Chemical and Petroleum Engineering, Vol. 42, Nos. 1–2, 2006

CRYOGENIC ENGINEERING, PRODUCTION AND USE OF INDUSTRIAL GASES. REFRIGERATION EQUIPMENT FREEZING OF DROPS OF SOLUTIONS IN CRYOGENIC GRANULATORS

M. B. Generalov and N. S. Trutnev

The following steps are examined in the freezing of drops in a liquid coolant: initial, crystallization of the solution, and formation of granules. A working diagram and system of equations describing the crystallization of a liquid drop of solution are presented. Investigations conducted on the freezing of saline solutions made it possible to determine the following kinetic parameters: the time and average cooling rate of drops to the crystallization temperature, as well as the time and crystallization rate of different sizes of drops of the solution, and the geometric dimensions and process parameters of the freezing plant.

The cryochemical method is widely used to produce nanomaterials from organic and inorganic salts. Freezing of solutions of material-forming components is a basic process of the cryochemical technology of solid-phase materials. Here, it is necessary to ensure the highest solidification rate of both the solvent, and also the dissolved substances, and also to retain the high chemical homogeneity of the solid phase, which is inherent to the initial solution. To produce ultra-disperse materials in accordance with cryochemical technology, the initial solution is atomized by some means or other in a container filled with a cryogenic liquid (for example, liquid nitrogen), where vigorous freezing of drops of the solution (DS) occurs. When DS of a certain radius strike the surface of the cryogenic liquid, a violent boiling process is initiated, and a vapor interlayer of the coolant liquid is formed along the surface wetted by the DS. Under the action of this vapor layer, granules are rotated about their own axis, and are chaotically displaced over the surface of the cryogenic liquid. In a number of cases, the DS become submerged in the cryogenic liquid on contact. Thus, the freezing of DS situated on the surface of a coolant with a boiling point appreciably lower than the crystallization temperature of the solution is rather complex. Owing to vigorous boiling of the liquid, which is caused by a high temperature differential ∆T = Td – Tl (Td and Tl are the temperatures, respectively, of the DS and cryogenic liquid), a vapor interlayer with a thickness h is formed between the surface of a sphere and the liquid (Fig. 1). The thermal flux from the warmer sphere is drawn away through the low-heatconducting layer of liquid vapor. According to experimental investigations, the following two regimes of coolant-vapor flow are possible: laminar and turbulent. Numerical assessment indicates [2] that for liquid nitrogen, a turbulent regime will set in when the radius of a drop Rd > 8 mm. Since appreciably smaller sizes of DS are used for the cryochemical method of porous-granule synthesis, it is possible to assume the presence of a laminar regime for the movement of vapor in the boundary film.

MGUIÉ, Moscow. Translated from Khimicheskoe i Neftegazovoe Mashinostroenie, No. 1, pp. 13–16, January, 2006.

0009-2355/06/0102-0023 ©2006 Springer Science+Business Media, Inc.

23

Fig. 1. Diagram showing freezing of drop in liquid nitrogen.

The density ρd of the DS is greater than the density ρl of the coolant; a drop is restrained from submergence by the excess vapor pressure that develops owing to the fact that the gaseous coolant has nonzero viscosity. To develop a sufficiently high ascentional force, the thickness h of the interlayer should be small (h < Rdθ, where θ is the angle of immersion, see Fig. 1). Theoretical investigations of the freezing kinetics of spherical DS in cryogenic liquids [3] have made it possible to establish that a granule with a radius Rd will float on the surface of the liquid under the following conditions: T f > Tl + Lv gρ vρl Rd3

(β − 1) 4 ; 18µ vλ vβ

 z θ = 2 arcsin   1 + cz 

 β  3  

1/ 4 

,  

where ρ β= d ; ρl

5b − 4 − 4 c z= ; 6b − 4 − 3c

β b=2 ; 27

 α c=   3

1/ 4

;

α=

18µ vλ v (T f − Tl ) Lv gρ vρl Rd3

;

Tf is the surface temperature of the granule; Lv is the enthalpy of coolant vaporization; ρv is the density of the coolant vapor; µv and λv are, respectively, the coefficient of dynamic viscosity and thermal conductivity of the coolant vapor; and g is the acceleration of free fall. If the vapor interlayer of coolant is uniformly distributed over the wetted surface, it is possible calculate its thickness from the equation [3] 2 ε 6 β / α − 3ε 4 (1 + ε ) + α (1 + ε )2 = 0 , where ε = h/Rd. As experimental investigations have shown [4], floating of DS with a size of 2.5 < 2Rd < 5.2 mm occurs when θ ≈ π. Figure 2 shows the variation in the coefficient of convective heat transfer α from the surface of the DS into the volume of the cooling liquid for these conditions as a function of drop diameter (D = 2Rd). It should be pointed out that the influence exerted by the angle of immersion θ of the DS on convective heat transfer is rather weak; variation in θ between the limiting values π/2 and π effects a change of no more than 40% in the coefficient α. Also, no significant influence exerted on the coefficient α by the parameter β, which is equal to the ratio of the densities of the solution of drops and coolant, is observed within the limits 1 < β < 2. Examination of Freezing Steps of Drop in Liquid Coolant. Theoretical solutions of the problem of the solidification of a liquid sphere in an infinite medium of coolants under various boundary conditions, which can be modeled by the single-component Stefan problem, have come into widespread use in procedures employed for various production processes. High-temperature differentials and crystallization rates of the solution are distinguishing characteristic features of cryocooling; this may lead to variation in the boundary conditions on the surface of phase transition, and the Stefan problem will become nonclassical. Nonuniform distribution of the temperature at which the crystallization step of the DS begins, and also 24

α, W/(m2·K) 2 200 3 1 100

2

3

4

5 D, mm

Fig. 2. Dependence of coefficient of convective heat transfer α on drop diameter D according to various authors: 1) [2]; 2) [4]; 3) [3].

the initial temperature distribution during final cooling of granules that have already solidified will exert a significant influence on the problem’s solution. Solution of the problem of the freezing of a liquid sphere with consideration of the effect of factors critical to the cryochemical methods employed for synthesis includes investigation of some of the steps of the process: the initial step – primary cooling of the liquid DS to time t1, when the temperature of its surface will reach the crystallization temperature; crystallization of the solution and formation of a solid granule – strictly, the phase transition from the initial temperature distribution at the end of the first phase and to time t2 of the termination of crystallization; and the closing phase – final cooling of the granule that has already solidified. Initial Cooling Step. The variation in temperature T along the current radius r (0 ≤ r ≤ R) as a function of time t (when 0 ≤ t ≤ t1) can be written in the following manner in conformity with the familiar solution of the problem of the cooling of a sphere of radius R with a constant initial temperature T0 over the entire volume (t = 0) in a medium with a constant temperature Tl and constant coefficient α on its surface [5]:  r 2(sin µn − µn cos µn ) sin  µn  T −T  R T* = 0 = 1− exp[ −µ2n Fo], T0 − Tl r n =1 (µn − sin µn cos µn ) µn R ∞

∑

(1)

where Fo = adt /R2 and ad is the thermal diffusivity of the solution of drops. In Eq. (1), the constant µ is a root of the characteristic equation, and will take on the form tanµ = –µ/(Bi – 1)

(2)

for a sphere, where Bi = αR/λd is the Biot number and λd is the thermal conductivity of the drop. Equation (2) is transcendental, and has innumerable roots for a given Bi value. The values of six roots of are calculated and tabulated in [5] for different Bi values. Since the value of µn in Eq. (1) will depend only on the Bi number, the equation of the temperature field can be represented as T * = F(r/R, Bi, Fo), (3) and then for the surface of the sphere (r = R), T *(R) = F1(Bi, Fo),

(4) 25

TABLE 1 T*(R) function is in numerator, and T*(0) function in denominator Bi number 0.05

0.10

0.15

0.20

0.25

0.30

0.04

0.01/0.00

0.02/0.00

0.03/0.01

0.04/0.02

0.05/0.03

0.06/0.04

0.10

0.04/0.00

0.05/0.01

0.07/0.02

0.08/0.03

0.01/0.05

0.02/0.06

0.40

0.12/0.01

0.18/0.03

0.22/0.06

0.25/0.10

0.30/0.15

0.34/0.20

1.0

0.26/0.02

0.36/0.06

0.44/0.15

0.50/0.25

0.57/0.34

0.62/0.40

4.0

0.62/0.03

0.75/0.15

0.82/0.33

0.87/0.50

0.90/0.60

0.93/0.07

and for the center of the sphere (r = 0), T *(0) = F2(Bi, Fo).

(5)

Values of the functions T *(R) and T *(0), which were calculated from expressions (4) and (5), respectively, for different Bi and Fo numbers, are presented graphically in [5], and are arrayed in Table 1. When t = t1, the temperature on the surface of a sphere T(R, t1) = Ts(c) is the temperature of initial crystallization along the liquidus line of a binary solution with a concentration c of the target product; in the case of the eutectic concentration of the binary solution, Ts = Te; then, T *(R, t1) = (T0 – Ts)/(T0 – Tl).

(6)

From the T *(R, t1) value determined from expression (6), and the Bi number from Table 1, or the plots presented in [5], it is possible to determine the value of Fo{T *(R), Bi} as a function of T *(R, t1) and Bi. As a result, t1 =

Fo{T * ( R, t1), Bi}R 2 . ad

(7)

The temperature Ti of the solution at the center of the sphere at time t = t1 can be found from the expression T *(0, t1) = = (T0 – Ti)/(T0 – Tl): Ti = T0 – T *(0, t1)(T0 – Tl). (8) The value of T *(0, t1) is found from the Bi and Fo(t1) = R2t1 /ad numbers from Table 1, or from the plots in [5]. The theoretical derivations can be illustrated in examples. Example 1. Let us determine the cooling time of a drop of water with a radius R =1⋅10–3 m. The temperatures of the drop are: initial T0 = 293 K, crystallization under normal conditions Ts = 273 K; and, the temperature of the liquid nitrogen Tl = 77 K. The thermophysical parameters of the drop are: thermal conductivity λd = 0.65 J/(m⋅sec⋅K), density ρd = 1000 kg/m3, heat capacity cd = 4230 J/(kg⋅K); and coefficient of convective heat transfer α = 100 J/(m2⋅sec⋅K). The thermal diffusivity of the water ad = λd /(cd ρd) = 0.65/(4230⋅103) = 1.54⋅10–7 m2/sec, and the coefficient of convective heat transfer from the surface of the drop to the coolant α = 160 J/(m2⋅sec⋅K). For Bi = 140⋅10–3/0.65 = 0.21 and T *(R) = (293 – 273)/(273 – 77) = 0.12, Fo ≅ 0.20 from Table 1. The cooling time of the drop in the initial period (prior to the start of movement of the crystallization front into the depths of the drop) t1 = Fo R2 /ad = 0.20⋅10–6/1.54⋅10–7 ≅ 1.29 sec. The average cooling rate wavg = (T0 – Ts)/t1 = (293 – 273)/1.29 = 15.7 K/sec.

26

Fig. 3. Working diagram of crystallization process: 1) solid phase; 2) liquid phase; ξ(t)) radius of solidification front; BPI) boundary of phase interface; SS) surface of sphere; CS) center of sphere.

Example 2. For the conditions of example 1, let us determine the temperature at the center of a drop of water with a radius R = 1⋅10–3 m at the end of preliminary cooling at time t = t1. For R = 1⋅10–3 m and t1 = 0.97 sec, Fo(t1) = 0.20 and Bi = 0.21. From Table 1, T *(0, t1) ≅ 0.03; based on formula (8), Ti = 293 – 0.03⋅(293 – 77) ≅ 287 K. Thus, the ratio Ts /Ti = 273/287 = 0.95, i.e., the gradient between the temperatures at the center and on the surface of the drop at the moment when it begins to crystallize is approximately 5%. Crystallization of Solution and Formation of Solid Granule. The second step of the solution of the problem of DS freezing is associated with determination of the position of the phase-transition front during crystallization of the solution, and with analysis of the temperature distribution in both the solid and liquid phases with the boundary condition of heat exchange on the surface of the sphere and the initial condition obtained as a result of solution of the problem of primary DS cooling. According to the working diagram (Fig. 3), the system of equations describing the crystallization of the liquid DS can be represented in the form of a heat-conduction equation: • for the solid phase  ∂2 T (r , t ) ∂T (r , t )  ∂T1(r , t ) 1 , = a1 + 1  ∂r 2 ∂t ∂r  

ξ(t ) < r < R , t1 < t < t2 ;

(9)

0 < r < ξ(t ) , t1 < t < t2 ,

(10)

• and for the liquid phase  ∂2 T (r , t ) ∂T (r , t )  ∂T2 (r , t ) 2 , = a2  + 2  ∂r 2 ∂t ∂r  

where a1 = λ1 /(c1ρ1) and a2 = λ2 /(c2ρ2) are the thermal diffusivities, λ1 and λ2 are the thermal conductivities, c1 and c2 are the heat capacities, ρ1 and ρ2 are the densities, and the subscripts 1 and 2 denote the solid and liquid phases, respectively. We then have: • on the boundary of the phase interface of the binary solution forming the eutectic λ1

∂T1 ∂r

r = ξ (t )

− λ2

∂T2 ∂r

r = ξ (t )

T1(ξ, t) = T2(ξ, t) = Te,

= ρ1qcr

dξ ; dt

t1 < t < t2;

(11)

(12) 27

• at the center of the sphere ∂T2 ∂t

= 0, r =0

t1 < t < t2 ;

(13)

• and on the surface of the sphere − λ1

∂T1 ∂r

r=R

= α (Ts − Tl ) ,

t1 < t < t2 ,

(14)

where qcr is the latent heat of crystallization and α is the coefficient of convective heat transfer on the interface between the coolant and solid surface. The initial conditions are: T2(t1, r) = Ti = const,

0 ≤ r ≤ R;

ξ(t1) = R;

ξ(t2) = 0.

(15)

The method of approximate solution consists in the fact that the functions T1(r, t) and T2(r, t) are selected so that they satisfy the differential equations of heat conduction, and the initial and boundary conditions. These functions are then substituted in the conjugation condition, and the differential equation obtained is solved for ξ(t). Based on quasi-stationary solution of Eq. (1) and using expressions (12) and (14), the temperature distribution in the solid layer is defined by the relationship T1(t , r ) = Te −

α (Te − Tl )ξR 2  1 1  − , αR( R − ξ) + λ1ξ  ξ r 

ξ(t ) < r < R , t1 < t < t2 .

(16)

Using expressions (12), (13), and (16), a particular solution of Eq. (10) will normally take on the form   6λ l r2  T2 (r , t ) = Te + (Ti − Te )  1 −  exp  −  2  ρ c ξ    l 1

 dt  , 2 ξ 0  t

∫

0 < r < ξ , t1 < t < t2 ,

(17)

where cl and λl are the heat capacity and thermal conductivity of the cryogenic liquid. From simultaneous solution of Eqs. (11), (16), and (17), Makeev [3] obtains a nonlinear integro-differential equation for determination of the law governing the motion of the solidification front ξ = ξ(t – t1), the simplified solution of which for determination of the duration of complete freezing of a spherical drop ∆t = t2 – t1 under conditions whereby solutions crystallize in cryogenic liquids can be represented as tcr = t2 − t1 =

ρ1qcr R 2 Bi1 + 2 , 6λ1(Te − Tl ) Bi1

(18)

where Bi1 = αcrR/λ1. When Bi1 → ∞, heat exchange with the sphere will take place under conditions of the internal problem (boundary conditions of the first kind), when the rate of external heat transfer is infinitely high as compared with that of internal transfer; in that case, the surface temperature of the sphere will be equal to the temperature of the cooling medium: Tf = Tl. For this hypothetical case, Eq. (8) will assume the form tcr =

28

ρ1qcr R 2 . 6λ1(Te − Tl )

(19)

TABLE 2 R·104, m

Bi

Fo

t1

tcr = t2 – t1

wavg

1.0

0.018

1.20

0.1623

0.3745

0.2670

1.5

0.027

1.06

0.2191

0.5620

0.2669

2.0

0.036

0.90

0.2597

0.7520

0.2659

3.0

0.054

0.70

0.3506

1.1290

0.2657

6.0

0.108

0.25

0.7013

2.2980

0.2610

10.0

0.180

0.15

1.2987

3.8453

0.2600

Equation (19) was obtained by S. S. Kovner from analysis of freezing and thawing (the equation is derived in [5]). When Bi1 → 0, heat exchange with the sphere will take place under conditions of the external problem, when the effect of the Bi number is diminished, and the temperature along the radius of the sphere is distributed uniformly (gradientfree heat transfer). Since the Bi number is always less than unity, but greater than zero (0 < Bi1 < 1) for conditions under which solutions are frozen in cryogenic liquids, Eq. (18) should be used to calculate the solidification time of DS. Let us address an example of the calculation of the freezing time of a spherical drop with a radius R = 1⋅10–3 m using this equation. The initial data are: crystallization temperature Te = 273 K, and the temperature of the liquid nitrogen Tl = 77 K. The thermophysical parameters of the ice are: thermal conductivity λ1 = 2.33 J/(m⋅sec⋅K); density ρ1 = 920 kg/m3; and latent heat of crystallization qcr = 3.34⋅105 J/kg. The coefficient of convective heat transfer α = 140 J/(m2⋅sec⋅K). The number Bi1 = = αR/λ1 = 140⋅10–3/2.33 = 0.06. From formula (18), tcr = [920⋅3.34⋅105⋅10–6/6⋅(273 – 77)⋅2.33]⋅[(2 + 0.06)/0.06] ≅ 3.85 sec. The average crystallization rate of spherical drops with a radius R in cryogenic liquids can be estimated from the formula wavg = R/tcr = R/(t2 – t1), (20) where the time of preliminary cooling t1 of a drop and the time ∆t of complete solidification are determined from formulas (13) and (18), respectively. Table 2 presents computed results of the time t1 of preliminary cooling, the duration of the crystallization process tcr = t2 – t1, and the average crystallization rate wavg from formulas (13), (18), and (20), respectively, as a function of the radius R of spherical drops of water for an initial temperature T0 = 293 K, crystallization temperature Te = 273 K, and temperature of liquid nitrogen Tl = 77 K. Computed results for the average crystallization rate of a drop of water in liquid nitrogen indicated (see Table 2) that the rate wavg remains essentially constant and is 0.26–0.27 mm/sec during the freezing of a drop with a radius R = 0.1–1.0 mm. The cooling rate or tempo of drops of solutions of cryogenic liquids is frequently estimated from the equation vcool = (T0 – Te)/t1.

(21)

In conformity with formula (21) and the data in Table 2, the cooling rate of drops of water in liquid nitrogen from an initial temperature T0 = 293 K to the crystallization temperature Te = 273 K varies from 125 K/sec for a drop with a diameter of 0.2 mm to 15 K/sec for a drop with a diameter of 2.0 mm. The investigations that we conducted on the freezing of saline solutions make it possible to determine the following kinetic parameters: the time and average cooling rate of a drop to the crystallization temperature, as well as the time and rate of crystallization of different sizes of drops of solution. The results obtained enable us to determine the geometric dimensions and production parameters of the freezing plant on the basis of known technical data on the dispersing plant (output, and size of the drops being dispersed). 29

REFERENCES 1. 2. 3.

4. 5.

30

S. M. Brazhnikov, M. B. Generalov, and N. S. Trutnev, “Vacuum-sublimation methods of producing ultra-disperse powders and inorganic salts,” Khim. Neft. Mashinostr., No. 12, 12–15 (2004). A. V. Klimenko and A. G. Sinitsyn, “Experimental study of film boiling on the surfaces of freely floating spherical particles,” Moscow Institute of Power Engineering, No. 91, Moscow (1986), pp. 128–137. A. A. Makeev, “Dispersion and freezing of solutions in plants producing inorganic materials via cryochemical synthesis,” Author’s Abstract of Dissertation for Candidate of Technical Sciences, Moscow Institute of Chemical Machine Building, Moscow (1983). A. V. Klimenko and A. G. Sinitsyn, “Heat transfer during film boiling on the surface of freely floating spheres,” Moscow Institute of Power Engineering, No. 161, Moscow (1988), pp. 78–87. A. V. Lykov, Theory of Heat Conduction [in Russian], Vysshaya Shkola, Moscow (1967).