A Study of Heat Exchange Processes within the Channels of Disk Pulse Devices

: The e ﬀ ect of basic parameters of the channels of disk pulse devices on the heat exchange e ﬃ ciency was studied both analytically and experimentally, especially in terms of pulse acting on the heat carrier. A methodology to determine the main parameters, namely the pressure and the temperature of the heat carrier as well as the pulse e ﬀ ect on the ﬂuid, was proposed. The mathematical models of the e ﬀ ect of the structural and technological parameters of the channels in the disk pulse device on the heat exchange e ﬃ ciency were developed. The models’ adequacy was proved based on a series of experimental studies involving devices with one-stage and multistage systems of pulsed heat carrier processing. This enabled the development, testing, and implementation of practical construction designs of pulse disk heat generators for decentralized heating of commercial and domestic buildings with one-stage and multistage systems of pulsed heat carrier processing. Taking into account the results of the mathematical modeling, the developed method of multistage pulse action was proved experimentally and implemented in regard to the structural design of a working chamber of the disk pulse heat

Abstract: The effect of basic parameters of the channels of disk pulse devices on the heat exchange efficiency was studied both analytically and experimentally, especially in terms of pulse acting on the heat carrier. A methodology to determine the main parameters, namely the pressure and the temperature of the heat carrier as well as the pulse effect on the fluid, was proposed. The mathematical models of the effect of the structural and technological parameters of the channels in the disk pulse device on the heat exchange efficiency were developed. The models' adequacy was proved based on a series of experimental studies involving devices with one-stage and multistage systems of pulsed heat carrier processing. This enabled the development, testing, and implementation of practical construction designs of pulse disk heat generators for decentralized heating of commercial and domestic buildings with one-stage and multistage systems of pulsed heat carrier processing. Taking into account the results of the mathematical modeling, the developed method of multistage pulse action was proved experimentally and implemented in regard to the structural design of a working chamber of the disk pulse heat generator. An efficient geometry of the working chamber of the disk pulse heat generator was specified for its further integration into the system of decentralized heat supply. One of the developed heat generators with the multistage pulse action on the heat carrier was integrated into the heating system of a greenhouse complex with a 0.86-0.9 efficiency coefficient.

Introduction
Production simplicity, commercial feasibility, equipment reliability, and reproducibility constitute essential factors in the development of environmentally friendly and efficient methods of energy generation [1][2][3]. Pulsed processing of process liquids is applied mainly as a method intensifying the hydrodynamic and mass-exchange processes. Currently, devices with pulse action on the liquids are used as heat energy generators [4]. The operating space of that type of aggregates has the form of a disk channel where one disk rotates and the other one remains immobile. The major part of the equipment has the following configuration of the operating space: centrifugal pumps, disk extruders, disk cavitators, and other devices [5,6]. The intensification of the gas dynamics and heat exchange processes within the disk pulse devices resulting from their operating space optimization is of great importance, especially in terms of improving the already available or developing new energy-efficient heat machines [4,[7][8][9][10][11]. Studies dealing with the improvement of the energy efficiency of disk devices, which involve renewable and environmentally safe energy sources, represent one of the most important trends of the energy complex development [12,13]. Current research in the field of pulse device improvement is aimed at optimizing technological parameters at the expense of changes in the working chamber geometry [14][15][16]. Researchers are considerably interested in analyzing the hydrodynamics and heat exchange in pulse devices in relation to different operating modes.
Analytical descriptions of particular cases of the equation of compressible fluid motion in the disk channel in a vector form with numerous assumptions that cannot be applied for all device structures with pulse action on the liquid are presented in the literature [16][17][18].
Nikolsky et al. [19] developed a mathematical model, which allowed analysis of gas dynamics processes in the disk gaps of rotor pulse devices. This model contains two nonlinear differential equations relative to some functions f and ς, which help express the radial V r and tangential V ϕ velocity components, respectively. In this context, the function ς and tangential V ϕ velocity component correspondingly has nonlinear dependence along the disk gap. However, for calculation of the basic technological parameters, e.g., device efficiency and power, this nonlinearity does not introduce any considerable errors. Moreover, the solution of this system of differential equations was in the approximation of voluntary flow, i.e., without considering the pressure gradient along the radial coordinate. In some cases, the latter assumption may significantly distort the final results.
In this paper, the analytical heat exchange processes of the contacting phases in the liquid-solid body system for disk pulse devices with external power sources were studied. For this purpose, the analytical studies of the efficiency of the heat exchange processes in terms of pulse action were carried out. The determination of the optimal geometrical characteristics and energy efficiency of the disk pulse heat generator were investigated. In addition, the design of the experimental disk pulse device with an energy-efficient working chamber was proposed.

Analytical Studies of the Heat Exchange Process and Its Computer-Based Representation
In Figure 1, the analytical model of a disk pulse device to develop a mathematical model of the heat exchange is presented. The following assumptions and boundary conditions are specified: - The process in the disk pulse device is axially symmetrical; we neglect the changes in physical values along the angular coordinate (∂/∂ϕ = 0); - The process is of a steady-flow nature (time changes in physical values are insignificant ∂/∂t = 0); -Mass forces are neglected (ρ·g i = 0); - The velocity component V z is neglected as h << R n ; Energies 2020, 13, 3492 3 of 14 - The temperature gradient at the expense of convective heat transfer along axis z is neglected; heat transfer along axis r is taken into consideration only in terms of heat convection.
-The velocity component Vz is neglected as h << Rn; -The temperature gradient at the expense of convective heat transfer along axis z is neglected; heat transfer along axis r is taken into consideration only in terms of heat convection.
Analyzing the temperature field in the working chamber of a disk pulse device, on the basis of its design and nature of hydrodynamic flows, we take into consideration heat transfer by means of convective heat exchange along the radial axis r ; while heat transfer at the expense of heat conductivity, along axes z and r , and the changes in temperature along the angular axis are neglected. Then, the equation of heat energy balance will be expressed by the following ratio: where T is the temperature; λ is the coefficient of heat conductivity; Fd(z,r) is the function of dissipation; Vr(z,r) is the radial velocity component in projections on the corresponding axes; ρ is thee fluid density; and Cp is the liquid heat capacity under constant pressure. The dissipation function makes it possible to define the heat power that is consumed in the working chamber of a disk pulse device. In order to do this, it is required to integrate the dissipation function in terms of the working zone volume.
In the case of Newtonian fluid, the dissipation function is as follows [20,21]: , where Vφ(z,r) is the tangential velocity component in the projections on the corresponding axes; η is the coefficient of dynamic fluid viscosity. The boundary temperature conditions along axis z are ∂T ∂z (0, r) = 0, ∂T ∂z (h, r) = 0. The initial conditions along axis r are T(z, R 1 ) = T n , ∂T ∂r (z, R 1 ) = 0. Analyzing the temperature field in the working chamber of a disk pulse device, on the basis of its design and nature of hydrodynamic flows, we take into consideration heat transfer by means of convective heat exchange along the radial axis r; while heat transfer at the expense of heat conductivity, along axes z and r, and the changes in temperature along the angular axis are neglected. Then, the equation of heat energy balance will be expressed by the following ratio: where T is the temperature; λ is the coefficient of heat conductivity; F d (z,r) is the function of dissipation; V r (z,r) is the radial velocity component in projections on the corresponding axes; ρ is thee fluid density; and C p is the liquid heat capacity under constant pressure. The dissipation function makes it possible to define the heat power that is consumed in the working chamber of a disk pulse device. In order to do this, it is required to integrate the dissipation function in terms of the working zone volume.
In the case of Newtonian fluid, the dissipation function is as follows [20,21]: where V ϕ (z,r) is the tangential velocity component in the projections on the corresponding axes; η is the coefficient of dynamic fluid viscosity. The function determining the tangential velocity component is expressed by the following ratio [8,22]: where ω is the angular velocity of disk rotation; h is the height of the disk gap; and z is the axial coordinate. The function determining the radial velocity component: will be determined by taking into consideration the pressure gradient from the following equation [19]: where ∂P ∂r is the pressure gradient along the radial coordinate. In terms of real operating modes of disk rotary-pulse devices, the complex of variables (5) is far less than the expression ω 2 h 2 ·z 2 ·r. Based upon the latter note, Equation (5) may be represented in the form of a differential equation with separable variables: It was assumed that the pressure gradient does not depend on the axial coordinate. The solution of Equation (6) taking into consideration zero boundary conditions for function f(z) will be as follows: In the latter equation, the pressure gradient along the radial coordinate ∂P ∂r remains unknown; in order to find this gradient, we will apply the continuity equation: where Q 1 is the consumption of the fluid coming through the inlet branch; Q 3 is the consumption of the fluid in the working chamber, kg/s; and Q 2 is the consumption of the fluid within the disk gap determined according to the following formula: Applying ratio (8) with the consideration of Equations (9) and (7): and having introduced (10) into (7), taking into account dependence (4), we will obtain the following expression:

Structural Features of the Experimental Disk Pulse Heat Generator
In order to test the adequacy of the obtained mathematical models determining the effect of geometrical parameters of the channels of disk pulse devices on the efficiency of the heat exchange processes, experimental studies of the incompressible liquid within the disk pulse device have been carried out. To do this, a disk pulse heat generator (see Figure 2) was designed. The heat generator is represented as a working chamber with the changeable geometry of disk channels and the working disk diameter (rotor stage). It was used to mount the experimental stand equipped with the measuring system for testing (see Figure 3). The scheme of the experimental setup for testing the disk pulse heat generator is presented in Figure 4.

Structural Features of the Experimental Disk Pulse Heat Generator
In order to test the adequacy of the obtained mathematical models determining the effect of geometrical parameters of the channels of disk pulse devices on the efficiency of the heat exchange processes, experimental studies of the incompressible liquid within the disk pulse device have been carried out. To do this, a disk pulse heat generator (see Figure 2) was designed. The heat generator is represented as a working chamber with the changeable geometry of disk channels and the working disk diameter (rotor stage). It was used to mount the experimental stand equipped with the measuring system for testing (see Figure 3). The scheme of the experimental setup for testing the disk pulse heat generator is presented in Figure 4.

Structural Features of the Experimental Disk Pulse Heat Generator
In order to test the adequacy of the obtained mathematical models determining the effect of geometrical parameters of the channels of disk pulse devices on the efficiency of the heat exchange processes, experimental studies of the incompressible liquid within the disk pulse device have been carried out. To do this, a disk pulse heat generator (see Figure 2) was designed. The heat generator is represented as a working chamber with the changeable geometry of disk channels and the working disk diameter (rotor stage). It was used to mount the experimental stand equipped with the measuring system for testing (see Figure 3). The scheme of the experimental setup for testing the disk pulse heat generator is presented in Figure 4.    Changes in the number of the shaft-located disk rotors (working wheels) facilitated the process of one-stage and multistage cavitation. The system analysis of the research results was used to determine the most efficient geometry of the disk pulse heat generator for its further integration into the system of decentralized heat supply.

Results and Discussion
The radial velocity distribution within the disk gap in terms of different values of the disk gap is presented in Figure 5. In this context, the following values are taken as the basis:  Changes in the number of the shaft-located disk rotors (working wheels) facilitated the process of one-stage and multistage cavitation. The system analysis of the research results was used to determine the most efficient geometry of the disk pulse heat generator for its further integration into the system of decentralized heat supply.

Results and Discussion
The radial velocity distribution within the disk gap in terms of different values of the disk gap is presented in Figure 5. In this context, the following values are taken as the basis: η = 5.471·10 −4 Pa·s; ω = 6.0 s −1 ; ρ = 988.1 kg/m 3 ; Q 1 = 8·10 −5 m 3 /s; Q 3 = 6·10 −5 m 3 /s.     The analysis of the dependences demonstrates that the maximum of the radial velocity component corresponds to disk channel height h = 4·10 −3 m and angular velocity ω = 20 s −1 . Taking into consideration the specified boundary conditions, the equation of heat energy balance (1) can be represented in the form of the following ratio: where V rc is the average value of radial velocity within the disk gap. Having completed the corresponding transformations, see Equation (2), the obtained expression for the dissipation function is presented as follows: In Figure 7, the distribution of the dissipation function within the disk gap is presented.
Energies 2020, 13, x FOR PEER REVIEW 8 of 14 The analysis of the dependences demonstrates that the maximum of the radial velocity component corresponds to disk channel height h = 4·10 -3 m and angular velocity ω = 20 s −1 . Taking into consideration the specified boundary conditions, the equation of heat energy balance (1) can be represented in the form of the following ratio: where rc V is the average value of radial velocity within the disk gap. Having completed the corresponding transformations, see Equation (2), the obtained expression for the dissipation function is presented as follows: 2 ) , ( In Figure 7, the distribution of the dissipation function within the disk gap is presented. The analysis of the dependences demonstrates that the maximum value of the dissipation function corresponds to h = 12 × 10 -3 m. In order to obtain the analytical solution of Equation (1), we use a transposition method (Fourier method) to write the following ratio: where Φ is the temperature representation and Θ is the heat flow representation. After corresponding re-grouping, substitution of (14) into (12), it results in the following expression: The analysis of the dependences demonstrates that the maximum value of the dissipation function corresponds to h = 12 × 10 −3 m.
In order to obtain the analytical solution of Equation (1), we use a transposition method (Fourier method) to write the following ratio: where Φ is the temperature representation and Θ is the heat flow representation. After corresponding re-grouping, substitution of (14) into (12), it results in the following expression: In analytical studies, it is accepted that ∂Φ ∂z = 0 i f z = h, so the factor of temperature change along the axis z ∂ 2 Φ ∂z 2 = 0, where a is the coefficient of temperature conductivity (a = λ/ ρ·C p ). The heat flow function varΘ is expressed by means of the following equation: Taking into consideration the two latter expressions, the function Φ will be determined from the following ratio: To solve Equation (16), we adopt the following boundary and initial conditions: The following ratio is introduced: The substitution of (19) into (16) results in the following expression: It should be emphasized that in Equation (20), the left side depends upon coordinate z, while the right side depends upon coordinate r. The left and right parts are divided by θ(z)·ξ(r) and the separate differential equations are divided by two in ordinary derivatives, having equated them to some constant (e.g., minus λ 2 ). Then, the following system of equations will be true: The solution of Equations (21) and (22) results in the following: Taking into account ratios (18) and (19), the constants of integration will have the following form: T n = [C 1 · sin(λ·z) + C 2 · cos(λ·z)]· exp −λ 2 · a |V rc | ·R 1 ·C 3 .
Energies 2020, 13, 3492 10 of 14 It results from expression (25) that the integration constant C 1 is equal to zero. Equation (26) indicates the correctness of the ratio: The latter ratio is met under the condition that: where n = 1,2,3 . . . Expression (27) is rewritten taking into consideration that C 1 is equal to zero: where C 23 = C 2 ·C 3 . It follows from Equation (30) that, in a general case, value C 23 depends upon coordinate z. To determine the constant of C 23 integration, the Taylor cosine approximation is applied: Hence: Based on ratio (32), the function Θ is determined from the following ratio: where β = n 2 ·π 2 h 2 · a |V rc | , ∂ 2 Θ ∂z 2 − |V rc | a · ∂Θ ∂r = 0. To determine function Φ in Equation (17), the following is assumed: 1. Since the derivative of the heat flow function in Equation (16) changes along the radial coordinate, then the partial derivative of that function may be replaced with the ordinary one. Consequently, Equation (16) will have the form of a simple differential equation of the first order with separable variables.
2. Express the initial conditions to determine function Φ, taking into account ratios (12), (14), and (18), as: Then, the solution of Equation (17) is as follows: Figure 8 represents the distribution of the temperature field within the disk gap.   Figure 9 shows the temperature distribution obtained for the same parameters as in Figure 5 within the disk gap outlet. In this context, the average temperature values within the gap outlet will be as follows: 324 K for curve 1, 308 K for curve 2, and 304 K for curve 3, respectively. The temperature distribution demonstrates that in terms of h = 4·× 10 -3 m, the average temperature within the disk gap outlet is 324 K. Along with the increasing dimensions of the disk gap, the average temperatures decrease to 304 K, corresponding to h = 8·× 10 -3 m. This is due to the fact that while calculating the temperature field distribution within the disk gap outlet, only friction forces were taken into account. The cavitation component of the heat exchange process (energy emitted during the collapse of cavities) was neglected. According to the experiments, in the real  Figure 9 shows the temperature distribution obtained for the same parameters as in Figure 5 within the disk gap outlet. In this context, the average temperature values within the gap outlet will be as follows: 324 K for curve 1, 308 K for curve 2, and 304 K for curve 3, respectively.
Energies 2020, 13, x FOR PEER REVIEW 11 of 14 Figure 8 represents the distribution of the temperature field within the disk gap.  Figure 9 shows the temperature distribution obtained for the same parameters as in Figure 5 within the disk gap outlet. In this context, the average temperature values within the gap outlet will be as follows: 324 K for curve 1, 308 K for curve 2, and 304 K for curve 3, respectively. The temperature distribution demonstrates that in terms of h = 4·× 10 -3 m, the average temperature within the disk gap outlet is 324 K. Along with the increasing dimensions of the disk gap, the average temperatures decrease to 304 K, corresponding to h = 8·× 10 -3 m. This is due to the fact that while calculating the temperature field distribution within the disk gap outlet, only friction forces were taken into account. The cavitation component of the heat exchange process (energy emitted during the collapse of cavities) was neglected. According to the experiments, in the real 6 x 10 -3 83 x 10 -3 160 x 10 -3 4 x 10 -3 2 x 10 -3 The temperature distribution demonstrates that in terms of h = 4·× 10 −3 m, the average temperature within the disk gap outlet is 324 K. Along with the increasing dimensions of the disk gap, the average temperatures decrease to 304 K, corresponding to h = 8·× 10 −3 m. This is due to the fact that while calculating the temperature field distribution within the disk gap outlet, only friction forces were taken into account. The cavitation component of the heat exchange process (energy emitted during the collapse of cavities) was neglected. According to the experiments, in the real process, the average temperature within the outlet of the cavitation chamber of a heat generator reaches 350 K for h = 8·× 10 −3 m.
In Figure 10 a graph demonstrating heating the energy-efficient structures of one-stage and two-stage disk pulse heat generators is given. process, the average temperature within the outlet of the cavitation chamber of a heat generator reaches 350 K for h = 8·× 10 -3 m. In Figure 10 a graph demonstrating heating the energy-efficient structures of one-stage and twostage disk pulse heat generators is given. Figure 10. Graph of the heat carrier heating within the one-stage-1 and two-stage-2 disk pulse heat generator.