Entropy generation in condensation in the presence of high concentrations of noncondensable gases

https://doi.org/10.1016/j.ijheatmasstransfer.2012.05.014Get rights and content

Abstract

The physical mechanisms of entropy generation in a condenser with high fractions of noncondensable gases are examined using scaling and boundary layer techniques, with the aim of defining a criterion for minimum entropy generation rate that is useful in engineering analyses. This process is particularly relevant in humidification-dehumidification desalination systems, where minimizing entropy generation per unit water produced is critical to maximizing system performance. The process is modeled by a consideration of the vapor/gas boundary layer alone, as it is the dominant thermal resistance and, consequently, the largest source of entropy production in many practical condensers with high fractions of noncondensable gases. Most previous studies of condensation have been restricted to a constant wall temperature, but it is shown here that for high concentrations of noncondensable gases, a varying wall temperature greatly reduces total entropy generation rate. Further, it is found that the diffusion of the condensing vapor through the vapor/noncondensable mixture boundary layer is the larger and often dominant mechanism of entropy production in such a condenser. As a result, when seeking to design a unit of desired heat transfer and condensation rates for minimum entropy generation, minimizing the variance in the driving force associated with diffusion yields a closer approximation to the minimum overall entropy generation rate than does equipartition of temperature difference.

Introduction

Because a truly reversible process is unachievable in finite time, in good thermal design, one seeks a configuration in which entropy production, or equivalently exergy destruction, is minimized, while meeting practical cost and performance parameters. Several recent studies have highlighted the importance of entropy generation minimization in maximizing the performance of humidification-dehumidification (HDH) desalination systems [1], [2], [3], [4], [5], [6]. Of particular relevance here is the conclusion that the greatest source of entropy generation in an HDH system is usually the condenser, or dehumidifier, where large fractions of noncondensable gas (typically 80–90%) control the overall heat transfer and condensation rates. This paper is a fundamental study of entropy generation minimization during condensation at high fractions of noncondensable gas.

In order to give the reader context, a brief overview of the HDH system, an intended application of this study, is provided. HDH functions very much like nature’s rain cycle. The system consists of three main components: a humidifier, a dehumidifier, and a heater. In the humidifier, warm seawater is sprayed over a packed bed, where dry air evaporates pure water vapor from the falling film of seawater. The warm, moist air then enters a dehumidifier, where the pure vapor condenses on coils cooled by cold, incoming seawater. The seawater is preheated in the process. A water heater between the humidifier and dehumidifier provides the heat input to the system. This particular embodiment of HDH is known as a closed air, open water (CAOW) cycle; there are several others that have been studied in detail [1], [7], but will not be discussed further here.

Much literature has addressed condensation of vapor from mixtures containing noncondensable gases. In particular, the problem of condensing water vapor from an air-steam mixture has received considerable attention. In that process, an air-steam mixture is exposed to a cold surface with a temperature lower than the local saturation temperature. As vapor condenses on the cold surface, the mixture is pulled convectively toward the surface, increasing the concentration of noncondensable gas near the wall. A concentration gradient is established, and the gas diffuses in opposition to the convective motion of the mixture. Temperature and vapor concentration gradients are both significant, and, especially in the case of high fractions of noncondensable gas, both the diffusional and thermal resistances impede the condensation process.

An early attempt at predicting heat transfer coefficients in these mixtures was performed by Colburn [8], who noted when even small amounts of air were present in steam condensers, condensation rates were significantly lower than those predicted by Nusselt theory [9]. A significant body of work was developed by Sparrow and coworkers using laminar boundary layer techniques to evaluate the effects of noncondensable gases, vapor superheating, interfacial resistance, and other phenomena on condensation in external flow in multiple geometries [10], [11], [12]. Denny, Mills, and Jusionis [13], [14] studied condensation of a number of species of vapor in forced, laminar flow using boundary layer equations. The work by Wang and Tu [15] is an early example of an analysis of falling film condensation in a vertical tube with noncondensable gas; the authors found that the effects of noncondensables are more pronounced in enclosures because the concentration of noncondensables increases as condensation proceeds. Various resistance network models have been developed to provide accurate ways to correlate experimental data on in-tube condensation with steam-air, steam-helium, and other mixtures [16], [17], [18].

A major application of these studies is in predicting heat transfer coefficients in steam condensers with relatively small amounts of noncondensable gas, such as result from leakage or dissolved gases. Lacking has been the study of condensation in the presence of high concentrations of noncondensable gases in temperature ranges above those normally encountered in HVAC systems (e.g., for which dehumidifiers have been studied in detail). These temperature ranges are of primary interest in HDH desalination systems, for example. In a study that does enter the HDH range, Rao et al. [19] used boundary layer techniques, and their results showed, as expected, that high fractions of noncondensable gas decrease the rate of condensation and heat transfer significantly.

One approach to entropy generation minimization in a heat exchanger is the technique of balancing. Key to understanding the connection between balancing and entropy generation minimization is the concept of remanent irreversibilities, or “flow imbalances” [20]. Entropy generation minimization by minimizing flow imbalances is in distinct contrast to minimizing entropy generation in heat transfer, say, by minimizing the driving temperature difference across which the heat travels. A simple, well-understood illustration is perhaps the best way to identify this contrast: the balanced, counterflow heat exchanger. When the capacity rates, m˙cp, and the heat transfer coefficients of both streams are approximately constant, the driving temperature difference will be constant along the flow path; this results in a minimization of remanent irreversibilities, even though there still exists a finite temperature difference by which entropy is produced. Indeed, it can be shown analytically that this configuration results in the minimum entropy production for a given set of inlet temperatures and heat exchanger effectiveness [2].

In the case of a heat and mass exchanger (HME), however, m˙cp does not fully define the axial temperature slope of each stream, owing to latent heat effects, and thus m˙cp does not define the variation in stream-to-stream driving temperature difference. A more general criterion for minimum entropy production of a fixed duty, fixed volume system undergoing any number of simultaneous transport processes (heat transfer, mass transfer, etc.) is given by Tondeur and Kvaalen [21]. They showed that for a transport process that obeys both the linear relations for entropy generation and Onsager’s relations [22], [23] (that is, it obeys the principle of microscopic reversibility, or is not too far removed from thermodynamic equilibrium), the criterion for minimum entropy production when any number of simultaneous transport processes occur is that the local, volumetric rate of entropy generation be constant in space and time. The theoretical result is known as the theorem of minimal dissipation, or equipartition of entropy production (EoEP). When the phenomenological coefficients-equivalently the heat and mass transfer coefficients-are constant, the equipartition of entropy production is characterized by an equipartition of thermodynamic driving force (EoF).

Johannessen et al. [24] allowed the conjugate heat transfer resistance to vary in a heat transfer process and showed that the equipartition of force is within 1 % of the true minimum, the EoEP, for most practical heat exchangers. Balkan [25] showed that equipartition of temperature difference (EoTD) in a counterflow heat exchanger with a constant overall heat transfer coefficient is a very good representation of the minimum entropy production state.

In the case of a saturated air-steam mixture undergoing a simultaneous, nonzero, heat and mass transfer, however, there cannot exist a process in which the heat and mass transfer driving forces will both be constant over a finite volume. This results from the exponential increase of saturation pressure with interfacial temperature. Saturation temperature and concentration are related monotonically, but not linearly. Hence, the magnitude of the concentration change caused by a given temperature change will be greater if the absolute temperature is greater. This result means that, in contrast to a heat exchanger, entropy generation minimization in an HME fundamentally relies on three parameters: (1) the ratio of the mass flow rates of each stream, (2) the bulk concentration of the diffusing species, and (3) the magnitude of the heat and mass transfer driving forces. As will be shown in the present work, the mean and variance in heat and mass transfer driving forces embody these three criteria completely, unlike the ratio of the minimum to maximum m˙cp.

If, therefore, one cannot achieve the equipartition of all driving forces, it is desirable to identify the dominant source of entropy production and design a flow geometry that results in an equipartition of the driving force associated with that dominant source of entropy generation. In the present analysis, expressions governing entropy production in terms of driving forces and associated fluxes are given, and then applied to a heat and mass exchanger in a general scaling analysis. Next the equations are applied directly in a laminar boundary layer analysis, where several boundary conditions are compared to identify the configuration that results in the true equipartition of entropy production and identify a set of criteria to approximate that minimum. Selected conditions representative of condensers in HDH desalination and HVAC systems are studied. An HDH system has higher rates of mass transfer than contemplated in previous boundary layer analyses of noncondensable gas problems [26], and it involves condensation in the presence of much higher concentrations of noncondensables. However, the majority of the work presented here could be applied to any binary mixture with a single species diffusing out of the control volume of interest.

Section snippets

Equations for entropy generation in a boundary layer

Let e be the mass specific internal energy of a mixture consisting of several components i. For a mixture in thermodynamic equilibrium, the canonical relationship states thatde=Tds-Pdv+igidmi,where T denotes absolute temperature, s the specific entropy, P the pressure, v the specific volume, gi the partial specific Gibbs energy of the ith species, and mi the mass fraction of species i. Now consider a perturbation in any number of the thermodynamic properties of the mixture. Assuming local

Scaling analysis

As will be discussed in detail in Section 4, the vapor-gas boundary layer is both the dominant resistance in a condenser with high fractions of noncondensable gas and the location of greatest entropy generation rate. The expressions derived in Section 2 are first scaled to identify which individual transport process dominates the entropy generation rate under given conditions. Again it is assumed that the transport processes of interest occur only in one dimension, which is representative of

Laminar boundary layer analysis

Next, the equations developed in Section 2 are applied directly in a laminar boundary layer analysis. In this section, the model geometry, equations, and code validation are presented first. The section concludes with a discussion of the entropy generation results obtained from the boundary layer analysis.

Conclusions

Mechanisms of entropy generation in condensers with high fractions of noncondensable gas have been examined with the aim of providing a set of criteria useful in engineering analyses for designing toward a minimum entropy production rate. In the present analysis, the following major conclusions have been demonstrated:

  • 1.

    From an entropic perspective, balancing for any heat and mass exchanger is fundamentally the manipulation of (1) the enthalpy rate of a stream and (2) the heat and mass transfer

Acknowledgements

The authors would like to thank the King Fahd University of Petroleum and Minerals in Dhahran, Saudi Arabia, for funding the research reported in this paper through the Center for Clean Water and Clean Energy at MIT and KFUPM under project R4-CW-08. Gratitude is also extended to the MIT Energy Initiative and its partner Eni, who supported the present work through a fellowship granted to the first author.

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