On the Influence of the Specific Dissipation Rate Distribution on the Efficiency of Mass Transfer in Apparatuses with Liquid-Phase Media

Abstract

A theoretical analysis of the influence of the distribution of the local specific energy dissipation rate on the specific interfacial area, the surface and volumetric mass transfer coefficients in apparatuses with heterophase processes and a liquid continuous phase, as well as the quality of mixing in apparatuses with homophase reactions in the liquid phase, is performed. It is shown that the average value of the specific energy dissipation rate over the volume of the device is not a full-fledged criterion for assessing the useful effect since it does not take into account, on the one hand, the local level of energy dissipation in the active zones and, on the other hand, the features of the flow structure and the local residence time in the active zones, depending on the geometry of the device and the method of energy input into it. Limiting cases are discussed: (1) uneven energy distribution in the presence of a small volume with a high specific dissipation rate and (2) ideally uniform energy distribution throughout the entire volume of the device. In the first case, a significant part of the volume is used inefficiently; in the second case, an excessive amount of energy is spent. In this regard, the concepts of dosed distributed energy input for long-term processes and maximum energy concentration in a microvolume for fast-flowing processes are considered.

1. Introduction

According to modern concepts, the specific energy dissipation rate ε (W/kg) is a defining characteristic of chemical or biotechnological equipment, both in traditional large-scale (including that used in absorbers, in biotechnological processes, and for wastewater treatment), for which the key mixing levels are macromixing and mesomixing [1,2], and for microscale devices, where micromixing is a key factor [1,3].

The specific energy dissipation rate is defined as the ratio of the power N dissipated per unit mass m of the processed medium (single-phase or multi-phase):

$$\epsilon ={\displaystyle \frac{N}{m}}={\displaystyle \frac{N}{\rho V}}$$

(1)

where ρ is the density of the processed medium, kg/m³; V is the volume of the processed medium, m³.

The local value ε_loc reflects the energy dissipation rate in a fixed elementary volume, which can formally be determined in the infinitesimal volume ΔV → 0:

$${\epsilon }_{\mathrm{loc}}=\underset{\Delta V\to 0}{\mathbf{lim}}{\displaystyle \frac{\Delta N}{\rho \Delta V}},$$

(2)

where ΔN is the elementary energy released per unit of time in the volume ΔV of the processed medium, W.

The average value of the specific energy dissipation rate ε_av is:

$${\epsilon }_{\mathrm{av}}={\displaystyle \frac{1}{{\rho }_{\mathrm{av}}V}}{\displaystyle \underset{V}{\int }{\epsilon }_{\mathrm{loc}}\rho dV}={\displaystyle \frac{N_{tot}}{{\rho }_{\mathrm{av}}V}},$$

(3)

where ρ_av is the average density of the processed medium by the volume of the apparatus, kg/m³; N_tot is the total power consumption in the volume V of the processed medium, W.

$$N_{\mathrm{tot}}={\displaystyle \underset{V}{\int }\rho {\epsilon }_{\mathrm{loc}}dV}.$$

(4)

The aim of this work is to demonstrate the ambiguous influence of the average specific energy dissipation rate ε_av on the hydrodynamic and mass transfer characteristics of chemical and biochemical reactors and to formulate ways to improve the efficiency of devices through rational energy distribution over their volume. In particular, when analyzing the literature, it was shown that for identical values of ε_av, the spread of the effect indicator values (mass transfer intensity, mixing time, etc.) can reach two decimal orders.

In addition, it is shown that the average specific energy dissipation rate as an average indicator does not reflect the specificity of the impact on heterophase systems, both with a non-deformable (solid) and with a deformable (liquid or gas) dispersed phase, taking into account the need to form an interphase surface. A classification of processes by three features is proposed in this paper: (1) by time; (2) by the number of phases participating in the process; and (3) by the level of the specific energy dissipation rate (local and average). As a result of the analysis, specific proposals have been developed for the rational energy distribution introduced into the apparatus by their volume.

In this article, due to space limitations, it is impossible to consider all possible systems and processes in which the energy distribution over the volume of the apparatus plays a key role; therefore, two processes were chosen: (1) homophase, used, for example, in co-precipitation processes in microreactors and (2) heterophase, considered in gas–liquid reactions, a typical process going on in chemical and biochemical reactors.

It should be noted that the mass transfer intensification discussed in this paper is crucial for fast or relatively fast reactions; in this case, mass transfer limitations play a critical role. In this paper, the regions of the apparatus with low energy dissipation are also discussed. Their role is (i) to decrease the total energy consumption of the apparatus and (ii) to perform slow reactions (if that is the case), see discussion below.

Let us consider some examples in which the average value of the specific energy dissipation rate ε_av is used as the only characteristic of the intensity of the input energy.

Example 1.

Mini and microreactors for the synthesis of nanosized and submicron particles.

Mini and microreactors have been increasingly used in the last 20–25 years for the synthesis of various inorganic substances and materials [4,5,6,7,8,9], metal–organic frameworks [10,11,12], and organic products [13]. The difference between mini and microreactors is rather arbitrary and is tied to a formal boundary of about 1 mm; in reality, the predominance of viscous and surface forces over inertial and gravitational forces for systems with the properties of water occurs up to a diameter of ~4.5 mm. The increased interest in mini and microreactors for solving these problems is due to the fact that high quality micromixing is achieved in microchannels [14,15], which ensures the formation of particles with a given stoichiometry (i.e., without impurity phases), and the short residence time in the mixing zone prevents the formation of agglomerates and aggregates.

Ottino et al. [16] proposed a multilayer model of a liquid–liquid system, and for the analysis of the energy efficiency of mixing, they introduced an indicator determined by the equation

$$\eta ={\displaystyle \frac{\dot{\gamma }}{{\dot{\gamma }}_{\mathrm{av}}}}={\displaystyle \frac{\dot{\gamma }}{\sqrt{{\epsilon }_{\mathrm{av}}/\left(2\nu \right)}}},$$

(5)

where $\dot{\gamma }$ is the part of the shear rate that is spent on mixing, s⁻¹; ${\dot{\gamma }}_{\mathrm{av}}$ is the average shear rate that occurs in the liquid flow, s⁻¹; ν is the kinematic viscosity of the liquid, m²/s. In the work of Baldyga et al. for a twin-screw extruder [17], it was found that the mixing efficiency index η does not exceed 1%.

One of the directions of process intensification is equipment miniaturization [9]. Microreactors have proven themselves especially well for fast reactions due to the high quality of micromixing, high uniformity of concentration, and temperature field distribution. In [18], eight types of micromixers were compared: (1) T-shaped round cross-section (T-mixer), (2) T-shaped planar (T-mixer), (3) with tangential solution injection (tangential IMTEK), (4) with a tortuous channel (caterpillar IMM), (5) with slit channels (slit interdigital IMM), (6) with radial flow segmentation (K-M micromixer), (7) with longitudinal flow segmentation (triangular interdigital Microglas), (8) with radial movement in flat layers (Starlam IMM).

Falk and Commenge [18] state the impossibility of theoretically calculating the energy efficiency of mixing since it is practically impossible to determine the part of energy spent on mixing. They obtained a formula for experimentally determining the time of micromixing caused by diffusion and shear (leading to a decrease in the diffusion path in the vortex element of the liquid) for a cylindrical channel in a laminar flow regime:

$$t_{diff+shear}=\sqrt{{\displaystyle \frac{\nu }{2{\epsilon }_{\mathrm{av}}}}}\mathbf{ln}\left(1.52\mathrm{Pe}\right).$$

(6)

Taking into account the definition of the indicator (5), Formula (6) for the laminar flow regime was adjusted in work [18] and acquired the form:

$$t_{diff+shear}={\displaystyle \frac{d}{8U\eta }}\mathbf{ln}\left(1.52 \mathrm{Pe} \eta \right),$$

(7)

where U is the average flow velocity in the microchannel, m/s.

Taking into account that the Peclet number Pe is in the range from 10³ to 10⁶, and η is several percent, the final result is [18]:

$$t_{diff+shear}~{\displaystyle \frac{d}{8U}}{Pe}^{0.15}{\mathsf{\eta }}^{-0.85} .$$

(8)

Expression (8) shows that the micromixing time is practically inversely proportional to the mixing efficiency index η. Thus, “for a given microchannel size and liquid velocity, the design of the micro-mixer (microreactor) is decisive in maximizing the mixing efficiency, i.e., minimizing the mixing time”, as noted in [18]. Thus, in the mentioned work [18], the decisive role of the geometry of the apparatus in the efficient transformation of energy in microreactors is recognized.

In [18], a boundary corresponding to Formula (7) is shown, which is valid for Re < 1000 in the case of very large Schmidt numbers Sc >> 1 for the diffusion-shear transfer mechanism. Such a transfer mechanism assumes the deformation of liquid elements by shear stresses in a laminar flow, leading to the stretching of liquid elements and a decrease in the diffusion path. It should be noted that the experimental results presented in [18] deviate from the theoretical line (t_m/d²) = f(Re) up to one decimal order in the region of large values and up to two orders of magnitude in the region of smaller values. Such a significant deviation may be due to the fact that the Reynolds number in devices of different geometries is not an unambiguously determining dimensionless parameter; in addition, in devices of different geometries at equal Reynolds numbers (and even at an equal specific energy dissipation rate ε_av), the mixing conditions differ significantly.

Information on the specific energy dissipation rate was compared with data on the quality of micromixing obtained in the same work [18] using the iodide-iodate method [19,20,21,22].

A comparison of the given data demonstrates the relationship between the Reynolds number Re, the micromixing time t_m, and the segregation index X_s. For example, at Re = 1000 and microchannel diameters d = 1 mm and d = 0.1 mm, along the line presented in [18], it was found that t_m ≈ 1 s and t_m ≈ 0.01 s, respectively. According to the curve in [18], for these values of the micromixing time, the segregation index is X_s ≈ 0.08 and X_s ≈ 0.001.

For comparison: in microreactors with impinging jets MRFIJ with a jet diameter of 0.5–2 mm, values of X_s ≈ 0.001 have been obtained [18,19], and in microreactors with intensively swirling flows MRISF-1, MRISF-2 with a characteristic diameter (neck diameter) of 2.4 mm, values of X_s ≈ 0.002 were achieved [23], which contributed to the synthesis of nanoparticles of various inorganic substances [24], and in a microreactor with impinging swirled flows (counter-current axial velocities) (micro-VJA-ISF-CC, or MRISF-CC) X_s ≈ 0.0047 (at other concentrations of sulfuric acid). The high quality of micromixing has been confirmed in a number of syntheses of nanosized and submicron particles of inorganic materials [25]. Comparison of flows with single swirl and counter swirling flows has shown that in the latter case, under comparable conditions, the segregation index X_s is approximately 10 times lower, i.e., the quality of micromixing is higher.

Thus, the MRFIJ, MRISF-1, MRISF-2, and MRISF-CC microreactors make it possible to achieve high quality micromixing even with a transverse size of the narrow part of the apparatus of about 2.0–2.4 mm.

Example 2.

Gas–liquid reactors.

The specific interfacial area during massive bubbling (based on the theory of locally isotropic homogeneous turbulence) [26] is defined by Equation (9):

$$a={\displaystyle \frac{S}{V_{app}}}=0.5{\left({\displaystyle \frac{{\rho }_L^3{\epsilon }_{\mathrm{av}}^2}{{\sigma }^3}}\right)}^{0.2}\cdot {\phi }^{0.35}\cdot {\left({\displaystyle \frac{{\mu }_L}{{\mu }_G}}\right)}^{0.25},$$

(9)

The surface mass transfer coefficient during bubbling can be determined by Equation (10) [26]:

$$k_L=0.24\sqrt{D_L\cdot {\left({\displaystyle \frac{{\rho }_L{\epsilon }_{\mathrm{av}}}{{\mu }_L}}\right)}^{0.5}}.$$

(10)

Although the literature presents a fairly large number of relationships linking the specific surface area a and mass transfer coefficients k_L, k_La (or the Sherwood number Sh) with other similarity numbers—Reynolds, Weber, Schmidt, Formulas (4) and (5) directly reflect the relationship between the specific energy dissipation rate and mass transfer coefficients.

The paper [1] presents a visual diagram of three mixing levels: macro (the scale of the apparatus as a whole), meso (the scale of intermediate vortices), and micro (the level of minimal vortices determined by Kolmogorov theory). It is shown that at all three levels, the specific energy dissipation rate has a direct impact on the mixing characteristics.

It should be noted that in airlift devices, the average bubble diameter (for systems like water–air) is d₃₂ = 14–18 mm, and in devices with mechanical mixing devices—turbine mixers, d₃₂ = 4.5 mm [27], which, in accordance with Formula (9), characterizes the energy characteristics of these devices.

On the other hand, for devices with a stirrer—for example, one of the most frequently used for dispersing gas in liquid, a Rushton turbine stirrer—it was shown in [28] that the distribution of the specific energy dissipation rate by volume is extremely nonuniform (Figure 1): the maximum value of ε_loc (ε_i in Figure 1), divided by the average ε_av is defined as a parameter of relative turbulent energy dissipation φ_i = ε_loc/ε_av = ε_i/ε_av and reaches φ_i = 34 for the zone immediately around the stirrer, while the volume fraction of this zone V_rel is only 0.0073, i.e., less than 1%. At the same time, in three peripheral zones with a volume fraction V_rel of 0.29, 0.343, and 0.195 φ_i = 0.56, 0.092, and 0.073, respectively. In the zones around and near the stirrer (red, orange, and yellow zones in Figure 1), with a total relative volume V_rel of 0.092 (9.2%), 70.7% of the total energy (in W) is dissipated.

The high unevenness of energy distribution over the volume of the apparatus with a Rushton turbine mixer also causes an uneven distribution of bubble sizes over the volume of the apparatus: bubbles that fall into the zones of powerful action are most intensively dispersed, but outside these zones they coalesce. The same applies to the mass transfer coefficient, as follows from Formula (10): as the bubbles move away from the mixer zone, the k_La level decreases significantly.

Thus, even in apparatuses with highly effective mechanical stirrers, in which the hydrodynamic situation is considered to be quite intense, the high unevenness of the distribution of dissipated energy ε = f(x, y, z) is the reason for the rather low values of the volumetric mass transfer coefficients k_La.

Several modern studies are devoted to some new methods and apparatuses of mass transfer intensification in gas–liquid systems.

The HiGee microbubble generator (HMG) was studied in [29]. The average specific energy rate ε increased from 0 to 320 W/kg when the rotational speed of the HiGee microbubble generator increased from 0 to 1400 r/min. The authors found the HMG to be a promising tool for the generation of microbubbles for enhanced gas–liquid mass transfer. It should be noted, however, that even the small bubbles have a tendency to coalesce outside of the HMG, i.e., in the average reactor volume, the specific interfacial area and mass transfer intensity could decrease significantly.

An extended study concerning power consumption in unbaffled stirred tanks was conducted in [30].

Two regimes have been distinguished: non-aerated conditions (sub-critical regime) and aerated (super-critical) conditions. The latter case is characterized by the free surface vortex touching the impeller and the gas phase being ingested through this vortex and dispersed inside the reactor. At rotational speeds involving air entrapment and dispersion inside the reactor (super-critical regime), a steep reduction in power number N_p and power consumption P is observed. A general correlation for power number prediction, applicable to both sub-critical and super-critical regimes, was proposed.

Mass transfer in G-L systems for unbaffled stirred tanks with a height-to-diameter aspect ratio of H/T = 1.0, 2.0, and 3.0 was investigated in the work [31]. Different slopes for dependence Np = f(P/V) for sub-critical and super-critical regimes were found (here P is power consumption, V is the volume of the reactor). The specific energy dissipation rate did not exceed 1.3 W/kg, and the highest k_La for the super-critical regime was 0.02 s⁻¹. The authors concluded that “maximum under-critical k_La values obtained (1.7·10⁻³ s⁻¹) indicate that this kind of bioreactor is fully able to satisfy the oxygen transfer rate demand of typical animal cell cultures, independently of the liquid aspect ratio configuration”. It was also found that better energy efficiency is obtained at relatively small height-to-diameter ratios (H/T = 1.0). This fact demonstrates that at the high non-uniformity of the energy dissipation rate typical for H/T > 1.0, larger energy consumption is needed to achieve the same mass transfer intensity.

A new type of static mixer for the absorption of gas (carbon dioxide) in liquid (alkaline solution) has been studied in [32]. Four-Kenics segment static mixer (FKSM) was compared with the Ross LPD static mixer in a 304 stainless steel tube with an inner diameter of 32 mm [32]. The average specific energy dissipation rate was in the range of 4.3 to 6.1 W/kg, depending on the two-phase flow velocity. No data on the mass transfer coefficient have been provided, but the ratio of CO₂ absorption to the initial CO₂ concentration was found.

In the first half of the reactor, the optimal condition is achieved at an energy consumption of 31.27 kWh/tCO₂ under the conditions U_L = 0.410 m/s and U_G = 0.169 m/s, and for the whole length of the reactor, the optimal condition is obtained at a cost of 60.1 kWh/tCO₂ under the conditions U_L = 0.362 m/s and U_G = 0.169 m/s.

The scrubbing factor (S_F) is proposed to compare the absorption efficiency of different reactors based on the NaOH-CO₂ system. It is defined as the ratio of CO₂ absorption to the molar flow rate of NaOH and to the volume of the reactor. The value of S_F was comparable with bubble columns and higher than that in rotating packed beds as well as in structured packed beds. The authors conclude that “the FKSM and Ross LPD have the excellent ability to enhance removal efficiency and the potential to become efficient gas absorption equipment”.

CO₂ absorption in a rotating packed bed was the focus of the paper [33]. The specific gas–liquid interfacial area was found as a function of the radius of the packed bed, and its value was in the range 5–10 m⁻¹ in the rotor zone and close to zero outside of the rotor (in inner and outer cavities), demonstrating a high level of non-uniformity even inside of the rotating packed bed. Turbulent dissipation rate ε (W/kg) grows quite rapidly along the radial direction (approx. from 5 to 35 W/kg for rotational velocity 800 rpm and approx. from 1 to 5 W/kg for 400 rpm), and then steeply falls outside of the rotating packed bed.

Jet Loop Reactors (JLR), both at the lab scale and at the pilot scale, have been the object of studies in [34].

An average specific energy dissipation rate ε was in the range of 1.02 to 4.70 W/kg, and the Sauter mean diameter of bubbles in a pilot-scale reactor has been in the range of 3.1 to 4.2 mm, very slightly depending on the ε value. The gas hold-up was quite low (in the range of 0.5% to 3.7%). The volumetric mass transfer coefficient k_La for the studied pilot-scale reactor was in the range of 18 to 92 h⁻¹ (i.e., from 5.0·10⁻³ s⁻¹ to 25.5·10⁻³ s⁻¹). In the majority of cases, lab-scale jet loop reactors demonstrated a smaller effect.

The characteristics of a gas–liquid vortex reactor with an original design have been investigated in [35].

Experiments are carried out over a wide gas–liquid volumetric flow ratio ranging from 150 to 3500.

The average specific energy dissipation rate ε was in the range of 25 to 100 W/kg, and the surface mass transfer coefficient k_L was determined in this paper to be in the range of 0.012 to 0.017 m/s.

A micronozzle having geometry close to a Venturi pipe with a neck diameter of 0.5 mm has been elaborated, and its mass transfer characteristics are studied in [36] by means of the resazurin color change technique.

Volumetric mass transfer coefficient k_La remains at a constant level (about 9 s⁻¹) for low flow rates and energy input (ε in the range from 20 to between 500 and 600 W/kg). Once turbulent bubble breakup begins at approximately (ε/ν)^0.5 = 28.8 s⁻¹, k_La increases up to a top-level value of 60 s⁻¹ at ε ≈ 10,000 W/kg. The reference straight microchannel yields mass transfer coefficients in the range of 7–12 s⁻¹ for slug and bubbly flow, with higher values for the latter as having more interfacial area per unit volume.

Note that such high values of ε are caused by the extremely small volume of the studied device. One of the significant limitations of micronozzles is their microsize, which is not applicable for macro-level apparatuses.

An original contacting apparatus—a rotating liquid redistributor—has been studied in [37]. This device is similar to the rotating packed bed but has several rotors fixed one over the other on one shaft. Liquid droplets in gas are formed while flowing through the holes with a diameter of 0.57 mm. Obviously, this apparatus is better suited for processes with a high gas-to-liquid flow rate ratio.

Interestingly, for energy dissipation rate ε (per one stage) in the range of 5 to 100 W/kg, the mass transfer coefficient k_GLa_GL grows rapidly from 0.05 to 0.25 s⁻¹, and for larger ε values (ranging from 100 to 350 W/kg), the k_GLa_GL level is almost constant (around 0.3 to 0.35 s⁻¹).

Due to the growing need for intensive dissolution of gases in liquids, for example, in wastewater treatment processes during aeration in aeration tanks, in biotechnological processes where bacterial mass consumes dissolved gases (in particular, oxygen), mass exchange between gas and liquid is a key stage limiting the overall rate of the process. In other words, the rate of gas dissolution limits the intensity of microorganism growth. In this regard, the development of effective methods for intensifying the processes of bubble dispersion and gas dissolution in liquid is an urgent task, as is ensuring high-quality micromixing in microreactors.

This article discusses general approaches to intensifying mixing and rational energy distribution in devices where mixing, dispersion of droplets and bubbles, and a high mass transfer rate are determining factors.

This paper is focused on the influence of energy dissipation rate non-uniformity on the specific interfacial area and mass transfer intensity (k_La in combination of these two factors), because the energy consumption in total defines the overall energy consumption per 1 kg of product, and energy uniformity defines how effectively the energy is distributed along all the microvolumes of the apparatus.

2. Theoretical Part

As a result of the analysis carried out in the introduction section of this article, in order to achieve high efficiency of equipment where mixing plays a significant role, in our opinion, it is necessary to ensure that at least the energy parameters inherent, on the one hand, to a specific process and, on the other hand, the characteristics of the equipment used to carry out this process, are taken into account.

In our opinion, to analyze the hydrodynamic situation, it is necessary to distinguish between local ε_loc and volume-average ε_av values of the specific energy dissipation rate. For convenience, we can also distinguish, for example, three energy levels: (1) low, (2) medium, and (3) high.

Many homophase processes do not require special measures to improve efficiency; for example, gas mixing occurs quite quickly due to high gas diffusion coefficients and high velocities causing convective mixing. However, there are homophase processes that require increased mixing intensity, including at the micro level. These include solution synthesis methods, among them: the sol–gel method, the chemical coprecipitation method, the hydrothermal method, and the solvent replacement method. A number of studies have shown that in the chemical coprecipitation method, improving the quality of micromixing is a key factor determining the purity of the resulting products (i.e., the absence of impurity phases) and the size of the resulting particles.

For heterophase processes, in addition to mixing in the continuous phase, there is a need for intensive mixing in the dispersed phase (for L-L and L-G systems), suspension (for L-S systems), as well as intensive interphase mass transfer.

It should be noted that the levels of local ε_loc and volume-averaged ε_av specific energy dissipation rates can differ significantly (by several orders of magnitude) in the volume of the same apparatus under fixed conditions [9,23,28,38].

Usually, in the literature, the average residence time in the apparatus τ_res is used as the “external” time. At the same time, it is well known [39] that the residence time distribution curves can have a very large dispersion, especially in the presence of intense return flows in the volume of the apparatus. This means that the average residence time is not a complete and adequate characteristic of the apparatus.

In general, τ_proc is affected by both the diffusion component (convection is also meant here and further as the other way of mass transfer) and the reaction kinetics. In practice, these two factors—diffusion and kinetic—must be of the same order. Otherwise, if the diffusion (and convection) process rate is insufficient, the rate of substances supply to the reaction zone will be insufficient, which not only limits the overall process rate but can also lead to the formation of by-products (as a result of different diffusion rates for different reagents). If the rate of diffusion processes is significantly higher than the rate of chemical reactions (the so-called kinetic region), this means, at a minimum, an excessive rate of convective transfer, the creation of which requires excess energy.

This analysis shows that the comparison of τ_proc and τ_res is rather superficial and not very suitable for most fine processes that have high requirements for yield, selectivity, energy consumption, and other indicators.

For this reason, in our opinion, along with the average residence time τ_res, when assessing the characteristics of the equipment, it is necessary to use another time scale—the duration of intensive action τ_act (for homophase processes—the time of intensive mixing τ_mix). This indicator, in turn, is associated with the energy dissipation rate, and this relationship is discussed below.

Let us analyze nine combinations of levels ε_av and ε_loc.

(1)

At a low level of ε_av, it is advisable to carry out processes that do not require maintaining an intensive effect on the liquid phase throughout the volume:

1.a. In the kinetic region, the level of ε_loc can also be low;

1.b. With moderate kinetics (occurring in the diffusion-kinetic region), in this case the level of ε_loc should be elevated;

1.c. Fast-flowing, i.e., with high process rates (for example, fast and ultrafast reactions); in this case, the level of ε_loc should be the highest.

(2)

At an average level of ε_av, it is advisable to carry out processes when the volume of equipment is not very large (otherwise it would be necessary to spend excess energy), according to the levels of local energy dissipation rate:

2.a. At a low level of ε_loc—auxiliary operations, for example, preparation of solutions;

2.b. At an average level of ε_loc—operations requiring an increased concentration of energy in a local volume, for example, suspension of a solid phase;

2.c. At a high level of ε_loc—processes in which it is necessary to overcome a high potential barrier, for example, processes of extraction or dispersion of highly viscous liquids.

(3)

At a high level of ε_av, it is advisable to carry out processes when we are talking about mini- and microscale equipment (active zone volume in the range from 10¹ mL to 10⁻¹ mL).

3.a. At a low level of ε_loc—operations in which no requirements are imposed on k_La (for example, gas–liquid processes with a single dispersion of gas or liquid);

3.b. At an average level of ε_loc—operations in which moderate requirements are imposed on k_La, for example, chemical and biochemical reactions accompanied by dispersion and redispersion of droplets and/or bubbles;

3.c. At a high level of ε_loc—processes in which high demands are placed on k_La, for example, fast chemical and biochemical reactions, including heterophase ones, coupled with liquid extraction.

3. Results and Discussion: Examples of the Analysis of the Dissipated Energy Distribution and Energy Efficiency of the Apparatus

Let us consider the nature of the distribution of dissipated energy and its influence on the efficiency of three types of apparatus: (1) a “standard” or typical apparatus (using a bubble column as an example and some similar apparatuses widely used in industry); (2) “ideal” with a uniformly high energy distribution throughout the entire volume; and (3) improved in terms of energy saving and increased efficiency.

Example 3:

A bubble column (case (1) - typical gas–liquid reactor).

Figure 2 represents a diagram of a bubble column fermenter [40], Figure 3 contains a diagram of a plate column fermenter, and Figure 4 includes a diagram of a gas lift loop column fermenter.

The main difference in a gas-lift fermenter is the presence of a circulation cylinder (pipe), due to which a stable circulation flow is organized in the apparatus.

From the point of view of the distribution of the specific energy dissipation ε_loc, all three devices shown in Figure 2, Figure 3 and Figure 4 are characterized by extremely high non-uniformity of ε_loc, which is shown in all diagrams in the form of two typical zones—a zone of intensive gas dispersion with ε_max and a zone with free movement of the gas–liquid flow with ε << ε_max. A consequence of this is the non-uniformity of the distribution of the mass transfer coefficient from gas to liquid: in the zone of intensive gas dispersion, (k_La)_loc can be higher by an order of magnitude or more than the average value for the volume (k_La)_av.

Figure 5a shows the nature of the dependence of the specific energy dissipation rate ε on the residence time in the apparatus τ, typical for real apparatuses. The zone τ_act corresponds to the most intense local energy dissipation ε_loc.max, for example, in the zone directly above the bubbler at the moment of bubble formation, in the immediate vicinity of the mechanical mixing device, injector, jet, or holes in the grid of the plate contact element, depending on the design of the apparatus.

The gas enters the zone with the highest values of the specific energy dissipation rate ε_loc.max (the active zone), where its intensive fragmentation and the most intensive mass transfer occur. According to [41], 70–90% of the total amount of transferred substance is transferred in the formation zone and at some distance from it. These values confirm the fact that the active zone plays the most significant role in mass transfer, and the remaining volume of the apparatus has a weak effect on the process.

The amount of energy dissipated in the active zone of a typical reactor per unit mass of the medium being processed (J/kg) is defined as

E_max0 = ε_loc.max0·τ_act,

(11)

where ε_loc.max0 is the local (maximum) value of the specific energy dissipation rate in the active zone, W/kg, τ_act—residence time in the active zone, s.

The amount of energy dissipated in the passive zone is defined similarly as

E_pas0 = ε_pas0·τ_pas,

(12)

where ε_pas0 is the value of the specific energy dissipation rate in the passive zone, W/kg, τ_pas—residence time in the passive zone, s.

On average, over the reactor volume, the amount of dissipated energy per unit mass (J/kg) is defined according to Equation (13):

E_tot0 = ε_av0·τ_res.

(13)

The specific power (specific energy dissipation rate) per unit volume (W/m³), dissipated per unit volume, is determined by the following expressions:

-

for the active zone:

N_Vmax0 = ρε_loc.max0·,

(14)

-

on average by reactor volume:

N_Vtot0 = ρε_av0 << N_Vmax0.

(15)

The specific volumetric power N_Vmax0 in the active zone is significantly higher than the average volume N_Vtot0.

Let us consider several indicators characterizing the uneven distribution of dissipated energy throughout the volume of the apparatus:

(1)

Active zone volume fraction

γ_act = V_act/V_tot

(16)

At γ_act → 1, in almost the entire volume of the apparatus, energy is dissipated at a high rate (Figure 5b); at γ_act → 0, the active zone occupies an extremely small part of the volume.

(2)

The degree (index) of uneven energy distribution is the ratio of the maximum value of specific energy dissipation E_max to the average value of E_tot for the volume of the apparatus:

IRE1 = E_max/E_tot.

(17)

The higher the IRE1 value, the more significantly the energy level in the active (energy-saturated) zone differs from the average value (Figure 5a). From the point of view of organizing processes in multiphase media, excessively high IRE1 values obviously mean a decrease in the local intensity of mass transfer.

(3)

Alternatively, a second indicator of uneven energy distribution (IRE2) can be used—the ratio of the maximum value of specific energy dissipation E_max to the minimal energy, i.e., to the specific energy dissipation in the passive zone E_pas:

IRE2 = E_max/E_pas.

(18)

The ratio of the energy introduced into the active zone of the apparatus to the total energy consumption in the apparatusis determined by the formula:

$${\eta }_E={\displaystyle \frac{E_{max}{\rho }_{act}{V}_{act}}{E_{tot}{\rho }_{av}{V}_{tot}}}\approx {\displaystyle \frac{E_{max}{V}_{act}}{E_{tot}{V}_{tot}}}={\gamma }_{act}{\displaystyle \frac{E_{max}}{E_{tot}}}={\gamma }_{act}IRE1$$

(19)

The final formulation in relation (19) is written under the assumption that the liquid densities in the active zone and on average over the volume of the apparatus are approximately the same. The denominator of Formula (19) is the amount of energy dissipated on average over the volume, i.e., the sum of the specific energies dissipated in the active and other (passive) zones of the apparatus. Note that the value of η_E is not always an informative parameter since it is the product of the indicators determined by Formulas (16) and (17), the first of which can be significantly less than one (γ_act << 1) and the second, on the contrary, significantly higher than one (IRE1 >> 1). The product of such values, γ_act and IRE1, leads to η_E ≈ 1, which is inconvenient for a quantitative comparison of two arbitrary macroscale apparatuses.

The value of η_E is aimed at demonstrating the role of the active zone from an energetic point of view. This parameter indicates the contribution of the active zone to the total energy dissipation. The values presented in

Table 1. Parameters of energy distribution by volume for a bubble column at three values of gas velocity in the bubbler holes (results of calculations).

wbarb, m/s	15	20	25
Emax0, J/kg	138	327	638
Etot0, J/kg	1.75·10−3	4.13·10−3	8.05·10−3
IRE1	7.856·104	7.902·104	7.919·104
IRE2	7.716·106	1.829·107	3.572·107
ηE	0.990	0.996	0.998
kLa, s−1	1.98·10−3	2.45·10−3	2.90·10−3

show that for bubble columns, the majority of energy (99.0% to 99.8%) is dissipated in the active zone. It means that the passive zone has a very low impact on the total power balance (from 0.2% to 1.0%) for common macroscaled reactors, for instance, gas–liquid reactors.

On the contrary, microreactors demonstrate much better energy distribution; see [15] for details. For example, in microreactors with intensely swirling flows, the main energy dissipation takes place in the neck (η_E = 65–75% depending on flow rate), i.e., the role of passive zones is around 25–35%.

Calculation for a bubbling column with a liquid height of 10 m at an average bubble velocity of 0.23 m/s gives a residence time of τ_res = 43.5 s and the time τ_act = 0.122 s, i.e., τ_res/τ_act = 357.

In this case, E_pas0 = 1.785·10⁻⁵ J/kg and E_max0 = 138, 327, and 638 J/kg at a gas velocity in the bubbler holes of 15, 20, and 25 m/s, respectively (Table 1).

The value of the average specific energy for the volume of the apparatus was calculated using Formula (20):

E_tot0 = γ_pas E_pas0·+ γ_act E_max0.

(20)

For a bubbling apparatus, the fraction of the active zone volume γ_act is extremely small (according to our estimate, 1.26 × 10⁻⁵), and the value of E_tot0 = 1.75 × 10⁻³, 4.13 × 10⁻³, and 8.05 × 10⁻³ J/kg.

Thus, for the bubbling apparatus, the level of energy consumption in the main volume (E_tot0) is significantly, almost 5 orders of magnitude lower than the energy consumption in the bubbler zone (E_max0) (IRE1 ≈ 7.9·10⁴). This means that a significant part of the apparatus volume (more than 99.99%) is not used to perform useful work, and the mass transfer processes are significantly slowed down, i.e., the main volume of the bubbling column is used extremely inefficiently. As follows from Formulas (9) and (10), the specific surface is proportional to a ~ ε_av^0.4, and the surface mass transfer coefficient is k_L ~ ε_av^0.25; therefore, k_La ~ ε_av^0.65. Note that similar relationships can be written for local values of a, k_L, and k_La in the corresponding zones of the apparatus—active and passive.

The k_La values presented in the last line of Table 1 were found using Formula (10) and correspond to the data published in [42].

A similar situation is observed for other typical devices, where the maximum dissipated energy is concentrated in the gas input zone—airlift, plate, with mechanical mixing devices—in the devices equipped with ejectors.

The IRE2 values for bubble columns presented in Table 1 demonstrate how much the maximum value of specific energy dissipation E_max is higher than the specific energy dissipation in the passive zone E_pas: for the considered case, the ratio is around 10⁷ (ten million folds!). This variable unambiguously indicates the huge non-uniformity of energy dissipation in bubble columns.

Example 4:

“Ideal” gas–liquid reactor with uniformly high energy distribution throughout the entire volume (case (2)).

The diagram of an “ideal” gas–liquid reactor, in which the input energy is distributed uniformly throughout the entire volume, is shown in Figure 5b.

Such a device is called “ideal” conditionally, since a high level of input energy also implies a high level of mass transfer, distributed throughout the entire volume.

In this case, the specific power per unit volume (W/m³), dissipated in the volume of the entire device, is determined by the expression:

N_Vtot1 = N_Vmax1 = ε_av1·ρ = ε_max·ρ.

(21)

The amount of energy dissipated in the active zone of an “ideal” reactor per unit mass of the processed medium (J/kg)

E_tot1 = ε_av1·τ_res = ε_max·τ_res = E_max1.

(22)

This means that such a device will consume significantly more energy than a “standard” reactor (e.g., a bubble column, an airlift column, or a mechanically stirred reactor). Since, as follows from the IRE1 values in Table 1, the main part of the energy falls on the zone with maximum dissipation ε_max, compared to a bubbling device, the increase in energy consumption for such a reactor is determined by an increase in the average value of E_tot over the device volume to the level of E_max (E_tot = E_max), i.e., IRE1 = 1. This means that the total increase in specific energy will be approximately 7.9·10⁴ times compared to a bubbling column.

That is, instead of the values in the range of 2·10⁻³–8·10⁻³ J/kg found for the bubbling apparatus (E_tot0 values in Table 1), the ideal apparatus will consume about 140,000–638,000 J/kg (E_max0 values in Table 1). With a liquid mass of 7854 kg, we obtain an energy consumption of 1–5 MJ (about 0.3–1.4 kW·h) for one cycle τres of gas movement through the volume of the apparatus.

Of course, such a high level of energy spent per unit of the obtained product is unacceptable for industrial equipment. In this connection, the question arises about synthesizing an apparatus that ensures a reduction in energy costs per unit of the obtained product practically without reducing the quality, i.e., achieving a high level of mass transfer.

Example 5.

Gas–liquid reactors with improved energy-saving characteristics and increased efficiency (case (3)).

One of the effective approaches to solving this problem, in our opinion, is to use the principle of the discretization of the energy introduced into the device: instead of the continuous dissipation of energy with a high level (Figure 5b), creating a pulsed input of energy in the flow of a multiphase medium (Figure 6), with the localization of a high energy level ε_loc.max in small-volume zones where energy dissipation will occur, and in the intervals between them, maintaining a minimum level of energy consumption ε_pas.

For processes with the formation of an interphase surface (liquid–gas and liquid–liquid systems), along with the distribution of the specific energy dissipation rate over the volume of the apparatus, the residence time of the dispersed phase (bubbles, drops) between the active zones, which can be called the “relaxation” time τ_rel, has a significant effect on the processes of dispersion and mass transfer. It is shown below that with periodic (pulse) action on a heterogeneous system, the relaxation time τ_rel, i.e., the time for the coalescence of bubbles or drops, determines the time and distance between two contact devices. The calculation of this time is determined by the physicochemical properties of the media, the specifics of the apparatus, and the conditions of action on the gas–liquid system and is not included in the task of this work.

At first glance, it may seem that the conditions shown in Figure 6 are reproduced in tray column apparatuses (Figure 3). In reality, the hydrodynamic conditions in such apparatuses are significantly better than in a bubbling apparatus since the bubble dispersion process is repeated on each plate (the ε_max zone). At the same time, due to the fact that the dissipated energy is a fraction of the kinetic energy of the introduced gas, the ε_max level in a bubbling apparatus, as in a tray one, is quite low, leading to the formation of large bubbles (d₃₂ = 14–18 mm) with a low specific surface area (a = 27–34 m⁻¹); the same applies to k_L. In this regard, the expected mass transfer effect from replacing a bubbling apparatus with a tray one is proportional to the number of plates and is not significant enough in absolute value, taking into account the low k_La values during bubbling.

The hydrodynamic conditions in the gas-lift loop column apparatus (Figure 4) are significantly better, since the quality of bubble dispersion is determined by the high kinetic energy of the liquid flow supplied to the gas ejection unit. An essential circumstance for this type of apparatus is the intense coalescence of bubbles in sections 4 (zones with free movement of the gas–liquid flow, Figure 4) between the ejectors, which requires the more frequent arrangement of ejectors along the length of the apparatus. In addition, the dispersed introduction of liquid through the ejectors leads to an abrupt change in the reduced flow velocity in sections 4 behind each ejector, and hence to a change in the conditions of bubble fragmentation and mass transfer in sections 4. A possible solution here is a stepwise increase in the cross-section of the apparatus in each subsequent section, but an excessive increase in the diameter of the apparatus will lead to an uneven distribution of phases over the cross-section.

Indeed, the energy dissipation rate (in W) is known from the fluid dynamics theory as N = ΔpQ, where Δp is pressure drop and Q is flow rate. For turbulent flows (common for macroscale apparatuses), Δp ∝ v² and Q ∝ v. Therefore, N ∝ v³, i.e., proportional to the third power of the average velocity v. For the given flow rate Q, an excessive increase in the diameter of the apparatus D results in a decrease in the average axial velocity v = 4Q/πD², then N ∝ v³ ∝ D⁻⁶. On the other hand, the volume of liquid V where the energy is dissipated increases with growing D, as V ∝ D³.

Hence, the increase in D results in a rapid decrease in dissipated power (N ∝ D⁻⁶) and an even greater decrease in specific energy dissipation rate: ε = N/ρV ∝ D⁻⁶/D³ = D⁻⁹.

To sum up, for the fixed liquid density and fixed flow rate, the specific energy dissipation rate is inversely proportional to the ninth power of the diameter D. For instance, if the diameter is increased twice, the local energy dissipation N drops 2⁶ = 64 times, whereas the energy dissipation rate ε decreases 2⁹ = 512 times (note that the energy dissipation rate here is related to the extended volume, which is increased 8-fold). It is worth noting that in the recent research [15], the decrease in energy dissipation after flow expansion in the microreactor (turbulent flow regime) was from 60 to 100, which is comparable with theoretical value of 64 obtained here theoretically.

In our opinion, one of the available solutions to the problem of mass transfer intensification is the use of a pulsating flow-type apparatus (PFA) in liquid–gas [43,44] and liquid–liquid [45,46] systems. In particular, the effect of doubling the specific contact surface of phases in an emulsion by introducing a small amount of inert gas into it has been demonstrated [47].

The PFA diagram is shown in Figure 7. The main part of pulsating flow-type apparatus consists of a pipe with a variable cross-section 6, including several elements 8 of the Venturi pipe type, and sections 9 with a constant cross-section alternating with them. Several pipes 6 can be connected in parallel to the pump 5. The preferred opening angles of the confusers 10 are within the range of 10° to 40°, and the angles of the diffusers 12 are from 4° to 20°.

The plots presented in Figure 8, Figure 9 and Figure 10 demonstrate the preferences of the PFA compared to the other types of apparatuses: bubble columns, stirred vessels, and static mixers.

A comparison of the dependence of bubbles Sauter mean diameter d₃₂ (m) on energy dissipation rate ε (W/kg) is presented in Figure 8. The dependence of the mass transfer coefficient k_La (1/s) in static mixers and in PFA is shown in Figure 9.

There are many types of static mixers studied in the literature. We have compared the efficiency of PFA with a variety of static mixers for liquid–liquid systems in the paper [45]. It has been shown that PFA exceeds the majority of static mixers used for comparison and is comparable with Sulzer SMV and SMX static mixers. Similar Sauter mean diameter values d₃₂ for the same energy dissipated per mass of the mixture (J/kg) were obtained for PFA, SMV, and SMX. The only information on Lightnin static mixers was found in the open literature for liquid–gas systems, and it was used for comparison in the present work. A comparison of mass transfer coefficient k_La (1/s) dependence on the average specific energy dissipation rate ε (W/kg) for gas–liquid reactors: bubble columns, stirred vessels, static mixers, and PFA is shown in Figure 10.

The pulsating flow-type apparatus provides periodic energy input according to the scheme shown in Figure 6. As follows from the plots presented in Figure 8, Figure 9 and Figure 10, due to the impulse energy input into the flow-type pulsating apparatus, it is possible to achieve the following results:

(1)

To concentrate the energy in a small volume of the necks, i.e., provide values up to ~1 kW/kg (an order of magnitude higher than in Lightnin static mixers, and two orders of magnitude higher than in stirred tanks, see Figure 8 and Figure 10);

(2)

To create a developed interface with d₃₂ ≈ 0.55 mm, while the specific surface area is a = 6φ/d₃₂ = 654 m⁻¹ (value found at φ = 6%), i.e., almost an order of magnitude higher than in devices with stirrers and ~3 times higher than in Lightnin static mixers (where the average bubble size was 1.5 mm), see Figure 8 and Figure 10;

(3)

To achieve a value of the volumetric mass transfer coefficient k_La 3–12 times higher than in Lightnin static mixers (see lines 1–3 and 4 in Figure 9).

In this case, in a pulsating flow-type device, disintegration is repeated in each of the elements (redispersion), and a periodic process (impulse energy input) is implemented to save energy.

In [50], a stirrer has a shaft paddle equal to the liquid column height in the tank (a paddle stirrer). The main idea was to distribute the gas along the height of the liquid through a central vortex around the stirrer. A large amount of experimental data has been processed, and a general dependence for volumetric mass transfer coefficient k_La (h⁻¹) on average specific energy rate ε (W/kg) and specific interfacial area a (m⁻¹) has been proposed in the form of Equation (23) for the range of ε from 0.02 to 20 W/kg, and:

k_La = 30 (ε^0.27 a^0.36),

(23)

and the corresponding range for k_La is from 3.0·10⁻³ s⁻¹ to 2.7·10⁻¹ s⁻¹. For k_La in “s⁻¹” units, the coefficient 30 in Equation (23) should be replaced by the coefficient 0.00833 = 30/3600.

It should be noted that in Equation (23), the exponent at a specific energy rate (0.27) is lower than the value predicted by Kolmogorov turbulence theory (0.65). The obtained results (the range of both ε and k_La) are very close to those shown in Figure 10 (area 2 for stirred tanks).

4. Conclusions

In this paper, the influence of the method of distributing the energy introduced into the apparatus on the efficiency indicators of its operation was analyzed. For the convenience of analyzing the efficiency of the apparatus, several criteria were introduced: the share of the active zone volume γ_act, two indices of uneven energy distribution—IRE1 and IRE2, and the share of energy costs in the active zone of the apparatus in the total energy costs in the apparatus η_E.

The analysis of these indicators demonstrated that in the bubble column, the level of energy costs in the main volume is extremely low (almost 5 orders of magnitude lower than the energy costs in the bubbler zone), i.e., a significant part of the apparatus volume (more than 99.99%) is not used to perform useful work, and the mass transfer processes in it are significantly slowed down. The calculated values of the mass transfer coefficient in the bubble column were k_La ~ 2–3·10⁻³ s⁻¹; the results of the calculated k_La are consistent with the data published in [42].

An analysis of the “ideal” gas–liquid reactor, in which the input energy is distributed uniformly throughout the volume (Figure 5b), showed an excessive level of energy consumption—the overall increase in specific energy was up to ~10⁵ times compared to a bubbling apparatus. For instance, for a liquid mass of 7854 kg, the energy consumed is 1–5 MJ (about 0.3–1.4 kW h) per cycle of gas movement through the volume of the apparatus.

As an alternative, it is proposed to use gas–liquid reactors with discrete energy input, for example, the reactor shown in Figure 4, which has two drawbacks: gas accumulation along the length of the apparatus and intense coalescence of bubbles in the zones between the ejectors.

According to our opinion, one of the most promising for liquid–liquid and liquid–gas systems is the pulsating flow-type apparatus (Figure 7) with a pulsed energy input into a two-phase system in the throat zone (where the specific energy dissipation rate can reach about 1 kW/kg). Periodic energy dissipation leads to a decrease in overall energy costs (compared to an “ideal” apparatus with uniform energy distribution). As follows from Figure 8 and Figure 9, the use of PFA allows for a finer dispersion of bubbles, maintaining a developed specific interface throughout the entire volume of the apparatus [44], and also significantly increasing the volumetric mass transfer coefficient—from 3 to 12 times compared to static mixers of the Lightnin type and to an even greater extent compared to apparatuses with stirrers, bubbling columns, and airlift apparatuses (see Figure 10).

The value of energy as one of the most universal parameters of the process and the reactor where this process takes place is an undeniable fact mentioned in the majority of monographs and textbooks. That is why the energy dissipation and its non-uniformity in its “pure” form, without any details concerning the structural characteristics and flow characteristics of the reactor, have attracted our attention. There are a variety of papers discussing the idea of the influence of energy dissipation rate on the performance characteristics of the reactors (see, e.g., [18,28]).

An attempt has been made in this paper to demonstrate the preferences of the impulse energy input into the reactor without going deep into the structural characteristics. Such an energetic approach was used in several papers because it allows to reveal some general dependences for a wide range of processes typical for chemical reactors, i.e., some similarity, for example, for droplet or bubble sizes and average mass transfer coefficients in turbulent flows.

In our future research, the next steps could be performed in order to study in detail the structural characteristics and flow characteristics of reactors with and without impulse energy input and to compare them.