Poiseuille Number Research Articles

Numerical calculations on liquid-metal magnetohydrodynamic flow in rectangular channel with wavelike uneven walls

Numerical calculations have been performed on liquid-metal magnetohydrodynamic (MHD) flows in rectangular channels with wavelike uneven walls in order to propose a flow channel in which the MHD pressure drop decreases. The calculations have been carried out for a Hartmann number of 10,000, a wall conductance ratio of 0.05, and channel aspect ratios of 1/4 and 1/9, in order to simulate typical conditions proposed to fusion reactor blankets. In the range examined in the present calculations, the Poiseuille number, which is the product of pressure gradient and Reynolds number, decreases to about half at maximum of that in a corresponding base case, i.e. a rectangular channel with flat walls.

Effect of the hotter groove on the capillary flow enhancement with nanofluids in a microgrooves wick

This work studied the capillary flow enhancement with Fe3O4-water nanofluids in a microgrooves wick when the groove is hotter than the fluid. Capillary-rise experiment was conducted to derive three parameters, ∆Pcap∙K, capillary pressure and Poiseuille number for evaluating the flow on flow driving force and resistance. Results show that the hotter groove promotes the enhancement by making it easier for the nanofluids to induce friction reduction. When the temperature difference between the groove and liquid pool, ∆T is about 15 °C, the nanofluids lead to a 76% increase in ∆Pcap∙K and 42% decrease in Poiseuille number on average, which is 30 and 15 percentage points larger than that for ∆T of zero respectively. Furthermore, it requires optimizing the concentration and ∆T for raising the enhancement with the nanofluids. However, the capillary pressure is slightly affected by the nanofluids when the groove is hotter. At the optimal concentration, the capillary pressure even becomes almost independent of ∆T. The mechanism is explained with how the hotter groove can change the particle migrations that exist in the groove cross-section and determine the wetting and friction reduction of a nanofluid.

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.

On the forced convective flow inside thermal collectors enhanced by porous media: from macro to micro-channels

PurposeAiming at finding the velocity distribution profile and other flow characteristic parameters such as the Poiseuille (Po) number, this study aims to focus on the three-dimensional forced convective flow inside rectangular ducts filled with porous media commonly used in air-based solar thermal collectors to enhance the thermal performance. The most general model for the fluid flow (i.e. the non-linear Darcy–Brinkman–Forchheimer partial differential equation subjected to slip and no-slip boundary conditions) is considered.Design/methodology/approachThe general governing equations are solved analytically based on the perturbation technique and the results are validated against numerical simulation study based on a finite-difference solution over a non-uniform but structured grid.FindingsThe analytical velocity distribution profile based on exponential functions for the above-mentioned general case is obtained, and accordingly, expressions for the Po are introduced. It is found that the velocity distribution tends to be uniform by increasing the aspect ratio of the duct. Moreover, a criterion for considering/neglecting the nonlinear drag term in the momentum equation (i.e. the Forchheimer term) is proposed. According to the sensitivity analysis, results show that the nonlinear drag term effects on the Nusselt number are important only in porous media with high Darcy numbers.Originality/valueA general analytic solution for three-dimensional forced convection flows through rectangular ducts filled with porous media for the general model of Darcy–Brinkman–Forchheimer and the general boundary condition including both no-slip and slip-flow regimes is obtained. An analytic expression to calculate Po number is obtained which can be practical for engineering estimations and a basis for validation of numerical simulations. A criterion for considering/neglecting the nonlinear drag term in the momentum equation is also introduced.

Effect of interface curvature on isothermal heat transfer in a hydrophobic microchannel with transverse ribs and cavities

Superhydrophobic microchannels have been found to deliver remarkable frictional flow resistance reduction owing to entrapped air on its surface micro texture. However, the entrapped air may adversely affect the heat transfer performance of the channel due to the increased thermal resistance offered by air. Hence the suitability of superhydrophobic microchannels in micro-scale heat transfer applications can only be ascertained through a combined thermo-hydraulic performance analysis which provides the trade-off between the decreased frictional resistance and increased thermal resistance. In this work, the thermo-hydraulic performance of a superhydrophobic microchannel consisting of transverse ribs and cavities is numerically investigated, at isothermal wall conditions, using CFD software Ansys-Fluent. The ribs and cavities that make up the channel wall can be represented by alternate no-slip (isothermal rib) and shear-free (adiabatic meniscus) zones. The liquid-gas interface formed at the cavity that separates the core flow and entrapped air forms convex, flat or concave menisci depending on the Laplace pressure. The effect of menisci shape on the combined thermo-hydraulic performance is studied for different gas-fractions and protrusion angles. The Nusselt number (Nu), Poiseuille number (fRe) and its ratio (Nu/fRe), are estimated for each case under convective heat transport settings. Increased shear-free fraction leads to an increase in the thermo-hydraulic performance. The concave menisci exhibited a uniform Nu irrespective of the protrusion angle, since the liquid near the interface is almost stagnant between successive no-slip riblets. In contrast, the convex interface demonstrated a drop in Nu with the rise in protrusion angle. The ribbed hydrophobic channel exhibited better performance than a no-slip classical channel having uniform heating for a protrusion angle range of -20°<0<+20° at different shear-free fractions. Moreover, the maximum thermo-hydraulic performance is observed for a nearly flat profile, which is in-line with the previous experimental observations. The thermal boundary layer development over the ribs and cavities is found to significantly influence the assessment of Nu and consequently the typical usage of a periodic boundary condition induces appreciable error in Nu estimation. For instance, a hydrophobic microchannel formed with 20 repeating units of ribs and cavities exhibited an error of 27% in Nu estimation in comparison with the asymptotic Nu, at a moderate Reynolds number of 100. Furthermore, two-phase VOF simulations are performed to compare the heat transfer characteristics of ribbed channels with and without the presence of entrapped air inside the cavities. Finally, the suitability of shear-free and adiabatic conditions is underscored through multi-phase simulations.

Hydrodynamic analysis of the nanofluids flow in a microchannel with hydrophobic and superhydrophobic surfaces

We studied the hydrodynamic features of the nanofluids flowing inside a straight microchannel with hydrophilic, hydrophobic, and superhydrophobic walls by the finite volume method. The single-phase method is applied for the prediction of hydrodynamic properties of water/Al2O3 nanofluids. The nanofluid is based on deionized water with particle concentrations of 0.1%, 0.5%, and 1.0%. Interpretations for velocity profiles, Poiseuille number, pumping power, and friction coefficients are presented for different hydrophobicities of the walls. We considered various flows with Reynolds numbers in the range of 50 to 400 and constant heat flux of 10,000 W/m2. The obtained results depict that the implementation of nanofluids leads to increase pumping power which is greater for higher volume fractions. The Poiseuille number shows that the effects of the entry region decrease when the superhydrophobic walls are applied because of the reduction in velocity slopes at the wall. For hydrophilic walls, nanofluid increases the pumping power by 74%. However, superhydrophobic walls with the nanofluid flow cause a 47% decrease in pumping power compared to the water flow with the usual walls. Using the superhydrophobic wall with nanofluids resolves the disadvantages of pressure drop rises significantly.

Poiseuille-Number-Based Kozeny–Carman Model for Computation of Pore Shape Factors on Arbitrary Cross Sections

Porous media characterization is crucial to engineering projects where the pore shape has impact on performance gains. Membrane filters, sportswear fabrics, and tertiary oil recovery are a few examples. Kozeny–Carman (K–C) models are one of the most frequently used to understand, for instance, the relation between porosity, permeability, and other small-scale parameters. However, they have limitations, such as the inability to capture the correct dependence of permeability on porosity, the imperfect handling of the linear and nonlinear effects yielded by its fundamental quantities, and the insufficiency of geometrical parameters to predict the permeability correctly. In this paper, we cope with the problem of determining shape factors for generic geometries that represent sundry porous media configurations. Specifically, we propose a method that embeds the Poiseuille number into the classical K–C equation and returns a substitute shape factor term for its original counterpart. To the best of our knowledge, the existing formulations are unable to obtain shape factors for pores whose geometry is beyond the regular ones. We apply a Galerkin-based integral (GBI) method that determines shape factors for generic cross sections of pore channels. The approach is tested on straight capillaries with arbitrary cross sections subject to steady single-phase flow under the laminar regime. We show that shape factors for basic geometries known from experimental results are replicable exactly. Besides, we provide shape factors with precision up to 4 digits for a class of geometries of interest. As a way to demonstrate the applicability of the GBI approach, we report a case study that determines shape factors for 19 generic individual pore sections of a laboratory experiment involving flow rate measurements in an industrial arrangement of a water-agar packed bed. Porosity, flow behavior, and velocity distributions determined numerically achieve a narrow agreement with experimental values. The findings of this study provide parameters that can help to design new devices or mechanisms that depend on arbitrary pore shapes, as well as to characterize fluid flows in heterogeneous porous media.

Pressure Losses at Moderate Reynolds Numbers in Diamond-Shaped Cylinders Arrays: Application to Microregenerators

Abstract Cylinders with an elliptical, oblong, lenticular, sinus, or diamond transveral shape are very interesting geometries for the design of compact heat exchangers. This work investigates the role of the porosity and of the apex angle of diamond-shaped cylinders networks on the pressure losses, at moderate Reynolds numbers, inside microheat regenerators. The design of the geometry under test has been chosen so that the cross section of the flow remains almost constant along the path of the flow between cylinders. Experiments have been performed at 1 ⩽ Re ⩽ 30 and a porosity range 0.40&amp;lt;ε&amp;lt;0.90 for an apex angle of α=33deg. Numerical simulations have been conducted using the same Reynolds and porosity ranges but varying the apex angle 33deg ⩽ α ⩽ 90deg. Experimental measurements and dimensional analysis have shown that the friction factor can be affected by the porosity. Two-dimensional numerical simulations confirmed that the friction factor increases with the porosity but also with the apex angle. An analysis at the scale of a channel flanked by adjacent cylinders has provided an original correlation able to describe easily the evolution of the Poiseuille number and the collective effects on the drag coefficient as a function of α and ε. Such a diamond-shaped design is found to induce much lower Poiseuille numbers than those expected from conventional stacked spheres, woven wires, and circular cylinders arrays. The findings of this study can help for better understanding the optimization of low pressure drop regenerators and how to reduce associated hydraulic power.

An analytical study on fluid flow characteristics for flow within a microchannel of rectangular cross section

The present work considers fluid flow analysis within a microchannel of rectangular cross section with different exact and approximate analytical techniques. Thus, velocity slip at the solid wall of the channel is considered. The slip flow is transformed to a no-slip flow with some minor modifications by applying an average slip velocity. The flow is represented by the Navier-Stokes equation exposed to slip boundary conditions. The governing equation is solved considering separation of variables method (SOV) as an exact method and, Integral and Variational methods as approximate methods. The Poiseuille number and slip coefficient are also determined for each method. The present prediction agrees well with available literature. To predict accuracy level of all the techniques, the results are presented in a comparative scale along with numerical prediction. It is observed that the accuracy level for the velocity profile obtained in each technique depends on the aspect ratio where as for the prediction of Poiseuille number and slip coefficient, all the technique show less dependency on the aspect ratio.

Pumping Power Performance and Frictional Resistance of Textured Microchannels in Gaseous Slip Flows

The investigation of pressure drop characteristics along the microchannels with surface texturing is crucial to arrive at a desired structural size and performance of a microchannel. This study presents the comparison of flow friction and pumping power for two different Reynolds numbers (Re = 1.01 and 50.31) for rarefied gas flows in textured microchannels. The three-dimensional microchannel is computationally modeled with ratios of the height of texture to the height of the microchannel taken as 20, 40, and 60 and texture shapes as rectangular, circular, and triangular. All simulations are conducted in the slip flow regime for the Knudsen number ranging from 1.10 A� 10-3 to 2.07 A� 10-2 and tangential momentum accommodation coefficient (TMAC) ranging from 0.1 to 0.9. The time- and cost-effective Taguchi statistical analysis of the L18 orthogonal array is carried out to find the efficiency of the textured microchannels in terms of Poiseuille number and pumping power per unit mass flow rate. It is found that the height of texture is the most important control factor for efficiency, while the shape is the least important factor for both Reynolds numbers. The second most significant factor for Re = 50.31 is the Knudsen number and that for Re = 1.01 is the TMAC. The present numerical work can be applied for designing micropiping systems and diffusers with lower pumping power, lower frictional resistance, and higher efficiency. ©

Laminar fluid flow in concentric annular ducts of non-conventional cross-section applying GBI method

Fluid flow in concentric or eccentric annular ducts have been studied for decades due to large application in medical sciences and engineering areas. This paper aims to study fully developed fluid flow in straight ducts of concentric annular geometries (circular with circular core, elliptical with circular core, elliptical with elliptical core, and circular with elliptical core) using the Galerkin-based Integral method (GBI method). The choice of method was due to the fact that in the literature it is not applied in ducts of cross-sections of the annular shape with variations between circular and elliptical. Results of different hydrodynamics parameters such as velocity distribution, Hagenbach factor, Poiseuille number, and hydrodynamic entrance length, are presented and analyzed. In different cases, the predicted hydrodynamic parameters are compared with results reported in the literature and a good concordance was obtained.

Numerical Investigation of Thermally Developing and Fully Developed Electro-Osmotic Flow in Channels with Rounded Corners

Electro-osmotic flow, that is, the motion of a polar fluid in microducts induced by an external electric field, is one micro-effect which allows fluid circulation without the use of mechanical pumping. This is of interest in the thermal management of electronic devices, as microchannels with cross sections of almost arbitrary shape can easily be integrated on the chips. It is therefore important to assess how the geometry of the channel influences the heat transfer performance. In this paper, the thermal entry region and the fully developed electro-osmotic flow in a microchannel of rectangular cross section with smoothed corners is investigated for uniform wall temperature. For the fully developed region, correlations for the Poiseuille and Nusselt numbers considering the aspect ratio and nondimensional smoothing radius are given, which can be used for practical design purposes. For thermally developing flow, it is highlighted how smoothing the corners increases the value of the local Nusselt number, with increases up to 18% over sharp corners, but that it also shortens the thermal entry length. It is also found that Joule heating in the fluid may cause a reversal of the heat flux, and that the thermal entry length has a linear dependence on the Reynolds number and the hydraulic diameter and on the logarithm of the nondimensional Joule heating.

Effect of groove depth on hydrothermal characteristics of the rectangular microchannel heat sink

A three-dimensional numerical simulation is conducted to study the characteristic of fluid flow and heat transfer in the rectangular microchannel heat sink with arc grooves. The numerical model of the microchannel is validated with the experimental data at various Reynolds numbers. The groove configurations in the present analysis are grooves on the floor of the microchannel, grooves on both the sidewalls and grooves on the sidewall, and the floor of the microchannel. The groove depth is varied from h = 20 μm to h = 80 μm for all different configurations. The simulations are carried out for Reynolds number (Re) 200, 900, and 2000, with constant bottom wall temperature. Nusselt number, Poiseuille number, and performance factor are analyzed for the above cases to quantify the effect of groove configuration and groove depth on the hydrothermal characteristics of the microchannel. It is observed that the addition of grooves on the surface of the microchannel, results in the formation of pseudo secondary flow which enhances the heat transfer in the microchannel at the expense of increased Poiseuille number. The groove depth only affects the performance of the microchannel at lower Reynolds number, and at higher Reynolds number, the groove depth does not have any role in the performance enhancement. The groove configuration also plays a major role in the performance enhancement of the microchannel.

Pore-scale simulation of flow in minichannels with porous fins

<p indent="0mm">Because of its high specific surface area, metal foam is expected to further enhance the flow and thermal performance of a minichannel heatsink. In this study, fully developed flow in minichannels with porous fins was studied. The porous geometry was constructed based on Laguerre-Voronoi tessellation and simulated in a pore scale, which eliminated the influence of parameter selection in a macroscale method. The influence of porosity, filling ratio, and Reynolds number on velocity field and pressure drop was analyzed. The general relationship between the Poiseuille number and filling ratio was determined, and a correlation was obtained using the study results. The results of this study were compared with those of the macroscale method, and finally, the influence of effective viscosity was discussed.

Poiseuille number behavior in an adiabatically choked microchannel in the slip regime

The influence of subsonic adiabatic choking on frictional resistance inside three-dimensional (3D) microchannels has not been studied for rarefied gas flows. In the present work, the variation of the Poiseuille number with respect to the Mach number has been documented for a 3D microchannel of aspect ratio (width/height) 0.49. Measurements of mass flow rate, static pressure, and temperature have been conducted with nitrogen in highly compressible and slightly rarefied (slip flow) regime: outlet Mach number (0.43–0.99), outlet Knudsen number (4.04 × 10−3–7.04 × 10−3), and pressure ratio (8.17–8.72). The present 3D measurements are compared with available analytical solutions for isothermal and adiabatic flows. A maximum deviation of only 4.8% from the adiabatic slip flow solution points toward the adiabatic nature of the exit choked state, which is being experimentally demonstrated for the first time in the highly compressible slip flow regime. Furthermore, the influence of losses of microchannel end manifolds on the overall pressure drop is calculated to be negligible. We further propose the ranges of the area ratio, Reynolds number, and Knudsen number for which these losses continue to be unimportant for gaseous slip flow. This study gives insights into the influence of subsonic choking on the frictional resistance at various mass flow rates and is relevant for future space expeditions and in certain biological applications.

Gas Flow in Microchannels and Nanochannels With Variable Cross Section for All Knudsen and All Mach Number Values

AbstractThe analytical solution for steady viscous pressure-driven compressible isothermal gas flow through micro- and nanochannels with variable cross section for all Knudsen and all Mach number values is presented in this paper. The continuum one-dimensional governing equations are solved using the friction factor that is established in a special way to provide solutions for mass flow rate, pressure, and velocity distribution through the microchannels and nanochannels in the entire rarefaction regime. The friction factor, defined by the general boundary condition and generalized diffusion coefficient proposed by Beskok and Karniadakis (1999, “A Model for Flows in Channels, Pipes, and Ducts at Micro and Nano Scales,” J. Microscale Thermophys. Eng., 3, pp. 43–77), spreads the solution application to all rarefaction regimes from continuum to free molecular flow. The correlation between the product of friction factor and Reynolds number (Poiseuille number) and Knudsen number is established explicitly in the paper. Moreover, the obtained solution includes the inertia effect, which allows the application of the solution to both subsonic and supersonic gas flows, which was not shown earlier. The presented solution confirms the existence of the Knudsen minimum in the diverging, converging, and microchannels and nanochannels with constant cross section. The proposed solution is verified by comparison with experimental, analytical, and numerical results available in literature.

Laminar drag reduction in microchannels with liquid infused textured surfaces

Liquid infused surfaces (LIS) achieve non-wetting properties through asperity structures containing pockets of a lubricating liquid rather than air. Although studies have demonstrated several potential applications of LIS, the effect of liquid infused texture on the flow of liquid in microchannels has not been systematically addressed. The present work focuses on the effect of liquid-infused texture geometry and the infused liquid properties on the flow characteristics of a bulk fluid in rectangular and round microchannel geometries. Considering four different textured geometries of transverse ridges, longitudinal ridges, in-lined square posts and staggered squares, a detailed parametric study is presented to assess the resulting drag reduction and friction factor, expressed in terms of the Poiseuille number, on the flow in microchannels. The effect of geometrical parameters is investigated for different asperity solid fraction (ϕs), asperity height (hs) and channel thickness (H) or tube diameter (D), while the effect of the infused fluid is studied in terms of a ratio of its viscosity relative to the viscosity of the flowing bulk fluid. Based on the results of the parametric study an extension of the conventional Moody diagram for non-wetting surfaces is developed. Further, a design of experiments study is conducted to relate the drag reduction and Poiseuille number to the various governing parameters, using which conditions that maximize drag reduction are identified.

A NOVEL FRACTAL SOLUTION FOR LAMINAR FLOW RESISTANCE IN ROUGHENED CYLINDRICAL MICROCHANNELS

In this work, a novel fractal model for the laminar flow in roughened cylindrical microchannels is proposed. The average height of rough elements is derived using the fractal theory. The effects of relative roughness on the friction factor and the Poiseuille number are discussed. It is found that the Darcy friction factor and the Poiseuille number increase with the increase in the relative roughness in the cylindrical microchannel. Besides, it is observed that the Darcy friction factor decreases with the increase in the Reynolds number. Each parameter of the proposed model has a clear physical meaning. The present model can properly reveal some mechanisms that affect the laminar flow in roughened cylindrical microchannels. The present model improves the understanding of the physical mechanisms of fluid flows through roughened cylindrical microchannels. Our model predictions are compared with the existing experimental data, and good agreement can be found.

Electro-osmotic non-isothermal flow in rectangular channels with smoothed corners

Microchannel heat sinks are able to provide high cooling capabilities in terms of heat flux rates. This makes them particularly interesting for the thermal management of electronic components such as CPUs, which have high power density and small dimensions. Pressure drop of the coolant across the microchannels may, however, be significant and give rise to viscous heating, thereby preventing the practical use of these devices. When the coolant is a polar fluid and the channel walls possess a net electric charge, an alternative means of moving the fluid is through an applied external electric field. The flow which originates is called electro-osmotic (EOF). EOF does not require moving parts, is free of vibrations and does not need lubrication, but is subject to Joule heating of the fluid and has flow and heat transfer characteristics which differ from those of pressure-drive flows. In spite of several previous investigation on EOF, no attention has been paid to the changes in velocity and temperature distributions caused by modifying the base cross-section of the channels which may be circular, rectangular or polygonal, thanks to the current capabilities of microfabrication. This work investigates numerically the influence of smoothing the corners of the cross-section at fixed hydraulic diameter on the values of the Poiseuille and Nusselt numbers for the laminar, steady and fully developed, electro-osmotic flow in a rectangular channel subject to uniform heat flux and Joule heating. Several aspect ratios are considered, as are different values of the ratio of Joule heating to heat flux through the walls. The results highlight a very slight increase of the Poiseuille number with the radius of curvature, whereas the Nusselt number experiences a significant improvement. Correlations are obtained for both the Poiseuille and Nusselt number as a function of the radius of curvature, aspect ratio and Joule heating-to-heat flux ratio.

Analytic study on pressure drop and heat transfer characteristics for low reynolds number flow in spirally finned tubes

Analytic solutions of the velocity and temperature profiles for a low Reynolds number flow in a spirally finned tube are obtained. The cross-sectional shape of the unit channel of the spirally finned tube is described by an annular sector with an inner radius ri, an outer radius ro, and an apex angle 2α. The channel is twisted about its longitudinal axis with a twist length H. To simplify the problem, it is assumed that the swirling effect is negligible, which is valid for low Reynolds number flows. By using an appropriate coordinate transformation, the fully-developed flow in the spirally finned tube can be treated as a quasi-2-dimensional flow. The perturbation method is used to solve the transformed momentum and energy equations for forced convection in the tube subject to the uniform heat flux condition. The Poiseuille and Nusselt numbers are obtained by using the analytic solutions for the velocity and temperature profiles. The values of fRe and Nu are presented in terms of the geometrical parameters, 2α, Ri=ri/ro, and ω=2πro/H. The results obtained from the analytic solutions show good agreement with numerical results for ω ≤ 0.5 and Re<21/ω. These analytic solutions are useful for optimizing the thermal performance of the spirally finned tube under various constraints. To illustrate their usefulness, the thermal resistance of a spirally finned tube under the constraint of fixed pumping power is evaluated. Based on the analytic solutions, the optimum fin geometry for which a minimum thermal resistance is attained can be analytically determined for a given non-dimensional pumping power. Furthermore, correlations of the optimum fin number and optimum twist ratio are presented in this study. According to the results, for relatively low pumping power conditions (Ppump*≤Ppump,crit*), the straight finned tube has better thermal performance than that of the spirally finned tube, and the optimum fin number increases as the pumping power increases. On the other hand, For relatively high pumping power conditions, (Ppump*>Ppump,crit*),the spirally finned tube has better thermal performance than that of the straight finned tube. The optimum twist ratio and the fin number increase as the pumping power increases.