Influence of the developing region of the thermal boundary layer on heat transfer during vapor condensation on horizontal tube bundles

. The purpose of this publication is to describe a phenomenon that is fundamentally important and, at the same time, hardly elucidated in the literature. In the process of film condensation of vapor on the bundles of horizontal tubes, a developing region of the thermal boundary layer is formed on each tube of the bundle; the role of this region in heat transfer is important, and in some cases decisive. The paper presents experimental results on the influence of various contributions of the developing region on heat transfer during film vapor condensation on tube bundles. Based on the data obtained, an algorithm for calculating a condenser during condensation of a stationary vapor without non-condensable impurities is proposed.


Vapor condensation on a bundle of smooth tubes
The process of vapor condensation is widely used in devices of various industries [1].To date, film vapor condensation on bundles of horizontal tubes is the most common and studied type of condensation [2][3][4][5][6][7][8][9][10].However, the issue of the influence of the developing region of the thermal boundary layer remains insufficiently studied.In this paper, the authors touched upon some important points concerning the influence of the developing region on heat transfer during condensation of stationary vapor without noncondensable impurities on tube bundles, using the results obtained earlier by I.I.Gogonin.The schematic diagram of the experiment (see [11]) and distributions of liquid temperatures in space between the tubes are plotted in Fig. 1 (a more detailed diagram is given in [9,11]).It can be seen that the liquid flows onto the underlying tube in the bundle at the saturation temperature.
The separated supercooled drops or jets of liquid warm up very quickly because intense condensation of vapor occurs on their surface.In terms of physics this means that condensation is not possible on the upper part of the cylinder located below, since the vapor temperature is equal to the temperature of liquid.On this part of the tube with length LD, heat is transferred by convection from liquid with temperature t L = t S to a cold wall with temperature t W . Vapor condensation begins at some distance from the point where liquid falls on the tube. .The value of angle  at which the thermal boundary layer reaches the film thickness can be calculated from the solution of the problem proposed by Rogers et al. [10]:  According to the solution [10], heat transfer in the developing region is determined by dependence: which uses the Nusselt number, plotted from viscous- , where It follows from relation (2) that heat transfer depends very weakly on the Reynolds number and there is a noticeable influence of the cylinder diameter through the Ga criterion.
The control series of experiments were carried out with condensation of freon R12 [11].In one series, the variable parameter was the diameter of tubes, where condensation took place, and in the other, it was the velocity of the free fall of condensate flowing onto the experimental tube (which was achieved by changing the ratio of the intertubular distance to the tube diameter S/D).Data on heat transfer on the bundles of tubes with a diameter varied from 3 to 45 mm are presented in Fig. 3 [11].It is obvious that heat transfer on the bundles of tubes with D = 3 and 6 mm is noticeably more intense than that on the tubes of a larger diameter.The increase in the heat transfer intensity is explained by the influence of capillary waves generated by the detachment of condensate drops, see more detailed comment in [11].
In addition, the influence of the developing region of the thermal boundary layer on small diameter tubes, where the initial section length is comparable to the tube half-perimeter, increases significantly.This is clearly seen if the results are presented in coordinates (Fig. 4): If the length of the developing region exceeds the tube half-perimeter, then only convective heat transfer occurs there, determined by the dependence: In the general case, heat transfer on the i-th tube of the bundle is determined by formula: here Nu C is the Nusselt number at condensation for the given film flow, see [9].The experimental Nusselt numbers are compared in Fig. 5 with those calculated by (4) depending on the Galileo criterion.The data given for small diameter tubes (D=3 and 6 mm).It can be seen that on tubes of a small diameter, convective heat transfer is the determining.The result obtained is fundamentally important, since it allows more accurate calculation of film condensation in industrial devices.
Data shown in Fig. 6 demonstrate that doubling of the free fall velocity has almost no effect on heat transfer during condensation on tube bundles of different diameters.

Vapor condensation on a bundle of finned tubes
The flow of liquid irrigating the bundle of finned tubes is presented in Fig. 7.
The forces of surface tension exceed significantly the forces of gravity and determine the regime of bundle irrigation: a) There is capillary retention of liquid between the fins from a minimum level to complete flooding of the valley, depending on parameter  a : According to our the optimal data of  a is within  0.6 0.8 a   .b) The arrows in Fig. 7 show the trajectory of liquid movement along the fin.Such a movement was observed when the tube bundle was irrigated with ethyl alcohol, and the emerging microbubbles illustrated the movement of liquid, which irrigated the fin.
с) The main heat-releasing surface of finned tubes is the fin surface, and it is in no way related to the irrigation density.The main amount of liquid flows down along the valley between the fins, as it is shown in Fig. 7.
Experiments on heat transfer confirm the above.The effect of irrigation density on heat transfer on a finned tube bundle (calculated per the total surface of finned tubes) is shown in Fig. 8.It can be seen that the Nusselt number for the first and last tubes of the bundle is almost unchanged.There is a threefold intensification of heat transfer in comparison with a bundle of smooth tubes.
It is necessary to distinguish two Re numbers when irrigating a bundle of finned tubes.One of them is plotted from the total irrigation density (Re * ), the other determines hydrodynamics of the film flowing down the fin surface (Re h ).
The heat transfer intensity on bundles of finned and smooth tubes, first published in [14] are compared in Fig. 9.It can be seen that at  0.6 0.8 a   , the conditional heat transfer coefficient increases by an order of magnitude.Here, the experiments on tubes with a finned surface factor of 3.6 are presented.
The total increase in the heat transfer coefficient consists of the following components: 1) An increase in the heat transfer surface due to finning.
2) A decrease in the characteristic linear size ( ).For the finned tube, the characteristic size is the fin height.
3) Capillary forces form hydrodynamics of the film shown in Fig. 7.The main area of the fin turns out to be unflooded and is the main part of the finned tube, involved most intensively in heat transfer.

Features of steam condensation on a tube bundle
It is generally accepted that when steam condenses, tube finning is not effective.However, our experiments have shown [14] that even with a small finned surface factor, the heat transfer intensity increases noticeably.
The results of heat transfer measurements during steam condensation on smooth (points 1) and finned tubes (points 2-3) are shown in Fig. 10.The experiments were carried out on cunial tubes (λ ≈ 20 W/(m•K)), the finned surface factor k for tubes No. 2, 3 and 4 was, respectively: 1.36, 1.57 and 1.63.
It can be seen that the total heat removal on the tube of bundle No. 4 is almost two times higher than at the same Re number on the bundle of smooth tubes.
The effect of irrigation density is very weak.

Fig. 2 .
Fig. 2. Scheme of liquid flow in a bundle of horizontal tubes.

Fig. 4 .
Fig. 4. Relative change in heat transfer during condensation on single cylinders: 1 -Re=5; and tube bundles: 2 -Re = 150[11].In the case of condensation on a bundle, the influence of the tube diameter is noticeably stronger than in the case of condensation on single tubes.

Fig. 7 .
Fig. 7. Scheme of liquid flow over the bundle of finned tubes: D1 -outer diameter of the tube including the fin height; Douter diameter of the tube; h -fin height; H -height of capillary liquid retention;  -film thickness in a valley; 1incoming condensate; 2 -direction of liquid flow over the lateral surface of a fin; 3 -permanent liquid layer between the fins.

Fig. 8 .
Fig. 8.The effect of irrigation density on heat transfer during condensation on bundles of smooth and finned (calculated per the total surface) tubes of various geometries: 1 -calculation by the Nusselt formula; 2 -bundle of smooth tubes; 3,4 -bundle of finned tubes,  9.2 a  ; 5,6 -bundle of

Fig. 9 .
Fig. 9. Comparison of heat transfer intensity during vapor condensation on the bundles of finned and smooth tubes [14]: R12, Re = 400.Here 2 is heat transfer coefficient on the bundle of finned tubes,  is heat transfer coefficient on the bundle of smooth tubes.