Basic Processes in Condensation

Condensation is a process in which the removal of heat from a system causes a vapor to convert into liquid. Condensation plays an important role in nature, where it is a crucial component of the water cycle, and in industry. Condensation processes are numerous, taking place in a multitude of situations. In view of their diversity, a classification of condensation processes is helpful. Classification can be based on various factors, including the following:

• Mode of condensation: homogeneous, dropwise, film, or direct contact.

• Conditions of the vapor: single-component, multicomponent with all components condensable, multicomponent including noncondensable component(s), etc.

• System geometry: plane surface, external, internal, etc.

There are of course overlaps among the categories from different classification methods. Classification based on mode of condensation is probably the most useful, and modes of condensation are now described.

Homogeneous Condensation

Homogeneous condensation can happen when vapor is sufficiently cooled below its saturation temperature to induce droplet nucleation, it may be caused by mixing of two vapor streams at different temperatures, radiative cooling of vapor–noncondensable mixtures (fog formation in the atmosphere), or sudden depressurization of a vapor. In fact, cloud formation in the atmosphere is a result of adiabatic expansion of warm and humid air masses that rise and cool.

The scale effect in two-phase flow and the classification of channel sizes were discussed in Section 3.6.2. The discussions in this chapter will primarily deal with channels with hydraulic diameters in the range 10 μm ≲ DH ≲ 1mm, where the limits are understood to be approximate magnitudes. For convenience, however, channels with 10 μm ≲ DH 100 < μm will be referred to as microchannels, and channels with 100 μm ≲ DH ≲ 1mm will be referred to as minichannels. The two categories of channels will be discussed separately, furthermore, because as will be seen there are significant differences between them.

Single-phase and two-phase flows in minichannels have been of interest for decades. The occurrence of flashing two-phase flow in refrigerant restrictors formed the impetus for some of the early studies (Mikol, 1963; Marcy, 1949; Bolstad and Jordan, 1948; Hopkins, 1950). The number of investigations dealing with two-phase flow in minichannels is relatively large, but two-phase flow in microchannels is a more recent subject of interest.

Two-phase flow in mini- and microchannels comprises a dynamic and rapidly developing area. Some attributes of two-phase flow in mini- and microchannels are not fully understood, and there are inconsistencies among experimental observations, phenomenological interpretation, and theoretical models. This chapter is therefore meant to be an outline review of the current state of knowledge.

Two-Phase Flow Regimes in Minichannels

Two-phase flow regimes in minichannels under conditions where inertia is significant have been experimentally investigated rather extensively. Table 10.1 summarizes some of the published studies.

Introduction

Internal-flow condensation is encountered in refrigeration and air-conditioning systems and during some accident scenarios in nuclear reactor coolant systems. Internal-flow condensation leads to a two-phase flow with some complex flow patterns. The condensing two-phase flows have some characteristics that are different from other commonly encountered two-phase flows. Empirical correlations are available for pure vapors condensing in some simple basic geometries (e.g., horizontal circular channels). Heat transfer (condensation rate) and hydrodynamics are strongly coupled and are sensitive to the two-phase flow regime. The two-phase flow regimes themselves depend on the orientation of the flow passage with respect to gravity.

Internal–flow condenser passages are usually designed to support vertical downward flow, inclined downward flow, or horizontal flow. Configurations that can lead to unfavorable hydrodynamics (e.g., countercurrent flow limitation and loop seal effect) are avoided in these systems. As a result, most of the published experimental studies and analytical models cover vertical downflow, and horizontal flow. Condensation in unfavorable configurations can be encountered during off–normal and accident conditions of many systems, however.

Shell-side phenomena in shell-and-tube-type condensers will not be discussed in this chapter. Complex three–dimensional flow is encountered in large power plant condensers. In these condensers the condensing fluid (steam) typically flows in the shell side of the shell–and–tube-type heat exchangers, with the secondary coolant flowing inside the tubes. The shell-side flow and condensation processes have certain common features with both internal and external condensing flows. Marto (1984, 1988) has written some useful reviews of these condensers.

The Concept of Drift Flux

The drift flux model is the most widely used diffusion model for gas–liquid two-phase flow. It provides a semi-empirical methodology for modeling the gas–liquid velocity slip in one-dimensional flow, while accounting for the effects of lateral (cross-sectional) nonuniformities. In its most widely used form, the DFM needs two adjustable parameters. These parameters can be found analytically only for some idealized cases and are more often obtained empirically. These empirically adjustable parameters in the model turn out to have approximately constant values or follow simple correlations for large classes of problems, however.

Recall that the diffusion models for two-phase flow only need one set of momentum conservation equations, often representing the mixture. Knowing the velocity for one of the phases (or the mixture), one can use the model's slip velocity relation (or its equivalent) to find the other phasic velocity. When used in the cross-section-average phasic momentum equations, the DFM thus leads to the elimination of one momentum equation. The mixture momentum equation can be recast in terms of mixture center-of-mass velocity. The elimination of one momentum equation leads to a significant savings in computational cost. Also, using the DFM, some major difficulties associated with the 2FM (e.g., the interfacial transport constitutive relations, the difficulty with flow-regime-dependent parameters, and numerical difficulties) can be avoided. These advantages of course come about at the expense of precision and computed process details.

Consider a one-dimensional flow shown in Fig. 6.1. Assume all parameters are time averaged.

Introductory Remarks about Two-Phase Mixtures

The hydrodynamics of gas–liquid mixtures are often very complicated and difficult to rigorously model. A detailed discussion of two-phase flow modeling difficulties and approximation methods will be provided in Chapter 5. For now, we can note that, although the fundamental conservation principles in gas–liquid two-phase flows are the same as those governing single-phase flows, the single-phase conservation equations cannot be easily applied to two-phase situations, primarily because of the discontinuities represented by the gas–liquid interphase and the fact that the interphase is deformable. Furthermore, a wide variety of morphological configurations (flow patterns) are possible in two-phase flow. Despite these inherent complexities, useful analytical, semi-analytical, and purely empirical methods have been developed for the analysis of two-phase flows. This has been done by adapting one of the following methods.

1. Making idealizations and simplifying assumptions. For example, one might idealize a particular flow field as the mixture of equal-size gas bubbles uniformly distributed in a laminar liquid flow, with gas and liquid moving with the same velocity everywhere. Another example is the flow of liquid and gas in a channel, with the liquid forming a layer and flowing underneath an overlying gas layer (a flow pattern called stratified flow), when the liquid and gas are both laminar and their interphase is flat and smooth. It is possible to derive analytical solutions for these idealized flow situations. However, these types of models have limited ranges of applicability, and two-phase flows in practice are often far too complicated for such idealizations.

2. […]

Introductory Remarks

Gas–liquid two-phase mixtures can form a variety of morphological flow configurations. The two-phase flow regimes (flow patterns) represent the most frequently observed morphological configurations.

Flow regimes are extremely important. To get an appreciation for this, one can consider the flow regimes in single-phase flow, where laminar, transition, and turbulent are the main flow regimes. When the flow regime changes from laminar to turbulent, for example, it is as if the personality of the fluid completely changes as well, and the phenomena governing the transport processes in the fluid all change. The situation in two-phase flow is somewhat similar, only in this case there is a multitude of flow regimes. The flow regime is the most important attribute of any two-phase flow problem. The behavior of a gas–liquid mixture – including many of the constitutive relations that are needed for the solution of two-phase conservation equations – depends strongly on the flow regimes. Methods for predicting the ranges of occurrence of the major two-phase flow regimes are thus useful, and often required, for the modeling and analysis of two-phase flow systems.

Flow regimes are among the most intriguing and difficult aspects of two-phase flow and have been investigated over many decades. Current methods for predicting the flow regimes are far from perfect. The difficulty and challenge arise out of the extremely varied morphological configurations that a gas–liquid mixture can acquire, and these are affected by numerous parameters.

Low Reynolds number aerodynamics is important to a number of natural and manmade flyers. Birds, bats, and insects have been of interest to biologists for years, and active study in the aerospace engineering community has been increasing rapidly. Part of the reason is the advent of micro air vehicles (MAVs). With a maximal dimension of 15 cm and nominal flight speeds of around 10 m/s, MAVs are capable of performing missions such as environmental monitoring, survelliance, and assessment in hostile environments. In contrast to civilian transport and many military flight vehicles, these small flyers operate in the low Reynolds number regime of 105 or lower. It is well established that the aerodynamic characteristics, such as the lift-to-drag ratio of a flight vehicle, change considerably between the low and high Reynolds number regimes. In particular, flow separation and laminar–turbulent transition can result in substantial change in effective airfoil shape and reduce aerodynamic performance. Because these flyers are lightweight and operate at low speeds, they are sensitive to wind gusts. Furthermore, their wing structures are flexible and tend to deform during flight. Consequently, the aero/fluid and structural dynamics of these flyers are closely linked to each other, making the entire flight vehicle difficult to analyze.

The primary focus of this book is on the aerodynamics associated with fixed and flapping wings. Chapter 1 offers a general introduction to low Reynolds number flight vehicles, considering both biological flyers and MAVs, followed by a summary of the scaling laws, which relate the aerodynamics and flight characteristics to a flyer's size on the basis of simple geometric and dynamics analyses.

Bird, bat, and insect flight has fascinated humans for many centuries. As enthusiastically observed by Dial (1994), most species of animals fly. There are nearly a million species of flying insects, and of the living 13,000 warm-blooded vertebrate species (i.e., birds and mammals), 10,000 (9000 birds and 1000 bats) have taken to the skies. With respect to maneuvering a body efficiently through space, birds represent one of nature's finest locomotion experiments. Although aeronautical technology has advanced rapidly over the past 100 years, nature's flying machines, which have evolved over 150 million years, are still impressive. Considering that humans move at top speeds of 3–4 body lengths per second, a race horse runs approximately 7 body lengths per second, a cheetah accomplishes 18 body lengths per second (Norberg, 1990), a supersonic aircraft such as the SR-71, “Blackbird,” traveling near Mach 3 (~2000 mph) covers about 32 body lengths per second, it is amazing that a common pigeon (Columba livia) frequently attains speeds of 50 mph, which converts to 75 body lengths per second. A European starling (Sturnus vulgaris) is capable of flying at 120 body lengths per second, and various species of swifts are even more impressive, over 140 body lengths per second. The roll rate of highly aerobatic aircraft (e.g., the A-4 Skyhawk) is approximately 720°/s, and a Barn Swallow (Hirundo rustics) has a roll rate in excess of 5000°/s. The maximum positive G-forces permitted in most general aviation aircraft is 4–5 G and select military aircraft withstand 8–10 G.

Physics of Choking

Choking can happen when a fluid is discharged through a passage from a pressurized chamber into a chamber that is at a significantly lower pressure. When a flow passage is choked, it supports the maximum possible fluid discharge rate for the given system conditions.

Choking can be better understood by the simple experiment shown in Fig. 17.1, where a chamber containing a fluid at an elevated pressure P0 is connected to another chamber that is at a lower pressure Pout by a flow passage. Suppose that the upstream conditions are maintained unchanged in the experiment, while the pressure in the downstream chamber, Pout, is gradually reduced, and the mass flow rate is continuously measured. It will be observed that the mass flux increases as Pout is reduced, until Pout reaches a critical value Pch. Further reduction of Pout will have no impact on mass flux or anything else associated with the channel interior.

The physical explanation of critical flow is as follows. A flow is critical (choked) when disturbances (or hydrodynamic signals) initiated downstream of some critical cross section cannot propagate upstream of the critical cross section. In single-phase flow, infinitesimally small disturbances (hydrodynamic signals) travel with the speed of sound. In a straight channel often the critical cross section occurs at the exit. In nozzles and other converging–diverging channels, the throat acts as the critical cross section.

General Description

Countercurrent flow limitation (CCFL), or flooding, refers to an important class of gravity-induced hydrodynamic processes that impose a serious restriction on the operation of gas–liquid two-phase systems. Some examples in which CCFL is among the factors that determine what we can and cannot be done are the following:

1. a) the emergency coolant injection into nuclear reactor cores following loss of coolant accidents,

2. b) the “reflux” phenomenon in vertically oriented condenser channels with bottom-up vapor flow, and

3. c) transport of gas–liquid fossil fuel mixtures in pipelines.

In the first example the coolant liquid attempts to penetrate the overheated system by gravity while vapor that results from evaporation attempts to rise, leading to a countercurrent flow configuration. The rising vapor can seriously reduce the rate of liquid penetration, or even completely block it. In the third example, the occurrence of CCFL causes a significant increase in the pressure drop and therefore the needed pumping power. CCFL represents a major issue that must be considered in the design and analysis of any system where a countercurrent of a gas and a liquid takes place.

To better understand the CCFL process, let us consider the simple experiment displayed in Fig. 9.1, where a large and open tank or plenum that contains a liquid is connected to a vertical pipe at its bottom. The vertical pipe itself is connected to a mixer before it drains into the atmosphere. Air can be injected into the mixer via the gas injection line.

General Remarks

The design and analysis of systems often require the solution of mass, momentum, and energy conservation equations. This is routinely done for single-phase flow systems, where the familiar Navier–Stokes equations are simplified as far as possible and then solved. The situation for two-phase flow systems is more complicated, however. The solution of the rigorous differential conservation equations is impractical, and a set of tractable conservation equations is needed instead. To derive tractable and at the same time reasonably accurate conservation equations, one needs deep physical insight (to make sensible simplifying assumptions) and mathematical skill. Fortunately, the subject has been investigated for decades, and at this time we have well-tested sets of tractable two-phase conservation equations that have been shown to do well in comparison with experimental data.

Generally speaking, conservation equations can be formulated and solved for multiphase flows in two different ways. In one approach, every phase is treated as a continuum, and all the conservation equations are presented in the Eulerian frame (i.e., a frame that is stationary with respect to the laboratory). This approach is quite general and can be applied to all flow configurations. In another approach, which is applicable when one of the phases is dispersed while the other phase is contiguous (e.g., in dispersed-droplet flow), the contiguous phase (the gas phase in the dispersed-droplet flow example) is treated as a continuum and its conservation equations are formulated and solved in the Eulerian frame.

As already mentioned, there are several prominent features of MAV flight: (i) low Reynolds numbers (104–105), resulting in degraded aerodynamic performance, (ii) small physical dimensions, resulting in much reduced payload capabilities, as well as some favorable scaling characteristics including structural strength, reduced stall speed, and impact tolerance, (iii) low flight speed, resulting in an order one effect of the flight environment such as wind gust, and intrinsically unsteady flight characteristics. The preferred low Reynolds number airfoil shapes are different from those typically used for manned aircraft in thickness, camber, and AR. In this chapter, we discuss low Reynolds number aerodynamics and the implications of airfoil shapes, laminar–turbulent transition, and an unsteady free stream on the performance outcome.

Schmitz (1942) was among the first to investigate the aerodynamics for model airplanes in Germany, and he published his research in 1942. His work is often considered to be the first reported low-speed wind-tunnel research. However, before him, experimental investigations of low Reynolds number aerodynamics were conducted by Brown (1939) and by Weiss (1939), in the first two (and only) issues of The Journal of International Aeromodeling. Brown's experiments focused on curved-plate airfoils, made by using two circle arcs meet at maximum camber point of 8%, at varying locations. The wing test sections were all of 12.7 cm × 76.2 cm, giving an aspect ratio of 6. In all cases, the tests were conducted at a free-stream velocity of 94 cm/s.