The classical Graetz-Nusselt problem is extended to describe heat and mass transfer over heterogeneously slippery, superhydrophobic surfaces. The cylindrical wall consists of segments with a constant temperature/concentration and areas that are insulating/impermeable. Only in the case of mass transport do the locations of hydrodynamic slip and mass exchange coincide. This makes advection near the mass exchanging wall segments larger than near the heat exchanging regions. Also the direction of radial fluid flow is reversed for heat and mass transport, which has an influence on the location where the concentration or temperature boundary layer is compressed or extended. As a result, mass transport is more efficient than heat transfer. Also the influence of axial diffusion on the Nusselt and Sherwood numbers is investigated for various Péclet numbers Pe. When Pe < 102, which is characteristic for heat transfer over superhydrophobic surfaces, axial conduction should be taken into account. For Pe ≥ 102, which are typical numbers for mass transport in microfluidic systems, axial diffusion can be neglected.

The classical Graetz-Nusselt problem considers transport of heat between a flowing medium and a tube, beginning at some location in the hydrodynamically developed flow region. The tube entrance is insulated, and the wall temperature of the heated or cooled section is constant but different from that of the entering fluid temperature. The fluid has constant physical properties, and viscous dissipation and axial heat conduction are neglected. Graetz and Nusselt solved this problem for a uniform1 velocity profile (no-shear at the wall) and a parabolic2,3 velocity profile (no-slip at the wall). The amount of local heat transfer is commonly expressed by the local Nusselt number Nux, which is a dimensionless heat transfer coefficient. Because of the analogies between heat and mass transfer, the Graetz-Nusselt solutions are also used to describe mass transfer processes.4 The mass transfer equivalent of Nux is the local Sherwood number Shx.

Previously, we investigated the Graetz-Nusselt problem for homogeneously slippery surfaces.5 Because homogeneous or intrinsic slip lengths are only of the order of tens of nanometres,6,7 heat/mass transfer can only be enhanced significantly in nanofluidic systems. Intrinsic slip lengths are too small to reduce flow friction substantially in larger systems.8 Superhydrophobic or micro-structured systems, however, can generate effective slip lengths on the order of micrometres, which reduce drag considerably in microscale systems (see, for instance, the review by Rothstein7).

The boundary conditions in the Graetz-Nusselt problem, which are no-slip/no-shear and a constant temperature/concentration at the wall, are not appropriate in for example membrane systems or superhydrophobic surfaces. Keller and Stein9 recognised the heterogeneous nature of a membrane and investigated the effects of pore size and spacing on diffusional transport. Juhasz and Deen extended this by considering convective transport.10,11 They described how mass transfer for a non-slippery surface with discrete active mass exchanging areas is affected by the local Péclet number. When the Péclet number increases, convection becomes more important and concentration variations along the surface are minimised. As a result, the surface increasingly resembles a homogeneous surface with a constant wall concentration.

Ling12 described heat transfer from a single, isothermal strip in a uniform shear flow. Three zones were identified, namely the leading edge, the trailing edge, and the zone connecting these edges. For sufficiently large strips or local Péclet numbers, edge effects (axial diffusion) can be neglected and the similarity solution of Lévêque13 can assumed to be valid everywhere. Ackerberg, Patel, and Gupta14 investigated this problem when heat/mass transfer is dominated by edge effects. This was extended to two-dimensional domains for small Péclet numbers by Phillips15 and for arbitrary Péclet numbers by Stone.16 All authors found that heat/mass transfer is not uniform over the strip or domain and that more shear results in increased transfer rates.

When studying transport over heterogeneous surfaces, scientists have often derived and modelled effective boundary conditions that represent a heterogeneous wall.17–19 Recently, Shah and Shaqfeh20 studied heat and mass transport in shear flow over a heterogeneous surface containing reactive domains. By using an effective medium theory, they derived expressions for the effective surface reaction rate to replace the details of the heterogeneous surface with a single boundary condition at a certain distance from the non-slippery, reactive wall. When the Péclet number is large, local variations in boundary conditions are screened and heat/mass transport is enhanced.

All examples mentioned so far involve non-slippery surfaces. We previously investigated for a specific type of superhydrophobic surface, which is heterogeneous regarding both momentum and mass transport, how effective wall slip affects mass transport.21 We showed that convective transport near the surface, as a result of slip, can considerably enhance interfacial mass transfer.

For mass transport over slippery, superhydrophobic surfaces, a generalised study such as an extended Graetz-Nusselt problem is lacking. This is not the case for heat transfer. Maynes, Webb, and Davies22 were the first to investigate (numerically) combined momentum and heat transport in the thermally developed regime over a superhydrophobic surface at constant temperature. They found that thermal transport through a gas-filled cavity to the flowing liquid is at least two orders of magnitude smaller than transport from the solid wall to the liquid. This suggests that the (slippery) gas-liquid interface can be treated as adiabatic. This results in a Nusselt number that is smaller than that for a classical no-slip surface at constant wall temperature. When the Péclet number increases, convective transport becomes more important and the average Nusselt number approaches that for a classical no-slip wall. These observations were confirmed in a semi-analytical work for transverse flow over a superhydrophobic surface containing ribs and cavities at constant heat flux.23 Later, this work was extended by assessing the influence of axial conduction on thermal transport.24 The results show that this cannot be neglected for Péclet numbers smaller than approximately 102, in particular, when the heterogeneity is large compared to the typical channel width.

Enright et al.25 and Maynes and Crockett26 studied isoflux thermal transport over superhydrophobic surfaces using effective hydrodynamic and thermal slip lengths. The effective thermal slip length describes an apparent temperature jump between the wall and the liquid, which accounts for the adiabatic part of the heterogeneous surface. Assuming that heat transport near the surface is purely diffusive, Enright et al. derived an expression for the thermal slip length, which was confirmed by simulations. Apparent hydrodynamic and thermal slip lengths behave similarly and have comparable values.25–27 Steigerwalt Lam et al.28 used hydrodynamic and thermal slip lengths to study thermally developing Couette flow. Moreira and Bandaru29 used an effective medium approach to study the thermal conductivity of superhydrophobic surfaces.

It appears little research has been done on convective heat/mass transport over heterogeneous and slippery surfaces, as recognised by others.20,22 The Graetz-Nusselt problem for superhydrophobic surfaces has only been studied for thermal transport. Mass transfer was not considered, while in that case the no-slip segment instead of the no-shear area is impermeable, and the no-shear region instead of the non-slippery part of the wall is exchanging mass with the fluid flow.21 Often unidirectional flow is assumed, thereby neglecting transverse velocity components present near heterogeneously slippery surfaces. Besides that, most research has been directed towards thermally developed transport.

This study aims to be a natural extension of our previous work5 by changing the boundary conditions from uniform to heterogeneous, for both momentum and heat/mass transport. As far as we know, it is the first study to discuss the Graetz-Nusselt problem for mass transfer over superhydrophobic surfaces and to compare this with thermal transport over such surfaces. We also investigate the influence of axial conduction on heat/mass transport for these systems.

In Fig. 1 a schematic representation is given of the system considered in this study. The flow in the tube is hydrodynamically fully developed and has a uniform temperature T0 when it enters the heated or cooled section of the tube. The wall temperature T1 of the no-slip wall segments is constant, but different from that of the entering fluid temperature T0. The fluid has constant physical properties, and viscous dissipation is neglected. Analogously, when considering mass transport, the inlet concentration is c0, and the wall concentration c1c0 of the no-shear wall segments is constant.

FIG. 1.

Illustration of the various physical systems studied. For the figures shown here, L = R and Pe = 10. The streamlines in the plots of the velocity magnitude $u ̃$ and the radial velocity component $u ̃ r$ have a fixed starting position. The contour lines in the dimensionless temperature/concentration plots are given for Θ = 0.1p with p = {1, 2, …, 9}.

FIG. 1.

Illustration of the various physical systems studied. For the figures shown here, L = R and Pe = 10. The streamlines in the plots of the velocity magnitude $u ̃$ and the radial velocity component $u ̃ r$ have a fixed starting position. The contour lines in the dimensionless temperature/concentration plots are given for Θ = 0.1p with p = {1, 2, …, 9}.

Close modal

We assume that the Reynolds number Re = uavD/ν is small, where uav is the average liquid velocity, D is the tube diameter, and ν is the kinematic viscosity. In that case momentum transport is described by the continuity and Stokes equations,

$1 r ∂ ∂ r r u r + ∂ u x ∂ x = 0 ,$
(1)
$0 = − ∂ p ∂ r + μ ∂ ∂ r 1 r ∂ ∂ r r u r + ∂ 2 u r ∂ x 2 ,$
(2a)
$0 = − ∂ p ∂ x + μ 1 r ∂ ∂ r r ∂ u x ∂ r + ∂ 2 u x ∂ x 2 ,$
(2b)

where p is the pressure, μ is the dynamic viscosity, and ui is the velocity in i-direction.

Stationary heat transport in an axisymmetric cylindrical system, under the assumption of constant density ρ, heat capacity Cp, and thermal conductivity k, is described as

$u r ∂ T ∂ r + u x ∂ T ∂ x = α 1 r ∂ ∂ r r ∂ T ∂ r + ∂ 2 T ∂ x 2 ,$
(3)

where α = k/(ρCp) is the thermal diffusivity. Axial conduction $( ∂ x 2 T )$ can be neglected when axial conduction is much smaller than axial convection. When replacing T for the concentration c, and α for the diffusion coefficient $D$, which is also constant, this equation describes stationary mass transport.

The continuity and Stokes equations are made dimensionless using $x ̃ =x/R$, $r ̃ =r/R$, $u ̃ i = u i / u a v$, and $p ̃ =pR/μ u a v$, which then read

$1 r ̃ ∂ ∂ r ̃ r ̃ u ̃ r + ∂ u ̃ x ∂ x ̃ = 0 ,$
(4)
$0 = − ∂ p ̃ ∂ r ̃ + ∂ ∂ r ̃ 1 r ̃ ∂ ∂ r ̃ r ̃ u ̃ r + ∂ 2 u ̃ r ∂ x ̃ 2 ,$
(5a)
$0 = − ∂ p ̃ ∂ x ̃ + 1 r ̃ ∂ ∂ r ̃ r ̃ ∂ u ̃ x ∂ r ̃ + ∂ 2 u ̃ x ∂ x ̃ 2 .$
(5b)

The convection-diffusion equation is non-dimensionalised using Θ = (T1T)/(T1T0), giving

$u ̃ r ∂ Θ ∂ r ̃ + u ̃ x ∂ Θ ∂ x ̃ = 2 Pe 1 r ̃ ∂ ∂ r ̃ r ̃ ∂ Θ ∂ r ̃ + ∂ 2 Θ ∂ x ̃ 2 .$
(6)

The Péclet number, which gives the ratio of advective to diffusive transport, is defined as

$Pe = Re Pr = u a v D α ,$
(7)

where Pr = ν/α is the Prandtl number. The domain in $x ̃$-direction runs from zero to any finite value.

The rate of heat transport can be described with a dimensionless correlation of the form $Nu x = f 1 Re , Pr , x / L$. Based on the analogies between heat and mass transport,4 the mass transfer rate for the same geometry and analogous boundary conditions can be described by $Sh x = f 2 Re , Sc , x / L$. Thus, replacing Θ in Eq. (6) for the dimensionless concentration Θm = (c1c)/(c1c0) yields the dimensionless convection-diffusion equation for mass transport. In that case, Pe = Re Sc, with $Sc=ν/D$ being the Schmidt number.

The local Nusselt number Nux is defined as

$Nu x = h x D k ,$
(8)

where hx is the local heat transfer coefficient. Using Fourier’s law of thermal conduction, the local heat transfer coefficient hx can be written as $h x =−k/ T − T 1 ∂ r T r = R$. The temperature gradient can be rewritten in dimensionless form using the flow-averaged temperature $Θ = ( T 1 − T ) / ( T 1 − T 0 )$. This yields

$Nu x = − 2 Θ ∂ Θ ∂ r ̃ r ̃ = 1 .$
(9)

The equivalent of the Nusselt number for mass transport is the Sherwood number

$Sh x = m x D D .$
(10)

Here, $m x =−D/ c − c 1 ∂ r c r = R$ is the local mass transfer coefficient. Using the flow-averaged concentration $Θ m = ( c 1 − c ) / ( c 1 − c 0 )$ we find for the local Sherwood number

$Sh x = − 2 Θ m ∂ Θ m ∂ r ̃ r ̃ = 1 .$
(11)

If $x ̃ →∞$, NuxNu, and ShxSh when averaged over the heterogeneities at the wall.

Various cases are studied here, all being characterised by a different set of boundary conditions (see Fig. 1). The boundary conditions for momentum transport are always the same, though. These are rotational symmetry in the middle of the tube,

$∂ u ̃ r ∂ r ̃ = 0 at r ̃ = 0 ,$
(12)

and a regular pattern of no-slip and no-shear (a surface free from stress) at the wall (see Fig. 1 for the $x ̃$-positions),

${ u ̃ = 0 at r ̃ = 1 and x ̃ = { no-slip areas } , u ̃ ⋅ n = 0 , − p ̃ I + ∇ u ̃ + ∇ u ̃ T n ̃ = 0 at r ̃ = 1 and x ̃ = { no-shear areas } .$
(13)

In all cases the inlet temperature or concentration is constant, being

$Θ ( r ̃ , 0 ) = 1 .$
(14)

Case C. In the first case, the wall temperature is constant (C), as could be the case in for instance surfaces containing patches that are infused with a lubricating liquid.30 This gives the following boundary condition at $r ̃ =1$:

$Θ ( 1 , x ̃ ) = 0 .$
(15)

Case T. This case describes the extended Graetz-Nusselt problem for thermal (T) transport. The slippery patches correspond to gas present in the micro-structures of the slippery, superhydrophobic surface. Because the thermal conductivity of a gas is much smaller than that of a solid, the slippery (gas) patches can be considered as insulating.22 Therefore,

${ Θ = 0 at r ̃ = 1 and x ̃ = { no-slip areas } , ∂ Θ / ∂ r = 0 at r ̃ = 1 and x ̃ = { no-shear areas } .$
(16)

Case M. The last case describes the extended Graetz-Nusselt problem for mass (M) transport. This means that the boundary conditions are reversed compared to case T. The heterogeneous wall could be considered as a membrane, used to contact a gas and liquid with each other.21 In that case, mass transfer takes place only between the liquid and the gas. The non-slippery solid wall can be considered as impermeable. Then

${ ∂ Θ m / ∂ r = 0 at r ̃ = 1 and x ̃ = { no-slip areas } , Θ m = 0 at r ̃ = 1 and x ̃ = { no-shear areas } .$
(17)

Regarding the positioning of the no-shear and adiabatic/impermeable wall segments, one can either maintain the same position of the zero-flux regions among all three cases or keep the location of the no-shear regions the same. These two options give the same result when the number of gas/solid modules becomes very large. Here it is decided to choose for the first option, implying that the positions of the no-shear and no-slip segments are reversed when switching from heat to mass transport.

Wall slip is usually described using Navier’s slip condition,7,31

$u b = − b ∂ u ∂ r r = R ,$
(18)

where ub is the slip velocity at the wall, and b is the slip length. In the case of a homogeneously slippery wall, this slip length has a uniform value, whereas for a heterogeneously slippery wall like a superhydrophobic surface, the local value of b varies. The slip length required to obtain a homogeneous slip flow with the same average velocity uav and pressure gradient ∂p/∂x as a flow over a heterogeneously slippery surface is referred to as the effective slip length b. For Stokes flow we find for the average velocity uav of a homogeneous slip flow through a cylinderthat

$u a v = − ∂ p ∂ x R 2 8 μ + b R 2 μ .$
(19)

Rewriting this equation and making it dimensionless using $b ̃ =b/R$, we obtain for a flow with $u ̃ a v =1$ the following expression for the (effective) slip length $b ̃$:

$b ̃ = − 1 4 8 ∂ p ̃ / ∂ x ̃ + 1 .$
(20)

The analytical solution derived by Davis and Lauga32 for a heterogeneously slippery surface can be used to estimate the (maximum) amount of slip. As derived before,21 for flat bubbles

$2 b ϵ L g ≈ π 8 .$
(21)

In order to rewrite this, we derive that Lg = ϵR/n, since the length of a no-slip (Ls) and a no-shear (Lg) wall segment is Ls + Lg = R/n. Here, n is the number of no-slip/no-shear units in a tube of length R. Substitution gives

$b ̃ ≈ π 16 ϵ 2 n .$
(22)

This means that for our system, with ϵ = 1/2 and n = 1, the amount of slip is $b ̃ =π/64≈0.05$. This corresponds closely to what is obtained from the simulations ($b ̃ =0.0542$).

COMSOL Multiphysics 5.0 was used to compute the velocity and temperature/concentration profiles. For solving the flow profile, the laminar flow module was used. Heat/mass transport was modelled utilising the convection-diffusion equation. Standard relative tolerance was 1 × 10−3. P2 + P1 discretisation (second order elements for velocity and first order elements for pressure) was used to solve the Stokes equations. Quadratic elements were used for solving the convection-diffusion equation.

From $x ̃ = [ 0 , 1 0 − 2 ]$ a logarithmically spaced mesh was used. The length of this mesh was varied. For very short domains, the logarithmic mesh ran from $x ̃ = [ 1 0 − 7 , 1 0 − 2 ]$, whereas for long domains this was $x ̃ = [ 1 0 − 3 , 1 0 − 2 ]$. The mesh in the $r ̃$-direction was defined as $r ̃ =1− 0 . 99 1 − r ̃ 0 2 + 0 . 01 1 − r ̃ 0$, with $r ̃ 0$ being a linearly spaced vector with 101 grid points. This mesh was also defined at $x ̃ =1$. For $x ̃ = 1 0 − 2 , L / R$ a free triangular mesh was used, the mesh being refined near the boundaries with 8 boundary layers. The standard mesh size was set to “extra fine,” resulting in a total number of approximately 29 × 103 domain elements and 670 boundary elements for a domain of length L/R = 1. When solving the model for very small no-shear areas, the meshing was further refined by setting the mesh size to “extremely fine.” This gave about 80 × 103 domain elements and 840 boundary elements for a domain of size 1 × 1.

Since the fluid flow is fully periodic and does not depend on the temperature/concentration field, the flow field was solved first for $u ̃ a v =1$. Subsequently, using the obtained velocity profile, the temperature/concentration profiles were computed. To obtain developed Nusselt/Sherwood numbers, the outlet temperature/concentration profile of domain j was used as the inlet profile of domain j + 1. This procedure was repeated until the Nusselt/Sherwood numbers in the domain did not change anymore. Subsequently the developed Nusselt Nu and Sherwood Sh numbers were computed.

In Fig. 1 the three cases considered in this study are displayed. The Péclet number is the same for all cases, which is Pe = 101. We observe that the velocity profiles are different for the three cases. At the start of the heat/mass exchanging wall segments, radial advection is directed inwards for case T but is directed outwards for case M. As will be discussed later, this affects the transfer rate of heat/mass, depending on the Péclet number. The rate of heat/mass transfer at each position $x ̃$ is expressed by the local Nusselt Nux or Sherwood number Shx. These numbers, which are proportional to the gradient at the wall and the inverse of the flow-averaged temperature/concentration, are obtained from the temperature/concentration profiles. Although these profiles for case T and case M look very similar, there are some differences. These differences can be related to the different boundary conditions and become more pronounced for increasing Péclet number Pe.

In Fig. 2(a) the local Nusselt Nux and Sherwood Shx numbers are plotted. First, the fluid flow normal to the surface leads to large variations in the curve for constant wall temperature. The profile for case C even crosses the classical Graetz-Nusselt profile for no-slip flow. We also observe that locally Nux or Shx is much larger than the transport number for homogeneous (slip) flow with a constant wall temperature/concentration. These variations have been observed before,22,26 although not when the flow is described as a uni-directional slip flow.23–26,28 Although case C does not represent a superhydrophobic surface, the set of boundary conditions does describe slippery, liquid-infused surfaces with patterned wettability30 when the surface and the infusing liquid have comparable thermal conductivity.

FIG. 2.

For all graphs, Pe = 101, $b ̃ =0.0542$ and there is no axial diffusion. In (a), the local Nusselt Nux and Sherwood numbers Shx are plotted for all three cases on a logarithmic scale. The solid lines are the classical Graetz-Nusselt solutions for no-slip and no-shear flow. The dashed curve is the profile for a homogeneous slip flow with a constant wall temperature/concentration and having the same slip length $b ̃$ as the superhydrophobic surface. Both Nusselt and Sherwood number are proportional to the temperature/concentration gradient and the inverse of the flow-averaged temperature/concentration, i.e., $Nu x , Sh x ∝ ∂ r Θ r ̃ = 1 / Θ$. The flow-averaged temperatures $Θ$ and concentration $Θ m$ are shown in (b), while the temperature/concentration gradients ∂rΘ at $r ̃ =1$ are given in (c). In (d) the ratios of both local and average Nusselt (case T) to Sherwood number (case M) are plotted. For the ratio Nux/Shx the data points above the adiabatic/impermeable wall segments are omitted, as division of very small numbers (Nux, Shx → 0) gives spurious results.

FIG. 2.

For all graphs, Pe = 101, $b ̃ =0.0542$ and there is no axial diffusion. In (a), the local Nusselt Nux and Sherwood numbers Shx are plotted for all three cases on a logarithmic scale. The solid lines are the classical Graetz-Nusselt solutions for no-slip and no-shear flow. The dashed curve is the profile for a homogeneous slip flow with a constant wall temperature/concentration and having the same slip length $b ̃$ as the superhydrophobic surface. Both Nusselt and Sherwood number are proportional to the temperature/concentration gradient and the inverse of the flow-averaged temperature/concentration, i.e., $Nu x , Sh x ∝ ∂ r Θ r ̃ = 1 / Θ$. The flow-averaged temperatures $Θ$ and concentration $Θ m$ are shown in (b), while the temperature/concentration gradients ∂rΘ at $r ̃ =1$ are given in (c). In (d) the ratios of both local and average Nusselt (case T) to Sherwood number (case M) are plotted. For the ratio Nux/Shx the data points above the adiabatic/impermeable wall segments are omitted, as division of very small numbers (Nux, Shx → 0) gives spurious results.

Close modal

Second, the slope of the profiles in the developing regime, i.e., for $x ̃ <1 0 − 1$ where the flow-averaged temperature/concentration is essentially unity ($Θ =1$), is different for heat and mass transfer. Whereas the exponent of the slope is −1/3 for heat transport (no-slip wall adjacent to entrance), it is −1/2 for mass transport (no-shear wall adjacent to entrance). This is in agreement with the analytical values for the exponent β found by Lévêque, who predicted that $Nu x , Sh x ∝ x ̃ − β$ in the developing regime.5,13

Finally, Fig. 2(a) reveals that for case T, heat transport is enhanced when the flow is about to enter a no-shear wall segment where ∂rΘ = 0. For case M the situation is exactly opposite, as mass transport is retarded when the flow enters a no-slip region where ∂rΘm = 0.

To investigate the influence of the different set of boundary conditions on the rate of heat/mass transport in more detail, we plotted the flow-averaged temperature $Θ$ and concentration $Θ m$ in Fig. 2(b), and the temperature/concentration gradient −∂rΘ at $r ̃ =1$ in Fig. 2(c). The profiles in Fig. 2(b) show that above the adiabatic/impermeable wall segments the amount of energy/massin the flow does not change, as no heat/mass is exchanged with the wall. The distribution of heat/mass above these segments does change, however, as the plots in Fig. 1 reveal. We also observe that the set of boundary conditions for case C leads to the most efficient cooling of the liquid flow. Between case T and case M the differences are smaller, although the flow-averaged concentration decreases faster than the temperature. This already indicates that the differences in boundary conditions for case T and case M make mass transport more efficient than heat transfer between wall and fluid. On the other hand, however, Fig. 2(c) reveals that the difference between the gradients for case T and case M becomes larger with position $x ̃$, the gradient for case M being smaller than for case T. Since $Nu x , Sh x ∝ ∂ r Θ r ̃ = 1 / Θ$, this compensates for the increasing difference in flow-averaged temperature/concentration. For that reason, as Fig. 2(d) shows, the ratio of Nusselt to Sherwood number is repetitive in the developed regime ($x ̃ >1$ for Pe = 101).

The precise value of the Nusselt/Sherwood number above the heat/mass exchanging wall segments strongly varies. This has various physical origins. First, we observe that both flow-averaged temperature/concentration (Fig. 2(b)) and the gradients (Fig. 2(c)) for case T and case M are high at the beginning of a heat/mass exchanging wall segment, but drop very quickly, contrary to the profiles for case C. Inspection of Fig. 1 shows that above the adiabatic/impermeable regions homogenisation of heat/mass takes place, which leads to high gradients at the beginning of each heat/mass exchanging wall segment. This suggests that, apart from the entrance of the tube, transport above all these segments is characterised by a developing regime. Second, the heat exchanging area is non-slippery, while the mass exchanging wall segment is slippery. Wall slip promotes transport, in particular in the developing regime, as classical theories1–3,5,13 show and as in agreement with the profiles in Fig. 2(a). Convection near these segments is therefore larger for case M. This is also supported by Fig. 2(d), showing that above the heat/mass exchanging wall segments initially Nux/Shx < 1 (mass transport is faster). Third, as Fig. 1 reveals, the direction of radial advection at the beginning/end of each heat/mass exchanging wall segment is reversed for case T and case M. At the beginning of these segments, the concentration/temperature boundary layer is compressed for case M (leading to larger gradients), but extended for case T (resulting in smaller gradients). This may explain why at the beginning of these segments the ratio Nux/Shx drops, then reaches a minimum, and at the end increases to values of Nux/Shx > 1 (thermal transport is faster).

These three phenomena make the ratio of heat to mass transfer non-symmetric around the midpoints of the heat/mass exchanging wall segments, as Fig. 2(d) shows. Consider for instance the second segment located at $0.75< x ̃ <1.25$. For $0.75< x ̃ <1.1$ mass transport is faster (Nux/Shx < 1), while for $1.1< x ̃ <1.25$ heat transfer is faster (Nux/Shx > 1). As a result, the average Sherwood number is always larger than the average Nusselt number, i.e., Nuav/Shav < 1 (Fig. 2(d)), and the flow-averaged concentration is smaller than the mixing-cup temperature, i.e., $Θ m < Θ$ (Fig. 2(b)). We therefore establish that, considering case T and case M, on average mass transport is faster than thermal transport.

Fig. 3 shows that with increasing Péclet number, the variations in the local transport coefficient, in this case the local Nusselt Nux (case T) and Sherwood number Shx (case M), become more pronounced. Since axial diffusion is not taken into account here, an increasing Péclet number implies that the ratio of (radial) advection to radial diffusion increases. Because of the different flow profiles for case T and case M, this clarifies why the differences between Nux and Shx also increase with increasing Pe. However, as indicated in Table I, it should be noticed that for heat transport over superhydrophobic surfaces in a microfluidic platform typically Pe ≤ 101, whereas for mass transfer generally Pe > 101.

FIG. 3.

These figures show the influence of the Péclet number Pe on heat/mass transport. In (a) the local Nusselt number Nux and in (b) the local Sherwood number Shx are plotted versus the dimensionless position $x ̃$. For both figures, the length is L = 10R and the number of no-slip/no-shear units per length R is n = 1. The dashed curves are the Nusselt/Sherwood profiles for a homogeneous slip flow with a constant wall temperature/concentration and having the same slip length $b ̃$ as the superhydrophobic surface.

FIG. 3.

These figures show the influence of the Péclet number Pe on heat/mass transport. In (a) the local Nusselt number Nux and in (b) the local Sherwood number Shx are plotted versus the dimensionless position $x ̃$. For both figures, the length is L = 10R and the number of no-slip/no-shear units per length R is n = 1. The dashed curves are the Nusselt/Sherwood profiles for a homogeneous slip flow with a constant wall temperature/concentration and having the same slip length $b ̃$ as the superhydrophobic surface.

Close modal
TABLE I.

Typical values of relevant dimensionless numbers for a microfluidic system.

D (m) L (m) Re Pr Sc Pe
Thermal transport  10−4  10−2  10−1  101  …  100
Mass transport  10−4  10−2  10−1  …  103  102
D (m) L (m) Re Pr Sc Pe
Thermal transport  10−4  10−2  10−1  101  …  100
Mass transport  10−4  10−2  10−1  …  103  102

For a heterogeneous surface as considered here, the local transport number can be much higher than for a homogeneously slippery surface with a constant wall temperature/concentration.5,33,34 However, the adiabatic/impermeable wall segments have an adverse effect on the average Nusselt Nuav or Sherwood Shav number, which are plotted in Figs. 4(a) and 4(b) for case T and case M, respectively. For that reason, fluctuations in the average Nusselt/Sherwood numbers are not so large as in the local transport numbers. The fluctuations become smaller with increasing position $x ̃$, because local variations are averaged out.

FIG. 4.

Local average Nusselt Nuav and Sherwood Shav number for case T and case M for n = 1 and L = 10R as function of the Péclet number Pe. In (a) and (b) axial diffusion is neglected, whereas (c) and (d) include the effect of axial diffusion. The solid lines are the classical Graetz-Nusselt solutions for Hagen-Poiseuille flow. The dashed curves are the Nusselt/Sherwood profiles for a homogeneous slip flow with a constant wall temperature/concentration and having the same slip length $b ̃$ as the superhydrophobic surface.

FIG. 4.

Local average Nusselt Nuav and Sherwood Shav number for case T and case M for n = 1 and L = 10R as function of the Péclet number Pe. In (a) and (b) axial diffusion is neglected, whereas (c) and (d) include the effect of axial diffusion. The solid lines are the classical Graetz-Nusselt solutions for Hagen-Poiseuille flow. The dashed curves are the Nusselt/Sherwood profiles for a homogeneous slip flow with a constant wall temperature/concentration and having the same slip length $b ̃$ as the superhydrophobic surface.

Close modal

Fig. 4(a) shows that the difference between the average Nusselt number Nuav and the corresponding classical Graetz-Nusselt solution becomes smaller with increasing Péclet number Pe. This is supported by the data for case T (no axial diffusion) in Fig. 5, where the developed Nusselt number Nu is plotted as function of the Péclet number. Nu may even be slightly larger than the classical value of 3.66. This requires very large Pe, though. It can be attributed to slip near the wall, because for non-slippery surfaces the Nusselt profile would approach the classical Graetz-Nusselt solution.10,11 The reason why transport increases with Péclet number is that advection in both axial and radial direction increasingly dominates over diffusion. For large Pe, the no-shear, adiabatic regions are “less effective at interrupting the thermal and hydrodynamic boundary layers.”22 Heat is “being swept downstream” so fast that the effect of the “tangential flow is to smooth out concentration variations.”10

FIG. 5.

Developed Nusselt and Sherwood number as function of the Péclet number Pe. The solid line indicates the classical value for no-slip flow, i.e., Nu, Sh = 3.66. The dashed line corresponds to the value for homogeneous slip flow with constant wall temperature and having the same slip length $b ̃$ as the superhydrophobic surface, i.e., Nu, Sh = 3.92.

FIG. 5.

Developed Nusselt and Sherwood number as function of the Péclet number Pe. The solid line indicates the classical value for no-slip flow, i.e., Nu, Sh = 3.66. The dashed line corresponds to the value for homogeneous slip flow with constant wall temperature and having the same slip length $b ̃$ as the superhydrophobic surface, i.e., Nu, Sh = 3.92.

Close modal

The behaviour of the average Sherwood number Shav is analogous to that of the average Nusselt number Nuav, as Fig. 4(b) shows. Comparison of the average Nusselt and Sherwood numbers reveals that Shav > Nuav and that the difference between heat and mass transfer becomes larger with increasing Péclet number. While Nuav approaches the classical Graetz-Nusselt profile for no-slip flow, Shav exceeds this profile and starts to overlap with the solution for homogeneous slip flow with constant wall concentration.5 This observation is supported by Fig. 5, showing that the developed Sherwood number is larger than the developed Nusselt number (no axial diffusion), i.e. Sh > Nu. This clearly illustrates that the difference in boundary conditions for heat/mass transfer affects the rate of transport. A coinciding location of hydrodynamic slip and mass transfer is beneficial for the overall transport rate, as was discussed in Sec. IV A.

The relative magnitude of radial convection to radial diffusion, for a given Péclet number, can be altered when changing the number of no-slip/no-shear units n per length R. The influence of this number n on mass transport is shown in Fig. 6. When n becomes larger, local fluctuations in the Sherwood number become smaller. According to Eq. (22), the effective slip length $b ̃$ decreases for increasing n, resulting in smaller transverse velocity components $u ̃ r$. Note that for large n the average Sherwood number becomes smaller than the classical Graetz-Nusselt solution for no-slip flow. When the size of one no-slip/no-shear unit compared to the tube radius becomes very small (n → ∞), the (slip) velocity $u ̃ b$ near the wall goes to zero ($b ̃ →0$). In that case advective transport near the wall vanishes, and the wall has, on the scale of the tube radius, a constant wall concentration. One would therefore expect that the Nusselt/Sherwood profiles overlap with the classical Graetz-Nusselt solution.10 The reason why this is not observed here is because axial diffusion has so far been neglected.

FIG. 6.

Local average Sherwood number Shav for Pe = 103 (with/without axial diffusion) for two different numbers of no-slip/no-shear units n per length R. The solid line is the classical Graetz-Nusselt solution. The dashed curves are the Sherwood profiles for a homogeneous slip flow with a constant wall concentration and having the same slip length $b ̃$ as the superhydrophobic surface.

FIG. 6.

Local average Sherwood number Shav for Pe = 103 (with/without axial diffusion) for two different numbers of no-slip/no-shear units n per length R. The solid line is the classical Graetz-Nusselt solution. The dashed curves are the Sherwood profiles for a homogeneous slip flow with a constant wall concentration and having the same slip length $b ̃$ as the superhydrophobic surface.

Close modal

To assess the influence of axial conduction/diffusion, the length of the tube could be a proper length scale. In that case PeL = Pe L/D, which gives, using the numbers given in Table I, for both heat and mass transport a value of at least PeL = 102 ≫ 1. Neglecting axial diffusion seems to be justified, even in case of heat transfer with a typical Péclet number of Pe = 100.

However, this compares the ratio of axial convection to axial diffusion on the scale of the tube length. When the average slip velocity ub near the wall becomes very small, close to the wall axial diffusion will be faster than convection on the scale of the heterogeneity of the superhydrophobic surface. One can assess the importance of local axial diffusion by looking at the local Péclet number

$Pe b = u b L s α = Re b Pr .$
(23)

Here, Reb is defined as

$Re b = u b u a v L s D u a v D ν = u b u a v L s D Re .$
(24)

In the case of Stokes flow, we can derive that $u ̃ b = u b / u a v =4 b ̃ / ( 1 + 4 b ̃ )$. Furthermore, we know that $b ̃ ≈π ϵ 2 /16n$ (Eq. (22)). For the length of the no-slip area, on which we assume heat/mass exchange takes place in this analysis, we find that Ls = Lg(1/ϵ − 1), where Lg = ϵR/n. This ultimately gives

$Pe b = π 4 ϵ 2 n / 1 + π 4 ϵ 2 n 1 − ϵ 2 n Re Pr .$
(25)

Because $b ̃ ≪1$ and ϵ2/n ≪ 1, in particular when n > 1,

$Pe b ≈ π 8 ϵ 2 − ϵ 3 n 2 Pe .$
(26)

In this study, ϵ = 1/2. This suggests that for heat transfer, Pebπ/(64n2) Pe < 1 for all n, because typically Pe ≤ 101. In this equation we used the effective slip velocity $u ̃ b$, which is larger than the actual velocity above the heat exchanging, no-slip area. This makes the required Péclet number Pe to let Peb > 1, suggesting that axial conduction can be neglected, even more stringent. Thus, on average, near the surface diffusion is faster than convection.

For mass transfer, generally Pe > 101. Then Peb = π/(64n2) Pe. For Pe = 102 this implies that n > 2 to let Peb < 1. Besides that, the actual velocity above the mass exchanging surface is larger than the effective slip velocity ub. Therefore it seems reasonable to neglect axial diffusion in case of mass transfer.

Fig. 6 shows that axial diffusion indeed conceals the effects of the surface heterogeneity on mass transfer. For n = 1, fluctuations in the Sherwood number hardly change in size when including axial diffusion: Peb ≫ 1. For n = 8 however, for which Peb < 1, fluctuations are very minor when including axial diffusion. Variations in concentration are smoothed out as a result of axial conduction,10,24 as predicted by Eq. (26). That for both n = 1 and n = 8 the average Sherwood number increases when including axial diffusion is because transport is still in the developing regime. In this regime, axial diffusion significantly affects the transport number.35,36 This difference disappears for Pe > 102 when the transition regime is approached.

Figs. 4(a) and 4(c) show the average Nusselt number obtained with and without including axial conduction in the calculations. As comparison of these two plots reveals, axial conduction cannot be neglected for heat transfer when Pe ≤ 102. For small Pe we observe that heat transport is significantly larger than without axial conduction. Also the fluctuations in Nuav become smaller, because axial conduction screens temperature variations along the wall. For Pe > 102 the influence of axial conduction diminishes, in particular when $x ̃ >10$.

For small $x ̃$, even for large Pe, axial diffusion extends the thermally developing regime. This is in line with the literature.35,36 The developed Nusselt number that is ultimately reached strongly depends on the Péclet number:35,36 for Pe = 100, Nu = 4.03, which rapidly decreases to the classical value of 3.66 when Pe > 101. This implies that in the case of heat transfer over superhydrophobic surfaces, for which typically Pe ≤ 101, axial conduction cannot be neglected. As an approximation, the classical Graetz-Nusselt solutions including axial diffusion could be used to describe heat transport.

Fig. 5 shows that the developed Nusselt number Nu for flow with axial conduction is minimum when 101 < Pe < 102. For low Pe, the developed Nusselt number corresponds to the classical problem including axial conduction, i.e., Nu = 4.03. Transport near the surface is fully dominated by conduction, and therefore Θ → 0 near the adiabatic wall segments. Nu < 4.03 when 101 < Pe < 102: both diffusion and advection are insufficiently large to level out all temperature variations along the wall. For Pe > 102 axial convection is so fast that the adiabatic regions cannot disturb the thermal boundary layer. Near the wall Θ ≈ 0. Besides that, for large Pe radial advection is large compared to radial diffusion. As a result, Nu increases and reaches values larger than the classical value for flow without axial diffusion (Nu = 3.66). Finally, the developed Nusselt numbers for flow with and without axial conduction converge towards the same value, as has been observed before.24 This confirms that only for large Péclet numbers axial diffusion can be neglected.

A similar analysis can be made for mass transport. Figs. 4(b) and 4(d) show that for Pe ≥ 102 the differences in the average Sherwood numbers Shav for flows with and without axial diffusion are small, in particular, in the developed regime. Inspection of Fig. 5 reveals that also the developed Sherwood numbers Sh (with/without axial diffusion) converge to the same value. Both observations suggest that in a typical microfluidic system where mass transfer takes place (see Table I), axial diffusion can be neglected. When Pe > 102, the average and developed Sherwood numbers approach those for a homogeneous slip flow without axial diffusion having a constant wall concentration and may even exceed them (Sh > 3.92 for Pe = 103). To simplify the description of mass transfer, the solution for homogeneous slip flow with a constant wall concentration could be used as an approximation.

Axial diffusion has greater impact on heat transfer than on mass transport, as comparison of the average and developed Nusselt and Sherwood numbers reveals. Fluctuations are damped more in Nuav than in Shav, as Figs. 4(c) and 4(d) show. Fig. 5 demonstrates that axial diffusion increases Nu relatively more than Sh. The reason is that heat exchange with the wall takes place where the liquid does not slip. Axial conduction, however, increases the temperature near the adiabatic shear-free wall. Since local slip velocities are high at these wall segments, this considerably enhances convective heat transfer. For mass transfer, the effect of axial diffusion is smaller: it increases the concentration near the impermeable regions. Since these wall segments are non-slippery, here convection is small compared to convection above the mass exchanging no-shear regions.

Note that for all data presented here the number of no-slip/no-shear units per length R is n = 1, resulting in an effective slip length of $b ̃ =0.0542$. With a typical tube diameter of D = 100 μm and a porosity of ϵ = 1/2, this means that the length of the no-shear segment is Lg = 25 μm. This is large, but experimentally feasible.37 Depending on how ϵ or n are changed, the slip length $b ̃$ may decrease. This affects the value of the Nusselt and Sherwood numbers found in microfluidic systems. When n → ∞ and thus $b ̃ →0$, fluctuations in Nusselt/Sherwood numbers arising from the heterogeneity of the superhydrophobic wall disappear. In that case, the tube wall can be considered as non-slippery and as having a constant temperature/concentration.

In this study we described the Graetz-Nusselt problem for heat and mass transport over slippery, superhydrophobic surfaces. The results revealed that the performance of heat and mass transfer can be very distinct. The subtle differences in boundary conditions arising from the heterogeneous nature of the wall should therefore be taken into account.

In the case of thermal transport, the no-slip wall has a constant wall temperature and the no-shear wall is adiabatic. For mass transfer, the no-shear wall has a constant wall concentration and the no-slip wall is impermeable. These differences make, on average, mass transfer more efficient than thermal transport, as expressed by the average Nusselt and Sherwood numbers. Only in the case of mass transfer do the locations of hydrodynamic slip and mass exchange coincide. Advection near the mass exchanging wall segments is therefore larger than near the heat exchanging regions. Also the direction of radial momentum transport is reversed, leading to a compression of the concentration/temperature boundary layer at the beginning of these segments for mass transport and at the end of these segments for heat transfer. Increasing the Péclet number Pe makes the difference in performance between heat and mass transport more pronounced.

Although it may simplify the description of heat/mass transport, axial diffusion cannot always be neglected. Our results show that for Pe < 102, which is typically the case for heat transfer in superhydrophobic microfluidic systems, axial conduction should be taken into account. This prevents underestimation of the length of the thermally developing regime and the value of the developed Nusselt number. For Pe ≥ 102, which are typical numbers for mass transport, axial conduction has only minor influence on the rate of mass transport and can be ignored.

R.G.H.L. acknowledges the European Research Council for the ERC starting Grant No. 307342-TRAM. We would like to thank Jeffery A. Wood for valuable discussions regarding various numerical aspects of this study.

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