Journal of Engineering Physics and Thermophysics, Vol. 86, No. 3, May, 2013
HEAT TRANSFER IN BOILING OF BINARY MIXTURES UNDER FREE-CONVECTION CONDITIONS I.I. Gogonin
UDC 536.423.4.535.5
A study has been made of the heat transfer in boiling of binary mixtures of refrigerant–refrigerant-type homogeneous liquids completely soluble in each other under free-convection conditions with allowance for the effects of diffusion and evaporation of the highly volatile component from a mixture. These effects are responsible for the complex dependence of the heat transfer on the concentration of the mixture at a constant heat flux. A criterial dependence satisfactorily describing the results of experimental investigations has been proposed. Keywords: boiling heat transfer, binary mixture, diffusion process, evaporation of a highly volatile component. Introduction. There are at least two reasons for the interest expressed in studying the heat transfer in boiling of binary mixtures. First, a complex dependence of the heat-transfer coefficient of the mixture on its concentration at a constant heat flux has not been described thus far even in criterial form. Second, the international agreement signed by the Government of the Russian Federation, among other states, has banned a number of ozone-unfriendly refrigerants. This brought about the problem of changing to ozone-safe refrigerants without substantial structural alterations of the available refrigerating equipment [1]. Recent years have seen the effort to replace the banned refrigerants R11, R12, and others by ozone-safe binary mixtures. A number of articles have appeared dealing with the boiling heat transfer of various refrigerant pairs under the conditions of variation in the parameters of the process (heat fluxes and pressure of the mixture and its concentration) within wide limits [2–8]. The characteristic dependence of the heat-transfer coefficient of an R22–R142b mixture on the concentration of its low-boiling component, obtained from experiments at ts = const and q = const, is shown in Fig. 1. Figure 2 gives the dependence of the difference of the equilibrium concentrations of the vapor and the liquid in this mixture on the concentration of its low-boiling component. The researchers note that there are two interrelated reasons for the sharp decrease followed by the growth in the heat-transfer coefficients of mixtures of refrigerants as a function of their concentrations. First, in boiling of the mixture, we have the evaporation of the low-boiling component from the mixture’s superheated wall layer (see Fig. 2), which causes the local pressure to change and the difference between the wall temperature and the mixture’s saturation temperature to grow. Second, the change in the concentration of the superheated wall layer of the mixture along the layer’s height results in diffusion on the liquid–vapor boundary. The indicated processes cause the heat-transfer intensity to decrease substantially in boiling of the mixture as demonstrated by the experimental results in Fig. 1. The minimum heat transfer in boiling of the mixture corresponds to the maximum difference in the concentrations of the vapor and the liquid. An effort was repeatedly made to describe the heat transfer in boiling of such mixtures [9, 10]. However, it is prematurely to consider this problem as being finally solved. The present work seeks to determine the basic parameters in boiling of binary mixtures that lead to a complex dependence of the heat transfer on the change in the concentration of a mixture. Also, this work seeks to show the decisive role of diffusion and of the change in the temperature head in the evaporation of the low-boiling component in the heat transfer in boiling of mixtures. Heat Transfer in Boiling of Single-Component Liquids. It follows from an analysis of the available experimental data that boiling heat transfer is a problem with conjugate boundary conditions [11, 12]. Such heat transfer is dependent not only on the thermophysical properties of the heat-transfer agent, but also on the properties of the heat-transfer wall and its roughness. In [13, 14], we have proposed a criterial dependence satisfactorily describing the results of experimental S. S. Kutateladze Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, 1 Akad. Lavrentiev Ave., Novosibirsk, 630090, Russia; email:
[email protected]. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 86, No. 3, pp. 646–651, May–June, 2013. Original article submitted November 30, 2011. 0062-0125/13/8603-0689 ©2013 Springer Science+Business Media New York
689
Fig. 1. Heat-transfer coefficient α of the R22–R142b mixture vs. concentration of its lowboiling component xlow c in boiling on a copper cylinder at D = 17.3 mm, Rz = 3.2 μm, ts = –15oC, and q = 15·103 W/m3 [7]. Fig. 2. Difference of the equilibrium concentrations Δx of the vapor and the liquid in the R22–R142b mixture vs. concentration of its low-boiling component xlow c at ts = –10oC [7].
investigations on the boiling of various liquids cooling walls from materials with different thermophysical properties and roughness
⎛ λC ρ liq ⎞ Nu *0 = 0.01 Re*0.8 Pr1/ 3 bK t0.4 Rz0.2 ⎜ ⎟ ⎝ λ w C wρ w ⎠
where
λC ρ liq λ w Cw ρ w
Pr1/ 3 bK t0.4 Rz0.2
where B = Pr b const) and Kt =
,
(1)
is the dimensionless group. Dependence (1) can be represented, for convenience, as
Nu*0 (λC ρ liq λ wC wρ w ) 0.2
1/3
−0.2
K t0.4 Rz0.2
(r ρ v ) 2
lσ
C PTs ρ liq σ
⎛ λC ρ liq ⎞ ⎜ ⎟ ⎝ λ w C wρ w ⎠
−0.2
⎡ and b = ⎢1 + 10 ⎢ ⎣
≡
Nu *0 = f (Re* ) , B
⎛ ρv ⎞ ⎜⎜ ⎟⎟ ⎝ ρ liq − ρ v ⎠
(2)
2/3⎤
⎥ are dimensionless variables (B = const at ts = ⎥ ⎦
is the thermal-similarity number.
As was shown in [13], dependence (1) should be considered as the limiting relation for description of data on the heat transfer in boiling of liquids on thick-walled surfaces when δ w hav > 1.0 . Rough evaluations enable us to state that this condition was observed unambiguously in the experiments of [2–8]. Figure 3 shows the processing of experimental data [2–8] obtained in boiling of R11, R12, R22, R113, R114, R134a, and R124a refrigerants on thick-walled tubes in the coordinates of (2). The experiments were conducted as the heat fluxes and pressures varied within wide limits in cooling copper cylinders of different diameters and roughnesses, and also platinum wire. This figure gives results of processing of data on the boiling of oxygen [15], acetone, n-hexane, carbon tetrachloride [16], n-butane [17], ethanol [18], and water [18, 19] at different pressures and heat fluxes on surfaces manufactured from various materials of different roughnesses. It is seen that the experimental data presented in the (2) coordinates are described satisfactorily by a unified dependence. This enables us to use dependence (1) as the scale for determination of the relative Nusselt number in the boiling of binary mixtures. Analysis of Experimental Data on the Boiling of Binary Mixtures. It must be emphasized that in the analyzed works experiments were conducted under different conditions. Thus, the condition P/Pcr = const was fulfilled in [3, 8]. In [3],
690
Fig. 3. Generalization of experimental data on the heat transfer in boiling of single-component liquids under free-convection conditions on copper (1), platinum (2), stainless steel (3), and nickel (4).
the value of Pcr was determined from the approximation expression obtained from experimental data. In [8], the value of Pcr for the mixture was determined according to the additivity rule. In [2, 4], experiments were conducted on the condition that P = const, whereas in [7], at ts = const. This means that it is the pressure and the temperature that were variable with variation in the concentration of the equilibrium mixture. The temperature of such a mixture was variable in [2, 4]; in [7], it is the pressure that varied with the concentration of the mixture. Different in the experiments were the geometric parameters of experimental portions, their surface finish, and the material from which they were manufactured. When experimental data are presented in dimensionless coordinates it is of fundamental importance to correctly determine the physical properties of a mixture and its concentration and diffusion coefficients. In the present work, we have calculated the physical and transport properties of the mixtures according to the International Program [20], whereas the diffusion coefficients, from the dependences recommended in [21]. It can be noted that theoretically determined properties of the mixtures of R22/R113, R22/R114, R22/R11, and R22/R124a refrigerants are in satisfactory agreement with the available experimental data for these mixtures. There are no experimental data on the physical properties of other mixtures. The diffusion coefficients of the mixtures under study were computed from the dependence [21]
0.0036 Dab20 =
(
1 1 + Ma Mb
3 1/ 3 ab ηb ϑ1/ a + ϑb
)
2
.
(3)
Here Ma and Mb are the molecular weights of the components a and b of the mixture, ηb is the coefficient of dynamic viscosity of the solvent of the mixture’s component b in centipoise, and ϑa and ϑb are the molar volumes of the components a and b. Dependence (3) enables us to compute the diffusion coefficient at a temperature of 20oC. The influence of temperature on the process in question can be determined from the relation
(
)
Dab = Dab20 ⎡1 + b1 t − 20D C ⎤ . ⎣ ⎦
(4)
In accordance with the recommendations of [21], it may be assumed that, for mixtures of the same type, the coefficients are a = b = 1. The coefficient b1 in dependence (4) is defined as
b1 =
0.2 ηb20 3ρ
.
(5)
b 20
For different pairs of refrigerants, the diffusion coefficient ranged within (1 ≤ Dab ≤ 5.5)·10–9.
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Fig. 4. Generalization of experimental data on the heat transfer in boiling of binary mixtures of refrigerants under free-convection conditions: 1) R134a–R113 [4], 2) R22–R113 [4]; 3) R22–R113 [4]; 4) R12–R113 [4]; 5) R11–R113 [5] 6) R142b–R11 [2]; 7) R22– R142b [7]; 8) R22–R114 [3]; 9) R22–R124a [8].
Gorenflo et al. ([3, 6]) described results of experimental investigations on the heat transfer in boiling of binary mixtures with the dependences proposed in [22, 23]. It was proposed in [22] that the temperature head ΔT = Tw – Ts in the boiling of a binary mixture at q = const is always larger than that in the boiling of a single-component mixture. For the temperature head, we propose the dependence 0.5 ⎡ ⎛ a ⎞ C P ⎛ dT ⎞⎤ ΔT = ( x1ΔT1 + x2 ΔT2 ) ⎢1 − ( y − x) ⎜ ⎟ ⎜ ⎟⎥ . r ⎝ dx ⎠⎥⎦ ⎝D⎠ ⎢⎣
(6)
Here ΔT1 and ΔT2 are the temperature heads in the boiling of the single-component mixtures from which a binary mixture has been prepared, x1 and x2 are the molar concentrations of each component of the mixture’s liquid phase, (y – x) is the difference dT in concentration of the low-boiling component in the vapor and liquid phases, and is the derivative with a positive sign. dx In [24], it was proposed that the relative heat transfer in the boiling of a binary mixture be determined from the dependence
α ΔT0 ΔT0 = = = N 7 /5 , α0 ΔT ΔT0 + ΔTeq
(7)
−1
0.5 ⎡ ⎛ a ⎞ C P ⎛ dT ⎞⎤ where N = ⎢1 − ( y − x) ⎜ ⎟ ⎜ ⎟⎥ , ΔT0 = x1ΔT1 + x2ΔT2, and ΔTeq is the difference of the equilibrium temperar ⎝ dx ⎠⎦⎥ ⎝D⎠ ⎣⎢ tures in the vapor and liquid phases at constant pressure. However, as was shown in [3], dependence (7) unsatisfactorily describes the experimental results in the boiling of mixtures of refrigerants. The dependence proposed in [23] is agreement with the experimental results of [3] only on selecting diffusion and mass-transfer coefficients. In accordance with [22], the variation in the relative coefficient of heat transfer in the boiling of binary mixtures is determined by the relation
Nu *e Nu *0
=
⎛ ⎞ αe 1 = f ⎜ ⎟⎟ 2 ⎜ α0 ⎝ Lu Ku ⎠
and depends on two additional similarity numbers: Lewis number Lu = D/a and Kutateladze number Ku =
692
(8) r . C P Δt
In generalizing the experiments conducted with variation in the pressure within wide limits, we must take into ρv . Then the relative account the change in the vapor density and supplement dependence (8) with the parameter ρ = ρ liq − ρ v coefficient of heat transfer in boiling of the mixtures will be defined as
Nu *e Nu *0
=
⎛ ρ2 ⎞ αe = f ⎜ ≡ f (M ) , ⎜ Lu Ku 2 ⎟⎟ α0 ⎝ ⎠
(9)
ΔT 2 λC P ρ 2v ρ 2v where M = = is the dimensionless number. Dr 2 ρliq (ρ liq − ρ v ) 2 Lu Ku 2 The concentrations of the liquid and the vapor in a binary mixture are assumed to be prescribed when the similarity numbers are computed from (1). The binary mixture is considered as a certain ideal liquid whose properties change only with concentration. The mechanism of heat transfer in boiling of the binary mixture is assumed to be the same as that in the boiling of homogeneous liquids. Boiling heat transfer is a complicated process and is described by numerous similarity numbers whose exponents are selected empirically in processing experimental data. The processing of experimental results in the relative coordinates of (9) enables one to substantially reduce the list of the number and to stress the role of diffusion and of the change in the temperature head during the evaporation of the low-boiling component. The Kutateladze number is computed from the temperature head determined experimentally in the boiling of the mixture. In the present work, we have processed experimental data on the boiling of binary refrigerant mixtures on cylinders manufactured from various materials and having different diameters at different heat fluxes and concentrations of the mixtures in a wide pressure range. Figure 4 gives results of processing of the experimental data of [2–8] on the boiling of binary mixtures of refrigerants in the (9) coordinates. It is seen that the experimental results are generalized by a unified dependence with a scatter of ±25%. The relative heat-transfer coefficient decreases three times as the determining parameter (diagnostic variable) M changes by four orders of magnitude. The data presented in Fig. 4 can be described by the criterial dependence
Nu *e Nu *0
= 0.17 M
−0.12
or
Nu *e
=
0.17Nu *0
⎛ ρ2 ⎞ ⎜⎜ ⎟ 2⎟ ⎝ Lu Ku ⎠
−0.12
.
(10)
From Fig. 4, it is clear that the experimental data obtained at P/Pcr = 0.8 [3] depart from the main body of other data at 10–3 ≤ M ≤ 10–2. It should be borne in mind that the error in determining the physical properties of the mixtures is the greatest as the critical pressure is approached. This regularity is observed when heat transfer during the boiling is measured. An error of ±20% is extremely difficult to overcome at P/Pcr = 0.8. In this case experimental data may depart from the theoretical dependence. Conclusions. We have proposed the criterial dependence which enables us to describe, with a satisfactory error, experimental results on the boiling of binary mixtures under different conditions, to reduce the list of numbers determining the heat transfer in boiling of the mixtures, and to reveal the role of diffusion processes and of the change in the temperature head in the evaporation of the low-boiling component from the mixtures.
NOTATION a, thermal diffusivity of the liquid, m2/s; CP, specific heat of the liquid at constant pressure, J/(kg·deg); Cw, heat capacity of the wall, J/(kg·deg); D, diffusion coefficient, m2/s; g, free-fall acceleration, m2/s; hav, chilling depth, m; lσ = σ g (ρ liq − ρ v ), capillary constant of the liquid, m; Nu* = σlσ , Nusselt number; n, density of steam-generation sites, m–2; λ qlσ ν Pr = , Prandtl number; P and Pcr, running and critical pressures, N/m2; q, specific heat flux, W/m2; Re* = , Reynolds r ρvν a number; R0, separating radius of a bubble, m; Rz, height of the asperities of the cooled wall, μm; r, latent heat of vaporization, Rz J/kg; Rz = 2 1/ 3 , dimensionless roughness; Ts and ts, saturation temperatures, K and oC; ΔT, temperature head, K; (ν g ) xlow c, concentration of the low-boiling component of the mixture; α0 and αe, coefficients of heat transfer of an ideal liquid and a mixture in boiling, W/(m2·deg); δw, thickness of the heat-transfer wall, m; δw/ hav, dimensionless chilling depth; λ and 693
λw, thermal conductivities of the liquid and the wall, W/(m·deg); ν, kinetic viscosity of the liquid, m2/s; (ν2/g)1/3, viscous gravitational constant, m; ρliq, ρv, and ρw, densities of the liquid, the vapor, and the wall, kg/m3; σ, surface tension of the liquid, N/m. Subscripts: liq, liquid; v, vapor; w, wall; *, number constructed from the capillary constant; e, experimental; av, average; s, saturation; eq, equilibrium; cr, critical.
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