Research papers
Generalized nighttime radiative deficits

https://doi.org/10.1016/j.jhydrol.2021.126971Get rights and content

Highlights

  • Radiative deficit model for dew water harvesting.

  • The model combines atmospheric/environmental incoming radiation.

  • The model includes near-horizon obstacles, cloud cover and water vapor transmittance.

  • Results broadly valuable for the field of radiative cooling and dew water harvesting.

Abstract

We derive a general, tilt-dependent, nighttime, radiative deficit model with an eye towards improved dew collection. The model incorporates atmospheric/environmental incoming radiation, a linear precipitable water vapor transmittance function dependent on local meteo data and the influence of near-horizon obstacles. A brief discussion of cloud cover is given. The model is then used more specifically to predict radiative deficits for an ideal blackbody emitter in an environment with an isotropic temperature. Knowing the tilt angle, near-horizon obstacles and local meteo-data, it is then possible to estimate the radiative deficit of a given emitter. We consider errors resulting from the assumption that the ground and obstacles are at the same temperature as the air. We also analyze the errors arising from the linear precipitable water vapor transmittance function by comparing the results against high-resolution, full-spectrum Modtran® data Modtran, 2021. We show that for typical tilt angles, the isotropic temperature model is a reasonable approximation as long as the above-horizon environmental heating is small. We believe these results will be broadly valuable for the field of radiative cooling where a general radiative treatment has yet to be made and in particular the field of dew water harvesting.

Introduction

While remaining a niche method within the water production landscape, passive dew collection has the distinct advantage of being local, renewable, easy to use, producing relatively clean water and not requiring electricity (Beysens, 2018, Tomaszkiewicz et al., 2015, Kaseke and Wang, 2018). While dew collection will not solve the world’s water problems (Global Risks Report, 2019, Mekonnen and Hoekstra, 2016), it can provide an important supplemental source in places where the right atmospheric conditions exist. Between gaseous, liquid and solid phases, there is approximately the equivalent of 12,900 km3 of liquid water in the atmosphere making it a prime source for exploration (Oki and Kanae, 2006).

Dew was unsuccessfully pursued as a water source in the early 20th century (Nikolayev et al., 1996). However, there has been recent renewed interest in dew as a supplemental water source (Muselli et al., 2002, Beysens et al., 2007, Sharan et al., 2011, Sharan, 2011, Tomaszkiewicz et al., 2015, Khalil et al., 2016) as our understanding of condenser physics (Nikolayev et al., 1996), radiative cooling (Head, 1959, Bliss, 1961, Nilsson et al., 1992, Hossain and Gu, 2016, Zeyghami et al., 2018, Zhou et al., 2018, Dong et al., 1961, Zhao et al., 2019a), and dew formation (Beysens, 1995, Agam and Berliner, 2006) have improved, and high-yield materials have come to bear (Beysens et al., 2003, Maestre-Valero et al., 2011, Maestre-Valero et al., 2012, Chen et al., 2016, Zhai et al., 2017, Pierre-Brice Bintein et al., 2019, Bao et al., 2017, Guan et al., 2013, Al-Khayat et al., 2017). In an effort to optimize dew yields, many parameters have been studied including: wind speed (Clus et al., 2008, Muselli et al., 2009), angle of the collector (Kidron, 2005, Beysens et al., 2003), shape of the apparatus (Jacobs et al., 2008, Ziatdinov et al., 2019, Beysens et al., 2013, Kotzen, 2015, Beysens et al., 2021) and scaling of dew collectors (Carvajal et al., 2018, Sharan et al., 2017) to name a few.

The enabling physical mechanism of passive dew collection is radiative cooling (Head, 1959, Bliss, 1961, Nilsson et al., 1992, Granqvist and Hjortsberg, 1981, Berdahl and Fromberg, 1982, Hossain and Gu, 2016, Dong et al., 1961, Chen et al., 2016, Zeyghami et al., 2018, Zhou et al., 2018, Zhao et al., 2019b, Zhao et al., 2019a). The atmosphere has a spectral transmission window in the long-wave infrared region roughly between 8 and 13 microns through which the Earth and bodies of similar temperature can undergo thermal exchange with space (treated as an infinite thermal reservoir at 2.7 K). A radiative deficit of a device is the cooling that comes from the differential between the blackbody radiated heat loss of the device and the blackbody atmospheric and environmental radiative heat gain back on to the device. If the deficit is large enough and care has been taken to limit other heating sources, the device can cool below the dew point and dew condenses on the surface.

While radiative cooling and dew theory are both fairly mature fields, there have been historical difficulties in linking them together. Optimal radiative deficits and optimal dew collection have competing demands and operate under different contexts. Ideal radiative cooling occurs with the emitter surface normal pointed at the zenith and with low path-integrated precipitable water vapor (PWV). PWV is defined as the liquid equivalent thickness of the water vapor column from the Earth’s surface to the top of the atmosphere. On the other hand, to overcome pinning forces of dew droplets on the emitters, it is necessary to tilt the emitter relative to the zenith, which decreases the cooling power. Further, high relative humidity, needed for high dew yields, can also imply high PWV, which lowers the atmospheric transmission and thus reduces the radiative deficit. Although radiative cooling is usually derived using an angularly-dependent sky emissivity, due to these above difficulties, in many dew theory models atmospheric heating is often described by a single, angle-independent, empirical, sky-emissivity parameter εatm (Nikolayev et al., 1996, Tomaszkiewicz et al., 2015, Beysens et al., 2016). Owing to the dynamic nature of the atmosphere, this requires constant surveillance of the atmosphere. Here, we derive a radiative model that incorporates PWV for an angle-dependent emissivity for emitters of arbitrary geometry, but focus on planar emitters tilted at arbitrary angles. The model further considers the effects from near-horizon obstacles as well as cloud cover.

We note that the field of radiative cooling lacks a generalized radiative treatment. The theory and most proof-of-concept emitter deficits are based on surface normals pointing at or near the zenith. However, as the need for radiative cooling increases and, for example paints, are used at non-zenith angles, a generalized treatement will be needed. By adding to this work the direct and indirect solar irradiance (Rayleigh scattered) contributions as well as the emitter absorbance properties in the visible and short wave infrared, a fully generalized treatment could be made.

This paper is organized as follows. We first express the basis for nightime radiative deficit for tilted emitters. Then we briefly describe the use of a popular atmospheric emissivity model. Next, we consider the effects of the tilt of a perfect blackbody emitter on the radiative deficit in an environment at the temperature of the air. Then, we incorporate a precipitable water vapor model into the atmospheric emissivity. Lastly, we discuss the results and potential errors in the assumptions in a Discussion section.

Section snippets

Nighttime radiative deficits for tilted emitters

The nighttime radiative deficit (excluding solar heating) of an emitter at temperature T is expressed by the equation:Pdeficit=Prad-Pin,where Prad represents the thermal power radiated from the emitter. For an arbitrary emitter geometry, it is found by integrating the emitted thermal radiation of the dew condenser at temperature T for each infinitessimal area dA of the emitter over all wavelengths λ and solid angles dΩ:Prad=AΩdA·dΩ0IBB(λ,T)ε(λ,θ,ϕ)dλ,where IBB is the ideal blackbody

Emissivity model

We break up the environmental emissivity parameter εenv into two main regions. For the solid angle of the emitter that is concerned with the downwelling radiation from the atmosphere we say that εenv=εatm. For simplicity, for all other solid angles, we will assume that the environmental emissivity is unity (an ideal blackbody). While there are numerous mathematical models to describe angle-dependent atmospheric emissivity (Awanou et al., 1998), a simple and popular (Beysens, 2018, Zeyghami et

Perfect blackbody in an environment of isotropic temperature

We now seek to solve the radiative deficit under the assumption that a planar emitter is a perfect blackbody, ε(λ,θ,ϕ)=1, in an environment where the temperature is isotropic (the same in all directions relative to the emitter). With this assumption, Eq. (3) reduces to σT4, where σ is the Stefan–Boltzmann constant.

Eq. (4) is somewhat complicated. A general solution requires considering four integration regions, two to represent the atmospheric radiation and two to represent the non-atmospheric

Atmospheric transmission vs precipitable water vapor

While water vapor is only a trace gas in the atmosphere, it has enormously significant effects on infrared atmospheric transmittance. In fact, the atmospheric transmittance in the region covering a wavelength range of approximately 8–13 μm is determined primarily by the water vapor content (and secondarily by the CO2 content) (Bliss, 1961). For determining radiative deficits, it is sufficient to know the real-time path-integrated zenith PWV to determine atmospheric transmission. PWV varies from

Discussion

Unlike other gases in the atmosphere, the water vapor scale height is location-dependent. For example, the tropics have very large scale heights. The implication then is that while there may be high relative humidity needed for dew condensation, the atmosphere is more opaque to the longwave IR making radiative cooling less effective. The geography surrounding the Mediterranean, for example, tends to have small scale heights, meaning there can be both high relative humidity and low PWV, making

Conclusion

In this paper, we have derived a clear sky, radiative theory that accounts for planar emitter tilt, precipitable water vapor and near-horizon obstacles and formulated an extension of the theory to non-zero cloud cover. We are able to make estimates of the radiative cooling power and thus dew yield given the surface temperature, relative humidity and atmospheric water vapor scale height. This is an important and necessary improvement of the former theories elaborated for horizontal radiative

CRediT authorship contribution statement

John C. Howell: Conceptualization, Methodology, Software, Supervision, Validation, Resources, Writing – original draft. Tomer Yizhaq: Formal analysis, Investigation. Nadav Drechsler: Formal analysis, Investigation. Yuval Zamir: Formal analysis, Investigation. Daniel Beysens: Conceptualization, Methodology, Validation, Writing – review & editing. Joseph A. Shaw: Methodology, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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