An integral method is a powerful and flexible technique for the approximate solution of boundary-layer problems. It is based on the integration of the boundary-layer conservation equations over the boundary-layer thickness and the assumption of approximate and well-defined velocity, temperature, and mass-fraction profiles in the boundary layer. In this way, the partial differential conservation equations are replaced with ODEs in which the dependent variable is the boundary-layer thickness. The solution of the ODE derived in this way then provides the thickness of the boundary layer. Knowing the boundary-layer thickness, along with the aforementioned approximate velocity and temperature profiles, we can then easily find the transport rates through the boundary layer. The integral technique is quite flexible and, unlike the similarity solution method, can be applied to relatively complicated flow configurations.

Integral Momentum Equations

Let us first consider the velocity boundary layer on a flat plate that is subject to the steady and uniform parallel flow of a fluid, as shown in Fig. 5.1. We define a control volume composed of a slice of the flow field that has a thickness dx and height Y. We choose Y to be large enough so that it will be larger than the boundary-layer thickness throughout the range of interest. The inflow and outflow parameters relevant to momentum and energy are also depicted in Fig. 5.1.

Miniature flow passages, defined here as passages with hydraulic diameters smaller than about 1 mm, have numerous applications. Some current applications include monolith chemical reactors, inkjet print-heads, bioengineering and biochemistry (lab-on-the-chip; drug delivery with ultrathin needles, etc.), microflow devices (micropumps, micro heat exchangers, etc.), and cooling systems for microelectronic and high-power magnets, to name a few. Miniature flow passages are an essential part of microfluidic devices, in which can be broadly defined as devices in which minute quantities of fluid are applied. Cooling systems based on microchannels can provide very large volumetric heat disposal rates that are unfeasible with virtually any other cooling technology. Their widespread future applications may in fact revolutionize some branches of medicine and industry.

The serious study of flow in capillaries (tubes with D ≈ 1 mm) goes back to at least the 1960s. The application of microchannels for cooling of high-power systems is relatively new, however (Tuckerman and Pease, 1981). The literature dealing with flow in microtubes is extensive. Useful reviews include those of Papautsky et al. (2001), Morini (2004), Krishnamoorthy et al. (2007), and Fan and Luo (2008). The field of flow in miniature channels, in particular with respect to very small channels (microfluidics and nanofluidics) is a rapidly developing one. In this chapter we review the flow regimes and size-based miniature flow passage categories, and we discuss the limitations of the classical convection heat and mass transfer theory with respect to its application to miniature flow passages.

Mixed convection refers to conditions when forced and natural (buoyancy-driven) effects are both important and neither one can be neglected. Situations in which forced and buoyancy-driven convection terms are of similar orders of magnitude obviously fall in the mixed-convection flow category. However, in many applications we deal with either a predominantly forced convective flow in which buoyancy-driven effects are small but considerable or a predominantly buoyancy-driven flow in which a nonnegligible forced-flow contribution is also present.

Mixed convection is relatively common in nature. In more recent applications, it occurs in rotating flow loops and in the cooling minichannels in the blades of modern gas turbines. In these flow loops, Coriolis centripetal forces arise because of the rotation. When the fluid is compressible, secondary flow caused by the centripetal effect contributes to the wall–fluid heat transfer.

Mixed-convection effects are not always undesirable. In some applications we may intentionally seek buoyancy effect in order to augment heat transfer. Some recent applications of supercritical fluids are examples to this point. The very large compressibility of these fluids, which is achieved without a phase change (although a pseudo–phase change does occur for near-critical fluids) is very useful.

In situations that are predominantly forced flow, buoyancy-driven effects have four types of impact on the overall flow field:

1. They contribute (assist, resist, or do both at different parts of the flow field) to the forced-flow velocity field.

2. They cause secondary flows. The secondary flows can enhance or reduce the heat transfer rate.

3. […]

General Remarks

In the previous chapters we noted that the dimensionless boundary-layer conservation equations for momentum, thermal energy, and mass species are mathematically similar. This similarity among these dimensionless equations suggests that the mathematical solution for one equation should provide the solution of the other equations. One may argue that the empirical correlations for friction factor, heat transfer coefficient, and mass transfer coefficient represent empirical solutions to the momentum, energy, and mass-species conservation equations, respectively. Thus a correlation for friction factor of the form f = f (Re) is the empirical solution to the momentum conservation equation for a specific system and flow configuration, whereas an empirical correlation of the form Nu = Nu (Re, Pr) for the same system is an empirical solution to the energy equation and an empirical correlation of the form Sh = Sh(Re, Sc). Thus, using the analogy arguments, knowing an empirical correlation for either of the three parameters f, Nu, or Sh for a specific system will allow us to derive empirical correlations for the remaining two parameters.

The usefulness of the analogy approach becomes clear by noting that measurement of friction factor is usually much simpler than the measurement of heat or mass transfer coefficients.

The conservation equations for fluids were derived in the previous chapter. Because of viscosity, the velocity boundary condition on a solid–fluid interface in common applications is no-slip. Velocity slip occurs during gas flow when the gas molecular mean free path is not negligible in comparison with the characteristic dimension of the flow passage. It is discussed in Chapter 13.

The complete solution of viscous flow conservation equations for an entire flow field, it seems, is in principle needed in order to calculate what actually takes place on the surface of an object in contact with a fluid. The complete solution of the entire flow field is impractical, however, and is fortunately unnecessary. The breakthrough simplification that made the analysis of the flow field at the vicinity of surfaces practical was introduced by Ludwig Prandtl in 1904. He suggested that any object that moves while submerged in a low-viscosity fluid will be surrounded by a thin boundary layer. The impact of the no-slip boundary condition at the surface of the object will extend only through this thin layer of fluid, and beyond it the fluid acts essentially as an inviscid fluid. In other words, outside the boundary layer the flow field does not feel the viscous effect caused by the presence of the object. It feels only the blockage caused by the presence of the object, as a result of which the streamlines in the flow field become curved around the object.

Laminar flow in low-viscosity fluids is relatively rare in nature and industry. Turbulent flow is among the most complicated and intriguing natural phenomena and is not well understood, despite more than a century of study. Nevertheless, out of necessity, investigators developed simple models that can be used for engineering design and analysis.

Turbulent flows at relatively high Reynolds numbers (fully turbulent flows) are characterized by extremely irregular fluctuations in velocity, temperature, pressure, and other properties. At each point the velocity and other properties fluctuate around a mean value.

Turbulent flows are characterized by eddies and vortices. Chunks of fluid covering a wide size range move randomly around with respect to the mean flow. Fluid particles move on irregular paths, and the result is very effective mixing. Even the smallest eddies are typically orders of magnitude larger than the molecular mean free path (MMFP) (in gases) and the intermolecular distances. Within the small eddies, molecular (laminar) transport processes take place, but the interaction among eddies often dominates the overall transport processes and make molecular transport effects unimportant.

With respect to analysis, the Navier–Stokes equations discussed earlier in principle can be applied to turbulent flow as well. However, to obtain a meaningful solution, these equations must be solved in such a way that the largest and smallest eddies in the flow field are resolved. This approach [direct numerical simulation (DNS)] is extremely computational intensive, and it is possible at this time only for simple flow configurations and low Reynolds numbers.

General Remarks

Near-wall phenomena in internal turbulent flow has much in common with external turbulent flow, and the discussions of property fluctuations and near-wall phenomena in the previous chapter all apply to internal flow as well. The confined nature of the flow field, however, implies that, unlike external flow in which the free-stream conditions are not affected by what happens at the wall, the transport phenomena at the wall do affect the mean flow properties.

Consider fully developed internal flow in a smooth pipe, shown in Fig. 7.1. Similar to external flow, the entire flow field in the pipe can be divided into three zones: the viscous sublayer, the buffer zone, and the turbulent core. The mean thickness of the viscous sublayer is equal to y+ = 5, where y+ = yUτ/v is the distance from the wall in wall units and the buffer zone extends to y+ = 30. Close to the wall, where the effect of wall curvature is small and the fluid is not aware that the overall flow field is actually confined, the universal velocity profile presented in Eqs. (6.5.1)–(6.5.3) apply. Only when we approach the centerline does Eq. (6.5.3) deviate from measurements. Similar observations can be made about noncircular ducts.