Meteorol Atmos Phys (2017) 129:1–15 DOI 10.1007/s00703-016-0449-x
ORIGINAL PAPER
Wet precipitation scavenging of soluble atmospheric trace gases due to chemical absorption in inhomogeneous atmosphere Tov Elperin1 • Andrew Fominykh1 • Boris Krasovitov1
Received: 28 October 2015 / Accepted: 15 March 2016 / Published online: 19 April 2016 Springer-Verlag Wien 2016
Abstract We analyze the effects of irreversible chemical reactions of the first and higher orders and aqueous-phase dissociation reactions on the rate of trace gas scavenging by rain in the atmosphere with non-uniform concentration and temperature. We employ an one-dimensional model of precipitation scavenging of chemically active soluble gaseous pollutants that is valid for small gradients of temperature and concentration in the atmosphere. It is demonstrated that transient altitudinal distribution of concentration under the influence of rain is determined by the partial hyperbolic differential equation of the first order. Scavenging coefficients are calculated for wet removal of chlorine, nitrogen dioxide and sulfur dioxide for the exponential and linear initial altitudinal distributions of trace gases concentration in the atmosphere and linear and uniform altitudinal temperature distributions. Theoretical predictions of the dependence of the magnitude of the scavenging coefficient on rain intensity for sulfur dioxide are in a good agreement with the available atmospheric measurements.
Nomenclature a c c(G) c(G) 0 c(G) gr c(L) cp HA I k k0
k1
k2 Responsible Editor: S. T. Castelli.
k 3 = k 1k 2
& Tov Elperin
[email protected]
k4
Andrew Fominykh
[email protected]
k 40
Boris Krasovitov
[email protected] 1
Department of Mechanical Engineering, The Pearlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, POB 653, 84105 Beer-Sheva, Israel
k 5 = k 2k 4
Raindrop radius, m Total concentration of soluble trace gas in gaseous and liquid phases, mole l-1 Concentration of soluble trace gas in gaseous phase, mole l-1 Concentration of soluble gas at the cloud bottom, mole l-1 Concentration of soluble gas at the ground, mole l-1 Concentration of dissolved gas inside droplet, mole l-1 Specific heat, kJ kmole-1 K-1 Henry’s law constant, mole l-1 atm-1 Rainfall rate, m s-1 Growth constant, m-1 Constant in a linear dependence of concentration from coordinate, mole l-1 m-1 Coefficient in linear dependence of solubility parameter on temperature, K-1 Coefficient in linear dependence of temperature on coordinate, K m-1 Coefficient in linear dependence of solubility on coordinate, m-1 Coefficient in linear dependence of the first order chemical reaction constant on temperature, s-1K-1 Coefficient in linear dependence of the n-th order chemical reaction constant on temperature, ln-1 mole1-n s-1K-1 Coefficient in linear dependence of a value of first order chemical reaction on coordinate, s-1 m-1
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2
k50 = k2k40
kch1 kchn K1;SO2 L m = HARgT m* = m/ (1 ? kch1sD) Rg qc
t T u UC UC0 UT z Z(t)
T. Elperin et al.
Coefficient in linear dependence of a value of n-th order chemical reaction on coordinate, ln-1 mole1-n s-1m-1 First order chemical reaction constant, s-1 N-th order chemical reaction constant, ln-1 mole1-n s-1 Dissociation constant, mole l-1 Distance between the ground and cloud bottom, m Dimensionless Henry’s law coefficient Effective dimensionless Henry’s law coefficient Universal gas constant, atm l mole-1 K-1 Mass flux density of dissolved gas transferred by rain droplets, mole m-2 s-1 Time, s Temperature, K Terminal fall velocity of droplet, m s-1 Wash-down front velocity, m s-1 Wash-down front velocity for physical absorption, m s-1 Velocity of temperature front propagation, m s-1 Vertical coordinate, m Coordinate of a scavenging front, m
Greek symbols b Coefficient of mass transfer, m s-1 sD = (am)/(3b) Characteristic time of a mass transfer, s / Volume fraction of droplets in air q Density, kg m-3 ðGÞ 1n Mole1-n ln-1 x¼ c K
Scavenging coefficient, s-1
Subscripts and superscripts 0 Value at the cloud bottom gr Value at the ground G Gaseous phase L Liquid phase
1 Introduction Atmospheric measurements performed during last decades revealed increase of concentrations of soluble trace gases in the atmosphere (see, e.g. Johansen 2009). These findings point to the importance of soluble trace gas scavenging by wet precipitation. Wet precipitation of trace gases in
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atmospheric transport modeling is becoming more significant in view of global warming. Indeed, the rise of atmospheric temperature causes the increase of the temperature of ocean–atmosphere interface and increases the rate of evaporation from the ocean surface that most likely enhances precipitation. Soluble atmospheric gases whose concentration in the atmosphere changes noticeably during rain of average intensity and duration can be divided into three groups. Wet precipitation scavenging of such gases as CO2, N2, O2, H2O2 is determined by physical absorption. Absorption of NH3, SO2, HNO3 is accompanied by aqueous-phase dissociation reactions of the dissolved species. Modeling of scavenging of these gases is usually conducted using equation of mass transfer with effective solubility parameter dependent on dissociation constants and pH (Sportisse and du Bois 2002). Absorption of Cl2, N2O4, O3, NO and NO2 by water droplets is accompanied by chemical reactions of the first and second orders in a liquid phase and washout of these gases can be modeled using mass transfer equation with a sink of the dissolved components in a liquid phase (Bin et al. 2001; Pareek et al. 2003). Due to a large number of admixtures in the atmosphere scavenging of some gases can be determined by absorption that is accompanied by chemical reaction of the order higher than two. Reactions of the order higher than two are discussed, e.g. by Klinzing et al. (1980) and Alvarez-Fuster et al. (1980). Some greenhouse gases are soluble in water, e.g. carbon dioxide (CO2), methane (CH4), ozone (O3), nitrous oxide (N2O), carbon monoxide (CO), sulfur hexafluoride (SF6) (Johansen 2009). Nitrogen trifluoride (NF3) is an example of a soluble greenhouse gas of anthropogenic origin. This gas is used in manufacture of flat-panel television sets, computer displays, microcircuits, and thin-film solar panels. The rate of increase in atmospheric carbon dioxide (0.5 percent a year) was about twice as slow as that of majority of greenhouse gases for several last decades. Carbon monoxide and nitrogen oxides, while not being greenhouse gases themselves, increase the amount of methane and ozone in the atmosphere. Tropospheric ozone is produced photochemically in the atmosphere by oxidation of carbon monoxide, methane, or other hydrocarbons in the presence of nitrogen oxides which act as catalysts. Atmospheric molecular chlorine is formed from sea salt particles generated by ocean wind waves. The main sources of Cl2 in the ocean surface layer are reactions of sodium chloride, the major component of sea-salt particles, with nitrogen oxides and photolysis of ozone in the presence of sea-salt particles (Oum et al. 1998; Liao et al. 2014). Another source of molecular chlorine in the atmosphere are volcanic emissions (Zelenski and Taran 2012). The major source of Cl2 in volcanic gases is catalytic oxidation of volcanic HCl by oxygen while certain amount of Cl2 is also produced by oxidative decomposition of Na, K and Mg
Wet precipitation scavenging of soluble atmospheric trace gases due to chemical absorption…
chloroferrates formed as a result of basalt acid leaching. Nitrogen oxides (NO and NO2) are produced naturally by bacteria, volcanic activity and lightning whereby natural sources outweigh anthropogenic emissions. Anthropogenic emissions are caused mainly by fossil fuel combustion from stationary sources, i.e. power generation, and mobile sources, i.e. transport. Other atmospheric contributions come from non-combustion processes, e.g. nitric acid manufacture, welding processes and the use of explosives. The most important sources of SO2 are volcanic activities, fossil fuel combustion, smelting, manufacture of sulfuric acid, conversion of wood pulp to paper, incineration of refuse and production of elemental sulfur. Coal burning is the single largest man-made source of SO2 accounting for about 50 % of annual global emissions, with oil burning accounts for additional 25–30 %. Vertical distribution of chemically active soluble atmospheric trace gases, such as NO2, NO, Cl2, SO2, O3, was studied by Meng et al. (2008), Fischer et al. (2006), Finley and Saltzman (2006), Baidar et al. (2012), Hua et al. (2013), Wong and Stutz (2010) and Riedel et al. (2013). Analysis of the results of atmospheric measurements of concentration profiles reveals that concentration of majority of soluble chemically active gases rapidly decreases with height. Development of advanced climate models requires information about spatial distribution of all these gases in the atmosphere (Johansen 2009). In-cloud and under-cloud wet scavenging introduces alterations to spatial distribution of soluble gases, including greenhouse gases and other gases, directly concerned with greenhouse gases. Depletion of soluble trace gases concentration in the atmosphere with altitude alludes to the significance of soluble trace gases scavenging by rain. Absorption of chlorine by water was investigated by Brian et al. (1966) and Danckwerts (1970) who determined the dependence of the constant of first-order chemical reaction on temperature. The constant of the second-order chemical reaction for NO2 dissolution in water was determined by Cheung et al. (2000). Constants of first-order chemical reactions for NO and N2O4 absorption were determined, e.g. by Komiyama and Inoue (1980). Transient solution for dispersed-phase controlled mass transfer during soluble gas absorption by a stagnant liquid droplet, accompanied by a first-order irreversible chemical reaction in a liquid phase, was obtained by Schwartz and Frieberg (1981). They showed analytically that concentration of soluble active gas in liquid droplet is always inhomogeneous and mass flux to a droplet is always finite even in a steady state. We present a solution of this problem taking into account mass transfer in both phases in ‘‘Appendix 1’’ of the present paper. Solution of a similar nonstationary problem for a single translating droplet was obtained by Ruckenstein et al. (1971) while a conjugate problem was
3
investigated by Kleinman and Reed (1996). Chemical absorption of soluble gases by atmospheric nanoaerosols taking into account mass transfer in both phases was analyzed by Elperin et al. (2013). Chemical reactions of the second and higher orders were investigated by Klinzing et al. (1980), Juncu (2002) and by Pareek (2003). Different aspects of chemical gas absorption in dispersed systems was studied by Chen et al. (2013), Elperin and Fominykh (1999, 2003) and Elperin et al. (2008). Note that in all the previous studies chemical gas absorption by falling liquid droplets was investigated for uniform distribution of soluble gas in a gaseous phase. Theoretical models which allow us to take into account a non-uniform distribution of concentration in the continuous phase during gas absorption with the first and second order reactions by a single falling droplet are presented in the ‘‘Appendixes 2 and 3’’, respectively, of this paper. In the present study we analyze dynamics of soluble chemically active gas scavenging by rain taking into account the effects which were neglected in the previous studies (see, e.g. Slinn 1974; Asman 1995; Wurzler et al. 1995; Zhang et al. 2006; Banzhaf et al. 2012), e.g. first-order and higher orders irreversible chemical reactions in a liquid phase, dependence of constants of chemical reactions on temperature and changing pH in the rain droplets during gas absorption. In our previous study (Elperin et al. 2015a) we investigated different regimes of scavenging of soluble trace gases by physical absorption. In particular, we investigated scavenging of moderately soluble trace gases, when the velocities of scavenging and temperature fronts are of the same order, UC0 * UT; scavenging of gases having low solubility, when UC0 \\ UT; and scavenging of gases having high solubility, when UC0 [[ UT. Note that scavenging velocity of a soluble gas due to chemical absorption is always higher than scavenging velocity of the gas having the same solubility due to physical absorption whereas velocity of temperature front propagation is the same. It will be showed later that UC1 = UC0 ? kch1/m0/k and UC2 ¼ UC0 þ ðGÞ ðGÞ c0 / kch2 m2 k1 = 1 c0 / kch2 m2 t ; where UC1 and UC2 are scavenging velocities due to absorption accompanied by the first-order and the second-order chemical reactions in a liquid phase. Remarkably, scavenging by chemical absorption proceeds in the homogeneous atmosphere in contrast to a scavenging process by physical absorption. The latter assertion is valid for initially saturated by soluble gas rain droplets. The goal of the present investigation is to determine the rate of soluble chemically active gas scavenging in the atmosphere under the combined effect of precipitation and changing temperature profile. Particular attention is given to the analysis of an influence of chemical reactions of the first and higher orders and changing pH on the rate of gas
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scavenging. Although the model of precipitation scavenging of chemically active gases developed in the present study is valid for gases having arbitrary solubility, numerical calculations are performed for highly soluble gas Cl2 (UC [[ UT) and for a gas having a low solubility, NO2 (UC \\ UT). Absorption of chlorine by water is accompanied by irreversible reaction of the first order, and absorption of nitrogen dioxide – by the irreversible reaction of the second order.
2 Description of the model 2.1 Scavenging in inhomogeneous atmosphere due to absorption of gases of arbitrary solubility accompanied by the irreversible first-order chemical reaction in a liquid phase We consider chemical absorption of soluble trace gases having arbitrary solubility by falling rain droplets from a mixture with inert gas in inhomogeneous atmosphere. The mathematical model developed in the this study is valid for arbitrary relation between the scavenging front velocity, UC, and velocity of temperature front propagation, UT. The performed analysis is pertinent to the below-cloud wet precipitation scavenging of active gases. Gas absorption is accompanied by the first-order chemical reaction inside the droplets. The initial vertical distributions of temperature and soluble trace gas concentration in the atmosphere are assumed to be known. Since the concentration of soluble chemically active trace gases in the atmosphere is very low (of the order of 1 ppbv), chemical gas absorption and inhomogeneous concentration distribution in the gaseous phase do not affect temperature distribution in the gaseous and liquid phases (see Elperin et al. 2015a). At the same time, the influence of the inhomogeneous temperature distribution in the atmosphere on the rate of chemical gas absorption by falling droplets is significant since the solubility parameter m and the constant of the first-order chemical reaction kch1 are temperature-dependent. The initial concentration of the dissolved trace chemically active gas in rain droplets is equal to the averaged over the droplet radius stationary concentration (see ‘‘Appendix 1’’, Eq. 45). This concentration corresponds to the concentration of the trace soluble gas in the below cloud atmosphere adjacent to the bottom of a cloud. Scavenging proceeds in a stagnant atmosphere (see, e.g. Walcek and Pruppacher 1984), and temperature and concentration distribution gradients in the atmosphere are assumed to be low (see, e.g. Elperin et al. 2014). Scavenging of the atmosphere removes soluble trace gases by rain droplets from the region below the cloud bottom to the ground. Depletion of
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the dissolved gas in rain droplets due to chemical reaction enhances scavenging. For low gradients of temperature in the atmosphere the instantaneous temperature of the falling rain droplet is close to the local atmospheric temperature at a given height, T ðLÞ ðzÞ T ðGÞ ðzÞ. The relation between concentrations of the active gas in liquid and gaseous phases at a given height for low gradients of concentration in the atmosphere is derived in ‘‘Appendix 2’’. Assume that the initial temperature distribution in the atmosphere is a linear function of the vertical coordinate, e.g. ðGÞ
T ðGÞ ¼ T0
þ k2 z;
ð1Þ
where z is directed from the bottom of the cloud to the ground. Then the temperature distribution in the gaseous phase during rain reads: ðGÞ
T ðGÞ ¼ T0
þ k2 ðz UT tÞ; ð2Þ cpG qG ð1 /L Þ þ cpL qL /L (for where UT ¼ I cpL qL details see Elperin et al. 2015a). Taking into account the linear dependence of the solubility parameter m on temperature, m = m0 - k1(T - T0) and using Eq. (2) we obtain the following formula for the solubility parameter: m ¼ m0 k3 ðz UT tÞ;
ð3Þ
where k3 = k1k2. Clearly, coefficient k3 vanishes when temperature distribution in the atmosphere is homogeneous or the solubility parameter is independent of temperature. Taking into account the linear dependence of the constant of the first order chemical reaction on temperature, kch1 = kch10 - k4(T - T0) and using Eq. (2) we obtain the following formula for the time and height dependence of kch1: kch1 ¼ kch10 k5 ðz UT tÞ;
ð4Þ
where k5 = k2k4, kch10 is the value of the constant of the first-order chemical reaction in the atmosphere in the vicinity of a cloud bottom. The total mass flux density of the dissolved gas transferred by rain droplets is determined by the following expression: qc ¼ / u cðLÞ ;
ð5Þ
where u is terminal fall velocity of droplets, c(L) is concentration of the dissolved gas in a droplet. In our previous studies (see, e.g. Elperin et al. 2011; Baklanov et al. 2013) we showed that accounting for the droplet size distribution does not lead to significant changes in the rate of gas scavenging in comparison with that calculated using the average radius approximation. Therefore, in the present study we performed calculations for the monosize droplets having the average rain droplet radius. Equations (5) and (49) imply that
Wet precipitation scavenging of soluble atmospheric trace gases due to chemical absorption…
qc ¼ m / u cðGÞ :
ð6Þ
Equation of mass balance for soluble trace gas in the gaseous and liquid phases reads: oc oqc ¼ / kch1 cðLÞ ; ð7Þ ot oz where / is volume fraction of droplets in air. For low gradient of concentration of soluble gas in the gaseous phase the expression for the total concentration of soluble gaseous pollutant in gaseous and liquid phases reads (see, e.g. Elperin et al. 2015a): c ¼ cðGÞ ½ð1 /Þ þ m ðt; zÞ /:
ð8Þ
Criteria for applicability of the approximation of low gradients of concentration and temperature were obtained by Hales (1972) and Elperin et al. (2014). Combining Eqs. (6)–(8) we obtain the following equation: ðGÞ
oc ½1 / þ m ðz; tÞ/ þ cðGÞ / k3 UT ot o cðGÞ þ cðGÞ / k3 u cðGÞ / kch1 ðz; tÞm ðz; tÞ ¼ m ðz; tÞ/ u oz ð9Þ ðGÞ
distribution of soluble trace gas concentration in the atmosphere determined by Eq. (11) we obtain: UT m0 ðGÞ þ UðnÞ ¼ c0 exp n / k3 u k3
ð13Þ k1ch0 m0 k5 k3 kþ þ : k3 u kch10 m0 Equations (12), (13) determine transient altitudinal distribution of soluble trace gas concentration in the atmosphere under the influence of rain. For absorption of highly soluble gases in the atmosphere (UC [[ UT) with nonuniform temperature distribution Eqs. (12), (13) yield:
k5 m0 kch10 k5 m20 / ðGÞ zþ cðGÞ =c0 ¼ exp þ t k3 uk3 u
m0 m0 m0 k5 kch10 þ þ kþ exp expð/uk3 tÞ z k3 k3 k3 u u
ð14Þ Equation (14) yields the following expression for the dependence of the coordinate of the scavenging front on time and for the scavenging coefficient: ZðtÞ ¼
ðGÞ
where c ¼ c ðz; tÞ. Taking into account that conditions k3 z=m0 \\1 and k5 z=kch10 \\1 are valid for all z, neglecting the terms of the higher order of smallness and using Eqs. (3), (4), Eq. (9) can be rewritten as follows: ocðGÞ ocðGÞ þ ð/ um0 / uk3 z þ / u k3 UT tÞ ot oz k k 5 3 ¼ cðGÞ t/ kch10 m0 UT þ / kch10 m0 : k10ch m0
k5 k3 þ z/ kch10 m0 þ kch10 m0 ðGÞ
ð10Þ
ð11Þ
The general solution of Eq. (10) is as follows (see Polyanin et al. 2002, p. 111, Eq. 11):
kch10 m0 k5 k3 k5 m20 / ðGÞ c ¼ UðnÞ exp þ t ; zþ k3 uk3 kch10 m0
h
i where n ¼ expð/ uk3 tÞ z mk30 /UkT3 u ð/ k3 u t 1Þ .
K¼
m0 k3
k m2 / ð1 expð/uk3 tÞÞ k þ mk03ku5 þ kch10 þ 5 k30 t u ; k5 m0 k10ch m0 k5 k1ch0 uk3 þ u expð/uk3 tÞ k þ k3 u þ u
ð15Þ
k5 m20 / m0 k5 m0 kch10 expð/ uk3 tÞ: kþ / uk3 z þ k3 k3 uk3 u
ð16Þ
The initial condition for Eq. (10) reads: t ¼ 0; cðGÞ ¼ c0 exp ðk zÞ
5
ð12Þ
The obtained solution is valid for an arbitrary dependence of initial concentration of soluble trace gas in the atmosphere on height. The explicit form of the function U ðnÞ in Eq. (12) can be determined from the initial condition. In particular, in the case of the exponential initial
The velocity of the scavenging front propagation is determined from Eq. (15) as UC = qZ(t)/qt. Detailed discussion of the concepts of ‘‘scavenging front’’ and ‘‘temperature front’’ can be found in Elperin et al. (2015b). Similarly, for gases having a low solubility when (UC \\ UT), Eqs. (12), (13) yield: cðGÞ ðt; zÞ " ðGÞ c expðk zÞ expððUC0 k þ / kch10 m0 Þ tÞ ¼ 0ðGÞ c0
z [ ZðtÞ
;
z\ZðtÞ
ð17Þ ZðtÞ ¼ UC0 t þ kch10 / m0 t=k;
ð18Þ
UC ¼ UC0 þ kch10 / m0 =k
ð19Þ
where UC0 = Im0 and k UC0 þ / kch10 m K¼ 0
z [ ZðtÞ z\ZðtÞ
ð20Þ
Here / ¼ 106 A I b , where 0.052 B A B 0.089, 0.84 B b B 0.94 and rain intensity is expressed in mm/h (see Pruppacher and Klett 1997, p. 38). Equation (20) implies that scavenging by gas absorption accompanied by
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the first-order chemical reaction proceeds in the homogeneous atmosphere in contrast to the scavenging by physical absorption. The latter assertion is valid when rain droplets are initially saturated by a soluble gas and for the isothermal atmosphere. When kch1 ? 0, Eqs. (17)–(20) recover the expressions that were derived previously by Elperin et al. (2014). 2.2 Absorption scavenging of gases having arbitrary solubility in inhomogeneous atmosphere and accompanied by the irreversible nth-order chemical reaction in a liquid phase In this Section we consider scavenging by gas absorption accompanied by the nth-order chemical reaction inside droplets. Expression for time derivative of the average concentration of dissolved gas in a single falling droplet reads: n dcðLÞ 1 ðGÞ ¼ mc cðLÞ kchn cðLÞ ; ð21Þ sD dt where c(L)—average concentration of the dissolved gas in a droplet. Equation (21) implies the following condition of equilibrium: n cðGÞ ¼ m1 cðLÞ þ sD kchn cðLÞ : ð22Þ For small values of kchn it can be assumed, that cðLÞ ¼ m cðGÞ . Taking into account the linear dependence of the constant of the n-th order chemical reaction on temperature, kchn ¼ kchn0 k40 ðT T0 Þ; and using Eq. (2) we obtain the following formula for the dependence of kchn on time and coordinate: kchn ¼ kchn0 k50 ðz UT tÞ;
ð23Þ
where k50 ¼ k2 k40 , kchn0 is a value of a constant of the n-th order chemical reaction in the vicinity of a cloud bottom. Equation of mass balance for soluble trace gas in the gaseous and liquid phases reads: n oc oqc ¼ / kchn cðLÞ : ð24Þ ot oz Combining Eqs. (6) and (24) we obtain the following equation: o cðGÞ ½1 / þ mðz; tÞ/ þ cðGÞ / k3 UT ot n o cðGÞ þ cðGÞ / k3 u cðGÞ / knch mn ; ¼ mðz; tÞ/u oz
ð25Þ
where cðGÞ ¼ cðGÞ ðz; tÞ, m = m (z, t) and kchn = kchn(z, t). Taking into account that conditions k3 z=m0 \\1 and k50 z=kchn0 \\1 are valid for all z, neglecting the terms of
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the higher order of smallness and combining Eqs. (3), (23) and we obtain the following equation: ocðGÞ o cðGÞ þ ð/ um0 / uk3 z þ / uk3 UT tÞ ot oz n ðGÞ ðGÞ ¼ c / k3 ðu UT Þ þ c 0 k5 k3 n / kchn0 mn0 þ / kchn0 mn0 þ z kchn0 m0
0 k5 k3 n / kn0 mn0 þ UT t : kchn0 m0
ð26Þ
The initial condition for Eq. (26) is determined by 1n , Eq. (26) Eq. (11). Using the new variable, x ¼ cðGÞ can be rewritten as follows: ox ox þ ð/ um0 /uk3 z þ / uk3 UT tÞ ¼ ð1 nÞ/ k3 ðu UT Þx ot oz 0 k5 k3 n þ z þ ð1 nÞ /kchn0 mn0 þ / kn0 mn0 kchn0 m0
0 k5 k3 n /kn0 mn0 þ UT t : kchn0 m0
ð27Þ The initial condition to Eq. (27) reads: ðGÞ 1n t ¼ 0; x ¼ c0 expðkð1 nÞzÞ:
ð28Þ
General solution of Eq. (27) is given by the following formula (see Polyanin et al. 2002, p. 138, Eq. 4): x ¼ exp½ð1 nÞ/k3 ðu UT Þt UðnÞ þ exp½ð1 nÞ/k3 ðu UT Þt 0 8 k5 < k mn kchn0 mn0 UT kchn0 þ km30n chn0 0 þ :k3 ðu UT Þ ð1 nÞ/k32 ðu UT Þ2
ð1 þ ð1 nÞk3 ðu UT Þ/tÞ Fðt; zÞ k
m
UT
0
5 Fðt; zÞ ¼ ð1 nÞ / kchn0 mn0 ðkchn0 þ km3 0nÞ
where z 0
ð29Þ
ð/ k3 u t1Þ
m0 U T
UT ð1þð1nÞk3 ðuUT ÞtÞ k3 / k3 u k3 /k3 u f/k3 uþð1nÞ/ k3 ðuUT Þ þ ð1nÞ / k3 ðuUT Þ þ ð1nÞ2 /2 k2 ðuUT Þ2 g 3 h i and n ¼ expð/ u k3 tÞ z mk30 /kU3Tu ð/ k3 u t 1Þ .
The general solution (29) is valid for arbitrary dependence of initial concentration of soluble trace gas in the atmosphere on height, e.g. c(G) = f(z) at t = 0. The explicit form of function U(n) can be determined from the initial condition. For the exponential initial distribution of soluble trace gas concentration in the atmosphere we obtain:
Wet precipitation scavenging of soluble atmospheric trace gases due to chemical absorption…
UT m0 ðGÞ 1n þ UðnÞ ¼ c0 exp kð1 nÞ n /k3 u k3 0 k5 n kchn0 m0 UT kchn0 þ km30n kchn0 mn0 k3 ðu UT Þ ð1 nÞ/k32 ðu UT Þ2 0 k5 k3 n þ ð1 nÞ/kchn0 mn0 þ kchn0 m0 ( ) m0 UT n k3 /k3 u þ /uk3 þ ð1 nÞ/k3 ðu UT Þ ð1 nÞ/k3 ðu UT Þ
ð30Þ Equations (29)–(30) determine the altitudinal and time evolution of soluble trace gas concentration distribution in the non-isothermal atmosphere under the influence of rain for a case of gas absorption accompanied by the nth-order chemical reaction, where n C 2 and n is an integer. For absorption of gases having a low solubility (UC \\ UT), accompanied by the second-order chemical reaction Eqs. (29), (30) yield: ðGÞ
c0
7
phase dissociation reactions were developed in the approximation of constant value of pH in the droplet (see, e.g. Elperin et al. 2014). In this study we take into account the change of pH in falling rain droplets due to gas absorption. For the dissolution of SO2 in water the effective -1 Henry’s law coefficient H** A [M atm ] is determined by the following expression (see e.g. Seinfeld and Pandis 2006): HSO ¼ HSO2 1 þ K1;SO2 10pH ; ð33Þ 2 where K1;SO2 ¼ 1:3 103 M and HSO2 ¼ 1:23 M atm1 at 298 K. The condition of electroneutrality for pH B 5.5 is [H?] = [HSO3-] (see Pruppacher and Klett (1997), p.754). Relation between concentrations of [HSO3-] inside a droplet and concentration of SO2 in the atmosphere in the equilibrium ðGÞ 1=2 (see Pruppacher reads: ½HSO 3 ¼ HSO2 K1;SO2 Rg Tc and Klett (1997), p. 754). For pH inside a droplet containing (L) dissolved sulfur dioxide we have: pH = - log{[H? 0 ] ? c },
þ
þ or pH ¼ log ½H0 þ ½HSO3 ¼ log ½H0 þ ðHSO2
ð32Þ
K1;SO2 Rg TcðGÞ Þ1=2 g. Here [H? 0 ] is concentration of ions inside a cloud droplet without dissolved sulfur dioxide. Taking into account that for typical concentrations of SO2 in the atmo ðGÞ 1=2 , Eq. (33) yields: sphere ½Hþ 0 \\ HSO2 K1;SO2 Rg Tc 1 0 ½H0þ K1;SO2 1 1=2 C B ðHSO2 K1;SO2 Rg TcðGÞ Þ C: B ¼ H 1 þ HSO SO2 @ 1=2 A 2 HSO2 K1;SO2 Rg TcðGÞ
where UC0 = I m. Equation (32) implies that in contrast to a scavenging by physical absorption, scavenging by gas absorption accompanied by the second-order chemical reaction proceeds in the homogeneous atmosphere. When kch2 ? 0, Eqs. (31), (32) recover the expressions that were derived previously by Elperin et al. (2013).
Assuming, that in the atmosphere (c - c(G) 0 )/ where c(G) is concentration of soluble trace gas 0 in the vicinity of a cloud bottom, Eq. (34) can be rewritten as follows:
cðGÞ ðz; tÞ ¼
ðGÞ c0 kch2
m2
;
exp ½k ðz UC0 tÞ þ / t ocðGÞ ðz; tÞ ot Kðz; tÞ ¼ cðGÞ ðz; tÞ ðGÞ k UC0 exp ðk ðz UC0 tÞÞ c0 / kch2 m2 ¼ ; ðGÞ exp ðk ðz UC0 tÞÞ c0 / kch2 m2 t
ð31Þ
ð34Þ (G)
c(G) 0 \\ 1,
0 K1;SO2 1
!
2½Hþ 0
1
2.3 Absorption scavenging of soluble gases accompanied by aqueous-phase dissociation reactions taking into account changing pH during gas absorption
B B B1 þ HSO ¼ H SO 2 B 2 @
Consider absorption scavenging of a soluble gas accompanied by aqueous-phase dissociation reactions from a gaseous mixture containing inert gas by precipitation whereby at time t = 0 the rain droplets begin to absorb gas from the isothermal atmosphere. Altitudinal distribution of concentration of the soluble trace gas in the atmosphere is assumed to be known. It is assumed also that the initial concentration of the dissolved trace gas inside the rain droplets is equal to the concentration of saturation in liquid corresponding to the concentration of trace soluble gas in the below cloud atmosphere adjacent to the bottom of a cloud. Previous models of gas absorption with aqueous-
For low gradient of concentration of soluble gas in the atmosphere the expression for the total concentration of soluble gaseous pollutant in the gaseous and liquid phases reads (see, e.g. Elperin et al. 2011): h i c ¼ cðGÞ ð1 /Þ þ m ð36Þ SO2 / :
1=2
ðHSO2 K1;SO2 Rg Tc0ðGÞ Þ ðGÞ 1=2 HSO2 K1;SO2 Rg Tc0
þ
C ðGÞ ½Hþ ðt; zÞ C 0 c 2 C C: ðGÞ A mSO c 2
0
ð35Þ
where m SO2 ¼ HSO2 Rg T. The total mass flux density of the dissolved gas transferred by rain droplets is determined by the following expression: ðGÞ qc ¼ m : SO2 / u c
ð37Þ
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8
T. Elperin et al.
Equation of mass balance for soluble trace gas in the gaseous and liquid phases reads (Elperin et al. 2011): oc oqc ¼ ; ot oz
ð38Þ
where z is directed from the cloud bottom to the ground. Combining Eqs. (35)–(38) we obtain the following equation: o cðGÞ ocðGÞ þ ða þ bcðGÞ Þ ¼ 0; ot oz
ð39Þ
where 0
2½Hþ 0
K1;SO2 1 1=2 B B ðHSO2 K1;SO2 Rg TcðGÞ 0 Þ B a ¼ m/uB1 þ ðGÞ 1=2 @ HSO K1;SO Rg Tc 2
2
0
!1 C C C; C A
ð40Þ
2/ u½Hþ 0 b¼ 2 : ðGÞ c0 General solution of Eq. (39) reads (see Polyanin et al. (2002), p. 312, Eq. 6): z ¼ ða þ bcðGÞ Þ t þ F cðGÞ : ð41Þ The explicit form of function F is determined by the initial conditions. If the initial distribution of the concentration of soluble trace gas in the atmosphere is a linear function of coordinate, e.g. ðGÞ
cðGÞ ¼ c0 þ k0 z; ð42Þ ðGÞ then F xðGÞ ¼ cðGÞ c0 = k0 and the dependence of c(G) on time and vertical coordinate is as follows: ðGÞ
cðGÞ ¼
c0 þ k0 ðz atÞ : 1 þ k0 tb
ð43Þ
The expression for the scavenging coefficient reads: h i ðGÞ k0 a ð1 þ k0 t bÞ þ c0 þ k0 ðz atÞ k0 b h i K¼ : ð44Þ ðGÞ ð1 þ k0 t bÞ c0 þ k0 ðz a tÞ
3 Results and discussion The above model of heat and mass transfer in the atmosphere allows analyzing the effect of transient altitudinal temperature distribution in the atmosphere on the rate of chemical absorption scavenging of soluble gas by precipitation. This model is valid for absorption of gases having arbitrary solubility and accounts for the dependence of the gas solubility
123
parameter and constants of chemical reaction on temperature. The suggested model takes into account the change of pH inside falling rain droplets during gas absorption with aqueous-phase dissociation reactions on the rate of gas scavenging. To illustrate the obtained results we considered scavenging of chlorine (Cl2), nitrogen dioxide (NO2) and sulfur dioxide (SO2) by rain using analytical approximation for the measured altitudinal distributions of these trace gases and temperature in the atmosphere. The growth constants in the exponential vertical distribution of soluble trace gases in (G) the atmosphere, k = L-1 ln (c(G) gr /c0 ) for chlorine and nitrogen dioxide were taken to be equal to the typical value equal to 10-3 m-1. The solubility parameter for chlorine for the temperature 298 K mCl2 = 2.2 (see, e.g. Sander 1999), the value of a constant of chemical reaction of the first order is kch1 ¼ 8:5 s1 at 15 C, k1 = 0.1 K-1and k4 ¼ 0:69 s1 K1 , where k1 and k4 are coefficients in a linear dependence of the solubility parameter and the constant of the first order chemical reaction on temperature (see Brian et al. 1966; Danckwerts 1970). In the case of the environmental lapse k2 = 7 10-3 K m-1, k3 = 7 10-4 m-1 and k5 = -4.9 10-3 m-1s-1 while k2 = -10-2 K m-1, k3 = -10-3 m-1 and k5 = 6.9 10-3 m-1s-1 in the case of nocturnal temperature inversion. For the nitrogen dioxide mNO2 ¼ 0:25 at 298 K and kch2 ¼ 4:25 107 mole1 l s1 (see Cheung et al. 2000). The results showed in Figs. 1, 2, 3, 4, 5, 6 and 8, 9 were obtained for T0 = 283 K and rain intensity 5 mm h-1. Inspection of Figs. 1 and 2 shows that after several hours chlorine in the atmosphere is washed out to the concentration equal to the chlorine concentration in the vicinity of the cloud bottom. Comparison of Figs. 1 and 2 shows that time required for scavenging of chlorine to the concentration in the interstitial air in a cloud is smaller for the temperature profile corresponding to the environmental lapse than for the temperature distribution in the nocturnal inversion. This behavior is explained by the dependence of gas solubility and constant of chemical reaction of the first order on temperature. The solubility of chlorine in water decreases with a growth of temperature as well as solubility of all atmospheric soluble gases. At the same time the constant of the chemical reaction of the first order for chlorine increases with the growth of temperature as well as the constant of chemical reaction for N2O4. When the initial temperature distribution in the atmosphere is determined by the environmental lapse, the falling rain droplet moves from a cooler region to a warmer region. Therefore, solubility of the soluble trace gas in a droplet decreases as it approaches the ground while the constant of the firstorder chemical reaction increases. When the initial temperature distribution in the atmosphere is determined by the nocturnal inversion, the situation is exactly reversed.
Wet precipitation scavenging of soluble atmospheric trace gases due to chemical absorption… 400
5 10
I=5 mm/hour
2
Cl
rain duration
350
Cl
2
t=0 min t=100 min t=200 min t=300 min t=400 min t=500 min
2
300
4 10
2
environmental lapse nocturnal inversion 3 10
2
2 10
2
1 10
2
z(t) [m]
250
height [m]
9
200
150
environmental lapse
100
50
0
0 1
1.1
1.2
1.3 (G)
c /c
1.4
1.5
0
5000
1 10
4
4
1.5 10
2 10
(G)
4
4
2.5 10
3 10
4
4
3.5 10
4 10
4
t [s]
0
Fig. 1 Evolution of chlorine distribution in the atmosphere caused by rain scavenging (Eq. 14). The initial distribution of the soluble trace gas in the atmosphere is exponential with the growth constant k = 10-3 m-1, the initial temperature distribution is determined by the environmental lapse (7 K km-1)
Fig. 3 Coordinate of the scavenging front vs time. The initial distribution of soluble trace gas in the atmosphere is exponential with the growth constant k = 10-3 m-1
-2
1.4 10
environmental lapse nocturnal inversion
400
rain duration
nocturnal inversion 350
Cl
300
Cl
2
-2
1.2 10
c
250
2
U [m/s]
I=5 mm/hour
height [m]
-2
1.3 10
t=0 min t=100 min t=200 min t=300 min t=400 min t=500 min t= 600 min
200
-2
1.1 10
150 1 10
100
50
-2
0
1 10
4
2 10
4
3 10
4
4 10
4
5 10
4
t [sec] 0
1
1.1
1.2
1.3 (G)
c /c
1.4
1.5
(G) 0
Fig. 2 Evolution of chlorine distribution in the atmosphere caused by rain scavenging (Eq. 14). The initial distribution of soluble trace gas in the atmosphere is exponential with the growth constant k = 10-3 m-1, the initial temperature distribution is determined by the nocturnal inversion (-10 K km-1)
Competition between these two opposite temperature dependencies—the decrease of solubility and the increase of the constant of chemical reaction with temperature growth—is responsible for the shapes of the curves in
Fig. 4 Dependence of scavenging velocity for gas absorption of Cl2 by rain droplets vs. time, I = 5 mm/h. The initial distribution of soluble trace gas in the atmosphere is exponential with the growth constant k = 10-3 m-1
Figs. 1, 2, 3, 4. Equation (20) implies that the rate of scavenging due to chemical absorption increases with the increase of solubility parameter and the constant of the first-order chemical reaction. Inspection of Figs. 1, 2, 3, 4 shows that the effect of the constant of the first-order chemical reaction growth with temperature increase is dominant for Cl2 scavenging in comparison with the effect
123
10
T. Elperin et al. 500
Cl
2
t=10 min
400
height [m]
t=100 min t=1000 min
300
environmental lapse 200
100
0 1.18 10
-5
-5
1.2 10
1.22 10
-5
1.24 10
-5
1.26 10
-5
-1
scavenging coefficient [s ]
Fig. 5 Dependence of scavenging coefficient vs. altitude for chlorine wash out by rain in the atmosphere with the environmental lapse (7 K km-1) temperature distribution. The initial distribution of soluble trace gas in the atmosphere is exponential with the growth constant k = 10-3 m-1
500
t=100 min t=200 min
400
Cl
2
explains the growth of the scavenging rate for absorption of Cl2 by rain droplets in the case of the temperature distribution corresponding to the environmental lapse 7 K km-1 (see Fig. 4). Secondary influence of the increase of solubility parameter with temperature decrease explains the decline of scavenging rate for the absorption of Cl2 vs. time for the nocturnal inversion temperature distribution (see Figs. 3, 4). Inspection of Figs. 5 and 6 shows that scavenging coefficient for Cl2 absorption decreases with altitude for the temperature profile corresponding to the environmental lapse 7 K km-1 and slowly increases with altitude for the nocturnal inversion temperature distribution. In the case of the environmental lapse temperature distribution the scavenging coefficient shows no significant dependence on time while in the case of nocturnal inversion temperature distribution the dependence of scavenging coefficient on time is weak. The shape of the curves in Figs. 5 and 6 is explained by the increase of scavenging coefficient with the increase of the solubility parameter and the constant of chemical reaction [see Eq. (20)], as well as by the temperature dependence of the solubility parameter and the constant of chemical reaction. Inspection of Fig. 7 shows that for the temperature profile corresponding to the environmental lapse rate 7 K km-1 the increase of scavenging coefficient with the increase of rain intensity for Cl2 is more pronounced than for the nocturnal inversion temperature distribution. Inspection of Fig. 8 shows that scavenging coefficient for gas absorption accompanied by the second-order chemical reaction in isothermal
height [m]
300
nocturnal inversion
-5
2.5 10
environmental lapse nocturnal inversion
200
-1
scavenging coefficient Λ [s ]
2 10
-5
100
0 1.6696
1.6697
1.6697
1.6698
1.6698 -5
1.6699
1.6699
-1
scavenging coefficient [10 s ]
Fig. 6 Dependence of scavenging coefficient vs. altitude for chlorine wash out by rain droplets in the atmosphere with nocturnal inversion (-10 K km-1) temperature distribution. The initial distribution of soluble trace gas in the atmosphere is exponential with the growth constant k = 10-3 m-1
Cl
2
-5
1.5 10
z=500 m
1 10
-5
5 10
-6
0
0
2
4
6
8
10
rain intensity [mm/hour]
of the decrease of solubility parameter with temperature increase. This assertion is also valid for scavenging of N2O4. The dominant influence of the constant of the firstorder chemical reaction growth with temperature increase
123
Fig. 7 Dependence of scavenging coefficient in the vicinity of the ground vs. rain intensity for chlorine wash out by rain droplets in the non-isothermal atmosphere. The initial exponential distribution of soluble trace gas in the atmosphere is exponential with the growth constant k = 10-3m-1
Wet precipitation scavenging of soluble atmospheric trace gases due to chemical absorption…
11
500 7 10
NO
2
6 10
height [m]
300
200
100
7 10
-10
-10
7.5 10
8 10
-1
-3
-1
-3
-1
-3
-1
NO
2
k=1.15*10 [m ] k=1.44*10 [m ]
5 10
-9
4 10
-9
3 10
-9
2 10
-9
1 10
-9
k=2.01*10 [m ]
0
5
10
15
20
rain intensity [mm/hour]
0 -10
-1
-3
k=1.05*10 [m ]
-9
0
6.5 10
-4
k=6.36*10 [m ]
-1
scavenging coefficient Λ [s ]
400
-9
-10
-1
scavenging coefficient Λ [s ]
Fig. 8 Dependence of scavenging coefficient vs. altitude for nitrogen dioxide wash out by rain. The initial distribution of soluble trace gas in the atmosphere is exponential with the growth constant k = 10-3 m-1
Fig. 10 Dependence of scavenging coefficient in the vicinity of the ground for nitrogen dioxide wash out vs. rain intensity for the exponential initial distribution of soluble trace gas in the atmosphere
10
sulfur dioxide
-4
-9
2.2 10
2
-9
1.8 10
z=500 m -9
1.6 10
-9
1.4 10
-9
1.2 10
1 10
3
-1
-9
c =18 μg/m
I=5 mm/hour
scavenging coefficient Λ [s ]
-1
scavenging coefficient Λ [s ]
2 10
(G)
NO
10
-5
experimental Maul (1978) experimental Martin (1984) k'=2*10
-9
10
-6
-10
6 10
-10
0
0.5
1
-1
k'=1.5*10 k'=1.0*10
-13
1.5
-14
2
-1
[mole*litre *m ]
-13
k'=5*10 8 10
-13
-1
-1
[mole*litre *m ] -1
-1
[mole*litre *m ] -1
-1
[mole*litre *m ] 2.5
3
3.5
4
rain intensity [mm/hour] 2
4
6
c
(G)
8
10
-10
[10 mole/litre]
Fig. 9 Dependence of scavenging coefficient in the vicinity of the ground for nitrogen dioxide wash out by rain vs. initial concentration of nitrogen in the vicinity of a cloud bottom. The initial distribution of soluble trace gas in the atmosphere is exponential with the growth constant k = 10-3m-1
atmosphere depends on altitude in contrast to the scavenging coefficients for physical absorption and for absorption accompanied by the first-order chemical reaction. Inspection of Fig. 9 reveals the increase of scavenging
Fig. 11 Comparison of the theoretical predictions with atmospheric measurements of Martin (1984) and Maul (1978) for sulphur dioxide scavenging by rain
coefficient for gas absorption accompanied by the second-order chemical reaction with the increase of the initial soluble trace gas concentration in the vicinity of a cloud bottom. Note that scavenging coefficients for physical gas absorption and for absorption, accompanied by the first-order chemical reaction do not reveal dependence on the initial soluble gas concentration. Scavenging coefficient for gas absorption accompanied
123
12
T. Elperin et al.
by the second-order chemical reaction increases with the increase of rain intensity and the growth constant in the initial exponential profile of concentration in the atmosphere (see Fig. 10). We compared theoretical expression for the scavenging coefficient of sulfur dioxide [see Eq. (44)] with the atmospheric measurements by Maul (1978) and Martin (1984). Inspection of Fig. 11 shows that Eq. (44) yields the same estimate for the scavenging coefficient during sulfur dioxide washout by rain as the experimental results reported by Maul (1978) and Martin (1984). Moreover, our theory predicts the same dependence of the magnitude of the scavenging coefficient on rain intensity as the experimental results.
3.
4.
4 Conclusions We developed a model for non-isothermal scavenging of chemically active soluble trace gases in the inhomogeneous atmosphere by rain that is applicable for low gradients of concentration of soluble trace gas and temperature. The developed model is valid for scavenging of gases having arbitrary solubility. A novel feature of this study is that we investigate the effect of irreversible chemical reactions of the first and higher orders inside rain droplets on the rate of soluble gas scavenging. We also analyze the influence of changing pH inside rain droplets on the efficiency of scavenging by gas absorption accompanied by aqueousphase dissociation reactions. We showed that absorption gas scavenging accompanied by the first and higher order chemical reactions and aqueous-phase dissociation in a case of low gradients of temperature and trace gas concentration distributions in the atmosphere is determined by the linear and non-linear hyperbolic partial differential equations of the first order. Using the developed model we calculated scavenging coefficient and the rates of scavenging of chlorine, nitrogen dioxide and sulfur dioxide by rain. The obtained results can be summarized as follows: 1.
2.
We showed that absorption scavenging accompanied by the first and second-order chemical reaction proceeds in the homogeneous atmosphere in contrast to the physical absorption scavenging. We demonstrated that in contrast to the scavenging coefficient for physical absorption and absorption accompanied by the first-order chemical reaction, scavenging coefficient for gas absorption in the isothermal atmosphere accompanied by the second order chemical reaction depends upon altitude and
123
5.
initial concentration of soluble trace gas in the atmosphere. We showed that competition between two opposite temperature dependencies—the decrease of solubility and the increase of the constant of chemical reaction with temperature growth—is responsible for the shapes of the curves presented in Figs. 1, 2, 3, 4, 5, 6 and 7. For example, the dominant influence of the constant of the first-order chemical reaction growth with temperature increase explains the growth of scavenging rate of gas absorption of Cl2 by rain droplets vs. time in the case of environmental lapse. We demonstrated that scavenging coefficient for Cl2 absorption decreases with altitude for the temperature profile corresponding to the environmental lapse rate and slowly increases with altitude for nocturnal inversion temperature distribution. In the case of environmental lapse, the increase of scavenging coefficient with rain intensity growth for Cl2 is more pronounced than for nocturnal inversion. The suggested model of scavenging by gas absorption with aqueous-phase dissociation reactions yields the same value of the scavenging coefficient during sulfur dioxide washout by rain and the same dependence of the magnitude of scavenging coefficient on rain intensity as the atmospheric measurements conducted by Maul (1978) and Martin (1984).
The suggested approach or parameterizations based on the obtained results can be integrated into the online coupled meteorology-chemistry models or climate-chemistry models, in particular in 3-D chemical transport model (see e.g. Knote and Brunner 2013).
Appendix 1: Gas absorption by a stagnant liquid droplet accompanied by the first-order irreversible chemical reaction Analytical solution of for gas absorption in the presence of inert admixtures by a stagnant liquid droplet accompanied of the first-order chemical reaction in a liquid phase can be derived by combining the methods suggested by Schwartz and Frieberg (1981) and Clift et al. (1978) p. 54. Schwartz and Frieberg (1981) solved dispersed-phase controlled mass transfer problem with chemical reaction while Clift et al. (1978) analyzed conjugate mass transfer problem of physical absorption in the presence of inert admixtures (see also Appendix A in Elperin et al. 2015b). The steady-state distribution of concentration of the dissolved gas in a droplet is as follows:
Wet precipitation scavenging of soluble atmospheric trace gases due to chemical absorption… ðGÞ
cðLÞ s ¼
c1 m a sinhðqr=aÞ ; 1 þ ðDL m=DG Þðq coth q= sinh q 1Þ r sinh q ð45Þ
where q = a(kch1/DL)1/2, DL and DG—coefficients of diffusion in liquid and gas phases, r—radial distance from the center of a droplet. Eq. (45) implies that distribution of concentration of the dissolved trace gas in a droplet is always non-uniform.
Appendix 2: Gas absorption by a falling droplet in the inhomogeneous atmosphere accompanied by the first-order chemical reaction in a liquid phase
13
cðLÞ usD grad cðGÞ ¼1 ðGÞ ðGÞ m c0 þ grad cðGÞ z c0 þ grad cðGÞ z 1 k1 þ 1 exp z sD u u In the case when sD udcðGÞ =dz \\1: ðGÞ cgr
ð50Þ
ð51Þ
Eq. (51) implies that c(L)(z, t) = m*c(G)(z, t). For small values of kch1 it can be assumed that ðLÞ c ¼ m cðGÞ . Physical gas absorption by falling droplets in the inhomogeneous atmosphere was considered, e.g. by Hales (1972) and Elperin et al. (2009).
The expression for time derivative of the average concentration of dissolved gas in a falling droplet reads: dcðLÞ 1 ðGÞ ¼ mc cðLÞ kch1 cðLÞ ; ð46Þ sD dt
Appendix 3: Gas absorption by a falling liquid droplet in the inhomogeneous atmosphere accompanied by the second-order chemical reaction in a liquid phase
where sD = (am)/(3b) is a characteristic time of a mass transfer, c(L)—average concentration of dissolved gas in a droplet, c(G)—concentration of a soluble trace gas in a gaseous phase, a—raindrop radius, b—mass transfer coefficient in a gaseous phase. The initial condition to Eq. (46) reads:
The expression for time derivative of the average concentration of dissolved gas in a falling droplet reads: 2 dcðLÞ 1 ðGÞ ¼ mc cðLÞ kch2 cðLÞ ; ð52Þ sD dt
ðLÞ
t ¼ 0 cðLÞ ¼ c0 :
ð47Þ
We assume a linear initial concentration distribution in the atmosphere: ðGÞ
cðGÞ ðzÞ ¼ c0 þ n z;
ð48Þ
where n = dc(G)(z)/dz. After substitution t = z/u, solution of Eq. (46) with the initial condition (47) can be written as follows: c ð LÞ ¼
ðGÞ mc0
1 þ kch1 sD
umsD grad cðGÞ 1 þ kch1 sD
1 þ m grad cðGÞ z þ e 1 þ kch1 sD
ðLÞ c0
kch1 1 s D uþ u
where sD = (am)/(3b) is a characteristic time of a mass transfer, b—mass transfer coefficient in a gaseous phase. Eq. (52) implies the following condition for equilibrium: 2 ðGÞ 1 ðLÞ c ¼m c þ sD mkch2 cðLÞ ð53Þ or c
ðLÞ
¼
1 þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 4sD mkch2 cðGÞ : 2sD mkch2
(1 ? 4sDmkch2c(G))1/2 = 1 ? If sDmkch2c(G) \\ 1, (G) 2sDmkch2c and cðLÞ ¼ m cðGÞ . The initial condition to Eq. (52) reads: ð LÞ
cðLÞ ðt ¼ 0Þ ¼ c0 :
z
! ðGÞ mc0 umsD ðGÞ þ grad c : 1 þ kch1 sD 1 þ kch1 sD
ð54Þ
ð55Þ
We assume a linear initial concentration distribution in the atmosphere: ðGÞ
ð49Þ Denote m* = m/(1 ? k1sD). In the case when c(L) 0 = (G) m*c0 Eq. (49) implies that
cðGÞ ðzÞ ¼ c0 þ nz;
ð56Þ
where n = dc(G)(z)/dz. After substitution t = z/u, Eq. (52) with the initial condition (55) can be rewritten as follows:
123
14
T. Elperin et al.
kch2 ðLÞ dcðLÞ c ¼ dz u
2
ðGÞ
cðLÞ mc0 m grad cðGÞ z þ þ ; sD u sD u sD u
ð57Þ
Introducing the new variable, g = -k2z/u, Eq. (57) can be rewritten as follows: dcðLÞ ðLÞ 2 cðLÞ mc0 m u grad cðGÞ g ¼ c þ þ : 2 dg sD kch2 sD kch2 sD kch2 ðGÞ
ð58Þ Eq. (58) is known as Riccati differential equation. Using R the substitution uðgÞ ¼ exp cðLÞ ðgÞdg (see, e.g. Polyanin and Zaitsev 2003) Eq. (58) can be transformed to the following ordinary differential equation of the second order: 00
0
ugg þ aug þ ðbg þ cÞu ¼ 0;
ð59Þ 2 b ¼ m ugrad cðGÞ =ðkch2 sD Þ,
a ¼ 1=ðkch2 sD Þ,
where
ðGÞ
c ¼ mc0 =ðkch2 sD Þ. Solution of Eq. (59) reads (see, e.g. Polyanin and Zaitsev 2003; p. 215, Eq. 12): pffiffiffi 1 2 pffiffiffi 3=2 2 pffiffiffi 3=2 bn bn n C1 J1=3 u ¼ exp ag þ C2 Y1=3 ; 2 3 3
ð60Þ where J1/3(x) and Y1/3(x) are Bessel functions of the first and second type, respectively, n ¼ g þ ð4c a2 Þ=ð4bÞ ¼ ðGÞ
g þ l, where l ¼ ð1 þ mc0 kch2 sD Þ=ð4mu grad cðGÞ sD Þ: The constant C2 near the second term in Eq. (60) must be set equal zero since Y1/3(n) ? -? when n ? 0. Solution (60) implies that
1 pffiffiffiffiffiffiffiffiffiffiffi 1 1 0 g l þ exp ag ðg lÞ1=2 ug ðgÞ ¼ a exp ag 2 2 2 2 pffiffiffi 3=2 1 pffiffiffiffiffiffiffiffiffiffiffi gl C1 J1=3 þ exp ag bn 3 2
3 pffiffiffi 2 pffiffiffi 1 2 pffiffiffi J1=3 ð bn3=2 Þ bðg lÞ1=2 C1 J4=3 ð bn3=2 Þ þ 2 3 3U 3
ð61Þ p ffiffi ffi where 23 bn3=2 ¼ U, g = -kch2z/u, ag = z/(sDu). Equations (60), (61) yield the expression for the dependence of concentration of the dissolved gas in a droplet on coordinate taking into account that c(L)(g) = -u-1 du/dg. The constant C1 can be found from the condition c(G)(z) = c(G) at z = 0. 0
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