Heat transfer during condensation of water vapour in the presence of non-condensable gas in vertical tube of small diameter

Case Studies in Thermal Engineering 40 (2022) 102519 Available online 29 October 2022 2214-157X/© 2022 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Heat transfer during condensation of water vapour in the presence of non-condensable gas in vertical tube of small diameter Filip Toman * , Petr Kracík, Jiˇ rí Pospíˇ sil Faculty of Mechanical Engineering, Brno University of Technology, Technick´ a 2896/2, 61669, Brno, Czech Republic ARTICLE INFO Keywords: Condensation Heat transfer Water vapour Non-condensable gas Vertical tube Experiment ABSTRACT The paper deals with a detailed analysis of published analytical relations for heat transfer during steam condensation in a vertical tube. The historical development of the approaches and the chronological development of the used analytical relations are presented. Attention is focused on the description of the processes at the gas phase-condensate film interface as well as on the inclusion of the influence of the presence of non-condensable gases. The analytical relations in 12 modifications are compared with each other for the same geometrical configuration of the vertical tube and identical boundary conditions. The authors performed comparative experimental measurements of the heat transfer coefficient during condensation of the vapour-gas mixture in vertical tubes of three diameters, namely 16, 20 and 26 mm. Based on a comparison of the experimental results and analytical relations, recommendations are made on the validity interval of the tested relations. Nomenclature c p specific heat capacity, J•kg −1 •K −1 d tube diameter, m D diffusion coefficient, m 2 •s −1 g gravity acceleration, m•s −2 h Enthalpy, J•kg −1 Δh c latent heat of condensation, J•kg −1 k condensation thermal conductivity, W•m −1 •K −1 l length of the tube, m L characteristic length, m ˙ M Mass flow rate, kg•s −1 Nu Nusselt number p absolute pressure, Pa Pr Prandtl number ˙ Q heat transferred, W R gas constant - 8314 J K −1 mol −1 Re Reynolds number, * Corresponding author. E-mail address: [email protected] (F. Toman). Contents lists available at ScienceDirect Case Studies in Thermal Engineering journal homepage: www.elsevier.com/locate/csite https://doi.org/10.1016/j.csite.2022.102519 Received 29 June 2022; Received in revised form 29 September 2022; Accepted 22 October 2022 Case Studies in Thermal Engineering 40 (2022) 102519 2 Sc Schmidt number, Sh Sherwood number, t temperature, ◦C T temperature, K u speed, m•s −1 U tot overall heat transfer coefficient, W•m −1 •K −1 v diffusion volume, y volume concentration, Greek letters α heat transfer coefficient, W•m −2 •K −1 β Mass transfer coefficient, m•s −1 Γ circumferential mass flow, kg•m −1 •s −1 δ condensate film thickness, m η dynamic viscosity, Pa•s λ thermal conductivity, W•m −1 •K −1 ν kinematic viscosity, m 2 •s −1 ρ density, kg•m −3 Τ shear stress, Pa Subscripts b mixture core cond condensation f film of codensate g gas in input part inner inside the pipe lam laminar mix mixture of air and vapour out output part outer outside the pipe tr turbulent v vapour w cooling water 1. Introduction Vapour condensation is a physical process used in many practical applications such as the cooling systems of heat circuits, air conditioning, nuclear power plants, and other devices for heat exchange [1]. Condensation can be defined as removing heat energy from a substance during the phase change from vapour to liquid. The removed heat energy is called latent heat. The condensation process is very intensive at the interface of the vapour-liquid phases. The width of this interface ranges from 1 to 2 diameters of the molecules [2]. The first detailed description of condensation was presented by Wilhelm Nusselt [3] in 1916, when he explained the principle of laminar film condensation. Nusselt’s description was later modified by Sparrow [4], who took into account the effect of a change in the momentum of the condensate on a vertical surface in the gravitational field. Gradually, the character of the condensate film flow started to be distinguished (laminar, turbulent), and the influence of the shear stress of the gaseous phase at the interface with the liquid, which causes the deformation of the film and the undulation of its surface, was taken into consideration. This was mathematically described, for example, by Aktershev, who describes in [5] the effect of flowing gas on liquid film in a parallel flow and counter flow arrangement. In article [6], using the finite element method, he describes the formation of natural waves and identifies the areas in which the film starts to be unstable. A description of water vapour condensation in a mixture with inert gas was formulated in 1956, when Knacke proposed a mathematical description of the condensation process at the gas-liquid interface in [7], known as the kinetic theory of condensation. Based on this theory, condensation is composed of two opposing phenomena, pure condensation and pure evaporation. The theory says that molecules pass from vapour to liquid through the gas-liquid interface, where heat and mass are transferred. As the phase changes from gaseous to liquid, water vapour dramatically reduces its volume, which causes a significant decrease in pressure at the gas-liquid interface. This results in the re-evaporation of part of the already condensed molecules. If non-condensable gases are present in the water vapour, they accumulate, due to the low local pressure, near this gas-liquid interface [2]. As a result of historical development, the different approaches to understanding film condensation evolved into various theories, one of which is the diffusion layer method used in this paper. The diffusion layer method is one of the most detailed theoretical descriptions for the calculation of film condensation in the F. Toman et al. Case Studies in Thermal Engineering 40 (2022) 102519 3 presence of inert non-condensing gases (NCG). The method was proposed by Colburn and Hougen in [8] and describes the process of vapour mass transfer in the presence of NCG due to diffusion. The diffusion layer, which forms a resistance to heat and mass transfer, is formed due to the accumulation of NCG at the gas-liquid interface during the condensation of water vapour in the steam-gas mixture. Fig. 1 shows that, due to an increasing concentration of non-condensable gases, concentration of vapour decreasing and partial pressure of water vapour (its saturation temperature) in the diffusion layer decreases as a result. Heat transfer from the core of the gasvapour mixture to the wall of the tube can be distinguished as heat transfer by condensation, transfer of the sensible heat of the mixture, and heat transfer through the condensate film. The transfer of latent heat is evaluated using the stagnant film model combined with a heat and mass transfer analogy. Since the Colburn and Hougen method requires the convergence of two unknown variables (temperature at the phase interface and the molar ratio of NCG), their form of calculation is complicated for practical use. During further development of this method, Peterson et al. [10] introduced a new physical quantity called condensation thermal conductivity, which is a parallel to the thermal conductivity coefficient and quantifies the transfer of latent heat through the diffusion layer. The condensation coefficient of heat transfer is then calculated based on the same formal relation as the coefficient of sensible heat transfer, only instead of the Nusselt number, the Sherwood number is used and instead of the characteristic dimension, the diffusion coefficient is used [11]. Peterson’s model was made more precise by including the already mentioned influence of the character of the film flow (undulation) and the suction effect. During condensation, the suction effect causes pressure differences in the gas-liquid boarder layer and, due to these differences, the normal component of the velocity of the gas-vapour mixture is not zero in relation to the surface of the condensate. This leads to the disturbance of the film surface and to its undulation and thus to the disturbance of this boarder layer. This phenomenon is called the suction effect, which, in general, increases heat transfer [12]. The suction effect primarily depends on the velocity of the flow and intensity of condensation. Its effect is expressed by the so-called blowing parameter [13]. Currently, there are more versions of condensation thermal conductivity based on the original Peterson’s model, which resulted in the publication of several variants of the Sherwood number. There are also multiple approaches to describe the condensation of water vapour in a non-condensable gas. In general, these theories can be divided into semi-theoretical and theoretical. In the current literature [14], probably the most elaborate theory of the diffuse layer is compared. Published experiments demonstrate that non-condensable gases in water vapour affect not only condensation heat transfer, but also heat transfer through the condensate film. The practical use of steam-gas mixture condensation dedicated to the literature [15], in which the condensation of vapors from biofuel gas was investigated. In addition to the non-condensablegas and the flow rate, the temperature of the mixture and the temperature of the cooling inlet medium are essential for condensation. Current studies in the field of heat transfer deal with the addition of nanoparticles to liquids to increase heat transfer [16]. It is clear from the above mentioned that the condensation of a gas-vapour mixture in a vertical tube became the focus of many authors.. The literature research revealed a large number of relations and calculation methods published in the past period. Many publications use different names for the individual heat transfers, which significantly affects the mutual comparison of the relations. For the reasons mentioned above, the authors of this paper focused on comparing the published analytical relations and comparing their results with the results of extensive measurements performed on a vertical tube. The experiments helped to identify the correspondence between the measurements and the calculation relations in the individual ranges of the parameters tested. The contribution of this article is in the comparison of available theoretical models for the condensation of water vapour in the presence of noncondensable gases in a vertical tube with a large amount of experimental data. The result is a suitable combination of analytical relations for a given capacitor geometry. The results can be regarded as valuable recommendations for the use of the relations in technical practice. Fig. 1. Illustration of the diffusion layer and the course of the temperatures, concentrations, and velocities inside a vertical tube [9]. F. Toman et al. Case Studies in Thermal Engineering 40 (2022) 102519 4 2. Analytical model The following chapter offers an analytical description of water vapour condensation in a mixture with air in a vertical tube. The total power of water vapour condensation in a gas-vapour mixture can be divided into three parts. The most important part is the release of latent heat during the phase change. Other processes include transfer of the sensible heat of the mixture and cooling of the condensate film. By analogy, we can express the overall inner heat transfer coefficient during heat transfer from the gas-vapour mixture core to the tube wall using three separate heat transfer coefficients based on Fig. 2 as follows [17]. α inner =1 1 α f+1 ( α cond+ α g) .(1) Here, α inner is the overall inner heat transfer coefficient, α f is the coefficient of heat transfer through the condensate film, α cond is the condensation heat transfer coefficient, and α g is the sensible heat transfer coefficient. The mathematical model uses empirical relations to describe the behaviour of the liquid film and the heat flow transfer. The solution involves several consecutive steps and an iterative procedure is followed. The following chapters include key equations that were used for developing the analytical calculation model, which produced the results presented in the next chapters. The relations were obtained through an extensive literature research. For some parameters, there are more variants of the relations published by different authors. In such a case, all the variants of the relations mentioned were tested in a parametric study whose results are presented in this paper. The sequence of the relations presented enables a detailed calculation of heat transfers inside a vertical tube. 2.1. Heat transfer in condensate film The condensate film exerts considerable resistance to heat transfer. To correctly determine heat transfer between a liquid and a wall, we must know the character of the film flow and determine its velocity. Considering the fact that the amount of condensate at the inlet to the vertical tube is zero, the relations for determining the film properties are related to the mean value between input and output of section. To determine the character of the film flow and the shear stress at the gas-liquid interface, we must know the velocity of the gas-vapour mixture, which can be obtained from the continuity equation as follows [18]. ug,out =4•˙ Mg ρ g• π •(din −2•δ)2.(2) The original Nusselt’s derivation of the thickness of the liquid film is a function of gravitational acceleration, which, however, did not include the influence of the shear stress between the gas and the liquid. Since the effect of the gas flow on the liquid film is considerable, Nusselt’s relation had to be extended with the effect of shear stress. For the given case, a relation that was derived using numerical simulations by Lee and Son [19] was used τ mix =3• ν mix •umix, din −2•δ.(3) To calculate the Nusselt number for turbulent flow, dimensionless shear stress must be determined as follows [20]. τ ∗ mix = τ mix g• ρ f•(1− ρ mix ρ f)•Γ .(4) Fig. 2. Diagram of heat transfer by the diffusion layer method F. Toman et al. Case Studies in Thermal Engineering 40 (2022) 102519 5 The film thickness can be obtained from the equation taking into account the circumferential flow, film thickness, and the influence of shear stress on the film by Rohsenow [21]. Γ=g• ρ f•( ρ f− ρ mix)•δ3 3• η f+ ρ f• τ mix•δ2 2• η f .(5) Blangetti’s model regards film flow as a combination of turbulent and laminar flow. Both types are then included under a single Nusselt number for heat transfer inside the film according to the ratio given by equation (6) [18]. Nuf=(Nu4 f,lam +Nu4 f,tr)0,25 ,(6) where the Nusselt number for laminar flow is based on equation (7) and the Nusselt number for turbulent flow is determined by equation (8) [12]. Nuf,lam =Lf δf ,(7) Nuf,tr =0,008663 •Re0,382 f•Pr0,5689 f•(1+e τ ∗ g0,541 ).(8) Where Lf is characteristic dimension for heat transfer in the film according to the literature [22]. Lf= υ 2 f g √.(9) The Reynolds number for the condensate film relating the inertial forces inside the film to its viscosity is determined as follows [23], Ref=4•˙ Mf π •din • μ f .(10) The heat transfer coefficient is defined by equation (11), α f=Nuf•λf Lf .(11) 2.2. Heat transfer in a gas-vapour mixture The relations for forced convection in a tube for a gaseous medium were used for the transfer of sensible heat from the gas-vapour mixture to the condensate film. The main criterion for determining the flow regime is the Reynolds number given by Equation (12). Reg=ug•din υ g .(12) With small diameter tubes, the film thickness must be considered and the free inner cross section of the tube must be reduced by this thickness. With larger diameter tubes, the influence of the film thickness on the free cross section is negligible. In the experiments, the values of the Reynolds number were in a turbulent flow regime. Therefore, the criterion equation for the Nusselt number for forced convection inside a circular tube in the turbulent flow regime according to the Dittus-Boelter equation was used as follows [24]. Nug=0,023 •Re0,8 g•Pr0,4 g.(13) From the Nusselt number follows the heat transfer coefficient of the sensible heat according to equation (14) α g=Nug•λg din (14) To calculate the condensation heat transfer coefficient, relations published in the last 20 years were used. As already mentioned, when calculating the latent heat transfer, Peterson introduced the so-called condensation thermal conductivity, which includes the effect of atomic diffusion through the diffusion coefficient and the differences in the concentrations of non-condensable gas inside the mixture flow and on the surface of the condensation film as follows [20]. kcond =Δh2 c•pmix •Mv•Mg•D R2•T3 mix,in •−ln(yg,b,in yg,f,in ) ln(1−yg,b,in 1−yg,f,in ).(15) This equation was further modified by Liao and Vierow, who proposed the following relation [20]. kcond =Δhc• ρ mix •D tmix,b−tmix,f•ln (1−yv,f,in 1−yv,g,in).(16) F. Toman et al. Case Studies in Thermal Engineering 40 (2022) 102519 6 The determination of the condensation heat transfer coefficient is based on the same empirical relations as the determination of sensible heat, only the Sherwood number is used instead of the Nusselt number. Below are three equations used for determining the Sherwood number by different authors. Sherwood number by Fr¨ ossling [25]. Sh =2+0,552 •Re0,5 mix •Sc1 3,(17) Sherwood number by Vdi Atlas [22]. Sh =0,023 •Re0,83 mix •Sc1 3,(18) Sherwood number by Kageyam [26]. Sh =0,021 •Re0,8 mix •Sc0,5 .(19) To correctly determine the diffusion properties of the gas-vapour mixture, the diffusion coefficient (sometimes also called the binary diffusion coefficient) must be calculated. It quantifies the willingness of the atoms of two gases to diffuse into each other. Below are two equations that were used in the mathematical model. The first one is a relation based on equation (20) according to VDI atlas [22]. D= 0,0143 •T1,75 mix (1 Mv+1 Mair)1 2 pmix  2 √(v 1 3 v+v 1 3 air)2,(20) where M signifies the molar mass of each component and v is their diffusion volume. The other relation is defined by equation (21) according to Maheshwari [12], who says that the diffusivity of two gases is given primarily by their temperature and pressure D=8,96038 •10−4•T1,5 mix pmix .(21) The condensation heat transfer coefficient is then defined by a general equation as α cond =Sh •kcond din .(22) 3. The experiment and the solution method 3.1. Experimental device The analytical procedures quantifying heat transfer during vapour condensation in a vertical tube were verified by experiments on Fig. 3. Diagram and real status of the testing apparatus F. Toman et al. Case Studies in Thermal Engineering 40 (2022) 102519 7 vertical tube-in-tube heat exchangers. Fig. 3 shows the basic diagram of the testing apparatus. The gas-vapour mixture goes through the inner tube and the heat is removed by cooling water running in the annular space of the exchanger shell. Both the coaxial tubes are made of copper. The experimental apparatus can change the direction of the cooling water flow and thus change the configuration of the flow of the condenser tube cooling from parallel flow to counter flow. The experimental section, where the water vapour condensation in a gas-vapour mixture is controlled, is 1 m long. The source of the water vapour is a steam generator with the maximum amount of steam generated being 35 kg/h. The power of the steam generator is adjustable with a step of 0.1 kg of steam per hour. The saturated vapour runs into the mixing chamber, where the vapour with a flow rate of ( ˙ Mv) can be mixed with any gas with a flow rate of (˙ Mg). The mixture then runs into a superheater, where it is heated to the required temperature. In the experiments, dry air was used as a non-condensable gas and the mixture was heated to a saturation temperature corresponding to its pressure and air concentration. At the inlet to the exchanger, the temperature (tmix,in) and pressure (pmix,in) of the mixture are measured, and, at the outlet of the exchanger, the output temperature (tmix,out) and pressure (pmix,out) of the mixture are measured. A cooling loop is used to maintain the required temperature of the cooling water at the inlet to the exchanger wall (tw,in) with a required mass flow rate ( ˙ Mw). At the outlet of the exchanger, the temperature (tw,out) is measured. Heat is removed from the cooling loop to the central cooling circuit using a plate exchanger. The temperatures in the key parts of the system are measured in duplicate. Specifically, type T thermocouples (coated, ungrounded) and type PT100 resistance probes were used. The pressures in the key parts of the system are measured by high precision pressure sensors (PXM) and the flow rate of the cooling water is measured by a FLOMAG induction flowmeter. The air parameters before the superheater were measured with an anemometric flow meter VA520, while the flow was regulated by a Bronkhorst mass regulator. The experimental apparatus was verified by measuring pure steam condensation in the literature [27]. 3.2. Experiment setting In this study, heat transfer during the condensation of water vapour in a non-condensing gas is measured for different inner diameters of pipes. Specifically, 16 mm, 20 mm and 26 mm. For the given diameters, the flow rates of the saturated water vapour and air before mixing were set so that the mass concentration of air in the mixture and the gas-vapour mixture velocity at the inlet to the heat exchanger would be achieved as given in Table 1. For each combination of air concentration and the gas-vapour mixture velocity at the Table 1 Measured combinations of the volume concentration of air in the water vapour and the gas-vapour mixture velocity. Velocity of the gas-vapour mixture at the inlet to the test section [m•s −1 ] 8.9 13.4 17.7 22.3 26.6 31.3 40 50 Non-condensable gases [% vol ] Inner diameter of the tube 16 mm 0 0 0 0 1.8 1.7 1.7 1.8 3.6 3.6 3.6 3.7 5.5 5.4 5.6 7.7 7.3 7.3 7.7 5.7 9.5 9.4 9.7 9.8 15.0 15.0 15.2 15.4 21.1 20.9 21.5 21.6 34.4 34.5 35.4 49.3 50.4 64.2 65.4 20 mm 0 0 0 0 0 0 1.8 1.8 1.8 1.7 1.7 1.8 1.8 3.6 4.5 3.6 3.5 3.6 3.7 3.7 5.4 5.4 5.4 5.4 5.6 5.7 5.9 7.4 7.4 7.4 7.5 7.5 7.7 8.0 9.4 9.5 9.6 9.5 9.6 9.8 10.2 15.1 15.1 15.1 15.1 15.1 15.6 16.0 20.9 21.0 21.0 21.1 21.3 21.7 22.4 34.1 34.6 34.4 34.6 34.8 35.5 37.1 48.8 48.9 49.2 49.1 49.9 51.0 52.9 63.3 63.6 64.1 64.5 26 mm 0 0 0 0 0 0 1.7 1.7 1.7 1.7 1.8 1.8 3.6 3.5 3.6 3.6 3.8 3.9 5.4 5.4 5.6 5.6 5.7 5.8 7.4 7.4 7.5 7.6 7.8 7.9 9.5 9.6 9.5 9.7 9.7 10.1 15.0 15.0 15.1 15.2 15.5 15.9 21.0 21.1 21.2 21.6 21.9 22.3 34.5 34.6 34.8 35.5 36.2 36.9 49.0 49.1 49.7 50.7 51.4 52.1 63.4 64.6 65.5 59.0 F. Toman et al. Case Studies in Thermal Engineering 40 (2022) 102519 8 inlet to the tube, two heat removal configurations were measured. The first configuration was parallel flow cooling, when the cooling water runs in the same direction as the vapour and the condensate. The other configuration is counter flow cooling when the cooling water runs in the opposite direction to the vapour and the condensate. The mass flow rate of the cooling water was adjusted so that the cooling intensity would be identical for all the options tested. All the conditions were measured after stabilization of all measured parameters for at least 10 min in order to minimize the effect of accumulation and other dynamic phenomena. 3.3. Determining the condensation heat transfer coefficient To determine the condensation heat transfer coefficient, we must know transferred power in the heat exchanger during condensation. Because the heat exchanger is insulated during the experiment, heat losses to the surroundings are not considered. The heat balance and heat power of the condensing process is therefore determined from the heating of the cooling water. The power of the cooling water is a function of the flow rate, pressure, and the input and output temperature of the cooling water as follows [28]. ˙ Qw=˙ Mw•(hw,out −hw,in ).(23) The overall heat transfer coefficient. It depends both on the intensity of external cooling and on the material and thickness of the tube wall. The overall heat transfer coefficient related to 1 m of the length of the tube is given by the equation [29]. Utot = ˙ Qw l•Δtln,tot .(24) ˙ Qw is the heat transferred to the water and l is length of the tube. The overall heat transfer coefficient includes all the individual heat transfers from the gas-vapour mixture flow to the water. If all the individual heat transfer coefficients inside the tube are combined into one based on equation (1), this coefficient can be expressed by equation (25), which includes the transfer of heat through the tube wall to the cooling water [27]. α inner =1 din • π •(1 Utot −1 dout• π • α w−ln dout din 2• π •λcu).(25) To calculate the condensation heat transfer coefficient by equation (1), we must determine the heat transfer coefficient in the film and the sensible heat transfer coefficient. To assess the character of the condensate film flow and of the gas-vapour mixture flow, we must know the flow rate of the mixture both at the inlet and outlet of the experimental section. In the inlet section, it is a simple sum of the mass flow rates of the constituent gases of the mixture ˙ Mmix,in =˙ Mv,in +˙ Mg.(26) To determine the output flow rate, we must know either the amount of the separated condensate in the exchanger or the concentration of vapour at the outlet of the exchanger. Since the two quantities are related to each other, it is an iterative process in which the variable is the amount of the condensate formed. The output amount of the gas-vapour mixture can be calculated, knowing the concentrations at the outlet, as follows ˙ Mmix,out = ˙ Mg 1 yv,b,out −1.(27) Here, yv,b,out is the concentration of water vapour in the core of the mixture at the outlet of the exchanger. The basis for a correct determination of the output amount of the condensate and the gas-vapour mixture is an energy balance between the heat given out by the gas-vapour mixture and the heat absorbed by the cooling water. The heat flow inside the tube is divided into three constituent parts [22], ˙ Qin =˙ Mf,out •Δhc+˙ Mmix,in •cp,g•(tmix,in −tmix,out)+˙ Mf,out •cp,f•(tmix,out −tf,out)(28) As stated above, the calculation of each power is an iterative process where the variable is the amount of the condensate formed and the resulting air concentrations and outputs, leading to the determination of the condensation heat transfer coefficient by equation (1). The calculation procedure involves the following steps. Step 1 Entering the input values. Step 2 In the first step of the iteration, the amount of condensate formed must be estimated. The next iteration steps calculate the amount of condensate Step 3 Determining the physical properties of the cooling liquid and the gas-vapour mixture. The physical properties are determined for mean temperature. Step 4 Determining the input volume concentration of water vapour by equation (29) and determining the volume of air in the mixture at the inlet to the exchanger. F. Toman et al. Case Studies in Thermal Engineering 40 (2022) 102519 9 yv,b,in = ˙ Mv,in ρ v(tv,in) ˙ Mv,in ρ v(tv,in)+ ˙ Mg ρ g(tg,in) .(29) Step 5 Power balance and recalculating the amount of the condensate formed. Step 6 Determining the water vapour concentration and the flow rate of the mixture at the outlet by equations (26) and (29). Step 7 Determining the heat transfer coefficient on the heat removal side and using the relations for the forced convection of the sensible heat transfer coefficient inside the tube. Fig. 4. Flowchart of the presented mathematical model F. Toman et al.