more than q_α = 0.4 of the estimator value, ${\overline{S}}_{\overline{x}}$. The q_α value is calculated as follows:

$$q_{\alpha }\stackrel{def}{=}\frac{t_{\alpha }}{\sqrt{n}}$$

(5)

where we have the following:

$$t_{\alpha }=\frac{\overline{x}-x_i}{{\overline{S}}_{\overline{x}}}\sqrt{n}$$

(6)

Tables were used [53] to specify that, for α = 1–0.9 and q_α = 0.4, the number of measurements rates are not less than 19. According to this tenet, it was specified that the number of experiments will be in the range of 19–120 for each minichannel.

To calculate the uncertainty of the measuring devices, a formula was used:

• Uncertainty of type u_B:

  $$u_B=\frac{{\mathsf{\Delta }}_g}{\sqrt{3}}$$

  (7)

  where ∆g is the limit of the error determined based on the instrument accuracy class, according to Equation (8):

  $${\mathsf{\Delta }}_g=\frac{Z\times K}{100}$$

  (8)

The A and B type standard uncertainties are used for the determination of the overall uncertainty of measurement and the overall standard uncertainty of measurement, which were determined as follows:

$$u_{Ł}=\sqrt{u_A^2+u_B^2}$$

(9)

where the standard uncertainty for the mean value takes the following form:

$$u_{\overline{y}}=\sqrt{{\displaystyle \sum _{j=1}^n\left(\frac{\partial y}{\partial x_i}\times u_{Ł}\right)}}$$

(10)

The standard uncertainty determined indirectly for y = f(x₁, x₂, x₃,…x_n) is obtained according to the Equation (11):

$$\mathsf{\Delta }y=\sqrt{{\left(\frac{\partial f}{\partial x_1}\mathsf{\Delta }{x}_1\right)}^2+{\left(\frac{\partial f}{\partial x_2}\mathsf{\Delta }{x}_2\right)}^2+{\left(\frac{\partial f}{\partial x_3}\mathsf{\Delta }{x}_3\right)}^2+\dots +\left(\frac{\partial f}{\partial x_n}\mathsf{\Delta }{x}_n\right)}$$

(11)

where ∆x = u_B, and the relative error is calculated according to the following:

$$\delta y=\frac{\mathsf{\Delta }y}{y}\times 100\%$$

(12)

When using a measuring card of a sampling frequency 10⁻⁶ Hz in measurements, it is possible to perform measurements in the amount of 10⁶/s. However, taking into account the limitations resulting from the technical parameters of the acquisition unit used in the tests, a maximum of 10³ measurements per second were included, which was a slowdown.

Endress–Hauser piezoelectric sensors with accuracy class k = 0.5% and range Z = 0 ÷ 1.6 MPa were used to determine pressure. The uncertainty of the measurements of u_A is 0.8 kPa:

$$\mathsf{\Delta }p=\sqrt{{800}^2+{\left(\frac{1600}{\sqrt{3}}\right)}^2}=1222.02 \mathrm{Pa}$$

(13)

Likewise, different values were calculated and presented in

Table 1. Summary of measurement errors.

Values Determined Experimentally	Standard Uncertainty of Measurement
Pressure, p	$\mathsf{\Delta }p=\sqrt{{800}^2+{\left(\frac{1600}{\sqrt{3}}\right)}^2}=1222.02 \mathrm{Pa}$
Mass flow, $\dot{m}$	$\mathsf{\Delta }\dot{m}=\sqrt[]{{0.005}^2+{\left(\frac{100}{\sqrt{3}}\right)}^2}\left(\frac{kg}{h}\right)$
Temperature, T	$\mathsf{\Delta }T=\sqrt[]{{0.05}^2\times {\left(\frac{0.05}{\sqrt{3}}\right)}^2}=0.06K$
Channel cross-sectional area A	∆A = 0.0000001 (m2)
Distance between sensors	∆l = 0.001 (m)
Indirectly determined figures	Calculated maximum error
Pressure differences, ∆p	$\mathsf{\Delta }p=\pm 0.0008\mathrm{Mpa}$
Movement speed signals change of pressure, vp, and temperature, vT	$\delta \mathsf{\Delta }v=\pm 6\%$
Density of the mass flux (G)	$\mathsf{\Delta }\left(G\right)=\pm 0.0022÷0.00042\frac{kg}{m^{2}s}$

.

6. Regression Model for Impulsive Instabilities Developed for the R404A, R507 and R448A Refrigerants

By using the regression function method, whose description is presented in the Appendix A, the formulas for the dimensionless velocity of pressure instabilities [16],

$$v_p^{+}=C\times {\mathrm{Re}}_{TPF}^a\times \mathsf{\Delta }{p}^{+}{}^b,$$

(14)

and temperature instabilities,

$$v_T^{+}=c\times {\mathrm{Re}}_{TPF}^a\times \mathsf{\Delta }{T}^{+}{}^{{}^b},$$

(15)

are proposed based on dimensional analysis. Equations (14) and (15) were reduced to linear form by the reciprocal logarithm:

$$log{v}_p^{+}=logC+a_1\times log{\mathrm{Re}}_{TPF}+b\times log\mathsf{\Delta }{p}^{+}$$

(16)

and

$$\mathit{log}{v}_p^{+}=\mathit{log}C+a_1\times \mathit{log}{Re}_{TPF}+b\times \mathit{log}\mathsf{\Delta }{T}^{+}$$

(17)

The calculation of the constant C and exponents a and b in Equations (14) and (15) was carried out by using a nonlinear regression model. To this end, computational modules available in the Statistica software (Koszalin University of Technology, Koszalin, Poland) license package were used. The adjustment calculus was performed for the equations built on the basis of the results of experimental research for the range of applied disturbances. With the use of Equation (14), the regression function was determined for the given refrigerant R404A and its substitutes, R507 and R448A. The constants and exponents were determined and are presented in

Table 2. Values for the regression model of pressure instabilities.

Model Concerns	Form of the Regression Model Formula
	Regression Model for Unit Pressure Instabilities		R404A	R507	R448A
	$v_p^{+}=C\times {\mathrm{Re}}_{TPF}^a\times {\left(\Delta p^{+}\right)}^b$
Sudden development of condensation process	Values of unknowns	C	302.96	4.67 × 109	327.8
		a	−0.48	−2.27	−0.52
		b	−1.05	1.33	−1.11
	r-Pearson significance coefficient		0.99	0.99	0.91
	Variance from σ2 population		98%	98%	96%
	Model compliance scope		±25%	±25%	±25%
Sudden disappearance of condensation process	Values of unknowns	C	85.95	34.6	16.46
		a	0.0095	0.12	0.18
		b	−1.034	−1.04	−0.80
	r-Pearson significance coefficient		0.97	0.96	0.97
	Variance from σ2 population		93%	91%	93%
	Model compliance scope		±20%	±20%	±20%

. Figure 5 presents these results in the form of relationships generated in Statistica software. It also includes the variance values and the significance coefficients obtained for the regression models of individual refrigerants considering the development and disappearance of the condensation process.

The dimensionless propagation velocity of pressure instabilities, $v_p^{+}$,is identified as the relation of the rate of change ∆p of the pressure signal to the propagation velocity of the refrigerant. Similarly, dimensionless temperature instabilities, $v_T^{+}$,are identified as the relation of the change ∆T of the duct wall temperature to the velocity of refrigerant travel in the minichannel.

For the variance σ² and r-Pearson significance coefficients of the given R404A, R507 and R448A refrigerants, the obtained deviation for calculation and experimental data was ±25%, as presented in Figure 6.

The calculated values of the unit temperature instabilities, $v_{Treg}^{+}$,were determined analogously to the pressure instabilities from Equation (15). The obtained values of the constants, exponents, σ² variance, r-Pearson significance coefficient and compliance range are presented in

Table 3. Values for the regression model of temperature instabilities.

Model Concerns	Form of the Regression Model Formula
	Regression Model for Unit Temperature Instabilities		R404A	R507	R448A
	$v_T^{+}=C\times {\mathrm{Re}}_{TPF}^a\times {\left(\Delta T^{+}\right)}^b$
Sudden development of condensation process	Values of unknowns	C	3.04 × 1014	7.58 × 1011	9.97 × 1011
		a	−2.49	−2.20	−2.28
		b	3.93	2.80	2.64
	r-Pearson significance coefficient		0.96	0.99	0.95
	Variance from σ2 population		92%	98%	96%
	Model compliance scope		±20%	±20%	±20%
Sudden disappearance of condensation process	Values of unknowns	C	101 × 109	4.67 × 109	1.54 × 109
		a	−2.47	−2.67	−2.13
		b	1.85	1.33	1.22
	r-Pearson significance coefficient		0.96	0.99	0.99
	Variance from σ2 population		89%	98%	98%
	Model compliance scope		±20%	±20%	±20%

.

Figure 7 presents the interdependencies of the parameters assumed from Equation (4), as generated from Statistica software. Additionally, for the variance σ² and r-Pearson significance coefficients of the given R404A, R507 and R448A refrigerants, the determined deviation of calculation and experimental data was ±20%, as presented in Figure 8.

7. Conclusions

This article presents the results of the research on modeling the propagation velocity of impulse dynamic instabilities during the condensation process in pipe minichannels obtained for the R404A, R507 and R448A refrigerants. The R507 refrigerant is a temporary sustainable substitute for R404A, with the prospect of it being replaced with R448A. An experimental study was carried out under the same conditions as those for R404A, and the following values were obtained:

• The pressure instability propagation velocity, v_p, remained in the range of 5–303 (m/s), which overlaps the range of 2–269 (m/s) obtained for the R404A refrigerant,
• For the results, a mathematical regression model was obtained that proposed a dimensionless number, $v_p^{+}$, which, when referring computational values to experimental values, provides satisfactory results within a 25% compliance interval at a variance of 94% and a significance coefficient of R = 0.97,
• The temperature instability propagation velocity, v_T, remained in the range of 1–4.9 (m/s), which overlaps the range of 1–5.2 (m/s) obtained for the R404A refrigerant,
• For the results, a regression mathematical model was proposed, obtaining dimensionless numbers describing the propagation velocity of pressure, $v_p^{+}$, and temperature impulsive instabilities, $v_T^{+}$, which, in relation to the experimental values, gives satisfactory results within the range of 20% to 25% of the compliance interval with the variance from 89% to 95% and the significance factor, R, in the range of 0.91 to 0.99.

In connection with the above findings, it can be concluded that the proposed models of regression functions sufficiently describe the velocities of impulse instabilities propagating in the minichannels both qualitatively and quantitatively.

Appendix A (below) presents the methodology for determining the values describing the regression function by means of dimensional analysis [53,54].