47 3 The Estimation of Time- Dependent (Relaxation) Processes Related with Condensation and Evaporation of Liquid Drop Mikhail V. Buikov CONTENTS Introduction 47 General Equations 48 The Solutions of the Equations 49 Relaxation of Salt Concentration 51 Thermal Relaxation 52 The Rate of Change of Drop Size 55 Intensive Evaporation of the Solution Drop 56 Summary and Conclusions 57 References 58 Nomenclature 58 INTRODUCTION The theory of condensational growth and evaporation of drops consisting of pure liquid or solution is a complicated thermodynamical problem that concerns many aspects of kinetic gas theory. A complete review of the theory of growth and evaporation of drops can be found elsewhere. 1-3 Here, only one problem will be investigated: the time-dependent processes as a result of which steady- state temperature, vapor, and salt concentration are reached. Usually, the adopted drop temperature is constant inside the drop volume and is determined by heat balance between phase transition and thermal conduction; the salt concentration equal to the average volume value, vapor concentration, and temperature outside drop obey steady-state distributions in the drop vicinity. This approach is applicable if the rate of change of droplet size is small enough and the temperature of the environ- ment varies slowly with time. The drop temperature can be defined as a psychometric one because it the same as that of aspiration psychrometer in the case of evaporation. The investigation of the transition to steady-state may be of some interest from a general point of view, and can find application if the environment temperature is changing fast enough and in the case of intensive drop growth and evaporation when the deviation of salt concentration near drop surface from mean L829/frame/ch03 Page 47 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC 48 Aerosol Chemical Processes in the Environment value results in substantial influence on the rate of change of drop size. Temperature relaxation has been investigated 4,5 ; the influence of inhomogeneous salt concentration on drop growth has also been studied. 6-8 Intensive evaporation of solution drop was considered by Buikov and Sigal. 9 Below are presented the main results of these research efforts. The exact formulation of the problem of growth (evaporation) of a drop of solution is presented below, the application of the heat potentials to the solution is described, and an analysis of the time-dependent process resulting in establishing the steady state is considered. In conclusion, the intensive evaporation of solution drop when the formation of a salt crust is possible at the drop surface is also considered. GENERAL EQUATIONS The equations that describe diffusion of some non-volatile salt in volatile dissolvant (e.g., water), heat conduction inside and outside the drop, and vapor diffusion in a gas-vapor environment in a spherical coordinate system are: (3.1) (3.2) (3.3) (3.4) The rate of change of drop radius is given by the conventional formula (3.5) but the effects under consideration will be displayed through the gradient of the vapor concentration. If a drop at t = 0 is placed into a gas-vapor environment, then the initial conditions are: (3.6) The boundary conditions are: (3.7) c( r,t ) and T 2 ( r,t ) must be finite at r = 0. Temperature, heat flux, and vapor concentration should be continuous at the drop surface ( r = R ( t )); that is, ∂ ∂ = ∂ ∂ + ∂ ∂ c t D c rr c r 1 2 2 2 ∂ ∂ = ∂ ∂ + ∂ ∂ T t K T rr T r 1 1 2 1 2 1 2 ∂ ∂ = ∂ ∂ + ∂ ∂ T t K T rr T r 2 2 2 2 2 2 2 ∂ ∂ = ∂ ∂ + ∂ ∂ q t D q rr q r 2 2 2 2 dR t dt D q r rRt () () = ∂ ∂ = 2 ρ cr c T r T T r T qr q R R(, ) , (, ) , (, ) , (, ) , ( ) .00 000 01 102 0 == === ∞∞ qrt q T rt T r(,) , (,) , .→→→∞ ∞∞2 L829/frame/ch03 Page 48 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC The Estimation of Time-Dependent (Relaxation) Processes 49 (3.8) (3.9) (3.10) Equation 3.10 takes into account the effect of dissolved salt in the simplest form. The total amount of salt inside the drop should not change with time: (3.11) Taking the derivative of both sides of Equation 3.11 and using Equation 3.1 we have (3.12) In a growing (evaporating) drop, the salt concentration decreases (increases) with r and does not change if R = const. THE SOLUTIONS OF THE EQUATIONS There are some specific (characteristic) time intervals in the problem. The first one ( t 1 ) is the time interval during which the steady-state field of the vapor is established in the vicinity of the drop. The time interval t 2 describes the relaxation of salt concentration . The establishment of the psycho- metric temperature can be reached after an elapsed time t 3 . The last time interval is connected with a substantial increase in drop size ( t 4 ); for example, for a solution drop growing in the saturated environment, it can be taken as the time when the salt concentration will be smaller than the initial value. It is well-known 1,3 that t 1 <<t 4 and t 1 << t 2 ( D 2 << D 1 ) and it will be clear later that t 1 <<t 3 ) . So, as conventionally adopted, one can use the simplified formula for q: (3.13) The solutions of Equation 3.1 to 3.4 can be presented as thermal potentials 10 (dimensionless variables): (3.14) Trt Trt r Rt 12 (,) (,), (),==at LD q r k T r k T r rRt 21 1 2 2 0 ∂ ∂ + ∂ ∂ + ∂ ∂ ==,(); at qrt q T AcRt r Rt s (, ) ( ) ( ()), ()=− {} =1 at 4 4 3 23 0 π π dr r c r t R c o ∫ =(,) . dR dt cRt D c r rR (,)+ ∂ ∂ = = 1 0 qrt q q q Rtr s (,) () . – =+− () ∞∞ 1 xx d K x∑=∂∂∂ ∫ (,) () (,, )τστ τ 0 Kx xZ xZ (, , ) () exp () () exp () () τ πτ ττ ∂= −∂ − −∂ () −∂ −− +∂ () −∂ 1 2 44 22 L829/frame/ch03 Page 49 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC 50 Aerosol Chemical Processes in the Environment (3.15) (3.16) (3.17) (3.18) (3.19) These solutions satisfy the equations and the initial and boundary conditions at great distances: (3.20) (3.21) The heat balance equation is transformed into: (3.22) The temperature continuity and salt conservation equations are, respectively, (3.23) (3.24) The subsidiary functions σ ( ∂ ), σ 1 ( ∂ ), σ 2 ( ∂ ) can be determined using other boundary conditions. xY x d K x 11 0 1 (,) () (,, )τστ τ =∂∂ ∂ ∫ Kx xZ xZ (, , ) () exp () () exp () () τ πβ τ βτβτ ∂= −∂ − −∂ () −∂ −− +∂ () −∂ 1 2 44 1 2 1 2 1 xY x d K x 22 0 2 (,) () (,, )τστ τ =∂∂ ∂ ∫ Kx xZ 2 2 2 2 1 2 4 (, , ) () exp () () τ πβ τ βτ ∂= −∂ − −∂ () −∂ Π ΠΠ (,) () x Z x s τ τ = − () ∞ dZ d d dx xZ τ δτ== Π () ∑= = = =YY 12 00, τ Yx 2 0→→∞, . Γ Π Γ 01 12 0 ∂ ∂ + ∂ ∂ + ∂ ∂ == x Y x Y x xZ, ( ).τ ZY d KZ d KZ()() () ( ,, ) () (,, ),ττ σ τ σ τ ττ =∂∂ ∂=∂∂ ∂ ∫∫ 0 11 0 22 dx x x Z Z 2 3 0 1 3 ∑=− − ∫ (,) .τ L829/frame/ch03 Page 50 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC The Estimation of Time-Dependent (Relaxation) Processes 51 RELAXATION OF SALT CONCENTRATION Concentration and thermal relaxations are not connected because the heat of dissolution is not taken into account. This is in agreement with Equation 3.24, which will be turned into the equation for σ ( θ ): the dependence on temperature is only through drop size, but not directly. If the dependence of Σ ( x, τ ) on x in Equation 3.24 is neglected, then we obtain the conventional formula for salt concentration as: (3.25) or, c ( r,t ) ≅ R –3 ( t ). To get the next approximation taking into account the difference between bulk and surface concentrations, the kernel K ( x, τ, θ ) is expanded into a series on x, keeping the terms ∼ x 3 : (3.26) Equation 3.25 is applicable if the following inequality is true: (3.27) Because (3.28) and (3.29) then, instead of Equation 3.27, we have that it is possible to neglect salt concentration relaxation if (3.30) This inequality cannot be satisfied for small τ ; but because τ 4 >> τ 2 , then (3.31) Z(τ) can be expanded for small τ and we obtain, from Equation 3.30, (3.32) It is a criterion of the applicability of the bulk concentration approximation, when only a small gradient of salt concentration exists inside a drop, that ∑≅ −(,) () – xZττ 3 1 ∑≅∑+∑(,) () ()xxττ τ 1 2 2 ∑>> ∑ ∑>> ∑ 1 2 21 2 xZ ( ) .or τ ∑≅ − − 1 3 1() () ,ττZ ∑)≅ ∑ 2 1 6 (τ τ d d 1 1 3 2 −>> − Z Z dZ d () () () τ τ τ τ dZ d ()τ τ τ≅+1 const. , τ>> >> 1 66 2 1 0 2 or t D R L829/frame/ch03 Page 51 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC 52 Aerosol Chemical Processes in the Environment (3.33) The gradient is negative for growth and positive for evaporation. For a 10-µm drop, t 2 is about 10 –2 s; for a 100-µm drop, t 2 is equal to some seconds; and for a 1-mm drop, t 2 is some minutes. In the opposite case, when τ << 1/6, it is possible to derive an approximate formula for the salt concentration, introducing Equation 3.14 into Equation 3.25, after integration, expanding for small τ, and adopting Z = 1, one can derive: (3.34) This means that (3.35) So, for the salt concentration, one has (3.36) The deviation of the salt concentration from the initial value takes place only in a very thin layer near the drop surface: (3.37) and deeper inside the drop volume c = c 0 . THERMAL RELAXATION Leveling of the temperature inside a drop is a more complicated process than salt concentration relaxation because it involves simultaneous heat exchange inside and outside the drop and is described by the heat balance equation (Equation 3.24), which can be transformed to the following form: (3.38) where (3.39) d dx x Z dZ d ∑ ≅− 2 () . ττ dZ∂∂=− − [] ∫ 0 3 3 1 τ σ π τ() () . σπ τ τ() ; .∂=− = =ςς dZ d 0 xx d x ∑≅−∂ − − ∂ ∂ ∫ (,) exp () τ π τ ς 0 2 1 4 2 x ≅−12τ, ΓΠ Γ Γ Ψ 01112 10(,) ( )() () () () ,ZYττστσττ++ + + + = Ψ() () () exp () / τ σ πβ τ β τ = ∂∂ −∂ − −∂ ∫ d 1 1 32 0 1 4 1 L829/frame/ch03 Page 52 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC The Estimation of Time-Dependent (Relaxation) Processes 53 Two more equations for subsidiary functions σ 1 and σ 2 should be added to Equation 3.38 to find the steady-state temperature Y(τ): (3.40) There are two complications in solving Equations 3.38 through 3.40: (1) non linearity due to the presence of Π s (Y, τ), and (2) time dependence through Σ, which depends on Π s (Y(τ), τ). The first difficulty can be easily overcome because temperature Y is, as a rule, small and Π s can be expanded using the first approximation. Because thermal relaxation is much faster than concentra- tion relaxation, and because drop size change is very small during thermal relaxation, one obtains: (3.41) So, instead of Equation 3.38, one has (3.42) Laplace transformation can be used to solve Equations 3.40 to 3.42; and for the Laplace transforms, one obtains (3.43) (3.44) (3.45) The complex roots of the equation (3.46) can be found for two cases: (1) Γ 0 << Γ 1 and (2) Γ 0 >> Γ 1 . The parameter λ = Γ 0 /Γ 1 in usual variables is (3.47) It is the ratio of two fluxes: condensation heat flux and thermal conductivity flux inside the drop. So, the inequality λ << 1 means that the real heat flux of phase transition heat is much smaller than the potential amount of heat that can be transferred by thermal conductivity in the drop. If λ << 1, there are branch point and two poles with small real parts in Equation 3.46, so the asymptotic formula for the surface temperature is Y dd () () () () () exp () τ σ πβ τ σ πβ τ βτ ττ = ∂∂ −∂ = ∂∂ −∂ − −∂ ∫∫ 2 2 0 1 1 0 1 22 1 1 ΠΠΠ s ZY(,) ()ττ≅− ∞ 1 10 11 0 111 1 22 1 +− () ++ () + () + () = ∞ −− ΓΓΠ ΓΠ Γ ΨY στβ στβ τ σ β β 1 1 1 1 2 12 () () exp ,s sY s s = −− () − σβ 22 2() (),ssYs= Ys ss () () .= ∞ ΠΓ Φ 0 Φ()s = 0 λ= ∞ Π LD q T R R Tk s20 0 0 01 () L829/frame/ch03 Page 53 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC 54 Aerosol Chemical Processes in the Environment (3.48) Y p is steady-state (psychometric) temperature of the drop surface: (3.49) (3.50) It can be shown that a more simple formula (Equation 3.51) is also applicable because it is possible that τ 3 >> τ >> 1. Temperature relaxation in this case is slow and regular enough. The primary reason for this is the slow growth or evaporation of the drop, which is determined by the low value of supersaturation Π ∞ . In the opposite case, when λ >> 1, there are no small poles in Equation 3.46 and thus, (3.51) (3.52) because the exponential term is absent in this case (the steady-state temperature is reached more quickly). In both cases, thermal relaxation inside the drop is slower than that outside because β 2 >> β 1 . To derive the conventional formula for the outside temperature from Equation 3.40, one obtains: (3.53) By substituting Equation 3.53 into Equation 3.16 after some transformation, it is easy to derive the following expression. (3.54) If β 2 τ >> (1 – x) 2 , then a formula similar to that for vapor concentration (Equation 3.18)) follows. (3.55) YY p ( ) expτ τ τ π βτ =−− − 1 2 3 0 2 Φ Y p = + ∞ ΠΓ Γ 0 1 , and τ β 3 1 2 31 = + Γ Γ() . YY p ( ) expτ τ τ =−− 1 3 YY p ()τ π βτ =− 1 2 0 2 Φ στ β π τ τ 2 2 0 2() () .= ∂ −∂ ∂ ∂ ∫ ddY d xY dz e Y x z x z 2 1 2 2 2 2 21 4 2 2 () () τ π τ β βτ =− − − ∞ − ∫ Yx Y x 2 (,) () τ τ = L829/frame/ch03 Page 54 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC The Estimation of Time-Dependent (Relaxation) Processes 55 The steady-state temperature Y p is obtained from Laplace transform (Equation 3.46) when s → 0; but the same formula can be obtained from Equation 3.42 when t → ∞; the last three terms vanish and the first two terms represent the conventional heat balance equation of the drop. From Equation 3.43, it follows that σ 2 (τ) can be connected with dY/dt by an integral equation similar to Equation 3.53, but with a different kernel. This kernel at large τ will be proportional to τ 1/2 , so asymptotically, this will be similar to Ψ(τ) and both these terms will be exponentially small (λ << 1) or as τ –1 when τ → ∞. More troublesome is the derivation of the formula for the temperature gradient inside the drop. An approach similar to that used in deriving Equation 3.27 for the salt concentration can be applied. Expansion of the kernel K 2 (τ, θ, x) in Equation 3.16 leads to: (3.56) (3.57) (3.58) It is natural that the second term that determines the gradient is proportional to temperature conductivity. For large τ, from Equation 3.58, one can obtain (3.59) The temperature gradient inside the drop is positive (negative) for the growing (evaporating) drop, smaller near the drop center, and very rapidly decreases with time. THE RATE OF CHANGE OF DROP SIZE Thermal relaxation can influence the rate of drop growth only through the saturatation vapor density in Equation 3.5 and Equation 3.13, which depend on surface temperature. If vapor supersaturation is small, the deviation of surface temperature from the steady-state value will result in some retardation of the rate of growth during relaxation (T 10 < T ∞ ) and will be greater than at steady- state. The mirror-reflected situation will take place in the case of evaporation. If the environmental temperature varies with time for the period (t ∞ ) much greater than the relaxation time (λ << 1), then the drop temperature will follow it. For the period smaller than the relaxation time, the drop will grow under average environment temperature. In the case λ >> 1, when there is no characteristic time, the latter case corresponds to t ∞ << R 0 –1 D 2 and the former to t ∞ >> R 0 –1 D 2 . Unlike thermal relaxation, deviation in the salt concentration can directly influence drop growth. For a saturated vapor environment and neglecting thermal relaxation, the following formula is derived for small time (usual variables). (3.60) Yx x 111 2 12 (,) ,τ= +ΠΠ Π 11 1 4 1 1 2 4 1 2 () ,τ π τβτ βτ =− () () ∞ − − ∫ ds e Y r s Π 12 1 1 4 2 2 1 1 2 2 3 14 1 2 () .τ β π τβτ βτ =− − () () ∞ − − ∫ ds e s s Y r s Yx Y 1 1 23 1 1 96 (,) () .ττ πβ τ =+ dR dt R t R Rt D =+− 0 0 5 2 0 32 1 1 2 3 ω ω π / L829/frame/ch03 Page 55 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC 56 Aerosol Chemical Processes in the Environment The second term in the brackets describes the growth at the initial salt concentration and does not depend on salt diffusivity. The third term decelerates the growth due to the decrease in salt concentration near the drop surface. The smaller the salt diffusivity, the greater the salt concentration gradient. This formula is valid during the first moments of concentration relaxation (t << t 2 ); during the last moments (t >> t 2 ), we have the equation to determine Z(τ): (3.61) For weak solutions, g << 1, and one obtains for the zero approximation: (3.62) The next iteration gives (3.63) INTENSIVE EVAPORATION OF THE SOLUTION DROP An example of the application of the theory of concentration relaxation is intensive evaporation of the solution drop. Salt concentration enhancement near the drop surface can be large and may result in the formation of a solid crust. The treatment of this problem was considered by Buikov and Sigal. Intensive evaporation takes place under high undersaturation in the environment and blowing of the drop with dry air, so it is possible to assume that the drop radius is a linear function of time; that is, (3.64) Using Equations 3.13 and 3.21, it is possible to obtain the integral equation for the subsidiary function σ(θ) as: (3.65) The salt concentration can be calculated from Equation 3.25. The following formula was derived for salt concentration at the drop surface: (3.66) Crystallization of salt will start when the solution near the surface is saturated (c(R(t s ), t s ) = c s ) and then the solid crust can grow at the drop surface. For the time for the formation of crust of thickness δ the following formula can be applied: Z g Zg 53 1 5 3 15() () .τττ−+ − [] = Zgt 0 15 1() ( ) . / τ≅ + ZZ g Z Z () () () () ττ τ τ ≅− − 0 0 2 0 1 1 Rt R bt() .=− 0 dKZ Z Z a∂∂ ∂ ∂=− − () =− ∫ 0 4 3 2 3 11 τ στ τ τ τ() (,) () () ; () . Σ() () () .τ τ πτ π =+ a ZDa 1 1 2 L829/frame/ch03 Page 56 Monday, January 31, 2000 2:06 PM © 2000 by CRC Press LLC [...]... complicated mathematical problems and complicates the solution In the research work reviewed in this chapter, a new approach is introduced: a heat balance equation on the drop surface that connects thermal processes inside and outside the drop with phase transition heat The application of thermal potentials to solving the system of equations of heat, vapor, and salt diffusion resulted in the integral equations... appropriate for use in numerical methods than the primary system of differential equations The approximate analytical analysis of the processes of drop growth carried out using these integral equations makes it possible to penetrate more deeply into heat and salt transfer inside the drop, as well as to follow the transition from the initial state of the drop to steady-state growth and steady-state fields of... (3. 67) Values of ts and tc calculated for the experimental conditions are given in Tables 3. 1 and 3. 2.11,12 SUMMARY AND CONCLUSIONS The classical formula for the condensation or evaporation rate of a liquid drop derived by Maxwell and modernized by Fuchs1 is based on some hypotheses of complete physical lucidity This formula is widely used in many branches of aerosol science Giving up the hypotheses...L829/frame/ch 03 Page 57 Monday, January 31 , 2000 2:06 PM The Estimation of Time-Dependent (Relaxation) Processes 57 TABLE 3. 1 The Time to Start Crystallization Velocity of Blowing (cm s–1) Experimental Value (s) Calculated Value (s) 40 90 160 0 .39 0 0.155 0.100 0 .35 0 0.145 0.046 TABLE 3. 2 The Time to Form Crust Substance Experimental (s) Calculated (s) 210 235 175 215 Na2SO4 NH4NO3 tc = t s + 2δρs... especially if the drop or environmental temperature varies with time rapidly enough © 2000 by CRC Press LLC L829/frame/ch 03 Page 58 Monday, January 31 , 2000 2:06 PM 58 Aerosol Chemical Processes in the Environment REFERENCES 1 Fuchs, N.A., Evaporation and Droplet Growth in Gaseous Media, Pergamon Press, New York, 1958 2 Mason, B.J., The Physics of Clouds, Clarendon Press, London, 1957 3 Pruppacher,... density Density of salt Kernels of thermal potentials ∑(x, τ) Π ∞ LD2 qs (T∞ ) T0 κ 1 R R0 0 [ ( ) q∞ 1 − Ac0 dqs (T∞ ) q∞ dT − = (C(r, t ) − C0 )C0 1 = ∫ 0 © 2000 by CRC Press LLC ] q∞ (T∞ )(1 − Ac0 ) − q∞ τ ∑1(τ) −1 Z 2 (θ) dθ σ 2 (θ) Z (θ) − 4( τ−θ ) e 2 π ( τ − θ) 3/ 2 L829/frame/ch 03 Page 60 Monday, January 31 , 2000 2:06 PM 60 Aerosol Chemical Processes in the Environment τ ∑2(τ) = ∫ 0 τ... vapor, and salt The asymptotical formulas for salt and temperature gradients are also derived Analytical studies resulted only in some corrections to the conventional formulas, but nevertheless it has been demonstrated that the procedure developed can be useful for intensive processes of drop evaporation It is hoped that the theory developed can find more wide application, namely for intensive growth... conductivity of gas-vapor mixture Phase transition heat Mass of salt in drop Molecular weight of salt (dissolvant) Vapor concentration Density of saturated vapor Radial coordinate Radius of drop Time = R02 D2–1 = R02 D1–1 Temperature of liquid inside drop © 2000 by CRC Press LLC L829/frame/ch 03 Page 59 Monday, January 31 , 2000 2:06 PM The Estimation of Time-Dependent (Relaxation) Processes T2 (r, t)... 4 Buikov, M.V and Dukhin, S.S., Diffusional and heat relaxation of evaporating drop, Eng Phys J., 5 (3) , 1962 (in Russian) 5 Buikov, M.V., Diffusional and heat relaxation of evaporating drop Part 2, Eng Phys J., 5(4), 1962 (in Russian) 6 Buikov, M.V., Time dependent growth of solution drop I Relaxation of concentration, Colloidny J., 24(6), 1962 (in Russian) 7 Buikov, M.V., Time-dependent growth of solution... Colloidny J., 25(1), 19 63 (in Russian) 8 Buikov, M.V., Some Problems of Growth and Evaporation of Drops in Gaseous Media, Thesis of candidate dissertation Kiev (in Russian), 19 63 9 Buikov, M.V and Sigal, V.I., Intensive evaporation of solution drop, Problems of Evaporation, Combustion and Gas Dynamics of Disperse Systems, Editor, V.A Fedoseev, Naukova Dumka Publ House, Kiev (in Russian), 1967 10 Smirnov, . (characteristic) time intervals in the problem. The first one ( t 1 ) is the time interval during which the steady-state field of the vapor is established in the vicinity of the drop. The time interval. L829/frame/ch 03 Page 49 Monday, January 31 , 2000 2:06 PM © 2000 by CRC Press LLC 50 Aerosol Chemical Processes in the Environment (3. 15) (3. 16) (3. 17) (3. 18) (3. 19) These solutions satisfy the equations. expanded into a series on x, keeping the terms ∼ x 3 : (3. 26) Equation 3. 25 is applicable if the following inequality is true: (3. 27) Because (3. 28) and (3. 29) then, instead of