Turbulent film condensation on a non-isothermal
horizontal tube—effect of eddy diffusivity
Sheng-An Yang
*, Yan-Ting Lin
Department of Mold and Die Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan Received 1 July 2004; received in revised form 1 January 2005; accepted 11 February 2005
Available online 14 April 2005
Abstract
A theoretical study is made into the process of heat transfer with vapor condensation on a non-isother-mal horizontal tube submerged in a forced vapor flow. The interfacial eddy diffusivity effect included in the energy relation is first considered here. The interfacial shear at the vapor condensate film is evaluated by employing the Colburn analogy. The condensate film flow and condensing heat transfer characteristics from a non-isothermal horizontal tube under the effects of the non-uniform wall temperature variation, the Froude number, sub-cooling parameter and system pressure parameter have been investigated. As the wall temperature variation amplitude increases, the value of the dimensionless mean heat transfer result, NuRe1=2, with the inclusion of the eddy diffusivity effect, decreases insignificantly.
Ó 2005 Elsevier Inc. All rights reserved.
Keywords: Turbulent; Film condensation; Eddy diffusivity effect; Horizontal tube
1. Introduction
The problem of film condensation on a horizontal tube submerged in a forced vapor flow is not only of theoretical interest, but also of great importance for industrial applications such as the
0307-904X/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2005.02.012
*
Corresponding author. Tel.: +1 886 7 3814526x5412; fax: +1 886 7 3835015. E-mail address:samyang@cc.kuas.edu.tw(S.-A. Yang).
Nomenclature
A the wall temperature variation amplitude CP specific heat of condensate at constant pressure D diameter of horizontal tube
F dimensionless inverse velocity parameter, 2 ugR2 1 h fgPr CPðTsatTwÞ Fr Froude number, u21=gR
f average friction coefficient f/ local friction coefficient Gr Grashof number, gRv23 l qlqv ql
g acceleration due to gravity h local heat transfer coefficient hfg latent heat of condensate
k thermal conductivity of condensate Nu local Nusselt number, h(D)/kl
n the power of Reynolds
Pr Prandtl number, v/a R radius of horizontal tube
Rel, Rev Reynolds number, u1D/vl, u1D/vv Re* shear Reynolds, Re+/Gr1/3
Re+ shear Reynolds parameter, Ru*/vl
S sub-cooling parameter, CPðTsat TwÞ=ðhfgPrÞ
St Stanton number
T temperature
T+ dimensionless temperature, ðT TwÞ=ðTsat TwÞ u velocity component in x-direction
ue the tangential vapor velocity at the edge of the boundary layer u* shear velocity, (sw/ql)1/2
y coordinate normal to the tube wall
y+ dimensionless distancenormal to the circular tube wall, yu*/vl Greek symbols
a thermal diffusivity
d local condensate film thickness d+ dimensionless film thickness, du*/vl eM momentum eddy diffusivity
eH thermal eddy diffusivity
u interfacial shear parameter, 2npC qv
ql vl vv n1 Grð3n1Þ=6 / angle measuring from top to tube
s shear stress
v kinematic viscosity
design of condensers for refrigeration, air conditioning and power plants. The main aim is to de-velop a simple model to predict the mean condensation heat transfer of flowing vapor onto a hor-izontal tube with variable wall temperature. To the authorsÕ knowledge, the study of the effect of non-uniform wall temperature variation on the turbulent film condensation of vapor flowing onto a horizontal tube has not been presented so far.
Forced convection film condensation from a saturated vapor flowing onto a horizontal isother-mal cylinder has received considerable attention. Earlier investigations of vapor flowing velocity effects including the interfacial shear boundary condition at the edge of the condensate film were regarded as an extension of NusseltÕs analysis[1]. As for flowing vapor, Shekriladze and Gomel-auri [2] first assumed that the shear stress at the liquid vapor interface is equal to the loss of momentum of the condensing vapor. Hsu and Yang [3], and Rose [4] modified the Shekriladze and Gomelauri model, and took into account the pressure gradient effect upon the same topic by using the potential flow theory. Both studies, which included the pressure gradient effect, found a small decrease in the mean heat transfer coefficient.
All the above-mentioned works relate a forced flowing condensation of an isothermal circular tube. Fujii et al.[5]investigated experimentally with the peripheral distribution of wall heat flux. Memory et al. [6]also found that the difference between the saturated vapor and the local non-uniform wall temperature usually varies with the ‘‘1 A cos /’’ profile for the forced convection film condensation problem. Afterwards, Yang[7]studied both the non-uniform wall temperature variation effect and the vapor superheated effect on the mean condensing heat transfer coefficients and showed that with the inclusion of the pressure gradient, the mean heat transfer coefficient de-creases appreciably with the non-uniform wall temperature variation effect.
When the vapor velocity is sufficiently high so that the effect of the associated shear stress on the motion of the condensate is comparable with that of gravity, Michael et al.[8]presented that the condensate film can be under a turbulent regime. For condensation on a circular tube, Sarma et al.[9]used KatoÕs model of eddy diffusivity in the condensate film and assumed that the friction coefficient at the liquid vapor interface is identical to that of air flow. Later on, Homescu and Pan-day[10] used a finite difference analysis to solve coupled boundary layer equations for turbulent flow condensation on a horizontal tube. They proposed an empirical equation for evaluating the
Subscripts c critical condition l condensate film m mean values v vapor d vapor–liquid interface w circular tube wall sat saturation
1 in the free stream Superscript
dimensionless mean heat transfer coefficient, which is mostly in good agreement with both Honda et al.Õs[11]data and Michael et al.Õs[8]data, except at very high vapor velocities. However, these coupled two-phase boundary layer equations are tedious numerical calculations. More recently, a simple model concerning the further effect of interfacial eddy diffusivity in the energy relation has been developed by Yang and Lin[12]. The dimensionless mean heat transfer coefficient NuRe1=2 shows a better agreement with the previous experimental data, especially in the very high vapor velocities regime. However, the above-mentioned studies focus on the isothermal wall case.
Basically, a simple and efficient model is developed to resolve the turbulent film condensation on a non-isothermal horizontal tube instead of a tedious two-phase model numerical calculation. The major purposes here are to consider both effects of the liquid vapor interfacial eddy diffusivity in the energy relation and the non-uniform wall temperature variation upon the mean condensing heat transfer characteristics and to improve the difference between the previous works Sarma et al. [9], Homescu and Panday[10] and experimental works in the very high vapor velocities regime.
2. Analysis
Fig. 1illustrates schematically a physical model and coordinate system of condensate film along an isothermal tube. The curvilinear coordinate (x, y) are aligned along the circular tube wall
face and its normal. Outside the condensate layer, is a pure saturated vapor at temperature Tsat and at uniform velocity u1. The tube wall surface temperature Twis below the saturation temper-ature. A thin condensate film is formed and flows under the influence of gravity, interfacial shear and physical properties. The following assumptions are made prior to the constitution of the basic equations.
(i) Velocities u; v, temperature T and thickness of liquid film d are time averaged values. (ii) Due to the film thickness being much smaller than the radius of a circular tube, the potential
stream function of a circular tube is assumed at the vapor–liquid interface. Further, the inter-face is assumed to be smooth and ripple free up to separation.
(iii) All physical properties of the condensate in the liquid film are assumed to be constant. (iv) The vapor density is much smaller than the liquid density.
(v) The inertia term and convective term of governing equations in the liquid film are negligible.
Using the above assumptions, the conservation equations governing condensate flow are uoT oxþ v oT oy ¼ d dy ða þ eHÞ dT dy ; ð1Þ sw¼ gdðql qvÞ sin / þ sd; ð2Þ d Rd/ Z d 0 qludy¼ kl hfg dT dyy¼0; ð3Þ
The boundary conditions are
x¼ 0; u¼ 0; ð4Þ
y¼ 0; u¼ v ¼ 0; T ¼ Tw; ð5Þ
y¼ d; ou
oy ¼ 0; T ¼ Tsat: ð6Þ
It is noted here, Eq.(2), shows a force balance of gravity, wall shear and interfacial vapor shear effects. Eq.(3)is an energy balance of the latent heat released through conduction in the conden-sate film between the interface and the horizontal tube wall surface.
For a horizontal tube in external flow, the mean Nusselt number can be stated in terms of Prandtl and Reynolds number by using a Hilpert semi-empirical model, as seen in Hilpert [13].
Nu¼ CRen vPr
1=3; ð7Þ
where the constants C and n are listed inTable 1. Further, using ColburnÕs analogy[14], one has
f
2 ¼ StPr
St¼ hv qvCpu1¼
Nu RevPr
: ð8bÞ
Note that St, namely, the Stanton number denotes the mean heat transfer coefficient in turbulent flow. From Eqs. (7) and (8a), we have the mean friction coefficient as
f ¼ 2CRen1v : ð9Þ
As mentioned in Homescu and PandayÕs[10]together with Sarma et al.Õs[9]assumptions, we may define the local friction coefficient in terms of a Sinusoidal function for a circular tube.
f/¼ CpðRen1v Þ sin /: ð10Þ
The local skin shear stress is defined as sd ¼
1 2ðqvÞðu
2
eÞf/: ð11Þ
Using potential flow theory for a uniform flow with velocity u1past a horizontal tube, one may derive the vapor velocity at the edge of boundary layer as
ue¼ 2u1sin /: ð12Þ
From Eqs. (10)–(12), the vapor–liquid interfacial shear stress can be obtained sd ¼ 2ðCÞðpÞðqvÞðu
2
1ÞðRen1v Þsin 3
/: ð13Þ
Finally, Eq. (2)is rewritten as follows sw¼ gdðql qvÞ sin / þ 2ðCÞðpÞðqvÞðu
2
1ÞðRen1v Þsin 3
/: ð14Þ
Since the condensate film is sufficiently close to the solid wall, the turbulent conduction across the condensate film is more significant than the convective component. As mentioned in Bejan [14], the energy equation (Eq. (1)) reduces to
d dy ða þ eHÞ dT dy ¼ 0: ð15Þ
Assuming the turbulent Prandtl number Prt (=eM/eH) = 1, Eq.(12) yields d dy 1þ eM vl Pr dT dy ¼ 0: ð16Þ Table 1
Values of C and n in Eq.(7)
Reynolds C n 0.4–4 0.989 0.330 4–40 0.911 0.385 40–4000 0.683 0.466 4000–40,000 0.193 0.618 40,000–400,000 0.0266 0.805
By introducing the dimensionless groupings, Eqs.(3), (14) and (16) can be restated as follows o o/ Z dþ 0 ðuþÞ dyþ¼ ðSÞðReÞ 1 þeM vl Pr ðGr1=3ÞoT þ oyþ yþ¼0 ; ð17Þ
ðReÞ3¼ dþsin /þ ðReÞðuÞðFrÞðnþ1Þ=2sin3/; ð18Þ
d dyþ 1þ eM vl Pr dTþ dyþ " # ¼ 0; ð19Þ
subject with dimensionless boundary conditions for Eq. (18)as follows
xþ ¼ 0; uþ¼ 0; ð20Þ yþ ¼ 0; uþ ¼ vþ ¼ 0; Tþ ¼ 0; ð21Þ yþ ¼ dþ; ou þ oy ¼ 0; T þ¼ 1: ð22Þ
Before proceeding to obtain the solution of Eq. (17) and thence to calculate the velocity pro-file in condensate film, according to Yang et al.Õs study [3,7] and assumption, the universal velocity distribution is being used in the estimation of the discharge rate of the condensate at any angular location. We may assume that sd is of the same order as sw and be considered as a valid approximation. As seen in Bejan [14], the velocity profile of condensate film can be obtained from ouþ oyþ ¼ 1þ eM vl 1 : ð23Þ
It is necessary to assume the eddy diffusivity of turbulent flow to solve the governing equa-tions (19) and (23). Kato proposed the following equation to predict the eddy diffusivity expression
eM vl
¼ 0:4yþ½1 expð0:0017yþ2Þ : ð24Þ
Then, specifying or fitting the wall temperature distribution from measured data, one can then calculate the mean wall temperature, as adopted in Hsu and Yang[3,7].
Tw¼ Z p
0
Twð/Þ d/=p ð25Þ
and derive the temperature difference across the film in the usual manner, as seen in Yang and HsuÕs model[3,7]; Memory et al.[6]
DT ¼ ðTsat TwÞð1 A cos /Þ ¼ DT ð1 A cos /Þ; ð26Þ
where A is a constant (0 6 A 6 1) and denotes the wall temperature variation amplitude. Combining Eqs. (19), (23), (24) and (26) for temperature gradient in the close vicinity of the tube yields
oTþ oyþ yþ¼0 ¼ ½1 A cos / 1 þ Pr eM vl Z dþ 0 dyþ 1þ Pr eM vl h i 2 4 3 5 8 < : 9 = ; 1 : ð27Þ
Considering the condensation heat transfer, FourierÕs law may be used to express the surface heat flux as kl dT dy y¼0 ¼ hðTsat TwÞ: ð28Þ
As in NusseltÕs theory, the dimensionless local heat transfer coefficient may be expressed as Nu¼ ReþdTþ dyþ yþ¼0 ; ð29Þ Nu Re1=2l ¼ ffiffiffi 2 p oTþ oyþ yþ¼0 Gr13 Fr !1 4 Re: ð30Þ
We are also interested in an expression for the overall mean heat transfer coefficient. The integral of Eqs.(29) and (30) read
Nu¼1 p Z p 0 Nud/; ð31Þ Nu Re1=2l ¼ ffiffiffi 2 p p Gr13 Fr !1 4Z p 0 oTþ oyþ yþ¼0 Red/: ð32Þ
As for the forced convection dominated case, Eq. (18)can be reduced to a simple form due to the negligible body force term
Re ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðuÞðFrÞðnþ1Þ=2sin3/ q
: ð33Þ
With the above equation, the relationship between Nu=Re1=2l and the given parameters can be ar-rived at Nu Re1=2l ¼ ffiffiffi 2 p oTþ oyþ yþ¼0 Gr13 Fr !1 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðuÞðFrÞðnþ1Þ=2sin3/ q : ð34Þ 3. Numerical method
To start the calculation, we assume the dimensionless condensate film thickness and the liquid film velocity at the upper stagnation point of a circular tube to be initially zero. Then, the liquid film thickness and velocity are obtained by using the dimensionless governing equations(17)–(19). For a given condition, a brief outline of the solution procedure is given as follows:
(i) Indicate the fixed physical environmental parameters of S, Fr and Gr.
(ii) Assume the dimensionless film thickness and velocity profiles are assumed to be zero (d+= 0, u+= 0) for an initial guess at the node i = 1 at the upper stagnation point.
(iii) Substitute D/ = p/500 and oTþ=oyþj
yþ¼0 into Eqs. (17) and (18). Iterate the NewtonÕs
method for different Dd+s till the following convergence criteria is satisfied. fðReÞ3
dþsin / ðReÞðuÞðFrÞðnþ1Þ=2
sin3/g 6 10e 6 and Re ¼ o o/ Z dþ 0 ðuþÞ dyþ=ðGr1=3Þ 1 þeM vl Pr ðSÞoT þ oyþ y¼0 ; where o o/ Z dþ 0 ðuþÞ dyþ¼Dd þ D/ u þ i þ uþi 2 ouþ oyþ yþ¼dþ " # :
(iv) Express the next film thickness for node i = 2 as dþiþ1¼ dþi þ Ddþand its corresponding angle as /i+1= /i+ p/500, when the increment of dimensionless film thickness (Dd+) for node i = i + 1 is convergent.
(v) Repeat the calculations from step (iv) to step (v) to find out all the needed values as /! p. Finally, both the dimensionless local and mean Nusselt numbers are determined from Eqs. (30) and (32).
4. Discussion of results
The present results cover the cases of laminar and turbulent film condensation of downward flowing vapor on a horizontal tube with variable wall temperature. From a practical point of view, besides the inverse vapor flow velocity F, the influencing parameters on both hydrodynamics of film flow and characteristics of heat transfer, such as A, S, u, Fr and Pr, are limited to a practical range. Two major topics: hydrodynamics of condensate film flow and characteristics of condens-ing heat transfer are presented and discussed in this section.
4.1. Flow hydrodynamics: condensate film profile
Characteristics of condensate film flow hydrodynamics are illustrated inFig. 2(a)–(c). Firstly, Fig. 2(a) shows that the condition of vapor flow, i.e. the range of Reynolds number, will affect the condensate film profile. As the value of n or Re becomes larger, the condensate will increase its thickness more abruptly. Thus, the separation point of condensate film flow must occur more upstream, as seen inFig. 2(a). According to a famous dynamics textbook by Schlichting [15], in the case of the vapor flow past a circular cylinder, separation occurs at / = 109.5°. In fact, the flow-ing vapor not only passes a movflow-ing-condensate liquid film but also condenses in the film. Hence, the separation point of vapor boundary layer /vmust occur at a more downstream point than the
liquid film layer /l. If /v< /l, due to very fast flows of vapor, the separation of vapor is accompa-nied by a sharp pressure rise, and could initiate an instability in the condensate film near /v.
Fig. 2(b) indicates the effect of the liquid sub-cooling upon the condensate film flow. According to the present study, it is found that the influence of condensate sub-cooling on the local film flow is appreciable for tubes with higher values of sub-cooling parameters, S. In general, the higher sub-cooling parameter is, the thicker the condensate film becomes. The condensate film separation will occur once the sub-cooling parameter reaches a critical value, Scr= 0.005, as seen inFig. 2(b). The separation phenomenon of the condensate film is attributed to the interfacial eddy diffusivity included here, which is not seen in Sarma et al. [9].
Basically, the higher the system pressure is, the thicker the condensate film becomes. Similarly, there also exists a critical system pressure value for telling if the condition of condensate film flow enters the turbulent zone. In other words, the higher system pressure will be more inclined to make the condensate film separate and then enter the turbulent zone. For the associated param-eters shown in Fig. 2(c), after system pressure u > 0.01, the condensate film flow becomes sepa-rated from the wall and enter the turbulent zone.
4.2. Characteristics of heat transfer 4.2.1. Effect of subcooling parameters, S
As mentioned inFig. 2(a), there exists a critical sub-cooling parameter value in which the densate film will abruptly become extra thick and then separate from the wall. Thus, its local con-densing heat transfer coefficient will decrease sharply after the separation point. This significant decrease will change the rate of the mean condensing heat transfer coefficient with an increase in S, as seen in Fig. 3. In other words, the dimensionless mean heat transfer coefficients
NuRe1=2 go down at two different rates. When S < Scr, since the condensate film flow is in the laminar zone, NuRe1=2 will decrease with S at lower rate. When S > S
cr, since the condensate film flow is in the turbulent zone, NuRe1=2 goes down with S at a higher rate.
4.2.2. Effect of system pressure, u
Fig. 4 illustrates the dependence of the dimensionless mean heat transfer coefficient on the dimensionless system pressure, u, and Prandtl number. Firstly, when u is approximately below 0.01, depending on Pr, the values of NuRe1=2 increase linearly with u, while the condensate film flow still belongs to the laminar or transition zone. This trend can be explained from the fact that Nu/Re0.5is increasing with u, as seen in Eq.(34). Once u is greater than 0.01, the condensate film thickness will abruptly become large and separation will occur, as seen inFig. 4. In other words, as the condensate film flow enter the turbulent zone, the mean heat transfer coefficient goes up slightly with u, as seen in Eq. (34).
4.2.3. Effect of the inverse vapor flow velocity, F
InFig. 5, when u < 0.01 and F is approaching 0.001, the condensate film is entering the turbu-lent zone. Hence, the value of NuRe1=2 is inversely proportional to F during laminar flow zone. When F < Fcr 0.001, the value of NuRe1=2 becomes proportional to F, because the condensate film flow enters turbulent zone. However, as for the two cases of u = 0.02 and u = 0.04, when F decreases to 0.08, the present curves turn proportional, which is in reasonable agreement with the previously reported experimental data. Note that, this proportional trend from transition to tur-bulent zone, is not seen in the studies of Homescu and Panday [10], and Sarma et al.[9]. This is attributed to their analysis without taking account of the eddy diffusivity at the liquid vapor inter-face energy balance.
It can be seen that there is a broad similarity between Michael et al. [8] data and the present result inFig. 6. In other words, the mean heat transfer coefficient is increasing with F for laminar flow. But for the transition zone, the mean heat transfer is increasing as F is decreasing. It reaches a climax near F = 0.01 for w = 0.01, and then comes down as F decrease further, which is located in the turbulent zone. Further, it is to be noted that the present results are not constant in contrast to a laminar model previously studied by Yang[7] when F is small.
Fig. 5. Dependence of mean Nusselt number on dimensionless vapor velocities F.
4.2.4. Effect of non-isothermal wall temperature variation amplitude, A
Fig. 7indicates that, in general, the dimensionless mean condensing heat transfer coefficient in-creases insignificantly with A, regardless the values of F. A similar trend occurs with the results for laminar model by Hsu and Yang[3]. Hence, if one ignores the effect of the wall temperature var-iation amplitude, A, the mean Nusselt number will be slightly over-predicted.
5. Concluding remarks
This is the first theoretical approach to resolve the forced convection film condensation on a horizontal tube under the simultaneous effects of the eddy diffusivity and the wall non-isothermal-ity. The effect of the eddy diffusivity at the liquid vapor interface makes the film separation occur at the critical values of corresponding parameters; system pressure u, subcooling parameter S, and the inverse vapor flow velocity, F. The major conclusions drawn here are
(1) The theoretical results indicate that the interfacial eddy diffusivity has an appreciable influ-ence on the local film thickness and heat transfer characteristics. The former can illustrate that the critical corresponding parameters play a role in telling the condition of the conden-sate flow, i.e. laminar or turbulent flow.
(2) The mean heat transfer coefficient is nearly unaffected by the wall non-isothermality, regard-less the values of F.
(3) At the very high vapor velocities, i.e. F 0.001, if one evaluates the mean condensing heat transfer coefficient NuRe1=2, without accounting for an interfacial eddy diffusivity, one will over-predict the coefficient, compared to the present results or experimental data.
Acknowledgment
Funding for this study provided by National Science Foundation, Taiwan, ROC under the grant number NSC 93-2212-E-151-002 is gratefully acknowledged.
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