Heat and Mass Transfer https://doi.org/10.1007/s00231-017-2250-x

ORIGINAL

Condensation of binary mixtures on horizontal tubes 1 ¨ A. Buchner

· A. Reif1 · S. Rehfeldt1 · H. Klein1

Received: 11 May 2016 / Accepted: 3 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017

Abstract The two most common models to describe the condensation of binary mixtures are the equilibrium model by Silver (Trans Inst Chem Eng 25:30–42, 1947) and the film model by Colburn and Drew (Transactions of the American Institute of Chemical Engineers 33:197–215, 1937), which is stated by Webb et al. (Int J Heat Mass Transf 39:3147–3156, 1996) as more accurate. The film model describes the outer heat transfer coefficient by subdividing it into two separate resistances against the heat transfer. The resistance of the liquid condensate film on the tube can be calculated with equations for the condensation of pure substances for the analogous flow pattern and geometry using the property data of the mixture. The resistance in the gas phase can be described by a thermodynamic parameter Z and the single phase heat transfer coefficient αG . In this work measurements for condensation of the binary mixtures n-pentane/iso-octane and iso-propanol/water on horizontal tubes for free convection are carried out. The obtained results are compared with the film model by Colburn and Drew (Transactions of the American Institute of Chemical Engineers 33:197–215, 1937). The comparison shows a rather big deviation between the theoretical model and the experimental results. To improve the prediction quality an own model based on dimensionless numbers is proposed, which describes the experimental results of this work significantly better than the film model.

List of symbols Latin symbols A area Bo Bond number C constant in Eq. 12 c molar concentration molar heat capacity C¯ p cp specific heat capacity Cn condensation number

mol m3 J kmol·K J kg·K

 A. B¨uchner

[email protected] A. Reif [email protected] S. Rehfeldt [email protected] H. Klein [email protected] 1

m2 − −

Institute of Plant and Process Technology, Department of Mechanical Engineering, Technical University of Munich, Boltzmannstr. 15, Munich, Germany

−

d D12 g h H¯ V hV Ja k L Le M˙ m ˙ N˙ n˙ Nu p Pr q˙ ˙ Q Rth Ra Re

diameter diffusion coefficient gravity constant fin height molar enthalpy of evaporation specific enthalpy of evaporation Jakob number overall heat transfer coefficient length Lewis number mass flow mass flux molar flow molar flux Nusselt number pressure Prandtl number heat flux heat flow thermal resistance Rayleigh number Reynolds number

m

m2 s m s2

mm J kmole J kg

−

W m2 ·K

m −

kg s kg m2 ·s kmol s kmol m2 ·s

− bar −

W m2

W m2 ·K W

− −

Heat Mass Transfer

film Reynolds number fin spacing fin thickness temperature logarithmic temperature difference flow velocity mole fraction of component i in liquid phase mole fraction of component i in yi gaseous phase Z thermodynamic parameter, ratio of heat fluxes q˙q˙G z spatial coordinate Greek symbols α heat transfer coefficient βth thermal expansion coefficient βG gas-side mass transfer coefficient  difference ζ Ackermann correction factor η dynamic viscosity λ thermal conductivity ξ friction factor π pi

density σ surface tension φ parameter in Ackermann correction High indices ∗ in equilibrium • corrected with Ackermann factor Low indices 1 component 1, lighter boiling component 2 component 2, higher boiling component b boiling Bulk bulk phase c condensation char characteristic cond condensate CW cooling water diff diffusive G gas phase in in, inwards inside inside L liquid phase out out, outwards outside outside Ph at phase boundary root fin root s saturation t tube tip fin tip total total W wall ReF s t T Tlog u xi

− mm mm ◦ C, K K m s

Abbreviations CS carbon steel SS stainless steel

1 Introduction

− − − m W m2 ·K 1 K m s

− −

kg m·s W m·K

− −

kg m3 N m

−

In a chemical plant condensation is one of the major tasks to deal with. Despite the fact that, in such a process, the stream to condensate is, in most cases, a mixture and might even carry some for the condensation inert gases, most experimental data on condensation in literature are for pure substances [4–9]. The condensation of pure components is thus, at least for smooth tubes, widely understood and the design of a technical condenser for a pure substance can be done properly. For mixtures this is different. The experimental work on binary mixtures is rather scarce and most of the time it is dealt with either a pure substance and inert gas [10–13] or with refrigerant mixtures not relevant to the process industry [14–16]. A broad comparison of experimental results with models has not been carried out yet. The two most popular theoretical models are the equilibrium model based on Silver [1] and the film model based on Colburn & Drew [2]. These two models have been compared by Webb et al. [3] concluding that the film model leads to better results. The description of this model and the physics behind the condensation of binary mixtures can be found in various text books and publications [17–19]. However, an actual evaluation of the model with experimental data of a binary mixture with two condensing components on the outside of a horizontal tube is still missing. This paper provides data of such measurements and compares the experimental results with the film model by Colburn & Drew to help evaluate the model.

2 Experimental set-up The experimental apparatus used for the condensation measurements is based on the apparatus described by Reif et al. [9] and B¨uchner et al. [20] used for pure component condensation measurements. Figure 1 shows the flow diagram of the apparatus. The apparatus consists of two separate loops. The cooling water is temperated by a thermostat (C) and pumped through the cooling water loop. A coriolis flow meter (1) records the mass and the volume flow of the cooling water before it enters at position (2) the tube which is built into the condenser (A). At the point of inlet (2) and at the outlet (3) the temperature of the cooling water is measured by Pt-1000

Heat Mass Transfer

E

D

P

T

P

T

T

2 1

3

A

L L F P T

B

C

A B C D E

Condenser Evaporator Thermostat Cold trap Vacuum pump

1 2 3

Coriolis flow meter Cooling water inlet Cooling water outlet

Fig. 1 Flow diagram of experimental apparatus

resistance thermometers. After leaving the tube the cooling water is sent back into the thermostat. The second loop is the measuring fluid loop. The liquid is heated in the evaporator (B). The gas phase then flows into the condenser where it condenses at the cold surface of the tube. To guarantee an equal composition in the gas phase over the whole length of the condenser, the gas enters the condenser at five inlets which are equally distributed over the length. The condensate trickles off the tube, is collected at the bottom of the condenser and flows back into the evaporator by gravity force. To verify the equal composition inside the condenser, three gas samples are drawn and analysed by gas chromatography to measure the composition of the gas phase. The measured values have an uncertainty of ± 0.5% in composition. This value is then used to calculate the temperature TBulk in the gas phase, where saturation is assumed. Additionally the pressure in the condenser is measured and the heating power of the evaporator is regulated by a set pressure of p = 1.013 bar in the condenser. Two different types of tubes are used in the condenser for the condensation measurements, smooth tubes and low-finned tubes called GEWA-K30 with rectangular fins. The tube materials are carbon-steel (CS, 1.0345) and stainless-steel (SS, 1.4404) and the tubes were all provided

by Wieland-Werke AG, Germany. The dimensions of the used tubes are listed in Table 1. The outer diameter doutside for the low-finned tubes is the diameter dtip at the fin tip. Two different mixtures were examined. One mixture is the very wide-boiling mixture of n-pentane and iso-octane. The other mixture is the mixture of iso-propanol and water. The (T , x, y)-diagrams of these two mixtures are shown in Fig. 2. The data for these diagrams as well as the properties of the mixtures were provided by the Linde AG, Germany.

Table 1 Dimensions of the used tubes

Outer diameter doutside in m Inner diameter dinside in m Tube length Lt in m Thermal conductivity λt in W/m · K Fin height h in mm Fin thickness t in mm Fin spacing s in mm

Low-finned tubes

Smooth tubes

CS

SS

CS

SS

0.01905 0.01384 2.00 55

0.01905 0.01440 2.00 15

0.01905 0.01384 2.00 55

0.01905 0.01440 2.00 15

0.9 0.3 0.55

0.9 0.3 0.55

– – –

– – –

Heat Mass Transfer Fig. 2 (T , x, y)-diagrams of the mixtures n-pentane/iso-octane (left) and iso-propanol/water (right)

The mixture n-pentane/iso-octane (left) is a wide-boiling mixture with a difference of about 63 ◦ C in boiling temperatures. There are no azeotropes in the mixture. The second mixture (right), iso-propanol/water, displays a different character. The boiling points of the pure substances differ by about 18 ◦ C. At a mole fraction of about xiso-propanol = 0.68 there is a minimum azeotrope at a temperature of about T = 80 ◦ C. For higher mole fractions of iso-propanol than xiso-propanol ≈ 0.60 the mixture has a very close-boiling range. For lower mole fractions the mixture is very wide-boiling. Table 2 summarizes the ranges of compositions for which the measurements were carried out. The component with the higher mole fraction is referred to as the main component and the other component as the secondary component.

3 Reduction of data According to Polifke & Kopitz [21] the transferred heat ˙ can be calculated from the temperature difference of flow Q the cooling water TCW, in at the inlet and TCW,out at the outlet, as well as the mass flow M˙ CW of the cooling water and the specific heat capacity cp,CW of the cooling water: ˙ = M˙ CW · cp,CW · (TCW,out − TCW,in ) Q

.

(1)

Table 2 Survey over the examined mixtures and their composition ranges Main component

N-pentane Iso-octane Iso-propanol Water

Secondary component

Iso-octane N-pentane Water Iso-propanol

Mole fraction main component

Smooth tube

Low-finned tube

– 0.990–0.505 0.960–0.580 0.975–0.825

0.960–0.600 0.965–0.680 0.985–0.575 1.000–0.805

The overall heat transfer coefficient k can be calculated from the transferred heat flow after Polifke & Kopitz [21] by: k=

˙ Q A · Tlog

.

(2)

Herein, Tlog is the logarithmic temperature difference which is, for an over the length of the condenser constant temperature TBulk in the bulk phase, calculated as Tlog =

TCW,out − TCW,in   TBulk −TCW,in ln TBulk −TCW,out

.

(3)

The temperature in the bulk gas phase is calculated in this work from the pressure and the composition measured in the condenser assuming saturation. The area A in Eq. 2 is the area the heat flux q˙ and the overall heat transfer coefficient k are referred to. For measurements of the outer heat transfer coefficient on smooth tubes this area is the outer area Aoutside of the cylindrical tube calculated with the outer diameter doutside . In this work, the reference area for lowfinned tubes is calculated as the cylindrical area of the tube with the diameter at the fin tip dtip = doutside , which is proposed as the standard reference area in B¨uchner et al. [22]. With the calculated overall heat transfer coefficient k, the outer heat transfer coefficient αoutside can be calculated using ) ln( ddoutside 1 1 inside = + k · Aoutside Ainside · αinside 2 · π · L t · λt 1 + . αoutside · Aoutside

(4)

Therefor the inner heat transfer coefficient αinside has to be calculated using the equation of Gnielinski [23]: Nu =

(ξ/8) · (Re − 1000) · P r αinside · dinside = √ λCW 1 + 12.7 ξ/8 · (P r 2/3 − 1)     dinside 2/3 · 1+ . Lt

(5)

Heat Mass Transfer

To calculate the dimensionless Nusselt number Nu in this equation, the length Lt and the inner diameter dinside of the tube, the dimensionless Reynolds number Re and the dimensionless Prandtl number P r of the cooling water are required. The friction factor ξ is calculated by using the equation of Konakov [24]: ξ = (1.8 · Re − 1.5)2

.

(6)

The thermal conductivity λt of the tube in Eq. 4 is a value provided by the tube manufacturer and listed in Table 1, as well as the inner diameter dinside and the outer diameter doutside of the tubes. For low-finned tubes the conduction of heat is calculated only for the solid part of the tube. Thus, in Eq. 4 the parameter doutside has to be substituted by the diameter at the fin root droot . The Reynolds number is defined as Re =

uCW · dinside · CW ηCW

(7)

with the flow velocity uCW , the dynamic viscosity ηCW and the density CW of the cooling water. The Prandtl number is calculated as ηCW · cp,CW (8) Pr = λCW with the specific heat capacity cp,CW and the thermal conductivity λCW of the cooling water. For all conducted measurements the Reynolds number Re of the cooling water in the tubes has been in between 8000 < Re < 10000 and the measurement uncertainties for all experiments are about 3%. A detailed analysis of the measurement errors can be found in [25].

4 Theoretical background The theoretical background of the condensation of a binary mixture consisting of two condensable components is described in this section. In Fig. 3 the profile of the temperature and the mole fraction of the lighter boiling component 1 close to the phase boundary is shown. This can also be found in various publications on this topic [3, 17, 18, 26]. In the bulk gas phase the composition y1,Bulk and the temperature TBulk of the gas are constant and assumed to be at saturation. TW is the temperature at the wall of the tube and is slightly lower than the temperature TPh at the phase boundary. This temperature decrease is caused by the thermal resistance Rth,cond of the condensate layer on the tube against the conduction of heat through it. Since convective mass transfer in the condensate layer is neglected, the mole fraction x1,W of component 1 at the wall equals the mole fraction x1∗ at the phase boundary.

Fig. 3 Profile of the temperature and the mole fraction of the lighter boiling component close to the phase boundary

There are two limiting cases for the condensation of binary mixtures: total condensation and equilibrium condensation. Usually, the condensation takes place in between these two cases. Thus, the temperature TP h at the phase boundary lies in between the boiling temperature and the condensation temperature of the mixture and is therefore lower than TBulk and a gradient in temperature in the boundary layer results. At the phase boundary material equilibrium is assumed. The mole fractions x1∗ and y1∗ are the mole fractions of component 1 at equilibrium at the temperature TPh . Therefore, a gradient in the mole fraction in the gas phase results. For component 1 this gradient is negative for the zdirection, since the mole fraction y1∗ at the phase boundary is, according to the (T , x, y)-diagram, higher than the mole fraction y1,Bulk in the bulk phase. Due to this gradient a diffusive mole stream N˙ 1,diff of component 1 results, which is directed from the phase boundary into the bulk phase and therefore slows down the transport of gas to the phase boundary. This inhibition is regarded as thermal resistance Rth,diff in the gas phase against the condensation heat transfer and is an additional thermal resistance added to the one caused by the condensate layer on the tube Rth,cond . Thus, the total resistance Rth,total against the heat transfer, which is the sum of all resistances Rth,total = Rth,cond + Rth,diff

,

(9)

is increased. This leads to a decrease in the outer heat transfer coefficient αoutside . For the condensation of mixtures the resistance Rth,diff in the gas phase tends to be the limiting factor for the condensation process.

Heat Mass Transfer

5 Results The results for the two different mixtures are shown below. The data is shown separately for each range of main component according to Table 2.

5.1 Mixture of n-pentane and iso-octane In Fig. 4 the results for the mixture n-pentane in iso-octane (with iso-octane as the main component) condensing on a smooth tube of carbon steel are shown. In the diagram the outer heat transfer coefficient αoutside is plotted against the heat flux q. ˙ The values for αoutside increase with increasing heat flux, but the more n-pentane is present in the gas phase, the lower the outer heat transfer coefficient becomes. This trend is typical for the condensation of a binary mixture. The higher the heat flux, the cooler is the actual wall temperature. A lower wall temperature leads to a higher mole fraction of n-pentane in the condensate. Therefore, the condensation rate of the lower boiling component n-pentane is higher, which leads to a decrease in the resistance in the gas phase and thus to a higher outer heat transfer coefficient. The higher the mole fraction of n-pentane at constant q, ˙ the higher the diffusive stream from the phase boundary into the bulk phase becomes. This leads to a higher resistance against the heat transfer and thus to lower outer heat transfer coefficients. In the diagram the curves for the Nusselt theory for npentane and for iso-octane are plotted, too. This theory is the commonly-accepted theory to calculate the outer heat transfer coefficient αoutside for the condensation of pure substances on a horizontal smooth tube. The equation was proposed by Nusselt [27] and can be found in common text

Fig. 4 Outer heat transfer coefficient αoutside versus heat flux q˙ for the condensation of the mixture n-pentane/iso-octane on a smooth tube compared with Nusselt theory according to Eq. 10

books, wherein the constant factor is slightly adjusted from 0.725 to 0.728 [17]: αoutside = αL



L · ( L − G ) · g · hV · λ3L = 0.728 ηL · (TBulk − TW ) · doutside

1/4 (10) .

This equation is for the outer heat transfer coefficient αoutside which is equal to the heat transfer coefficient αL which stands for the heat conduction through the condensate layer on the tube and relates to Rth,cond in Eq. 9. To calculate αoutside , the density of the gas phase G and of the condensate L are needed. Furthermore, the gravity constant g, the specific heat of evaporation hV , the thermal conductivity λL and the dynamic viscosity ηL of the condensate and the temperature TW of the tube wall are needed. It can be seen that the values for the condensation of the mixtures for all heat fluxes and mole fractions are always smaller than the ones for the pure substances. For heat fluxes above q˙ = 30 kW/m2 and low mole fractions of n-pentane, the measured data is approaching the theoretical curves. In this region, the resistance Rth,diff in the gas phase against the heat transfer becomes smaller and is thus in the same order of magnitude as the resistance Rth,cond of the condensate layer. In this region, the limitation of the general process shifts from Rth,diff to Rth,cond . For the condensation of the same mixture, n-pentane in iso-octane (with iso-octane as the main component) on a low-finned tube, the trend of the measured values in Fig. 5, looks quite identical. For all measured compositions the outer heat transfer coefficient αoutside increases with increasing heat flux. Additionally, as observed for the

Heat Mass Transfer Fig. 5 Outer heat transfer coefficient αoutside versus heat flux q˙ for the condensation of the mixture n-pentane/iso-octane on a low-finned tube compared with pure substance data of Reif et al. [9]

smooth tube, the values of αoutside become smaller for increasing mole fraction yn-pentane of the lower boiling component n-pentane. There are no general accepted prediction models for the outer heat transfer coefficients of pure hydrocarbons and alcohols on low-finned tubes. Thus, the experimental results are compared with measured data of the pure substances extracted from Reif et al. [9]. Again, the values for the mixture are lower than the values for the pure substances. As compared with the data for smooth tubes, it can be observed that the difference between the values of the pure substances and the mixtures is significantly higher for the low-finned tubes. Thus, even a small amount of n-pentane in the gas phase has a very high impact on the measured outer heat transfer coefficient αoutside for the low-finned tubes. This results from the very low resistance Rth,cond in the condensate layer on a low-finned tube compared to the

Fig. 6 Outer heat transfer coefficient αoutside versus heat flux q˙ for the condensation of the mixture iso-octane/n-pentane on a low-finned tube compared with pure substance data of Reif et al. [9]

smooth tube. Therefore, the resistance Rth,diff in the gas phase is the limiting step and has a big influence on the outer heat transfer coefficient. Concluding from the data shown in the Figs. 4 and 5 one could surmise, that the higher the mole fraction of the lower boiling component in a mixture, the lower the outer heat transfer coefficient αoutside will become. This assumption would lead to very small values for mixtures with a very high mole fraction of n-pentane and almost no iso-octane. The results of the condensation of the mixture iso-octane in n-pentane (with n-pentane as the main component) condensing on a low-finned tube made of stainless steel are shown in Fig. 6. It can be seen that, again, the outer heat transfer ˙ In this coefficient αoutside rises with increasing heat flux q. case, the higher boiling component iso-octane has the lower mole fraction and it can be seen that, with higher mole

Heat Mass Transfer

fractions of iso-octane in the gas phase, the measured values become lower. This means that, independent of the boiling point, the presence of a second component is responsible for the decrease in the outer heat transfer coefficient. It can be seen in Fig. 6 that, compared to the measured values of the pure substances by Reif et al. [9], the values for the mixture are significantly lower. This can again be explained by the very low resistance against the condensation Rth,cond caused by the condensate layer on the tube for the condensation on a low-finned tube. This leads to a limitation by the resistance in the gas phase Rth,diff and therefore a significant decrease in the measured αoutside even for only small mole fractions of iso-octane.

5.2 Mixture of iso-propanol and water The second binary mixture considered here is the mixture of iso-propanol and water. In Fig. 7 the measured values for the mixture water in iso-propanol (with iso-propanol as the main component) on a stainless steel smooth tube are shown for mole fractions of water between 0.035 < ywater < 0.395 including the azeotropic point at ywater ≈ 0.32. Contrary to the observations in Section 5.1, the outer heat transfer coefficient αoutside decreases with an increasing heat flux q˙ and there is no decrease in the measured value with an increasing mole fraction of water. This behaviour is similar to that of a pure substance. The comparison with the data from the Nusselt theory according to Eq. 10 for pure isopropanol (dotted line) supports this statement. The values for the mixture with the main component iso-propanol are higher than the values calculated with Eq. 10 for pure isopropanol. The trend of the measured values also follows the trend of Eq. 10. The values for pure water are much higher (solid line). If the mole fraction of water is increased, the values for the outer heat transfer coefficient actually increase, too. Fig. 7 Outer heat transfer coefficient αoutside versus heat flux q˙ for the condensation of the mixture water/iso-propanol on a smooth tube compared with Nusselt theory according to Eq. 10

In Fig. 8 the measured values for the same mixture on a low-finned tube of stainless steel are shown and compared to the data of Reif et al. [28] for the pure substances (a comparison of the data of the pure substances on smooth and low-finned tubes can be found in [28]). The values do look quite similar to those of the smooth tube. The outer heat transfer coefficient αoutside decreases with an increasing heat flux and an increase of the measured values with an increasing mole fraction of water can be observed. Alike the values for the smooth tube the measured outer heat transfer coefficients on a low-finned tube for the mixture show about the same or higher values and the same trend as the values of pure iso-propanol. The values for pure water are higher. Thus, the higher the mole fraction of water becomes, the more the measured values tend to the values of pure water. The high values of the outer heat transfer coefficient αoutside for the smooth tube and the low-finned tube can be explained by taking a closer look at Fig. 2. The mixture of iso-propanol and water has a very close-boiling range for mole fractions of iso-propanol higher than xiso-propanol ≈ 0.6. This means that, for a constant mole fraction, the difference between the boiling temperature and the condensation temperature is very small which leads to a very small temperature gradient in the gas boundary layer. This leads to a small difference of the mole fractions in the bulk phase and at the phase boundary. Since this gradient is responsible for the diffusive stream from the phase boundary into the bulk phase, this stream is negligible. Therefore, the resistance Rth,diff in the gas phase against the condensation is very small such that it can be neglected as for pure substances. According to this explanation, the results in the wideboiling range for this mixture with water as the main component should look like the values of the mixture n-pentane/iso-octane. To verify this, the results for this mixture (with water as the main component) for the condensation on a smooth tube are shown in Fig. 9.

Heat Mass Transfer Fig. 8 Outer heat transfer coefficient αoutside versus heat flux q˙ for the condensation of the mixture water/iso-propanol on a low-finned tube compared with measured data of Reif et al. [28]

There is a big difference between the values calculated with the Nusselt theory according to Eq. 10 for pure water and the measured outer heat transfer coefficients αoutside for the mixture with a mole fraction of iso-propanol of only 0.035. It can be observed that the more iso-propanol is added to the mixture, the lower the measured outer heat transfer coefficient becomes. For mole fractions higher than yiso-propanol = 0.055, the values also increase with increasing heat flux q. ˙ For lower mole fractions it seems as if there is no significant trend at all or maybe even a very small decrease. It is noticeable that most of the results for the mixture are higher than the calculated values for pure iso-propanol (dotted line). In Fig. 10 the values for the same mixture condensing on a low-finned tube are plotted. In this case, very low mole fractions of secondary component around yiso-propanol = 0.005 were examined, too. The values of αoutside are quite high and at about the same order of magnitude as the values for pure water, maybe even slightly above, and no trend

Fig. 9 Outer heat transfer coefficient αoutside versus heat flux q˙ for the condensation of the mixture water/iso-propanol on a smooth tube compared with Nusselt theory according to Eq. 10

can be observed. For mole fractions of iso-propanol higher than yiso-propanol = 0.025 the measured values are lower than pure water and increase over heat flux. The more isopropanol in the system, the lower the outer heat transfer coefficient becomes. Contrary to the results for the smooth tube, the measured values for mole fractions higher than yiso-propanol = 0.015 are also lower than the results for pure iso-propanol. Thus, the results with water as the main component for the smooth and for the low-finned tube both behave as stated above and in the wide-boiling range of the mixture the typical behaviour of the outer heat transfer coefficient for mixtures can be observed.

6 Discussion There are different models for the prediction of the outer heat transfer coefficient αoutside for wide- and close-boiling

Heat Mass Transfer Fig. 10 Outer heat transfer coefficient αoutside versus heat flux q˙ for the condensation of the mixture water/iso-propanol on a smooth tube compared with values by Reif et al. [28]

mixtures. Firstly, the data in the close-boiling range, water in iso-propanol, shall be compared to theory. Later, the data for the wide-boiling range of this mixture and the results for the mixture n-pentane/iso-octane shall be analysed. The mixture properties needed for the calculations in this chapter were provided by the Linde AG, Germany for all of the used substances. Details on the mixture properties can be found in [25].

6.1 Comparison with theory: close-boiling mixtures The outer heat transfer coefficient αoutside for close-boiling binary mixtures can be calculated using the equations for pure substances inserting the property data of the mixtures. It is assumed that the condensate film has the respective boiling temperature Tb of the mixture [17, 32]. The constraint for this procedure is set by Fullarton & Schl¨under [32], as well as by Baehr & Stephan [17] as: Tb − TW >2 . Tb − Tc

(11)

Tc is the respective condensation temperature of the mixture. The data measured for the mixture with water as the secondary component, fulfills this constraint and therefore the measured data shall be compared to this theoretical approach. For a smooth tube Eq. 10 can be used. For low-finned tubes, no commonly accepted equation exists. Reif et al. [28] show that equations given in literature [29–31] cannot predict their values for the condensation of pure substances on low-finned tubes, and postulate an own equation which fits significantly better: −1/3

Cn = C · ReF

· Bo−1.9843

.

(12)

The constant C is material-specific. For stainless steel tubes, it is C = 0.2308 and for carbon steel tubes it is C = 0.3452. Cn is the so-called “Condensation number”  1/3 2 ηL , (13) Cn = αoutside · λ3L ·  · L · g ReF the film Reynolds number defined as ReF =

2 · q˙ · doutside · π ηL · hV

Bo the Bond number σL Bo =  · g · s · doutside

and

(14)

(15)

including the surface tension of the condensate σL . After determining Re and Bo using Eqs. 14 and 15 the condensation number Cn can be calculated with Eq. 12. The outer heat transfer coefficient αoutside is then obtained by rearranging Eq. 13. Since the same tubes as Reif et al. [28] were examined and Eq. 12 was developed for the pure substances used for the here examined mixtures, Eq. 12 is used for predicting αL for low-finned tubes. The parity plot for the data of this mixture for the condensation on a smooth tube and on a low-finned tube is shown in Fig. 11. It can be observed that this approach fits quite well for the measured data. On the left-hand side the results for the smooth tube are shown. The deviation between the calculated and the measured values is quite small and is significantly lower than 20%. Thus, it can be concluded that, for the smooth tube, this approach fits very well. The parity plot for the low-finned tube is shown on the right-hand side. Here, the deviation between model and measurement is a bit higher. Almost all calculated values are about 20–40%

Heat Mass Transfer Fig. 11 Parity plot of the outer heat transfer coefficient αoutside for the theoretical approach for close-boiling mixtures for a smooth (left) and a low-finned tube (right) for the mixture water in iso-propanol

higher than the measured values. This deviation could occur because the property data of the composition in the gas phase is used, which differs from the actual composition of the condensate. Since, for example, the surface tension has a big influence on the outer heat transfer coefficient for pure substances on low-finned tubes, this could have an effect on the accuracy of the model. Another explanation could be that, due to the very high heat transfer coefficients αoutside of the low-finned tubes for pure substances, the resistance in the gas phase Rth,diff has a higher impact and significance for these tubes and cannot be neglected as it is proposed by this model. It can be concluded that, for the low-finned tube, this approach does not fit as well as it does for the smooth tube, but the calculated values are still in quite good agreement with the measurements. Thus, considering the data for the smooth and for the low-finned tube, it can be stated that the results obtained here can be described quite well with the presented theoretical approach.

6.2 Comparison with theory: wide-boiling mixtures The presented measured data for the wide-boiling mixtures shall be compared to the theoretical film model by Colburn & Drew [2]. According to Webb et al. [3] this model predicts the measured values with a higher quality than the equilibrium model by Silver [1]. The transferred heat flux q˙ can be written as q˙ = q˙G + n˙ · H¯ V = αoutside · (TBulk − TW ) .

(16)

Herein, n˙ is the mole flux vertical to the phase boundary, H¯ V the molar heat of evaporation and αoutside the outer heat transfer coefficient. q˙G is the heat flux in the gas phase and can be calculated using the following equation: • · (TBulk − TPh ) . q˙G = αG · ζ · (TBulk − TPh ) = αG

(17)

The heat transfer coefficient αG for the single-phase heat transfer in the gas phase is, in this equation, multiplied with ζ , the Ackermann correction factor, since Webb et al. [3]

state that by using the Ackermann correction [33] better • is the corrected single-phase results are obtained. Thus, αG heat transfer coefficient in the gas phase. The Ackermann correction factor is defined as ζ =

1 − e−

(18)

wherein the parameter is defined as

=

n˙ · C¯ p,G αG

(19)

with C¯ p,G being the molar heat capacity of the gas phase. The heat flux q˙L for the conduction of heat through the condensate is equal to the total heat flux q˙ and can be described by q˙L = q˙ = αL · (TPh − TW ) .

(20)

From the Eqs. 16, 17 and 20 it can derived that  q˙ =

q˙G

1 q˙ + • αL αG

−1 · (TBulk − TW )

= αoutside · (TBulk − TW )

.

(21)

By defining Z as Z=

q˙G q˙

(22)

the outer heat transfer coefficient αoutside can be written as 1 αoutside

=

1 Z + • αL αG

.

(23)

To calculate the outer heat transfer coefficient, it is necessary to calculate the heat transfer coefficient αL . For smooth tubes Eq. 10 can be used here, for low-finned tubes Eq. 12 is used. The property data used in Eq. 10 and for the dimensionless numbers in Eq. 12, need to be calculated for the temperature at the phase boundary TPh and the respective composition. The equation for the calculation of

Heat Mass Transfer

TPh is provided after the derivation of the therefor required parameters. The single phase heat transfer coefficient αG in the gas phase can be calculated for free convection by αG · Lchar λG  = 0.752 +

Nu =

0.387 · Ra 1/6 (1 + (0.559/P r)9/16 )8/27

2 .

(24)

This equation is proposed by Klan [34] for single phase heat transfer for free convective flow around a horizontal tube. In this equation Ra is the dimensionless Rayleigh number, defined as Ra =

g

· L3char

2 · G

· cp,G

ηG · λ G

· βth · (TW − TBulk )

1 TBulk

.

(25)

(26)

This set of dimensionless numbers is valid for the characteristic length of Lchar =

π · doutside 2

.

(27)

According to Webb et al. [3] and Baehr & Stephan [17], the mole flux n, ˙ which is needed for the parameter in the Ackermann correction in Eq. 19, can be calculated with   ∗ x1 − y1∗ . (28) n˙ = βG · cG · ln x1∗ − y1,Bulk In this equation the mass transfer coefficient βG is needed, but this parameter is not easy to access. Thus, in order to avoid the need of calculating βG , Fullarton & Schl¨under [32] propose the use of the LewisRelationship (Chilton-Colburn-Analogy) instead. Here, the Lewis number Le can be written as αG . (29) Lem = βG · cG · C¯ p,G With Eqs. 29 and 28, the parameter φ can be calculated as

 ∗ ∗  x1 −y1 ¯ · c · ln β ¯ G G n˙ · Cp,G x1∗ −y1,Bulk · Cp,G = φ = α αG G ∗ ∗  x1 −y1 ln x ∗ −y 1,Bulk 1 = . Lem

Le =

(30)

The exponent m, which is necessary in the above equation is given by Fullarton & Schl¨under [32] as m = 0.6.

λG D12 · cp,G · G

(31)

the actual calculation of the mass transfer coefficient βG is not necessary in this case. Only the diffusion coefficient D12 is necessary, which is easier to access. All property data • has to be derived at the gas bulk needed to calculate αG phase temperature and bulk composition. The last parameter needed in Eq. 23 to calculate the outer heat transfer coefficient is Z. It can be calculated by combining Eqs. 17 and 22 to get Z=

and P r is the Prandtl number calculated for the gas phase. In Eq. 25 βth is the thermal expansion coefficient which, according to Bird et al. [35], can be approximated for ideal gases as βth =

Since the Lewis number Le is defined as

• · (T αG Bulk − TPh ) q˙

.

(32)

The temperature at the phase boundary TPh can be calculated by combining Eqs. 16–19 TPh = TBulk −

q˙ − αG · φ · αG ·

hV cp,G

φ 1−e−φ

.

(33)

Since TPh is also needed for calculating αG , an iteration of TPh is necessary to be able to calculate the outer heat transfer coefficient αoutside . In Fig. 12, the parity plot for the measured outer heat transfer coefficient and the calculated αoutside is shown. It can be seen that the measured and the calculated values do agree with each other for some mixtures and tubes and do differ quite distinctly from each other for others. There is no obvious trend. While the values for n-pentane in iso-octane on the low-finned tube are predicted poorly by the model, the same mixture on the smooth tube does agree with the model almost within a maximum deviation of about 20–30%. On the other hand, the mixture of isopropanol in water on the smooth tube shows the highest negative deviation of the measured and the calculated values, whereas the same mixture on the low-finned tube agrees very well with the model. Since, in this parity plot, the outer heat transfer coefficient αoutside is plotted, one has to keep in mind that the calculated values include the value of the heat transfer coefficient αL , which describes the heat conduction through the condensate layer. However, since the equations used for αL are in quite good agreement with the pure substance measurements as shown by Reif et al. [28], this tends to improve the prediction quality of the model rather than leading to a bigger deviation. The Lewis-Relationship in Eq. 29 is valid for laminar and turbulent flow [36] and Steeman et al. [37] state that they do get good results with the Lewis-Relationship for free convection. Since the agreement of the model and the measurements is quite poor, there is a need for a better way to describe the measured values.

Heat Mass Transfer

Fig. 12 Parity plot of the outer heat transfer coefficient αoutside for the film model by Colburn & Drew [2] using the Lewis-Relationship

6.3 Modelling of the data for the wide-boiling range The initial point for an own modelling is Eq. 23. Since there are good and valid equations for αL and αG , there is no need to alter anything for these two terms. Thus, the only parameter which needs to be modified is the parameter Z. The approach of Colburn & Drew did not show good results, so an approach of describing Z with dimensionless numbers is applied here. Therefore, the relevant parameters need to be clarified and each of these accounted for in a dimensionless number. Since Z is the ratio of the heat flux in the gas phase q˙G and the total heat flux q˙ the specific heat capacity in the gas phase cp,G and the specific heat of evaporation hV need to be considered. These two values do appear in the dimensionless Jakob number J a, which is the ratio of latent to sensible heat: cp,G · (TBulk − TPh ) . (34) Ja = hV However, the parameter Z also depends on the heat flux q˙ itself, which is described by the Reynolds number Re m ˙ · doutside · π q˙ · doutside · π = Re = ηG · hV ηG

.

(35)

Furthermore the ratio of heat capacity and thermal conductivity is an important factor which is included in the Prandtl number P r ηG · cp,G . (36) Pr = λG

Fig. 13 Parity plot of the outer heat transfer coefficient αoutside for the model described with Eq. 37

The gas phase properties in these equations are calculated at bulk temperature TBulk . With these three dimensionless numbers, an equation to calculate Z based on the experimental results was fitted. The resulting equation is as follows: Z = 97.883 · Re−0.975 · P r −3.040 · J a 1.420

.

(37)

In Fig. 13, the parity plot of αoutside for this equation is plotted. It is obvious that with Eq. 37, the outer heat transfer coefficient αoutside can be predicted much better than with the film model by Colburn & Drew [2]. The maximum deviation between calculated and measured values for all measured mixtures and for all examined tubes is less than ± 20%. Most values show a lower deviation. Thus, Eq. 37 is much better suited for describing the experimental results obtained in this work.

7 Conclusion Condensation measurements of two different mixtures on smooth and low-finned horizontal tubes were carried out. The mixture n-pentane/iso-octane is a wide-boiling mixture with no azeotropes. The outer heat transfer coefficient increases with rising heat flux and decreases at a constant heat flux but with an increasing mole fraction of the secondary component. The values for the mixture are always lower than the values for the pure substances. The

Heat Mass Transfer

mixture iso-propanol/water has a very close-boiling region for low mole fractions of water, a minimum azeotrope at a mole fraction of xwater ≈ 0.32 and a wide-boiling region for higher water content. In the wide-boiling range, the measured values behave just like the results of the first mixture, but for the close-boiling region, the values behave completely different. They decrease with an increasing heat flux and with an increasing mole fraction of water in the gas phase the experimental results increase rather than decrease. The observed trend of αoutside over q˙ is very alike to pure substance condensation and can be explained due to the very small differences in temperature and mole fraction in the gas phase due to the close-boiling range. The comparison with the common model for condensation of mixtures, the film model by Colburn & Drew [2] using the Lewis-Relationship, did not lead to satisfying results. Despite the fact that, for some values, the model did provide good results, the model cannot be regarded as reliable since, for other data, the deviation between measured values and calculated values for the outer heat transfer coefficient differed to a great extent. Thus, it was tried to create an equation of dimensionless numbers, which is describing the thermodynamic parameter Z, such that the measured and the calculated values would fit better. Equation 37 was fitted for the experimental data except for the close-boiling range. A comparison shows that this equation can describe the data of the wide-boiling mixtures much better than the film model. The deviation of the model and the measured data is in any case less than ± 20%. This equation does, of course, need to be reviewed and checked with other mixtures and other regimes of gas flow to prove its correctness, but it is a first step towards improving prediction quality for the outer heat transfer coefficient for the condensation of binary mixtures on horizontal tubes. Acknowledgements The authors gratefully acknowledge the support of the German Federal Ministry of Education and Research (BMBF) and Wieland-Werke AG.

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