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Numerical estimation of the performance of a flow-type electrohydrodynamic heat exchanger with the streamlined electrode configuration
Journal of Electrostatics 97 (2019) 31–36
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Numerical estimation of the performance of a flow-type electrohydrodynamic heat exchanger with the streamlined electrode configuration
T
A.V. Gazaryan, V.A. Chirkov∗ St. Petersburg State University, 7/9 Universitetskaya nab, St. Petersburg, 199034, Russia
ARTICLE INFO
ABSTRACT
Keywords: Computer simulation Dielectric liquids Electrohydrodynamic flow Electro-thermo-convection Enhanced cooling Heat transfer
Electrohydrodynamic (EHD) devices show many advantages over classical ones and have found wide application in various fields of engineering, in particular, for the heat transfer enhancement. The paper studies numerically the flow-type electroconvective heat exchanger that bases on both EHD pumping and mixing near the heater. The problem of the computer simulation of a realistic EHD heat exchanger is resolved by virtue of reduced model resource intensity—a technique for 3D/2D model substitution for a novel streamlined electrode configuration is proposed. The integral characteristics of EHD heat exchanger at hand are presented.
1. Introduction The past decade saw electrohydrodynamics (EHD) making a great leap from the academic science over to the industrial applicability. EHD devices show many advantages over classical ones and have found wide application in various fields of engineering, e.g., for pumping dielectric liquids on meso- and microscale [1], electrostatic atomization [2], breaking water-oil emulsions [3], and the heat transfer enhancement [4–8]. The last application is highly promising, which makes for the continuous increase in the number of research works on the topic. There are two principal types of heat exchangers—single- and doublephase ones. The operation principle of the first-type devices bases on the heat transfer from the hot surface to the adjacent layer of the working fluid (owing to the thermal conduction) and further on to the cooler. In turn, double-phase devices provide heat removal by virtue of the latent heat and convective boiling of the fluid [9]. The present paper focuses on the single-phase electrohydrodynamic heat exchanger (EHDHE), though the other case is also of great interest and represents the competing technology. The heat transfer enhancement (without using convective boiling) is achievable in the two following ways. The first one bases on the raise of pumping flow rate in the cooling loop, which can be performed by the virtue of EHD flows and was widely investigated [1,10–12]. However, the increase in pumping intensity leads to the growth of fluid resistance, especially in the case of small cross-section dimensions of the system, and to the rise of the power consumption. The second approach
∗
suggests liquid mixing near the hot surface, which enables heating the whole bulk of the fluid rather than just its boundary layers, thus reducing the energy loss due to the viscous friction. To realize liquid mixing, EHD flow (or the so-called electroconvection) can be used, which creates an additional jet in the normal direction to the heater surface. The corresponding devices are called the flow-type mixing EHDHEs, and they are studied in the paper. Presently, the experimental development of a highly efficient EHDHE is an extremely complex issue, since the physical processes involved are highly complicated, with just few parameters being available for observation, and call for profound understanding of the underlying physics. Moreover, flow-type heat exchangers exhibit strong interaction between the external flow (established in the closed loop) and the electroconvection within the EHD mixing section. In turn, the computer simulation and especially the numerical evaluation of the performance of devices in question also pose a complicated problem. It can be split into the following parts: 1) the computer simulation of EHD flows; 2) taking account of heat transfer; 3) the estimation of electrophysical characteristics needed to be introduced into the model (especially the so-called injection function); 4) the computation of a realistic device in the 3D problem statement. All subproblems except for the last one have been solved. Thus, there is a method to estimate the injection function [13,14], and a number of papers have investigated EHDHEs as fluid mixing devices but in the simplified case when the heater and the cooler are arranged within one unit [5,15–17]. At the same time, the flow-type
Corresponding author. Peterhof, Ulianovskaya ul., d. 1., L-404, St. Petersburg, 198504, Russia. E-mail address: [email protected] (V.A. Chirkov).
https://doi.org/10.1016/j.elstat.2018.11.005 Received 15 October 2017; Received in revised form 20 July 2018; Accepted 17 November 2018 0304-3886/ © 2018 Elsevier B.V. All rights reserved.
Journal of Electrostatics 97 (2019) 31–36
A.V. Gazaryan, V.A. Chirkov
div(u) = 0
(4)
γCp ∂T/∂t + div(−k∇T + γCp uT) = 0,
(5)
Supplemented with the following definitions and expressions:
configuration of the device (Fig. 1), i.e., when the heater and the cooler are placed in different sections of the same closed loop, seems to be more realistic albeit less studied, especially by means of numerical simulation. Typically, the extended injecting electrode (a blade or wire) is directed across the external jet [18–20], which leads to the following problems. Firstly, the area of the heat transfer enhancement is limited by the cross width of the electroconvection cell. Secondly, two types of flow, the external one and the electroconvection, hinder each other, producing, in particular, the backflow region [19]. The paper proposes a novel configuration of a flow-type mixing EHDHE with streamlined electrode configuration, where the external flow is directed along the extended electrode (blade) while the EHD jet runs in the normal direction, and the technique for the numerical estimation of its performance. Such a design allows the EHD flow just to mix up fluid moving along the hot surface and improve the heat transfer from the heater to the bulk without holding up the external flow, as is demonstrated below. 2. Model description and features of EHD flows To simulate electro-thermo-convection of incompressible liquid, one needs to calculate the complete set of EHD equations [21] and the heat transfer one: (1)
∂ni/∂t + div(ji) = g(n1, n2)
(2)
γ ∂u/∂t + γ (u,∇)u = −∇P + ηΔu + ρE
(3)
(6)
ρ = e(n1 – n2)
(7)
ji = nibiE − Di∇ni + niu
(8)
g(n1, n2) = σ02/(e(|b1|+|b2|)εε0) − e(|b1|+|b2|)/(εε0)
(9)
Here E is the electric field strength, ρ is the space charge density, ε is the relative electric permittivity, ε0 is the vacuum permittivity, φ is the electric potential, n is the ion concentration, j is the ion flux density, b is the ion mobility, D is the diffusion coefficient, u is the fluid velocity, σ0 is the low-voltage conductivity, e is the elementary electric charge, γ is the mass density, P is the pressure, η is the dynamic viscosity, f is the volume force, T is the temperature, Cp is the specific heat at constant pressure, k is the thermal conductivity coefficient, t is the time; subscript i indicates the ion species. The computer model allows for two ion species only (i = 1, 2: positive and negative), both univalent. The computation uses software package COMSOL Multiphysics® based on the finite element method. It is worth noting that the set of equations disregards the field-enhanced dissociation (the Wien effect due to the latter being negligible for the used value of low-voltage conductivity [22]. Actually, the direct simulation of these processes is an extremely resource-intensive task, since it requires computing the complete set of EHD and heat transfer equations in the 3D problem statement, as opposed to the above-mentioned simplified configuration of EHDHE [5,15–17]. However, the 3D problem for the selected system can be made easier by assuming that the convection is established in the plane, perpendicular to the direction of the external pumping, and, thus, weakly interacts with the latter. The idea can be clarified basing on the results of the computer simulation of the electroconvection in the 2D problem statement for the blade-plane electrode configuration. Fig. 2a shows the model geometry and boundary conditions, where di(ni,E) describes the charge loss, which is believed to equal the total current density of ions arriving to the boundary from the bulk, and finj(E) is the injection function that was obtained in Ref. [13] for the interface between steel and polydimethylsiloxane-5:
Fig. 1. A schematic illustration of the flow-type heat transfer system.
div(E) = ρ/εε0
E = −∇φ
finj(E) = A1E + A2(E − Eth)2 θ(E − Eth) Here θ is the Heaviside step function, Eth is the threshold value of the electric field; A1 = 5.3 × 107 1/(m × V × s), A2 = 0.9 × 103 1/ (V2 × s), and Eth = 1.3 × 107 V/m. All properties of the numerically
Fig. 2. Geometry and boundary conditions of the 2D EHDHE model (a); contour plots of space charge density (b) and fluid velocity with flow streamlines (c), where ρmax = 1C/m3, umax = 1.1 m/s, and N denotes the normal to the boundaries. 32
Journal of Electrostatics 97 (2019) 31–36
A.V. Gazaryan, V.A. Chirkov
investigated liquid also correspond to polydimethylsiloxane-5: γ = 920 kg/m3, η = 5.9 × 10−3 Pa × s, σ0 = 2.4 × 10−12 S/m, |b1| = |b2| = 1 × 10−8 m2/(V × s), D1 = D1 = 1 × 10−8 m2/s, ε = 2.4, k = 0.12 W/(m × K) and Cp = 1.6 × 103 J/(kg × K). Consequently, the simulation results can be considered as very close data to those observable in a realistic experiment. It is worth noting that an inflated value of the diffusion coefficient for both ion species was used in the simulation to increase the stability of the numerical solution. The substantiation of the latter approach can be the small contribution of the diffusion component to the total flow as compared to the migration and convection ones; the chosen value of the diffusion coefficient helps avoiding numerical oscillations yet keeps the solution nearly undistorted. Besides, the finite-element grid is constructed considering the features of unknown quantity distributions and has a mapped structure in the most important areas (near both electrodes and within the central jet). The linear dimensions of finite elements in boundary layers near the blade tip and the heater surface are about 1 μm whereas the maximum element size is less than 0.1 mm; the total number of elements is 22 000, which corresponds to 300 000 degrees of freedom solved for. To ascertain the solution to be mesh independent, a separate investigation was undertaken, and reducing element sizes by a factor of 1.5 (i.e., approximately twofold increase in the number of elements) was found to change the solution by less than 1.5%. The values of relevant non-dimensional numbers are the following. The electric Reynolds number Reel, i.e., the ratio of the average fluid velocity to the average rate of ion migration Reel = u/(bV0/L), is approximately 200 for the central jet, which means that convective mechanism of charge transport prevails. The dimensionless density of the injection current, i.e., the ratio of the injection current and conduction densities kinj = efinj/(σ0E/2) at the blade tip is about 8000. The latter means that the EHD flow is governed by the injection charge formation rather than the electrical conduction. The Reynolds number for the incoming flow is 3 for the model described hereinafter, whereas it is approximately 34 for the EHD flow described hereinbefore (if the half-width of the EHD jet is used as the characteristic linear dimension). In the literature [13,14,22–26], there are a lot of experimental data on EHD flow structure for the blade-plane configuration. In general, the observed flows are quite stable though not always [24]. To test numerically whether the reflection symmetry can be used, EHD flows in the system were computed using the model without the symmetry and stable flows were obtained. Therefore, all further simulations consider just a half of the model. Fig. 2b and c presents space charge and flow velocity distributions obtained at voltage V0 = 25 kV. The injection charge formation takes place just near the blade tip, thus providing a very thin charged jet to propagate through the interelectrode gap. The maximum value of space charge density is as high as 1C/m3, which causes the emergence of a very intense EHD flow with velocity in excess of 1 m/s. In turn, it is noteworthy that the circulation of the fluid in the EHDHE closed loop consumes the most part of the energy; therefore, its intensity is to be reduced as much as possible to avoid efficiency drop. If a flow-type heat exchanger does not use mixing, the heat removal can be improved mostly only by increasing the external flow intensity. However, just a thin layer of the fluid warms up in the latter case and the most part of the liquid remains unheated. Otherwise, the fluid can be warmed up nearly completely when EHD mixing is used, which provides high heat removal even in the case of low intensity of the external pumping. Thus, the use of EHD flows secures making the fluid circulation in the closed loop less intense. The average velocity of the external flow, according to experimental studies on the EHD heat transfer [9,27], is about several cm/s and can be even less if the electroconvective mixing works well. Owing to the considerable difference in the velocities of electrohydrodynamic and external jets, the total flow could be suggested to represent a superposition of two nearly independent streams, which is
the main issue of the study. In such a case, the complete 3D model (with steady-state distributions) can be substituted with the 2D one (with the transient solution), where changing time t corresponds to the displacement along the channel axis for distance z = u0 × t (here u0 is the average velocity of the external flow). The scope of the applicability of the approach is described as follows. On one hand, the characteristic time for liquid to make one turn (in the cross-section) due to electroconvection has to be much smaller than time needed for liquid to flow past the entire heater. Otherwise stated, the EHD-induced velocity has to be much higher than the average external pumping one. On the other hand, the total flow has to be laminar (i.e., the Reynolds number is to be approximately less than 1000) to avoid liquid mixing due to a turbulence. However, the latter restriction is quite a soft one, since the application of the electroconvection is useful when there is no other way (in particular, a turbulence) to mix a non-isothermal fluid. It is worth noting that the 2D transient problem (with three degrees of freedom (DOF) {x,y,t}) is much less resource-intensive than the 3D steady-state one (with the same number of DOF {x,y,z}) due to the following reasons. First of all, the time (and the size of random access memory) needed to solve the problem using the finite-element method is proportional to the squared number of DOF [28], i.e., one time step in the 2D problem requires approximately l2 less time (where l is the number of finite-element mesh nodes along z-direction in 3D) and the RAM size than computing one iteration for the corresponding 3D problem. Consequently, the total computing time for the 2D problem with m time steps is, at least, l2/m less than that for the 3D steady-state case. Since the main goal of the study is to assess the performance of a flow-type mixing EHD heat exchanger with streamlined electrode configuration, which can be practically realized just in the 2D approach, the paramount objective is to substantiate the feasibility of the 3D/2D model substitution by comparing the results obtained in both models. However, the task is so resource-intensive that even single calculation seems to be almost unfeasible. Therefore, to make computations in the 3D case possible, the proposed computer model is simplified at the first stage on the basis of calculating merely the heat transfer and hydrodynamic equations (7)–(9), where the actual Coulomb force distribution is replaced with a zone of a volume force. In the case of the blade electrode, the cross-section of the volume force area is roughly rectangular. In earlier works on the EHD flow simulation, when there was no possibility yet to compute the complete set of EHD equations [29,30], a rectangular, uniformly charged (force) area was introduced in the model instead of solving ion transport equations. The width of the force area and space charge density were estimated basing on the in-depth study of the experimental data on the EHD flow structure. The corresponding simulation results reproduce satisfactorily the experimentally observed flow pattern [29,30], but the simulation technique needs the input data to estimate the width of the area and the value of space charge density. Otherwise stated, the simulation method has no predictive power (since there is no way to adjust the input data for modified external conditions) though can help reproducing experimentally observed flows in the model and can be used to compare the results of the 2D and 3D model computations. Fig. 3 presents the simplified heat exchanger under study, which is a rectangular-section channel containing two electrodes—a blade and a plane, respectively—with the latter acting as a heater as well. Since electrostatics and ion transport equations are disregarded at this step, the shape of the blade can be simplified with its cross-section represented as a rectangle and a circle at the end. The red-shaded rectangles represent the area of volume force that is directed from the blade to the heater and is f = 5 × 104 N/m3. The volume force value was chosen in such a way to obtain a typical EHD flow velocity, which ranges from several tenths up to several units of m/s. Thus, on one hand, the velocity value computed in the simplified model (about 0.5 m/s) is a representative one; on the other hand, it is half the velocity obtained in the complete model, thus providing the validation of the 33
Journal of Electrostatics 97 (2019) 31–36
A.V. Gazaryan, V.A. Chirkov
Fig. 3. Geometry and boundary conditions for the 3D (on the left) and the 2D EHDHEs (on the right), the widths of the blade and the volume force zone are represented not to scale.
Fig. 5. Temperature distributions for the 3D model at z = 20 mm (on the left) and the 2D model at Δt = 2 s (on the right) and flow streamlines (black lines).
proposed technique for 3D/2D model substitution to be conducted under more stringent conditions. Adiabatic (for the heat transfer) and no-slip (for hydrodynamics) boundary conditions are used at the sidewalls of the heat exchanger in both 2D and 3D cases. The heater is simulated by setting fixed temperature Th = 80 °C at the boundary (a line for the 2D case and a plane for the 3D one). In the 3D model, the liquid flows in through the rectangular cross-section with uniform velocity u0 = 1 cm/s and temperature Tin = 30 °C; further, it passes through a buffer area 60 mm long, where the Poiseuille flow develops nearly completely; condition P = 0 is set at the outlet. Eqs. (7)–(9) are calculated for ∂/∂t = 0 in the 3D case, while the transient problem statement is considered for the 2D model. Besides, in order to adjust the 2D problem to be as close as possible to the 3D one, the heater is turned on only when the fluid flow is in the steady-state; so, the boundary condition for the heater temperature changes at ts = 10 s from Tin to Th where steadying time ts was estimated according to [26,31]. The 2D model has the same number of finite elements and configuration of the mesh as the cross-section of the 3D model. It consists of 4550 elements, while the full-scale 3D model has 360 000 elements. A typical computation time for 3D steady-state problem (for 220 mm long model) is approximately 500 min (and 90 GB RAM is used), whereas the time is 2 min for 2D transient model (and 1 GB RAM is used). At the second step, a flow-type mixing EHDHE is simulated basing on both the complete set of Eqs. (1)–(9) and the technique for 3D/2D model substitution. The corresponding geometry and part of boundary conditions are shown in Fig. 2a, whereas boundary conditions for the heat transport equation matches those presented in Fig. 3.
3. Results and discussion The section presents the velocity and temperature distributions obtained in the simulation as well as line plots that allow a comparison of two models—2D vs. 3D. Fig. 4 shows the cross sections of the velocity field for different z coordinates and streamlines from the 3D model to illustrate the overall structure of the flow within the EHDHE at hand. The first to be noted are helix-like streamlines of the fluid, which are explicable by the interaction of two flows—fluid pumping along the blade and EHD flow from the blade. In fact, the central part of the vortex moves faster than the periphery, which is due to the presence not only of the EHD flow field, but also of the Poiseuille flow in the cross section of the heat exchanger; the latter results from the fact that the no-slip condition is set on the walls. This is expected to be the major reason for possible differences between 2D and 3D models. To compare the results between models, it is important to know how to relate z coordinate in the 3D model and time t in the 2D one. The issue can be solved easily if coordinate z of the analyzed section of the heat exchanger is linked to the average velocity of fluid pumping u0: tλ = zλ/u0, where λ indicates different cross-sections. So, Fig. 5 shows the temperature distribution and the fluid flow lines corresponding to the 3D section of the problem at z = 20 mm (on the left) and time instant t = 20 mm/10 mm/s = 2 s (on the right). The figure shows streamlines to be quite similar, which indicates a high degree of correspondence between the velocity fields in these two models. Temperature distributions near the heater are also quite comparable, while the results near the top wall in these two models differ.
Fig. 4. Velocity distributions (m/s) for 3 cross-sections of the 3D model and the streamlines (red curves) of the flow. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
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Journal of Electrostatics 97 (2019) 31–36
A.V. Gazaryan, V.A. Chirkov
Fig. 7. The dependence of the ratio of the heat power removal and the flow velocity on the time in the complete 2D EHDHE model.
Fig. 6. The dependence of the heat power removal on the length of the simplified 3D EHDHE with (3D EHD) and without mixing (3D NO EHD) and on the time for the simplified 2D EHDHE (2D EHD).
flow-type EHDHE with the streamlined electrode configuration, which was obtained using the complete set of Eqs. (1)–(9) and the technique for 3D/2D model substitution. The curve shows the dependence of the ratio of the total heat power removal and the flow velocity on the time span of fluid flowing along the heater for the specified configuration of EHDHE and liquid properties. The data can be easily used to estimate the length of mixing section for desired performance. For example, if one needs to remove 100 W and can provide fluid pumping with velocity 5 cm/s (i.e., W/u0 = 2000 W × s/m), the time span of fluid flowing along the heater has to be about 4 s, thus providing the length of EHDHE to be 20 cm.
The discrepancy is explainable by the following. In the 2D formulation, the fluid is supposed to flow with uniform velocity u0 in the normal direction to the cross-section, whereas there is something like Poiseuille velocity distribution in the 3D case. Therefore, given the flow rates are equal for both problem statements, the local values of the velocity in z direction differ for 2D and 3D tasks. In turn, the time span needed for a fluid element located near walls to flow past the entire heat exchanger is longer in the realistic geometry than suggested in the 2D formulation. It means finally that fluid near the upper wall in the 3D model has more time to warm up than that in the 2D problem statement (and vice versa for the central area of the cross-section). Even though the differences between local temperature distributions exist, the main parameter that enables assessing the feasibility of the 3D/2D model substitution is the heat power removal, which is to be quantitatively compared in two models. Calculation of the value in the 3D case poses no difficulties and is represented as the following integral: W3D = ∬ S q⊥dS
4. Conclusions The paper deals with a flow-type mixing EHD heat exchanger that gives an opportunity to place the heater and the cooler in different sections of the same closed loop. The proposed streamlined electrode configuration is more preferable in comparison to those presented in the literature and allows the electroconvection just to mix up fluid moving along the heater without holding up the external flow. The key point is that the performance of such an EHDHE can be quantified using 2D problem statement, where varying time t corresponds to the displacement along the channel axis for distance z = u0 × t; in the case at hand, the error lies within approximately 20%. The latter feature enables evaluating the performance of a realistic heat exchanger in the complete EHD formulation even with allowance for the actual electrophysical properties and their temperature dependencies.
(10)
Where q⊥ is the normal component of the heat flux, and S is the area of the heater. In the 2D case, an analogous value can be obtained from the following consideration: W2D = ∬ S q⊥(t)dS(t) = ∬ S q⊥(t)dz(t)dx = ∫ 0X∫ t_st_end q⊥(t)u0dtdx (11) Where X is the model width, and tend = Z/u0, here Z is the length of the 3D model. Fig. 6 shows the comparison of the values in Eqs. 10 and 11 as dependent on the length of the 3D model (and, correspondingly, the calculation time for the 2D one) as well as the heat power removal for the 3D case without the volume force, i.e., only in the presence of the external liquid pumping. A growth in these curves shows that the longer the 3D model, the higher the power removal, which is fairly understandable, since the area of the heater, S, increases. The curves for 3D and 2D models with the volume force lie within 20% accuracy, which is quite an acceptable result, since the transition to the 2D model reduces the computation time and computing requirements by a factor of hundreds. In turn, this allows investigating the EHDHE in the complete EHD formulation in the transient 2D problem statement even with taking into account the actual electrophysical properties, which is an extremely resource-intensive task when the 3D approach is used. Besides, it is worth noting that the presence of the EHD mixing enhances the heat removal manifold (nearly by an order of magnitude) comparing to the case without the electroconvection. Finally, Fig. 7 displays the evaluation of the performance of the
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in a plane layer of dielectric liquid, Numer. Heat Tran. 68 (2015) 471–500, https:// doi.org/10.1080/10407782.2014.986410. A.I. Zhakin, A.E. Kuzko, Electrohydrodynamic flows and heat transfer in the bladeplane electrode system, Fluid Dynam. 48 (2013) 310–320, https://doi.org/10. 1134/S0015462813030046 the blade-plane electrode system,” Fluid Dyn., vol. 48, no. 3, pp. 310–320, May 2013. S.S.T. Fadaki, N.A.H.M. Deylami, Numerical analysis of the EHD driven flow with heat transfer in a smooth channel using multiple collectors, Heat Mass Tran. (2017), https://doi.org/10.1007/s00231-017-1994-7. D. Testi, F. D'Ettorre, D. Della Vista, W. Grassi, Parabolic flight results of electrohydrodynamic heat transfer enhancement in a square duct, Int. J. Therm. Sci. 117 (2017) 1–13, https://doi.org/10.1016/j.ijthermalsci.2017.03.015. A. Castellanos, Electrohydrodynamics, Springer Vienna, Vienna, 1998, https://doi. org/10.1007/978-3-7091-2522-9. V.A. Chirkov, S.A. Vasilkov, Y.K. Stishkov, The role of field-enhanced dissociation in electrohydrodynamic flow formation in a highly non-uniform electric field, J. Electrost. 93 (2018) 104–109, https://doi.org/10.1016/j.elstat.2018.04.008. M. Daaboul, C. Louste, H. Romat, PIV measurements on charged plumes-influence of SiO2 seeding particles on the electrical behavior, IEEE Trans. Dielectr. Electr. Insul. 16 (2009) 335–342, https://doi.org/10.1109/TDEI.2009.4815161. P. Traoré, M. Daaboul, C. Louste, Numerical simulation and PIV experimental analysis of electrohydrodynamic plumes induced by a blade electrode, J. Phys. D Appl. Phys. 43 (2010) 1–8, https://doi.org/10.1088/0022-3727/43/22/225502. V.A. Chirkov, A.A. Sitnikov, Y.K. Stishkov, A technique for rapid diagnostics of dielectric liquids basing on their high-voltage conductivity, J. Electrost. 81 (2016) 48–53, https://doi.org/10.1016/j.elstat.2016.03.001. M. Daaboul, C. Louste, H. Romat, Transient velocity induced by electric injection in blade-plane geometry, J. Electrost. 67 (2009) 359–364, https://doi.org/10.1016/j. elstat.2009.02.003. F.S. Moghanlou, A.S. Khorrami, E. Esmaeilzadeh, H. Aminfar, Experimental study on electrohydrodynamically induced heat transfer enhancement in a minichannel, Exp. Therm. Fluid Sci. 59 (2014) 24–31, https://doi.org/10.1016/j.expthermflusci. 2014.07.019. O.C. Zienkiewicz, R.L. Taylor, P. Nithiarasu, The Finite Element Method for Fluid Dynamics, sixth ed., Elsevier Science, 2005. Y.K. Stishkov, V.A. Chirkov, Computer simulation of EHD flows in a needle-plane electrode system, Tech. Phys. 53 (2008) 1407–1413, https://doi.org/10.1134/ S1063784208110030. Y.K. Stishkov, V.A. Chirkov, Features of electrohydrodynamic flows in needle-plane electrode system, 2008 IEEE Int. Conf. Dielectr. Liq, IEEE, 2008, pp. 1–3, , https:// doi.org/10.1109/ICDL.2008.4622493. V.A. Chirkov, Y.K. Stishkov, Current-time characteristic of the transient regime of electrohydrodynamic flow formation, J. Electrost. 71 (2013) 484–488, https://doi. org/10.1016/j.elstat.2012.12.005.
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Numerical estimation of the performance of a flow-type electrohydrodynamic heat exchanger with the streamlined electrode configuration
T
A.V. Gazaryan, V.A. Chirkov∗ St. Petersburg State University, 7/9 Universitetskaya nab, St. Petersburg, 199034, Russia
ARTICLE INFO
ABSTRACT
Keywords: Computer simulation Dielectric liquids Electrohydrodynamic flow Electro-thermo-convection Enhanced cooling Heat transfer
Electrohydrodynamic (EHD) devices show many advantages over classical ones and have found wide application in various fields of engineering, in particular, for the heat transfer enhancement. The paper studies numerically the flow-type electroconvective heat exchanger that bases on both EHD pumping and mixing near the heater. The problem of the computer simulation of a realistic EHD heat exchanger is resolved by virtue of reduced model resource intensity—a technique for 3D/2D model substitution for a novel streamlined electrode configuration is proposed. The integral characteristics of EHD heat exchanger at hand are presented.
1. Introduction The past decade saw electrohydrodynamics (EHD) making a great leap from the academic science over to the industrial applicability. EHD devices show many advantages over classical ones and have found wide application in various fields of engineering, e.g., for pumping dielectric liquids on meso- and microscale [1], electrostatic atomization [2], breaking water-oil emulsions [3], and the heat transfer enhancement [4–8]. The last application is highly promising, which makes for the continuous increase in the number of research works on the topic. There are two principal types of heat exchangers—single- and doublephase ones. The operation principle of the first-type devices bases on the heat transfer from the hot surface to the adjacent layer of the working fluid (owing to the thermal conduction) and further on to the cooler. In turn, double-phase devices provide heat removal by virtue of the latent heat and convective boiling of the fluid [9]. The present paper focuses on the single-phase electrohydrodynamic heat exchanger (EHDHE), though the other case is also of great interest and represents the competing technology. The heat transfer enhancement (without using convective boiling) is achievable in the two following ways. The first one bases on the raise of pumping flow rate in the cooling loop, which can be performed by the virtue of EHD flows and was widely investigated [1,10–12]. However, the increase in pumping intensity leads to the growth of fluid resistance, especially in the case of small cross-section dimensions of the system, and to the rise of the power consumption. The second approach
∗
suggests liquid mixing near the hot surface, which enables heating the whole bulk of the fluid rather than just its boundary layers, thus reducing the energy loss due to the viscous friction. To realize liquid mixing, EHD flow (or the so-called electroconvection) can be used, which creates an additional jet in the normal direction to the heater surface. The corresponding devices are called the flow-type mixing EHDHEs, and they are studied in the paper. Presently, the experimental development of a highly efficient EHDHE is an extremely complex issue, since the physical processes involved are highly complicated, with just few parameters being available for observation, and call for profound understanding of the underlying physics. Moreover, flow-type heat exchangers exhibit strong interaction between the external flow (established in the closed loop) and the electroconvection within the EHD mixing section. In turn, the computer simulation and especially the numerical evaluation of the performance of devices in question also pose a complicated problem. It can be split into the following parts: 1) the computer simulation of EHD flows; 2) taking account of heat transfer; 3) the estimation of electrophysical characteristics needed to be introduced into the model (especially the so-called injection function); 4) the computation of a realistic device in the 3D problem statement. All subproblems except for the last one have been solved. Thus, there is a method to estimate the injection function [13,14], and a number of papers have investigated EHDHEs as fluid mixing devices but in the simplified case when the heater and the cooler are arranged within one unit [5,15–17]. At the same time, the flow-type
Corresponding author. Peterhof, Ulianovskaya ul., d. 1., L-404, St. Petersburg, 198504, Russia. E-mail address: [email protected] (V.A. Chirkov).
https://doi.org/10.1016/j.elstat.2018.11.005 Received 15 October 2017; Received in revised form 20 July 2018; Accepted 17 November 2018 0304-3886/ © 2018 Elsevier B.V. All rights reserved.
Journal of Electrostatics 97 (2019) 31–36
A.V. Gazaryan, V.A. Chirkov
div(u) = 0
(4)
γCp ∂T/∂t + div(−k∇T + γCp uT) = 0,
(5)
Supplemented with the following definitions and expressions:
configuration of the device (Fig. 1), i.e., when the heater and the cooler are placed in different sections of the same closed loop, seems to be more realistic albeit less studied, especially by means of numerical simulation. Typically, the extended injecting electrode (a blade or wire) is directed across the external jet [18–20], which leads to the following problems. Firstly, the area of the heat transfer enhancement is limited by the cross width of the electroconvection cell. Secondly, two types of flow, the external one and the electroconvection, hinder each other, producing, in particular, the backflow region [19]. The paper proposes a novel configuration of a flow-type mixing EHDHE with streamlined electrode configuration, where the external flow is directed along the extended electrode (blade) while the EHD jet runs in the normal direction, and the technique for the numerical estimation of its performance. Such a design allows the EHD flow just to mix up fluid moving along the hot surface and improve the heat transfer from the heater to the bulk without holding up the external flow, as is demonstrated below. 2. Model description and features of EHD flows To simulate electro-thermo-convection of incompressible liquid, one needs to calculate the complete set of EHD equations [21] and the heat transfer one: (1)
∂ni/∂t + div(ji) = g(n1, n2)
(2)
γ ∂u/∂t + γ (u,∇)u = −∇P + ηΔu + ρE
(3)
(6)
ρ = e(n1 – n2)
(7)
ji = nibiE − Di∇ni + niu
(8)
g(n1, n2) = σ02/(e(|b1|+|b2|)εε0) − e(|b1|+|b2|)/(εε0)
(9)
Here E is the electric field strength, ρ is the space charge density, ε is the relative electric permittivity, ε0 is the vacuum permittivity, φ is the electric potential, n is the ion concentration, j is the ion flux density, b is the ion mobility, D is the diffusion coefficient, u is the fluid velocity, σ0 is the low-voltage conductivity, e is the elementary electric charge, γ is the mass density, P is the pressure, η is the dynamic viscosity, f is the volume force, T is the temperature, Cp is the specific heat at constant pressure, k is the thermal conductivity coefficient, t is the time; subscript i indicates the ion species. The computer model allows for two ion species only (i = 1, 2: positive and negative), both univalent. The computation uses software package COMSOL Multiphysics® based on the finite element method. It is worth noting that the set of equations disregards the field-enhanced dissociation (the Wien effect due to the latter being negligible for the used value of low-voltage conductivity [22]. Actually, the direct simulation of these processes is an extremely resource-intensive task, since it requires computing the complete set of EHD and heat transfer equations in the 3D problem statement, as opposed to the above-mentioned simplified configuration of EHDHE [5,15–17]. However, the 3D problem for the selected system can be made easier by assuming that the convection is established in the plane, perpendicular to the direction of the external pumping, and, thus, weakly interacts with the latter. The idea can be clarified basing on the results of the computer simulation of the electroconvection in the 2D problem statement for the blade-plane electrode configuration. Fig. 2a shows the model geometry and boundary conditions, where di(ni,E) describes the charge loss, which is believed to equal the total current density of ions arriving to the boundary from the bulk, and finj(E) is the injection function that was obtained in Ref. [13] for the interface between steel and polydimethylsiloxane-5:
Fig. 1. A schematic illustration of the flow-type heat transfer system.
div(E) = ρ/εε0
E = −∇φ
finj(E) = A1E + A2(E − Eth)2 θ(E − Eth) Here θ is the Heaviside step function, Eth is the threshold value of the electric field; A1 = 5.3 × 107 1/(m × V × s), A2 = 0.9 × 103 1/ (V2 × s), and Eth = 1.3 × 107 V/m. All properties of the numerically
Fig. 2. Geometry and boundary conditions of the 2D EHDHE model (a); contour plots of space charge density (b) and fluid velocity with flow streamlines (c), where ρmax = 1C/m3, umax = 1.1 m/s, and N denotes the normal to the boundaries. 32
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investigated liquid also correspond to polydimethylsiloxane-5: γ = 920 kg/m3, η = 5.9 × 10−3 Pa × s, σ0 = 2.4 × 10−12 S/m, |b1| = |b2| = 1 × 10−8 m2/(V × s), D1 = D1 = 1 × 10−8 m2/s, ε = 2.4, k = 0.12 W/(m × K) and Cp = 1.6 × 103 J/(kg × K). Consequently, the simulation results can be considered as very close data to those observable in a realistic experiment. It is worth noting that an inflated value of the diffusion coefficient for both ion species was used in the simulation to increase the stability of the numerical solution. The substantiation of the latter approach can be the small contribution of the diffusion component to the total flow as compared to the migration and convection ones; the chosen value of the diffusion coefficient helps avoiding numerical oscillations yet keeps the solution nearly undistorted. Besides, the finite-element grid is constructed considering the features of unknown quantity distributions and has a mapped structure in the most important areas (near both electrodes and within the central jet). The linear dimensions of finite elements in boundary layers near the blade tip and the heater surface are about 1 μm whereas the maximum element size is less than 0.1 mm; the total number of elements is 22 000, which corresponds to 300 000 degrees of freedom solved for. To ascertain the solution to be mesh independent, a separate investigation was undertaken, and reducing element sizes by a factor of 1.5 (i.e., approximately twofold increase in the number of elements) was found to change the solution by less than 1.5%. The values of relevant non-dimensional numbers are the following. The electric Reynolds number Reel, i.e., the ratio of the average fluid velocity to the average rate of ion migration Reel = u/(bV0/L), is approximately 200 for the central jet, which means that convective mechanism of charge transport prevails. The dimensionless density of the injection current, i.e., the ratio of the injection current and conduction densities kinj = efinj/(σ0E/2) at the blade tip is about 8000. The latter means that the EHD flow is governed by the injection charge formation rather than the electrical conduction. The Reynolds number for the incoming flow is 3 for the model described hereinafter, whereas it is approximately 34 for the EHD flow described hereinbefore (if the half-width of the EHD jet is used as the characteristic linear dimension). In the literature [13,14,22–26], there are a lot of experimental data on EHD flow structure for the blade-plane configuration. In general, the observed flows are quite stable though not always [24]. To test numerically whether the reflection symmetry can be used, EHD flows in the system were computed using the model without the symmetry and stable flows were obtained. Therefore, all further simulations consider just a half of the model. Fig. 2b and c presents space charge and flow velocity distributions obtained at voltage V0 = 25 kV. The injection charge formation takes place just near the blade tip, thus providing a very thin charged jet to propagate through the interelectrode gap. The maximum value of space charge density is as high as 1C/m3, which causes the emergence of a very intense EHD flow with velocity in excess of 1 m/s. In turn, it is noteworthy that the circulation of the fluid in the EHDHE closed loop consumes the most part of the energy; therefore, its intensity is to be reduced as much as possible to avoid efficiency drop. If a flow-type heat exchanger does not use mixing, the heat removal can be improved mostly only by increasing the external flow intensity. However, just a thin layer of the fluid warms up in the latter case and the most part of the liquid remains unheated. Otherwise, the fluid can be warmed up nearly completely when EHD mixing is used, which provides high heat removal even in the case of low intensity of the external pumping. Thus, the use of EHD flows secures making the fluid circulation in the closed loop less intense. The average velocity of the external flow, according to experimental studies on the EHD heat transfer [9,27], is about several cm/s and can be even less if the electroconvective mixing works well. Owing to the considerable difference in the velocities of electrohydrodynamic and external jets, the total flow could be suggested to represent a superposition of two nearly independent streams, which is
the main issue of the study. In such a case, the complete 3D model (with steady-state distributions) can be substituted with the 2D one (with the transient solution), where changing time t corresponds to the displacement along the channel axis for distance z = u0 × t (here u0 is the average velocity of the external flow). The scope of the applicability of the approach is described as follows. On one hand, the characteristic time for liquid to make one turn (in the cross-section) due to electroconvection has to be much smaller than time needed for liquid to flow past the entire heater. Otherwise stated, the EHD-induced velocity has to be much higher than the average external pumping one. On the other hand, the total flow has to be laminar (i.e., the Reynolds number is to be approximately less than 1000) to avoid liquid mixing due to a turbulence. However, the latter restriction is quite a soft one, since the application of the electroconvection is useful when there is no other way (in particular, a turbulence) to mix a non-isothermal fluid. It is worth noting that the 2D transient problem (with three degrees of freedom (DOF) {x,y,t}) is much less resource-intensive than the 3D steady-state one (with the same number of DOF {x,y,z}) due to the following reasons. First of all, the time (and the size of random access memory) needed to solve the problem using the finite-element method is proportional to the squared number of DOF [28], i.e., one time step in the 2D problem requires approximately l2 less time (where l is the number of finite-element mesh nodes along z-direction in 3D) and the RAM size than computing one iteration for the corresponding 3D problem. Consequently, the total computing time for the 2D problem with m time steps is, at least, l2/m less than that for the 3D steady-state case. Since the main goal of the study is to assess the performance of a flow-type mixing EHD heat exchanger with streamlined electrode configuration, which can be practically realized just in the 2D approach, the paramount objective is to substantiate the feasibility of the 3D/2D model substitution by comparing the results obtained in both models. However, the task is so resource-intensive that even single calculation seems to be almost unfeasible. Therefore, to make computations in the 3D case possible, the proposed computer model is simplified at the first stage on the basis of calculating merely the heat transfer and hydrodynamic equations (7)–(9), where the actual Coulomb force distribution is replaced with a zone of a volume force. In the case of the blade electrode, the cross-section of the volume force area is roughly rectangular. In earlier works on the EHD flow simulation, when there was no possibility yet to compute the complete set of EHD equations [29,30], a rectangular, uniformly charged (force) area was introduced in the model instead of solving ion transport equations. The width of the force area and space charge density were estimated basing on the in-depth study of the experimental data on the EHD flow structure. The corresponding simulation results reproduce satisfactorily the experimentally observed flow pattern [29,30], but the simulation technique needs the input data to estimate the width of the area and the value of space charge density. Otherwise stated, the simulation method has no predictive power (since there is no way to adjust the input data for modified external conditions) though can help reproducing experimentally observed flows in the model and can be used to compare the results of the 2D and 3D model computations. Fig. 3 presents the simplified heat exchanger under study, which is a rectangular-section channel containing two electrodes—a blade and a plane, respectively—with the latter acting as a heater as well. Since electrostatics and ion transport equations are disregarded at this step, the shape of the blade can be simplified with its cross-section represented as a rectangle and a circle at the end. The red-shaded rectangles represent the area of volume force that is directed from the blade to the heater and is f = 5 × 104 N/m3. The volume force value was chosen in such a way to obtain a typical EHD flow velocity, which ranges from several tenths up to several units of m/s. Thus, on one hand, the velocity value computed in the simplified model (about 0.5 m/s) is a representative one; on the other hand, it is half the velocity obtained in the complete model, thus providing the validation of the 33
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Fig. 3. Geometry and boundary conditions for the 3D (on the left) and the 2D EHDHEs (on the right), the widths of the blade and the volume force zone are represented not to scale.
Fig. 5. Temperature distributions for the 3D model at z = 20 mm (on the left) and the 2D model at Δt = 2 s (on the right) and flow streamlines (black lines).
proposed technique for 3D/2D model substitution to be conducted under more stringent conditions. Adiabatic (for the heat transfer) and no-slip (for hydrodynamics) boundary conditions are used at the sidewalls of the heat exchanger in both 2D and 3D cases. The heater is simulated by setting fixed temperature Th = 80 °C at the boundary (a line for the 2D case and a plane for the 3D one). In the 3D model, the liquid flows in through the rectangular cross-section with uniform velocity u0 = 1 cm/s and temperature Tin = 30 °C; further, it passes through a buffer area 60 mm long, where the Poiseuille flow develops nearly completely; condition P = 0 is set at the outlet. Eqs. (7)–(9) are calculated for ∂/∂t = 0 in the 3D case, while the transient problem statement is considered for the 2D model. Besides, in order to adjust the 2D problem to be as close as possible to the 3D one, the heater is turned on only when the fluid flow is in the steady-state; so, the boundary condition for the heater temperature changes at ts = 10 s from Tin to Th where steadying time ts was estimated according to [26,31]. The 2D model has the same number of finite elements and configuration of the mesh as the cross-section of the 3D model. It consists of 4550 elements, while the full-scale 3D model has 360 000 elements. A typical computation time for 3D steady-state problem (for 220 mm long model) is approximately 500 min (and 90 GB RAM is used), whereas the time is 2 min for 2D transient model (and 1 GB RAM is used). At the second step, a flow-type mixing EHDHE is simulated basing on both the complete set of Eqs. (1)–(9) and the technique for 3D/2D model substitution. The corresponding geometry and part of boundary conditions are shown in Fig. 2a, whereas boundary conditions for the heat transport equation matches those presented in Fig. 3.
3. Results and discussion The section presents the velocity and temperature distributions obtained in the simulation as well as line plots that allow a comparison of two models—2D vs. 3D. Fig. 4 shows the cross sections of the velocity field for different z coordinates and streamlines from the 3D model to illustrate the overall structure of the flow within the EHDHE at hand. The first to be noted are helix-like streamlines of the fluid, which are explicable by the interaction of two flows—fluid pumping along the blade and EHD flow from the blade. In fact, the central part of the vortex moves faster than the periphery, which is due to the presence not only of the EHD flow field, but also of the Poiseuille flow in the cross section of the heat exchanger; the latter results from the fact that the no-slip condition is set on the walls. This is expected to be the major reason for possible differences between 2D and 3D models. To compare the results between models, it is important to know how to relate z coordinate in the 3D model and time t in the 2D one. The issue can be solved easily if coordinate z of the analyzed section of the heat exchanger is linked to the average velocity of fluid pumping u0: tλ = zλ/u0, where λ indicates different cross-sections. So, Fig. 5 shows the temperature distribution and the fluid flow lines corresponding to the 3D section of the problem at z = 20 mm (on the left) and time instant t = 20 mm/10 mm/s = 2 s (on the right). The figure shows streamlines to be quite similar, which indicates a high degree of correspondence between the velocity fields in these two models. Temperature distributions near the heater are also quite comparable, while the results near the top wall in these two models differ.
Fig. 4. Velocity distributions (m/s) for 3 cross-sections of the 3D model and the streamlines (red curves) of the flow. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
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Fig. 7. The dependence of the ratio of the heat power removal and the flow velocity on the time in the complete 2D EHDHE model.
Fig. 6. The dependence of the heat power removal on the length of the simplified 3D EHDHE with (3D EHD) and without mixing (3D NO EHD) and on the time for the simplified 2D EHDHE (2D EHD).
flow-type EHDHE with the streamlined electrode configuration, which was obtained using the complete set of Eqs. (1)–(9) and the technique for 3D/2D model substitution. The curve shows the dependence of the ratio of the total heat power removal and the flow velocity on the time span of fluid flowing along the heater for the specified configuration of EHDHE and liquid properties. The data can be easily used to estimate the length of mixing section for desired performance. For example, if one needs to remove 100 W and can provide fluid pumping with velocity 5 cm/s (i.e., W/u0 = 2000 W × s/m), the time span of fluid flowing along the heater has to be about 4 s, thus providing the length of EHDHE to be 20 cm.
The discrepancy is explainable by the following. In the 2D formulation, the fluid is supposed to flow with uniform velocity u0 in the normal direction to the cross-section, whereas there is something like Poiseuille velocity distribution in the 3D case. Therefore, given the flow rates are equal for both problem statements, the local values of the velocity in z direction differ for 2D and 3D tasks. In turn, the time span needed for a fluid element located near walls to flow past the entire heat exchanger is longer in the realistic geometry than suggested in the 2D formulation. It means finally that fluid near the upper wall in the 3D model has more time to warm up than that in the 2D problem statement (and vice versa for the central area of the cross-section). Even though the differences between local temperature distributions exist, the main parameter that enables assessing the feasibility of the 3D/2D model substitution is the heat power removal, which is to be quantitatively compared in two models. Calculation of the value in the 3D case poses no difficulties and is represented as the following integral: W3D = ∬ S q⊥dS
4. Conclusions The paper deals with a flow-type mixing EHD heat exchanger that gives an opportunity to place the heater and the cooler in different sections of the same closed loop. The proposed streamlined electrode configuration is more preferable in comparison to those presented in the literature and allows the electroconvection just to mix up fluid moving along the heater without holding up the external flow. The key point is that the performance of such an EHDHE can be quantified using 2D problem statement, where varying time t corresponds to the displacement along the channel axis for distance z = u0 × t; in the case at hand, the error lies within approximately 20%. The latter feature enables evaluating the performance of a realistic heat exchanger in the complete EHD formulation even with allowance for the actual electrophysical properties and their temperature dependencies.
(10)
Where q⊥ is the normal component of the heat flux, and S is the area of the heater. In the 2D case, an analogous value can be obtained from the following consideration: W2D = ∬ S q⊥(t)dS(t) = ∬ S q⊥(t)dz(t)dx = ∫ 0X∫ t_st_end q⊥(t)u0dtdx (11) Where X is the model width, and tend = Z/u0, here Z is the length of the 3D model. Fig. 6 shows the comparison of the values in Eqs. 10 and 11 as dependent on the length of the 3D model (and, correspondingly, the calculation time for the 2D one) as well as the heat power removal for the 3D case without the volume force, i.e., only in the presence of the external liquid pumping. A growth in these curves shows that the longer the 3D model, the higher the power removal, which is fairly understandable, since the area of the heater, S, increases. The curves for 3D and 2D models with the volume force lie within 20% accuracy, which is quite an acceptable result, since the transition to the 2D model reduces the computation time and computing requirements by a factor of hundreds. In turn, this allows investigating the EHDHE in the complete EHD formulation in the transient 2D problem statement even with taking into account the actual electrophysical properties, which is an extremely resource-intensive task when the 3D approach is used. Besides, it is worth noting that the presence of the EHD mixing enhances the heat removal manifold (nearly by an order of magnitude) comparing to the case without the electroconvection. Finally, Fig. 7 displays the evaluation of the performance of the
Acknowledgments Research was carried out using resources provided by the Computer Center of SPbU, Center "Geomodel" and Center for Diagnostics of Functional Materials for Medicine, Pharmacology and Nanoelectronics of Research park of St. Petersburg State University. The reported study was supported by RFBR, research project No. 15-08-07628. References [1] M.R. Pearson, J. Seyed-Yagoobi, Experimental study of EHD conduction pumping at the meso- and micro-scale, J. Electrost. 69 (2011) 479–485, https://doi.org/10. 1016/j.elstat.2011.06.003. [2] A. Kourmatzis, J.S. Shrimpton, Electrohydrodynamic inter-electrode flow and liquid jet characteristics in charge injection atomizers, Exp. Fluid 55 (2014), https://doi. org/10.1007/s00348-014-1688-6. [3] S.M. Hellesø, P. Atten, G. Berg, L.E. Lundgaard, Experimental study of electrocoalescence of water drops in crude oil using near-infrared camera, Exp. Fluid 56 (2015) 122, https://doi.org/10.1007/s00348-015-1990-y. [4] V.K. Patel, J. Seyed-Yagoobi, Long-term performance evaluation of microscale twophase heat transport device driven by EHD conduction, IEEE Trans. Ind. Appl. 50 (2014) 3011–3016, https://doi.org/10.1109/TIA.2014.2304613.
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