Circular Microtube Research Articles

Analysis of an evaporator with single horizontal circular micro-tube using FC-72 as working fluid for electronic cooling applications

Flow boiling in micro-channels is one of the most efficient methods for heat rejection in electronics, necessitating proper system design. Numerical simulations are the most suitable tool for this purpose, with most literature focusing on CHF situations, while few studies address conditions bellow it. This paper presents a parametric analysis of an evaporator with a horizontal circular micro-tube, maintaining a fixed outlet quality, using numerical simulation. The simulation employs a one-dimensional steady-state model with FC-72 as the working fluid, considering both subcooled liquid and two-phase lengths, computed using theoretical equations and correlations. The effects of inner tube diameter, heat flux, inlet pressure, and subcooling level on key evaporator design parameters are discussed in detail. Results indicate that to minimize hydraulic pumping power, fluid temperature difference, and inlet temperature are required, the evaporator should operate with the lowest heat flux (50kW∙m-2), the largest inner tube diameter (1.3mm), the highest subcooling level (30K), and the lowest inlet pressure (2bar). Under these conditions, hydraulic pumping power can be reduced by up to 86%. Furthermore, to achieve the minimum fluid temperature difference along the tube (approximately 4.9K) and avoid CHF situation, operation with the lowest subcooling level (5K) is necessary. Finally, the ratio of the heat flux to the critical heat flux, q"/CHF, increases with rising inlet pressure and subcooling level, and/or decreasing inner tube diameter. The minimum q"/CHF ratio of 43.9% was obtained at pin=2bar, q"=50kW∙m-2, D=0.9mm, and ΔTsub=5K. These findings provide insights into optimal design parameters for evaporators in micro-electronics cooling applications.

The electrokinetic flow of a micropolar fluid in a microtube with velocity and spin velocity slippage

This article examines a theoretical investigation of electrokinetic microstructure fluid flow in a circular microtube. It is based on a linearized Poisson–Boltzmann equation and a microstructure electrolyte of Eringen micropolar type. In our analysis, we assumed that the surface zeta potential was small due to the use of the Debye-Hückel approximation. The influences of velocity slip and spin velocity slip at the surface of the microtube, as well as overlapped electrical double layers are considered. The findings of the present study demonstrate that the micropolarity of fluid particles has a declining effect on volume transport, streaming potential, and energy transport efficiency, while velocity slip and spin velocity slip enhance these factors. It was found that when the ionic Péclet number decreases from 1 to 0.1, the volume transport decreases approximately 56%, the streaming potential decreases by around 334.6%, and the energy transport efficiency increases by approximately 88.2%. The physical quantities of this study are highlighted through graphs for different values of the relevant parameters and compared with the available data in the literature.

Time-periodic electromagnetohydrodynamic flow in a circular microtube with surface charge dependent slip

The time-periodic electromagnetohydrodynamic (EMHD) flow in circular microtubes is studied by considering slip boundary conditions. An oscillating electric field and a constant magnetic field orthogonal to each other are applied externally to drive the fluid flow. The slip length considered here depends on the surface charge of the channel, which is generally a more realistic situation as demonstrated by recent experimental work. The velocity is obtained by using the characteristics of the Bessel equation. The influences of surface charge density σs and electrical oscillating Reynolds number Re on velocity and flow rate are discussed in detail. Our results demonstrate the EMHD slip flow rate can be reduced by up to 8.5% at low Reynolds number (Re=1) by considering the surface charge effect on slip, but enhanced by up to 8% at high Reynolds number (Re=50). These theoretical results can provide some theoretical basis for experimental studies.

Entropy Generation and Poiseuille Number Link in Developing Isothermal Laminar Micro-Flow

Abstract It is well-known that Poiseuille number (Po, hitherto viewed mainly as a fluid mechanics parameter) decreases along a hydrodynamically developing flow, from infinity at inlet to a fixed value downstream. This study reveals that the dimensionless entropy generation rate per unit length due to fluid friction (S˙¯′gen,fr) varies exactly the same way; hence, Po and S˙¯gen,fr′ are jointly studied for their dependence. Laminar hydrodynamic development of isothermal flow of incompressible fluid (water) in a circular micro-tube (diameter, D) is examined. Results are obtained for a given flow velocity for different D, and then, numerical experiments are conducted for different flow velocities for the same D-values. Striking similarity in trends of Po and S˙¯gen,fr′ shows a unique linear relation between them for the hydrodynamically developing region. It is theoretically shown that Po is a direct measure of entropy generation due to fluid friction, which explains its numerically obtained linear relation with S˙¯gen,fr′. It is found that in hydrodynamically developing region, both Po and S˙¯gen,fr′, decrease with decreasing D, which is the identified micro-effect.

Two-phase analysis of heat transfer and entropy generation of water-based magnetite nanofluid flow in a circular microtube with twisted porous blocks under a uniform magnetic field

In the present numerical analysis, the forced convection of nanofluid flow is investigated in a microtube with twisted porous blocks while existing a uniform magnetic field based on the first and second laws of thermodynamics with various geometries. Using the finite volume technique and SIMPLE algorithm, the governing equations are discretized and solved. The governing equations are solved using a two-phase mixture model and second-order upwind technique. With the heat flux of 5000 W.m−2, the effects of Hartmann number, Reynolds number, the volume fraction of nanoparticles, as well as porosity percentage of twisted porous blocks are investigated on pressure drop, heat transfer rate, and flow fields. The values of the Hartmann and Reynolds numbers are different within the range of Ha = 0 to Ha = 20 and Re = 250 to Re = 1000, respectively, and the volume fraction of nanoparticles is different within φ = 0 to 2%. The findings indicate that placing the porous blocks in the microtube leads to protrusions in the local diagrams, moreover, by increasing the number of twisted porous block layers, the vertices value of these protrusions increments. By incrementing the Reynolds number, the pressure drop and convective heat transfer coefficient are increased, and the friction factor is decreased. Incrementing φ and the thermal conductivity of the fluid increase the heat transfer rate. Moreover, by increasing the Hartmann number, the local friction factor and the Nusselt number are increased. According to the results, by increasing the number of twisted porous block layers in the highest φ, the average pressure drop becomes 38.171 Pa. The performance evaluation criterion in Ha = 20 and microchannel with three twisted block layers possess the highest value of 1.717.

Alternative constitutive relation for momentum transport of extended Navier–Stokes equations* *Project supported by the National Natural Science Foundation of China–Outstanding Youth Foundation (Grant No. 51522903), the National Natural Science Foundation of China (Grant Nos. 11602276 and 51479094), and the Fund from the Key Laboratory for Mechanics in Fluid Solid Coupling Systems of the Chinese Academy of Sciences.

The classical Navier–Stokes equation (NSE) is the fundamental partial differential equation that describes the flow of fluids, but in certain cases, like high local density and temperature gradient, it is inconsistent with the experimental results. Some extended Navier–Stokes equations with diffusion terms taken into consideration have been proposed. However, a consensus conclusion on the specific expression of the additional diffusion term has not been reached in the academic circle. The models adopt the form of the generalized Newtonian constitutive relation by substituting the convection velocity with a new term, or by using some analogy. In this study, a new constitutive relation for momentum transport and a momentum balance equation are obtained based on the molecular kinetic theory. The new constitutive relation preserves the symmetry of the deviation stress, and the momentum balance equation satisfies Galilean invariance. The results show that for Poiseuille flow in a circular micro-tube, self-diffusion in micro-flow needs considering even if the local density gradient is very low.

The electroviscous flow of non-Newtonian fluids in microtubes and implications for nonlinear flow in porous media

This paper aims to interpret the low-velocity nonlinear flow occurring in low-permeability reservoirs based on the theories of electrokinetic transport and non-Newtonian rheology of fluids. To achieve this end, we simulate the steady-state electroviscous flow of Bingham-Papanastasiou (BP) fluids in circular microtubes by simultaneously solving the Poisson-Boltzmann and the modified Navier-Stokes equations. The induced electrical field strength Es, velocity profiles, and the transport capacity of the non-Newtonian fluid under the effects of various factors (such as capillary radius R, zeta potential ζ, yield stress τ0, and stress growth index m) were examined. The results show that the generated Es of the BP fluid is highly affected by the fluid rheology, which is quite different from that of the Newtonian liquid. The velocity profiles become lower and flatter as m or τ0 increases, and this is more remarkable in smaller microtubes. The apparent viscosity of non-Newtonian fluid declines monotonically with increasing c∞, yet non-monotonically with R, m, τ0, and ζ. In addition, the low-velocity nonlinear flow in microtubes can be successfully captured when considering the electrokinetic flow of the non-Newtonian fluid rheology. While for the Newtonian fluid, only involving the electroviscous effect fails to generate the nonlinear flow behavior. The contributions of electrokinetic parameters versus rheological properties to the degree of flow nonlinearity are also discussed. The impact of electrokinetic parameters (ζ, c∞) on the flow characteristics is significant at high-pressure gradients and becomes trivial when the pressure gradient is relatively low. In contrast, the fluid rheological parameters (m, τ0) greatly determine the magnitude of the flow nonlinearity occurring at the low-pressure gradients. In sum, the electroviscous flow of BP fluids in microchannels provides a possible explanation of the low-velocity non-Darcy flow in porous media.

The gravitational magnetic component and its magnetic effects in linearized theory of gravity

According to the General Relativity (GR), under the approximation of the weak field, the weak gravitational field not only contains the classical Newtonian gravitational field, but also contains a gravitomagnetic (GM) field analogous to the concept of magnetic field, and the name of GM field borrows the basic idea of the magnetic field in electrodynamics. In order to study the physical properties of GM field and its associated effects, firstly, we use a similarity transformation method to decompose the gravitomagnetic component from the second-order tensor field in linearized Einstein field equation and define the concept of the GM field. Then, on this basis, we consider a circular microtube model with uniform velocity fluids (or superfluids), and study the distribution characteristics of the far GM field by this special model. We use a simpler approximation approach to improve the previous calculation method of the GM field in these kind of rings model, the result shows that the distribution characteristic of the far GM field in this model is analogous to the magnetic field produced by a dipole, it is a good correspondence between this microtube model and the dipole model. After that, we have studied the dynamic characteristics of GM field by analogizing the properties of magnetic field, and studied the test particles’ tracks in the linear time-varying GM field and the cosine time-varying GM field in the first time. In addition, in order to improve the previous research methods of ‘gravitational induction’ and ‘frame dragging’ in the GR, we have designed a circular microtube model which has a double-layer structure and with the accelerating flow fluid (or superfluids), we use simpler mathematics than before to explain these two effects by this special model. In conclusion, this work provides some new methods for the study of GM field and its associated effects.

Deep learning-based accurate and rapid tracking of 3D positional information of microparticles using digital holographic microscopy

Digital holographic microscopy (DHM) encrypts three-dimensional (3D) volumetric information of a test sample in the form of diffraction patterns. Computationally intensive numerical reconstruction and autofocusing procedures, which require precise determination of some parameters, are essential for the extraction of 3D positional information. In this study, a method that can satisfy these requirements by combining digital in-line holographic microscopy (DIHM) with deep learning was proposed. Deep neural networks were trained with tens of thousands of defocused holograms of microparticles moving in a microtube. The in-plane positions of the spherical microparticles were detected by applying Segnet and circular Hough transform, and the depth positions of the microparticles were estimated using a convolutional neural network. The performance of the proposed method was verified by conducting a planar surface experiment and tracking the 3D motions of the particles in a circular microtube. Results indicated that the trained neural networks precisely determined the 3D positions and 3D trajectories of the particles, and the proposed method outperformed conventional methods in measuring 3D positional information in the in-plane and out-of-plane direction. The process time of the proposed method for 3D tracking was only 1.92% of that of the conventional method. The present deep learning-based DIHM method can facilitate the analysis of the 3D dynamic motions of particles or cells and 3D tracking of numerous samples in motion.

Electro-magneto-hydrodynamics Flows of Burgers' Fluids in Cylindrical Domains with Time Exponential Memory

This paper investigates the axial unsteady flow of a generalized Burgers’ fluid with fractional constitutive equation in a circular micro-tube, in presence of a time-dependent pressure gradient and an electric field parallel to flow direction and a magnetic field perpendicular on the flow direction. The mathematical model used in this work is based on a time-nonlocal constitutive equation for shear stress with time-fractional Caputo-Fabrizio derivatives; therefore, the histories of the velocity gradient will influence the shear stress and fluid motion. Thermal transport is considered in the hypothesis that the temperature of the cylindrical surface is constant. Analytical solutions for the fractional differential momentum equation and energy equation are obtained by employing the Laplace transform with respect to the time variable t and the finite Hankel transform with respect to the radial coordinate r. It is important to note that the analytical solutions for many particular models such as, ordinary/fractional Burgers fluids, ordinary/fractional Oldryd-B fluids, ordinary/fractional Maxwell fluids and Newtonian fluids, can be obtained from the solutions for the generalized fractional Burgers' fluid by particularizing the material coefficients and fractional parameters. By using the obtained analytical solutions and the Mathcad software, we have carried out numerical calculations in order to analyze the influence of the memory parameters and magnetic parameter on the fluid velocity and temperature. Numerical results are presented in graphical illustrations. It is found that ordinary generalized Burgers’ fluids flow faster than the fractional generalized Burgers’ fluids.

Geometry-Based Entropy Generation Minimization in Laminar Internal Convective Micro-Flow

Abstract In this theoretical study, fully developed forced convective laminar water flow is considered in circular micro-tubes, for the constant wall heat flux boundary condition. The change in entropy generation rate ( Δ S ˙ gen \Delta {\dot{S}_{\mathrm{gen}}} ) for N micro-tubes (each of diameter D N {D_{\mathrm{N}}} ) relative to a reference tube (of 1 mm diameter) was investigated towards the micro-scale, for different tube length (l). A given total heat flow rate is to be removed using a fixed total mass flow rate through N tubes. Hence, the wall heat flux for one of the N tubes decreases towards the micro-scale, which is “thermal under-loading”. For given l, Δ S ˙ gen \Delta {\dot{S}_{\mathrm{gen}}} due to fluid conduction decreases and Δ S ˙ gen \Delta {\dot{S}_{\mathrm{gen}}} due to fluid friction increases towards the micro-scale. There exists an optimum D N {D_{\mathrm{N}}} ( = D N , opt ={D_{\mathrm{N},\mathrm{opt}}} ) at which the change in sum-total S ˙ gen {\dot{S}_{\mathrm{gen}}} ( Δ S ˙ gen , tot \Delta {\dot{S}_{\mathrm{gen},\mathrm{tot}}} ) is minimum; where D N , opt {D_{\mathrm{N},\mathrm{opt}}} decreases with decreasing l. For given l, cooling capacity of the heat sink increases towards the micro-scale. A general criterion for minimization of Δ S ˙ gen , tot \Delta {\dot{S}_{\mathrm{gen},\mathrm{tot}}} is obtained in terms of Reynolds number, Brinkman number, and dimensionless l.

Notes on factitious shear work of slip flow in a channel

Gas slip flow is observed in a micro scale channel whose characteristic length is less than about 10 μm under atmospheric conditions. To analyze the slip flow, the energy equation with the viscous dissipation term is solved with a boundary condition which includes a factitious sliding shear work term due to the slip at the wall to compensate the energy balance. Recently, an alternate method which uses the boundary condition without inclusion of the factitious sliding shear work term has been proposed. However, it seems that physics of the slip flow has not been properly understood by researchers. In this paper to clarify the issue, a very simple configuration, a steady state laminar slip flow in a hydro-dynamically fully developed region of a circular micro-tube with an adiabatic wall, is considered. Also all thermo-physical properties of the fluid are assumed to be constant. Theoretical justification to the boundary condition of the energy equation is provided using the discontinuity of the velocity of the slip flow on the wall.

An experimental study of pressure distribution nonlinearity in a circular micro-tube

This paper presents an optical absorption method for determining the magnitude of pressure distribution nonlinearity in gas flowing through a circular micro-tube, based on the technique of tunable diode laser absorption spectroscopy (TDLAS), and its application to quantify the dependence on pressure ratio and rarefaction of the nonlinearity of such pressure distributions. This technique has the advantages of straightforward deployment and non-intrusive measurement, avoiding the complex fabrication of embedded mechanical sensors inside microfluidic devices used in previous studies. A dimensionless nonlinearity quantity is defined to examine the deviation of the pressure profile from a linear profile. The nonlinearity is measured for different pressure ratios between inlet and outlet. We also operate the system in a regime where the influence of pressure ratio on the nonlinearity is constant, in order to unambiguously quantify the nonlinearity as a function of mean Knudsen number (Kn). It is found that increased rarefaction changes the pressure profile from convex to linear. Our measurements were conducted over a range of mean Kn of gas flow from 0.02 to 6.6, covering the slip flow regime (0.01 < Kn < 0.1) and the bulk of the transition flow regime (0.1 < Kn < 10), extending the lower limit of pressure compared with previous investigations of pressure profiles in micron-scale channels. Furthermore, it is shown that existing theoretical models provide differing predictions of the spatial pressure profile, and this disagreement becomes significant in the transition and free molecule regimes. In particular, except for the continuum regime and the early slip regime, the existing models have not been experimentally verified, due to the lack of available experimental data. In this paper, the existing models are examined based on our experiments from the slip regime (Kn = 0.02) to the near free molecule regime (Kn = 6.6), with detailed discussions provided. Moreover, an empirical correlation is proposed that characterizes the nonlinearity as a function of both Knudsen number and pressure ratio. With this correlation, one can predict the nonlinearity of the pressure profile at a given inlet/outlet pressure condition.

Numerical simulation of flow patterns and the effect on heat flux during R32 condensation in microtube

Based on the fluid volume approach, a transient numerical model for the condensation heat transfer and flow features in a microchannel is proposed. The flow condensation of R32 in a circular microtube with 0.1 mm diameter was studied. Four typical flow patterns, annular, injection flow, slug flow and bubbly flow are simulated along a two-dimension calculational domain successively. The numerical mode is verified by the experiments from the literature. The numerical results discern that the increase of mass flux, wall temperature and saturation temperature affect the detachment point of vapor slug further toward the outlet with higher occurrence frequency, which can be attributed to the higher Weber number and Capillary number of tail vapor core respectively. The local heat flux and wall shear stress will decrease along the flow direction overall, and tend to be constant in the single phase liquid area. However, there exit some rebounds and oscillations of the local heat flux and wall shear stress during injection and slug flow. The transient oscillations of wall shear stress can induce fluctuations and even waves in the annular flow upstream, which can be concluded as “surface tension force affecting upstream”, is proposed based on the minimum potential energy theory. This mechanism could be a supplement for the traditional theory of “flow pattern transition at high mass flux being induced by fluctuations which grow up while flowing downstream”.

Electro-osmotic flow of Eyring fluids in a circular microtube with Navier's slip boundary condition

Electro-osmotic flows of non-Newtonian fluids play an important role in microdevices and microfluidic systems. A theoretical and numerical analyses are conducted to explore the characteristics of electro-osmotic flow of non-Newtonian Eyring fluids in a circular microtube under the Navier's slip boundary condition. The exact solution of velocity distribution as well as corresponding approximate solution is obtained. The effects of slip length, electrokinetic parameter and non-Newtonian characteristics on the electro-osmotic flows are discussed and plotted graphically.

Torsional wave in a circular micro-tube with clogging attached to the inner surface

In the present paper, we study the torsional wave propagation along a micro-tube with clogging attached to its inner surface. The clogging accumulated on the inner surface of the tube is modeled as an “elastic membrane” which is described by the so-called surface elasticity. A power-series solution is particularly developed for the lowest order of wave propagation. The dispersion diagram of the lowest-order wave is numerically presented with the surface (clogging) effect.

Diffusiophoresis of a charged particle in a microtube.

The diffusiophoresis of a charged sphere along the axis of a circular microtube filled with an electrolyte solution is studied theoretically. The tube wall may be either nonconductive and impermeable or prescribed with a linear electrolyte concentration distribution. The electric double layers at the solid surfaces are thin, but the diffuse-layer polarization effect over the particle surface is considered. The general solutions to the electrokinetic differential equations are expressed in spherical and cylindrical coordinates, whereas the boundary conditions at the particle surface are satisfied by a collocation technique. The collocation solutions for the diffusiophoretic velocity of the particle, which are in good agreement with the asymptotic formula derived from a reflection method, are obtained for various values of the radius ratio and zeta potential ratio between the particle and the microtube and of other relevant parameters. The contributions from the diffusioosmotic flow along the tube wall and wall-corrected diffusiophoretic driving force to the particle velocity can be superimposed due to the linearity. Although the diffusiophoretic velocity in an uncharged microtube is in general a decreasing function of the particle-to-tube radius ratio and can reverse its direction, it can increase with increases in this ratio due to the competition of the wall effects of possible electrochemical enhancement and hydrodynamic retardation to the particle motion. When the zeta potentials associated with the tube and particle are equivalent, the diffusioosmotic flow induced by the tube wall dominates the diffusiophoretic motion.

On Temperature Jump Condition for Slip Flow in a Microchannel With Constant Wall Temperature

The analytical solution in the fully developed region of a slip flow in a circular microtube with constant wall temperature is obtained to verify the conventional temperature jump boundary condition when both viscous dissipation (VD) and substantial derivative of pressure (SDP) terms are included in the energy equation. Although the shear work term is not included in the conventional temperature jump boundary condition explicitly, it is verified that the conventional temperature jump boundary condition is valid for a slip flow in a microchannel with constant wall temperature when both viscous dissipation and substantial derivative of pressure terms are included in the energy equation. Numerical results are also obtained for a slip flow in a developing region of a circular tube. The results showed that the maximum heat transfer rate decreases with increasing Mach number.

Electrophoresis of a colloidal sphere with double-layer polarization in a microtube

A theoretical study is presented for the electrophoretic motion of a spherical particle in an electrolyte solution along the axis of a circular microtube, whose wall may be either insulating or prescribed with the linear far-field electric potential distribution. The electric double layers adjoining the charged particle surface and tube wall are finitely thin, and the polarization of the diffuse layer at the particle surface is allowed. The general solutions to the electrostatic and hydrodynamic governing equations are constructed in combined cylindrical and spherical coordinates, and the boundary conditions are enforced on the tube wall by the Fourier transform and along the particle surface by a collocation method. The collocation results for the electrophoretic mobility of the confined particle, which agree well with the asymptotic formulas obtained by using a method of reflections, are obtained for various values of the particle, wall, and solution characteristics. An insulating tube wall and a tube wall with the far-field potential distribution affect the electrophoresis of the particle quite differently. Although the particle mobility in a tube with uncharged wall in general decreases with an increase in the particle-to-tube radius ratio a/b, it can increase with an increase in a/b as this ratio is close to unity for some cases because of the competition between the wall effects of hydrodynamic retardation and possible electrochemical enhancement on the particle migration. When the zeta potential of the tube wall is comparable to that of the particle, the electroosmotic flow of the bulk fluid induced by the tube wall dominates the electrokinetic migration of the particle.

Energy Equation of Gas Flow With Low Velocity in a Microchannel

The energy equation for constant density fluid flow with the viscous dissipation term is often used for the governing equations of gas flow with low velocity in microchannels. If the gas is an ideal gas with low velocity, the average temperatures at the inlet and the outlet of an adiabatic channel are the same based on the first law of the thermodynamics. If the gas is a real gas with low velocity, the average temperature at the outlet is higher or lower than the average temperature at the inlet. However, the outlet temperature which is obtained by solving the energy equation for constant density fluid flow with the viscous dissipation term is higher than the inlet gas temperature, since the viscous dissipation term is always positive. This inconsistency arose from choice of the relationship between the enthalpy and temperature that resulted in neglecting the substantial derivative of pressure term in the energy equation. In this paper, the energy equation which includes the substantial derivative of pressure term is proposed to be used for the governing equation of gas flow with low velocity in microchannels. The proposed energy equation is verified by solving it numerically for flow in a circular microtube. Some physically consistent results are demonstrated.