Effects of slip length and hydraulic diameter on hydraulic entrance length of microchannels with superhydrophobic surfaces

Abstract

This paper investigated effects of slip length and hydraulic diameter on the hydraulic entrance length of laminar flow in superhydrophobic microchannels. Numerical investigations were performed for square microchannels with Re ranging between 0.1 and 1000. It is found that superhydrophobic microchannels have a longer hydraulic entrance length than that of conventional ones by nearly 26.62% at a low Re. The dimensionless hydraulic entrance length slightly increases with the increasing slip length at approximately Re<10, and does not vary with the hydraulic diameter. A new correlation to predict the entrance length in square microchannels with different slip lengths was developed, which has a satisfying predictive performance with a mean absolute relative deviation of 5.69%. The results not only ascertain the flow characteristics of superhydrophobic microchannels, but also suggest that super hydrophobic microchannels have more significant advantages for heat transfer enhancement at a low Re.

Keywords

laminar flow / hydraulic entrance length / super hydrophobic surface / slip length / hydraulic diameter

Introduction

Since microchannels were first used for chip cooling by Tuckerman and Pease [¹] in 1981, the thermal performance improvement has constantly been the key in this field [²]. However, a big pressure drop is mostly inevitable due to the small channel size, which is an obstacle to further applications. Methods to reduce the pressure drop mainly include increasing hydraulic diameter, designing multi-ports [³], and adding polymers and surfactants [⁴]. Recent researches [⁵,⁶] suggested that microchannels with superhydrophobic surfaces might lead to a high thermal performance with a reduced pressure drop. Considering the fact that the slip effect generated by the superhydrophobic surface can effectively reduce the flow resistance and enhance condensation, which is beneficial to save energy for the applications including thermal management [⁷], water harvesting [⁸,⁹], desalination [¹⁰], industrial power generation [¹¹], and building heating and cooling [¹²,¹³].

Superhydrophobic surfaces were originally inspired by natural phenomena, such as the water-repellent properties of lotus leaf [¹⁴], colocasia leaves [¹⁵], and water striders’ legs [¹⁶]. An artificial superhydrophobic surface usually combines a hierarchical structure consisting of micro- and nano-scale material with a low surface energy. The superhydrophobic surface exhibits unique properties, including a large contact angle (>150°) and low contact angle hysteresis (<10°). Trapped air in the hierarchical structure is maintained due to the surface tension and pressure difference between liquid and air layer, namely the Cassie-Baxter state [¹⁷]. The stable air trapped is the main reason for the slip flow and drag reduction of superhydrophobic surface. The no-slip condition is usually imposed at a normal solid-liquid interface. In case of microchannels with superhydrophobic surfaces, the slip boundary condition should be assumed. With regard to the slip flow regime, the concept of a slip boundary condition was first defined by Navier [¹⁸], as shown schematically in Fig. 1. The wall velocity can be expressed as

(1)u_{\mathrm{w}}=l_{\mathrm{s}}{{\displaystyle \frac{\partial u}{\partial n}}}_{\mathrm{w}},

where l_s is the slip length, u_w is equal to the product of the slip length and the shear rate at the wall. Generally, a bigger slip length means a stronger superhydrophilicity. This model is considered valid when the shear rate is far lower than a critical value of 10¹¹ s^–1 [¹⁹].

Recently, superhydrophobic surfaces have been shown to have a great ability for drag reduction. Some studies [²⁰–²²] pointed out that pressure drop was significantly reduced in laminar flow due to slip flow. They achieved drag reductions greater than 40% in some cases, and revealed that the pressure drop decreased with the increased velocity at the wall surface. Guan et al. [²³] experimentally investigated the slip flow in superhydrophobic microtubes. The reduction in the pressure drop and the friction factor exceeded 60%. Slip lengths were obtained by combining a theoretical method and experimental data. Other Refs. [⁶,²⁴,²⁵] also arrived at the conclusion that superhydrophobic surface was good for drag reduction in microchannels. Choi et al. [²⁶] experimentally studied the slip effects in hydrophilic and hydrophobic microchannels. The slip length was found to vary approximately linearly with the shear rate, and a value of nearly 30 nm was obtained in hydrophobic microchannels at a shear rate of 10⁵ s^–1. On the other hand, slip flow also has a big influence on the heat transfer inside the microchannel [²⁷]. Ermagan and Rafee [⁵,²⁸] numerically compared the fluid flow and heat transfer of superhydrophobic and conventional microchannels. The results showed that superhydrophobic microchannels have a superior thermal performance under identical pressure drop. The thermal resistance was reduced by 20%. The goodness factors of superhydrophobic and conventional microchannels were compared by Cheng et al [⁶]. The numerical results suggested that the superhydrophobic microchannel has a better performance at Re= 100 and 1000 but not Re= 1.

It is well known that the thermal performance of microchannels is greatly affected by the flow condition (developing flow or fully developed flow), and developing flow is beneficial for heat transfer enhancement. Therefore, it is of significant importance to precisely evaluate the hydraulic entrance length of the laminar flow in microchannels [²⁹,³⁰]. Atkinson et al. [³¹] experimentally investigated the effect of Reynolds number (Re) on hydraulic entrance length for macroscale flows. The dimensionless hydraulic entrance length can be expressed as

L_{\mathrm{h}\mathrm{y}}/D_{\mathrm{h}}=C_1+C_2\mathit{Re},

where L_hy, D_h, and Re are the hydraulic entrance length, hydraulic diameter, and Reynolds number respectively; L_hy/D_h is the dimensionless hydraulic entrance length (called L⁺_hy); and C₁ and C₂ are fitting coefficients.

There is a linear relationship between the dimensionless hydraulic entrance length and Reynolds number. Chen [³²] proposed a new form of correlation for parallel plates:

L_{\mathrm{h}\mathrm{y}}^{+}=C_1/(1+C_2\mathit{Re})+C_3\mathit{Re}\mathrm{.}

When Re is high, an approximately linear curve is depicted (C₁/(1+ C₂Re)<<C₃Re). However, at a very low Re, L⁺_hy becomes constant and is independent of Re (C₂Re, C₃Re≈ 0).

Ahamad and Hassan [³³] experimentally investigated the adiabatic and developing flow in square microchannels using microparticle image velocimetry, and proposed an empirical correlation for predicting hydraulic entrance length with the hydraulic diameter of 100 mm, 200 mm, and 500 mm while Reynolds numbers range from 0.5 to 200. Duan and Muzychka [³⁴] established an analytical model for predicting the Poiseuille number and the hydraulic entrance length in developing gaseous flows and continuum flows in microchannels. They predicted that the deviations of the model are approximately 10% for most duct shapes. Renksizbulut and Niazmand [³⁵] numerically studied the laminar flow and heat transfer in the entrance region of trapezoidal and rectangular channels. The accuracy of the correlation for hydraulic entrance length was estimated to be 15% for 10≤Re≤1000 and rectangular channel aspect ratios ranging from 0.5 to 2. Ma et al. [³⁶] studied the effects of Reynolds numbers and aspect ratio on the hydraulic entrance length of rectangular microchannels in slip regime by using lattice Boltzmann simulation. A correlation for the hydraulic entrance length of slip condition was proposed for rectangular channels with the aspect ratio between 1 and 10 and 1<Re<1000. Hsieh and Lin [³⁷] experimentally studied the laminar flow and heat transfer in microchannels with hydrophobic and hydrophilic surfaces. It was found that the hydrophobic channels have a longer hydraulic entrance length than that of the hydrophilic channel at 5<Re<240. The correlation of L⁺_hy= 0.064Re (not involving slip length) for hydrophobic microchannels was presented. Chakraborty and Anand [³⁸] theoretically and experimentally investigated the implications of entrance region transport in hydrophobic parallel plates. The results showed that the hydraulic entrance length of hydrophobic microchannels only with a layer of nanobubbles can be increased by a maximum of 50%. Analytical solutions of velocity profiles, friction factor, and hydraulic entrance length were obtained. Ranjith et al. [³⁹] studied the hydrodynamics of a steady nonuniform developing flow between parallel plates with hydrophilic and hydrophobic surfaces by using the method of dissipative particle dynamics. The hydrophobic and hydrophilic surfaces were modeled using partial-slip and no-slip boundary conditions, respectively. The results suggested that hydrophobic surfaces can increase the hydraulic entrance length due to a smaller shear force compared with hydrophilic surfaces. Yu et al. [⁴⁰] studied the developing flow in parallel plates with transverse grooves and also obtained a similar conclusion as Ranjith.

In general, a good estimation of hydraulic entrance length can provide a useful guidance for the design and application of superhydrophobic microchannels. Previous studies of superhydrophobic microchannels focused on the drag reduction and heat transfer characteristics mainly in the fully developed flow. Some studies on the developing flow were mainly carried out for the parallel plates. Only a few researches on superhydrophobic rectangular or square microchannels (main shapes in applications) were conducted. The knowledge for the developing flow of superhydrophobic microchannels is still insufficient. Therefore, more in-depth investigations are needed.

In this paper, the effects of slip length and hydraulic diameter on the hydraulic entrance length were numerically studied for superhydrophobic square microchannels, with Reynolds numbers ranging from 0.1 to 1000 (including the Stokes flow) and hydraulic diameters between 100 mm and 500 mm. Besides, the differences between superhydrophobic and conventional microchannels were discussed. Moreover, a correlation to predict the entrance length in square microchannels with different slip lengths was developed.

Model description

Physical model

The geometry investigated in this paper is illustrated in Fig. 2(a), which is a square channel with a fixed length L of 1 cm. Different side lengths of 100 mm, 200 mm, and 500 mm are considered. The inlet and outlet of water are located at x = 0 and x = L. Considering that the temperature rise in the actual microchannel for thermal management is small, the assumption of constant properties is adopted during the research. The properties of water at 293 K are adopted: density ρ = 997 kg·m^–3, viscosity m = 0.000855 kg·(m·s)^–1. Inlet velocities (u_in) range from 0.001 to 8 m·s⁻¹, corresponding Reynolds numbers of 0.1–1000.

Superhydrophobic surfaces are applied to channel walls (y = 0 and a, z = 0 and b). In Qu’s study, a slip length of 20 mm was obtained in a rectangular microchannel with a = 2.54 mm and b = 127 mm [²⁰]. Xu and Li [⁴¹] pointed out that the slip lengths were mainly dependent on the interfacial parameters at a fixed apparent shear rate. Slip length is almost independent with the channel size exceeding a critical value within a couple of tens of molecular diameters. Williams [⁴²] and Solomon et al. [⁴³] reviewed the researches on slip length in detail, indicating that the slip length ranges from nanoscale to microscale. A maximum slip length of 400 mm was reported by Lee and Kim [⁴⁴]. In the present study, the prescribed slip lengths (l_s) of 2, 5, 10, 12, 15, and 20 mm were considered regardless of the morphology of the superhydrophobic surface and the Reynolds number for superhydrophobic microchannels [⁴⁵,⁴⁶]. Table 1 is a summary of numerical parameters.

Numerical model

A three-dimensional flow model was established by COMSOL Multiphysics 5.3a as shown in Fig. 2. The flow is assumed to be laminar, steady, and incompressible. The properties of water are assumed to be constant. The governing equations are given by

Continuity equation:

(2){\displaystyle \frac{\partial u_i}{\partial x_i}}=\mathrm{0}.

Momentum equation:

(3){\displaystyle \frac{\partial \left({{\displaystyle \rho u}}_i{{\displaystyle u}}_j\right)}{{{\displaystyle \partial x}}_i}}={\displaystyle \frac{\partial p}{{{\displaystyle \partial x}}_j}}+\mu {\displaystyle \frac{{\partial }^{\mathrm{2}}{{\displaystyle u}}_j}{{{\displaystyle \partial x}}_i^{\mathrm{2}}}}.

Boundary condition:

(4)\mathrm{At}\hspace{0.17em}x=\mathrm{0},\hspace{0.17em}{{\displaystyle u}}_x={{\displaystyle u}}_{\mathrm{in}}\mathrm{;}\hspace{0.17em}{{\displaystyle u}}_y,\hspace{0.17em}{{\displaystyle u}}_z=\mathrm{0.}

(5)\mathrm{At}\hspace{0.17em}x=L,\hspace{0.17em}p=\mathrm{0.}

At y = 0 and a, z = 0 and b, the Navier-slip boundary condition is applied on the channel walls, u_w = l_s(∂u/∂n)|_w, l_s = 2, 5, 10, 12, 15, and 20 mm are considered.

The hydraulic diameter D_h and Reynolds number Re are defined as

(6)D_{\mathrm{h}}={\displaystyle \frac{\mathrm{2}ab}{a+b}},

(7)\mathit{Re}={\displaystyle \frac{\rho u_{\mathrm{in}}{D}_{\mathrm{h}}}{\mu }}.

The hexahedral meshing grid scheme was used to mesh the domain area as depicted in Fig. 2(b). A highly compressed non-uniform grid near the channel walls was adopted in order to properly resolve viscous shear layers. Grid nodes were also concentrated along the flow direction in the inlet of the channel in order to properly resolve the developing flow region. A grid independence study was conducted using velocity profiles at the centerline as a criterion to ensure that the results are independent of the mesh. Taking the channel of 100 μm × 100 μm as an example, three different grid sizes of 200 × 20 × 20 (grid 1), 300 × 30 × 30 (grid 2) and 400 × 40 × 40 (grid 3) were used for x-y-z direction, respectively. The independence results were illustrated graphically in Fig. 3. The centerline velocity changed by 0.15% from grid 1 to grid 2, and by 0.04% upon further refinement to gird 3, with x ranging from 0 to 400 mm. Hence, the intermediate grid size of 300 × 30 × 30 was chosen in order to improve computational efficiency. Similar tests were performed for other channels.

As for the calculation of the hydraulic entrance length, which is theoretically defined as the distance from the inlet to a location where the velocity profiles have reached 99% of the fully developed flow, the hydraulic entrance length is theoretically defined as the distance from the inlet to a location where the velocity profiles have reached 99% of the fully developed flow. Referring to other methods of calculation [³⁰,³³,⁴⁷], the hydraulic entrance length is defined as the length from the inlet to the location where the centerline velocity reaches 99% of the corresponding one in fully developed flow. As a typical example, Fig. 4 demonstrates the hydraulic entrance length identification in the 100 μm × 100 μm channel at u_in = 0.5 m·s^–1. Figure 4(a) exhibits the centerline velocity profile along the flow direction. It shows that x = 450 mm is the end of the entrance region, i.e., L_hy = 450 mm. Figure 4(b) displays the velocity profile of plane xy at z = 50 mm.

Results and discussion

Numerical validations

To validate the model above, a flow between parallel plates with superhydrophobic surfaces is constructed. The channel has a dimension of H× L = 20 mm × 1 cm, and L is considerably larger than H to ensure that fully developed flow occupies most of the channel length. The pressure drop Dp was found for different inlet velocities ranging from 1.0–2.0 m/s and different slip lengths between 2–20 mm. The numerical results were validated by comparing with an empirical pressure drop correlation of a fully developed laminar flow between parallel-plate microchannels which are based on the experiments by Choi et al. [²⁴] and Kandlikar et al. [⁴⁸].

(8){{\displaystyle \frac{\mathrm{\Delta }{p}_{\mathrm{slip}}}{\mathrm{\Delta }{p}_{\mathrm{no-slip}}}}|}_Q={\displaystyle \frac{\mathrm{1}}{\mathrm{1}+\mathrm{6}{l}_{\mathrm{s}}/H}},

(9){{\displaystyle \mathrm{\Delta }p}}_{\mathrm{no-slip}}={\displaystyle \frac{\mathrm{48}{{\displaystyle \mu u}}_{\mathrm{in}}L}{{{\displaystyle D}}_{\mathrm{h}}^{\mathrm{2}}}},

where Dp_slip and Dp_no-slip are the pressure drop of the slip flow (superhydrophobic microchannels) and the no-slip flow (conventional microchannels), respectively. Equation (8) expresses the relationship between Dp_slip and Dp_no-slip when the flow rate is identical.

The pressure drops of the correlation and simulation were compared and plotted in Fig. 5. The lines represent the results calculated by Eqs. (8) and (9), while the points represent the results of the simulation. The pressure drops of the simulation are slightly larger than that of correlation, which may result from the result of the simulation including the hydraulic entrance region. A good agreement between the results indicates the ability of the present simulation to predict the hydraulic performance of superhydrophobic microchannels with different slip lengths.

Drag reduction

The superhydrophobic surface is well known as one of the effective drag reduction methods. A number of results have been published to support the drag reduction effect of superhydrophobic surfaces. In this paper, Fig. 6 shows the pressure drop and friction factor in the 100 μm × 100 μm channel at u_in = 1 m·s⁻¹ along the fluid flow direction. Both pressure drop and friction factor decrease with the increasing slip length, which means a better superhydrophobicity will lead to a large drag reduction. Significant reductions in the pressure drop and friction factor are obtained compared with the no-slip (l_s = 0) and the slip flow. For example, a reduction in pressure drop of more than 60% is achieved when the slip length changes from 0 mm to 20 mm in a 1 cm long channel.

Effect of slip length

The variation of L⁺_hy with Re at different slip lengths in the 100 mm × 100 mm channel is presented in Fig. 7, in logarithmic scale in order to clearly show the numbers at low Re numbers. For comparison, Fig. 7 includes the experimental results (black line) of Ahmad and Hassan [³³] for conventional channels. It is found that for superhydrophobic and conventional channels, the variation of L⁺_hy versus Re follows a similar law. However, superhydrophobic channels have a longer hydraulic entrance length than that of the conventional one by nearly 26.62% at a low Re (approximately Re<10), while the differences of both are very little at Re>10. Notably, the experimental results of Hsieh and Lin [³⁷] pointed out that the hydrophobic channel holds a longer hydraulic entrance length than that of the hydrophilic channel by nearly 9.52% at 5<Re<240. There are two reasons for the difference between the two conclusions. First, Hsieh analyzed and fitted the hydraulic entrance length correlations of the hydrophobic and hydrophilic channels within a big Reynolds range without special attention to the low Reynolds numbers. Secondly, the superhydrophobicity of the channels used in Hsieh’s experiments is unknown to us. Except for the difference, it is clear that superhydrophobic channels have a longer hydraulic entrance length.

Comparing superhydrophobic microchannels with conventional microchannels, when the inlet velocity is identical, the slip flow results in a nonzero wall velocity, and the centerline velocity decreases. This leads to a decreased velocity gradient and overall shear force, while the inertia force is basically the same. At a low Re, the flow development is mainly controlled by the shear force. The reduction of the shear force leads to an increase in the hydraulic entrance length. However, the dominant force of flow development changes from the shear force to the inertial force with the increase in Re, and consequently, the difference between superhydrophobic and conventional microchannels is very small at a large Re. Therefore, the change of the dominant force affecting the flow development causes the difference in hydraulic entrance length between superhydrophobic and conventional microchannels. For superhydrophobic microchannels, it is also found in Fig. 7 that L⁺_hy slightly increases with increasing slip length at a low Re, which results from the decreased shear force. In a word, the differences in L⁺_hy due to different slip lengths within the observed range are very small.

Effect of hydraulic diameter

Based on the analysis in the Section 3.3, the effect of slip length just has an influence on the hydraulic entrance length at low Reynold numbers. To save computational time, this section just takes superhydrophobic channels with a slip length of 2 mm as an example for further study.

L⁺_hy versus Re in square channels with different hydraulic diameters at l_s = 2 μm is shown in Fig. 8, including 100 μm × 100 μm, 200 μm × 200 μm and 500 μm × 500 μm channels. With the increase in Re, L⁺_hy increases slightly and almost maintains constant at low Reynolds numbers (Re<10), and then it increases linearly with Reynolds numbers at Re>10. This means that the Reynolds numbers influence the hydraulic entrance length at relatively higher Re values. In addition, it is obvious that the different square channels follow the same trend as shown in Fig. 8. Thus, the hydraulic diameter has no influence on L⁺_hy in square channels with superhydrophobic surfaces.

Correlation developments for superhydrophobic microchannels

The numerical data of Fig. 6 were compared with the predicted results calculated by using six correlations based on no-slip flow from the literature. Four of them are L⁺_hy correlations of rectangular channel, they are Ahamad and Hassan [³³], Galvis et al [³⁰], Han [⁴⁹] and Wiginton and Dalton [⁵⁰]. Two others are the correlations for the parallel plates, which are Atkinson et al. [³¹] and Chen [³²]. In Fig. 9, numerical and calculated L⁺_hy are taken as abscissa and ordinate under the logarithmic coordinate respectively. The comparison results are shown in Table 2. In Table 2, the parameters of mean absolute relative deviation (MARD), mean relative deviation (MRD), the percentage of points calculated within a deviation bandwidth and the standard deviation s_s are taken for the statistical analysis. Each parameter is defined as follows:

(10)\mathrm{M}\mathrm{A}\mathrm{R}\mathrm{D}={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \mathrm{\sum }_{\mathrm{1}}^n\left|{\displaystyle \frac{{\left(L_{\mathrm{hy}}/D_{\mathrm{h}}\right)}_{\mathrm{cor}}-{\left(L_{\mathrm{hy}}/D_{\mathrm{h}}\right)}_{\mathrm{sim}}}{{\left(L_{\mathrm{hy}}/D_{\mathrm{h}}\right)}_{\mathrm{sim}}}}\right|}\times \mathrm{100},

(11)\mathrm{M}\mathrm{R}\mathrm{D}={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \mathrm{\sum }_{\mathrm{1}}^n\left[{\displaystyle \frac{{\left(L_{\mathrm{hy}}/D_{\mathrm{h}}\right)}_{\mathrm{cor}}-{\left(L_{\mathrm{hy}}/D_{\mathrm{h}}\right)}_{\mathrm{sim}}}{{\left(L_{\mathrm{hy}}/D_{\mathrm{h}}\right)}_{\mathrm{sim}}}}\right]}\times \mathrm{100},

(12){\sigma }_{\mathrm{s}}=\sqrt{{\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^n{\left({\alpha }_k-\overline{\alpha }\right)}^{\mathrm{2}}}},

(13){\alpha }_k={\displaystyle \frac{{\left(L_{\mathrm{hy}}/D_{\mathrm{h}}\right)}_{\mathrm{cor}}-{\left(L_{\mathrm{hy}}/D_{\mathrm{h}}\right)}_{\mathrm{sim}}}{{\left(L_{\mathrm{hy}}/D_{\mathrm{h}}\right)}_{\mathrm{sim}}}},

(14)\overline{\alpha }={\displaystyle \frac{\mathrm{1}}{n}}{\displaystyle \mathrm{\sum }_{k=\mathrm{1}}^n{\alpha }_k}.

As shown in Fig. 9, compared with numerical data, Atkinson and Chen’s correlation tends to underestimate that Han and Wiginton & Dalton’s correlations produce a remarkable deviation (far more than ±40%) when Reynolds number is less than 1, but they have a relatively better predictive performance at Re>1. L⁺_hy is proportional to the Reynolds number by a fixed constant according to Han and Wiginton & Dalton’s correlations, which are not suitable for Stokes flow in superhydrophobic microchannels. Both Ahamad & Hassan and Galvis’s correlation show a relatively high predictive performance referring to Table 2. However, the predictive performance still needs to be improved for Stokes flow. It is necessary to develop a better correlation for superhydrophobic microchannels.

Based on the correlation of Galvis, a correctional form L⁺_hy = C₁/(1+ 0.09Re) + C₂Re is adapted to fit a new correlation for superhydrophobic microchannels with different slip lengths. The hydraulic entrance length at low Reynolds number flow is closely related to C₁, and the effect of slip length is most apparent at low Reynolds number. Therefore, C₁ takes in the following form to consider the effect of slip length.

(15)C_{\mathrm{1}}=a+b{\displaystyle \frac{l_{\mathrm{s}}}{D_{\mathrm{h}}}}+c{\left({\displaystyle \frac{l_{\mathrm{s}}}{D_{\mathrm{h}}}}\right)}^{\mathrm{2}}.

Finally, the correlation as follows can be obtained.

(16)L_{\mathrm{hy}}^{+}={\displaystyle \frac{\mathrm{0.77384}+\mathrm{0.93427}{\displaystyle \frac{l_{\mathrm{s}}}{D_{\mathrm{h}}}}-\mathrm{2.55824}{\left({\displaystyle \frac{l_{\mathrm{s}}}{D_{\mathrm{h}}}}\right)}^{\mathrm{2}}}{\mathrm{1}+\mathrm{0.09}\mathit{Re}}}+\mathrm{0.06832}\mathit{Re}.

The correlation is valid for the superhydrophobic square microchannels with slip length ranging from 2 mm to 20 mm, and Reynolds number ranging from 0.1 to 1000. As shown in Fig. 10, the results of simulation and the new correlation have a good agreement. Table 2 reveals that the performance of the new correlation is the best, MRD, MARD and σ_s are 0.05%, 5.69%, and 6.32%, respectively. In addition, the new correlation has an excellent predictive performance at Stokes flow. Based on the correlation, the increase in l_s/D_h will increase the hydraulic entrance length. The slip effect will become more and more obvious as the characteristic scale of the flow channel decreases.

The comparison with Hsieh’s experimental was conducted. At 5<Re<240 (the scope of Hsieh’s experimental research), the developed correlation in the current work can be simplified as L⁺_hy= 0.06832Re, which is closed to the correlation of L⁺_hy= 0.064Re fitted by Hsieh et al.

Discussion on practical application

During the numerical study, an effective slip length was assumed to study the flow and heat transfer in a superhydrophobic microtube. In reality, superhydrophobic surfaces usually have a heterogeneous texture and morphology, which leads to a different velocity profiles at different positions of the channel wall. This means that slip length depends on the surface morphology and varies with the surface position. The scale of the hierarchical structure is approximately 0.1 mm to 1 mm, which is several orders of magnitude lower than the channel scale (100 – 500 mm) in the present paper. Therefore, the size effects of superhydrophobic surfaces on micro flow behaviors are very limited. The size effects will become more obvious as the channel scale decreases, which should be taken seriously. Most of the previous numerical and analytical studies [⁶,⁵¹–⁵³] were performed for regular structural surfaces (including square holes, square posts, longitudinal and transverse grooves). However, some researches [⁴⁶,⁵⁴] adopted the effective slip length and temperature jump length. In experimental studies, slip length is usually determined by measuring the flow rate and pressure drop of superhydrophobic and smooth channels (the formula of DP_slip/DP_no-slip|_Q = 1/(1+ 6l_s/h) was used to calculated the slip length for parallel plates with superhydrophobic surfaces, where DP_slip/DP_no-slip|_Q represents the ratio of pressure drop of slip flow to no-slip flow at the same flow rate, and h is the distance of the parallel plates.), which can also be considered as the effective value of the entire superhydrophobic surface [⁵⁵]. Therefore, the assumption of an effective slip length is acceptable and widely used to study the flow and heat transfer characteristics of superhydrophobic channels. The new correlation can provide guidance for the optimal design of superhydrophobic microchannels.

Conclusions

A laminar flow model was established and validated for microchannels with superhydrophobic surfaces in this paper. The effects of slip length and hydraulic diameter on hydraulic entrance length were numerically studied, with Reynolds numbers ranging from 0.1 to 1000 and hydraulic diameters between 100 mm and 500 mm. The differences between superhydrophobic and conventional microchannels were found and discussed. A new correlation was explored to describe the entrance length for superhydrophobic microchannels. The main conclusions are as follows:

Due to the decrease in the shear force, superhydrophobic channels have a longer hydraulic entrance length than the conventional ones by nearly 26.62% at Re<10. However, the differences between both are so little that can be ignored at Re>10 because the inertial force is identical at the same inlet velocity. Notably, a longer hydraulic entrance length is good for heat transfer enhancement, which means superhydrophobic channels have more significant advantages for heat transfer enhancement at low Re values.

Hydraulic entrance length slightly increases with the increasing slip length at low Re values due to the decrease in shear force.

Hydraulic diameter has almost no influence on the dimensionless hydraulic entrance length in superhydrophobic square microchannels. With increasing Re, the dimensionless hydraulic entrance length increases slightly and almost maintains constant at low Reynolds numbers (Re<10), while increases linearly with Reynolds numbers at Re>10.

A new correlation involving the slip length is obtained, and has a favorable performance. The MARD of that is 5.69% for a wide range of Reynolds numbers.

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