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Mixed-convection film condensation along outside surface of vertical tube in saturated vapor with forced flow
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Applied Thermal Engineering 28 (2008) 547–555 www.elsevier.com/locate/apthermeng
Mixed-convection film condensation along outside surface of vertical tube in saturated vapor with forced flow Tong-Bou Chang
*
Department of Mechanical Engineering, Southern Taiwan University of Technology, Tainan County, Taiwan, ROC Received 12 January 2007; accepted 25 April 2007 Available online 5 May 2007
Abstract This study employs the nonsimilar transformation method to perform a comprehensive investigation into the steady-state mixed convection of a condensate film on an isothermal vertical tube in dry saturated vapor with a forced flow. The analysis takes account of the inertia and convection effects in the condensate layer and the resistance at the liquid–vapor interface. The numerical results indicate that the effect of the forced-flow parameter increases as the thickness of the liquid condensate layer increases. Moreover, it is found that a higher forced-flow intensity increases the condensate flow rate, but has no more than a marginal effect on the temperature profile in the condensate layer. In the extreme case of a tube with an infinite radius, the results show that neglecting the inertia term induces a significant error. Accordingly, inertia effects must be taken into account in condensation problems characterized by higher values of the forced-flow parameter (n) and Jakob number (Ja). 2007 Elsevier Ltd. All rights reserved. Keywords: Film condensation; Vertical tube; Mixed convection
1. Introduction The problem of heat transfer with phase change has received extensive attention in the literature due to its wide range of applications in such diverse fields as heat exchange systems, chemical processing plants, evaporative condensers, distillation facilities, chemical vapor deposition processes, and many other industrial applications. The heat transfer rates of laminar film condensation on vertical or nearly vertical surfaces were first predicted by Nusselt [1] in 1916. In Nusselt’s analysis, the condensate film was assumed to be thin and convective and to have negligible inertial effects. Furthermore, it was assumed that the shear stress at the liquid–vapor interface was zero and that the temperature profile within the condensate film was linear. Following the publication of Nusselt’s findings, many researchers attempted to improve the accuracy of *
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1359-4311/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.04.012
the original analysis by implementing more realistic assumptions [2–5]. Fujii [6] provides an excellent review of these studies in his investigation into pure forced and pure free condensation. Prior to 1980, very few analytical studies were presented relating to the problem of mixed-convection film condensation [7–9]. However, in those studies which were presented, the authors generally neglected the effects of inertia and convection in the condensate liquid phase and assumed a zero resistance at the liquid–vapor interface. In the 1990s, Shu and Wilks [10,11] studied the mixed convection of saturated vapor along a vertical plate using a perturbation series method and a modified Cebeci box method to solve the corresponding set of coupled differential equations. Winkler and Chen [12,13] investigated the condensation of saturated and superheated vapors along an isothermal vertical plate in forced-convection-dominated and freeconvection-dominated regimes. In their studies, the governing equations were transformed into dimensionless form by means of a group of transformation variables.
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T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
Nomenclature Cp f F g G Grx h hfg H Ja k m_ Nu Pr qw R Re r, x T Tsat
specific heat dimensionless stream function for condensate liquid layer (defined in Eq. (17)) dimensionless stream function for vapor layer (defined in Eq. (20)) gravitational acceleration or nonsimilarity parameter (defined in Eq. (44)) nonsimilarity parameter (defined in Eq. (44)) local Grashof number local heat transfer coefficient latent heat condensation number (defined in Eq. (33)) Jakob number (defined in Eq. (87)) thermal conductivity condensate mass flux Nusselt number Prandtl number surface heat flux viscosity ratio (defined in Eq. (32)) Reynolds number radial and vertical coordinates temperature saturation temperature
The present study performs a theoretical investigation into the problem of heat transfer in a condensate layer on a vertical isothermal tube under mixed-convection conditions. Importantly, the analysis takes account of the inertia and convection effects in the condensate layer and the resistance at the liquid–vapor interface. The complex system of governing equations is transformed into dimensionless form using a dimensionless transformation method, and the resulting equations are then solved using the local nonsimilarity method [14]. When the tube has an infinite radius, the problem becomes one of heat transfer in a condensate layer on a vertical isothermal plate. The results obtained for this extreme case are compared with those presented by Rohsenow [2] and Fujii and Uehara [8] for a vertical plate in order to determine the respective effects of the inertia and convection terms on the heat transfer coefficient.
Tw u, v
wall temperature vertical and radial velocity components
Greek symbols d liquid film thickness g dimensionless variable (defined in Eqs. (15) and (19)) h dimensionless temperature (defined in Eq. (18)) l dynamic viscosity n forced-flow parameter (defined in Eq. (32)) q density s shear stress m kinematic viscosity / nonsimilarity parameter (defined in Eq. (44)) w stream function (defined in Eqs. (17) and (20)) Superscript quantity in vapor layer * Subscripts 1 free stream condition i liquid–vapor interface w wall condition
Under steady-state conditions, the thickness of the liquid film boundary layer, d, is zero at the leading edge of the tube and increases gradually as the liquid flows downward along the tube surface. Furthermore, a vapor
r = ro
r
o
X
X
liquid vapor interface
δ
u∞ , Tsat
2. Analysis This study investigates the problem of condensation heat transfer for an isothermal vertical tube with wall temperature Tw positioned in a forced flow of saturated vapor. As shown in Fig. 1, the vapor has a downward velocity of u1 and a uniform temperature Tsat. If Tw is lower than Tsat, and the liquid wets the surface of the tube ideally, a thin film of condensate is formed on the tube surface.
Fig. 1. Schematic illustration of film condensation along isothermal vertical cylinder in mixed-convection conditions.
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
velocity boundary layer with a thickness of D is formed between the condensate film and the vapor bulk. In analyzing the current steady-state heat transfer problem, the following assumptions are made: 1. The condensate flow is steady and laminar. 2. The wall temperature and vapor temperature are both uniform and remain constant. 3. The properties of the dry vapor and the condensate remain constant. Applying these assumptions, and taking account of the effects of inertia and convection in the condensate layer and the resistance at the liquid–vapor interface, respectively, the equations of continuity, momentum and energy conservation can be expressed as follows: For the condensate film: continuity ou 1 oðrvÞ þ ¼0 ox r or
ð1Þ
momentum
ou ou q1 1 o ou u þv ¼g 1 r þm ox or r or or q
ð2Þ
energy
oT oT k 1 o oT þv ¼ r u ox or qC p r or or
ð3Þ
For the vapor boundary layer: continuity
ou 1 oðrv Þ ¼0 þ ox r or
ð4Þ
momentum ou ou q 1 o ou r þ v ¼ g 1 1 þ m u r or ox or q or
The boundary conditions for the conservation equations given in Eqs. (1)–(5) are as follows: At the tube surface (r = r0): u ¼ 0;
v ¼ 0;
T ¼ Tw
Note that the energy conservation equation is not required for the vapor boundary layer since the vapor phase is in a saturated condition. In the equations above, the x-coordinate is measured from the leading edge of the tube and the r-coordinate is measured from the center line of the tube. Meanwhile, u and v are the velocity components in the x- and r-directions, respectively, T is the temperature, g is the gravitational acceleration, m is the kinematic viscosity, k is the thermal conductivity, and q is the density. Finally, the asterisk symbol (*) indicates that the corresponding quantity relates to the vapor phase, while the subscripts w, i and 1 denote the cooling surface, the liquid–vapor interface and the free stream in the bulk vapor, respectively.
ð6Þ
In the vapor bulk at the free stream position (r ! 1): ou u ! u1 ; ¼0 ð7Þ or At the liquid–vapor interface (r = r0 + d): u ¼ ui ¼ ui T ¼ T i ¼ T sat ou ou l ¼ l or i or i m_ x ¼ ðqu dd=dx qvÞi ¼ ðq u dd=dx q v Þi oT k ¼ m_ x hfg or i
ð8Þ ð9Þ ð10Þ ð11Þ ð12Þ
where l is the dynamic viscosity, hfg is the latent heat of condensation and m_ x is the local condensation mass flux. Eqs. (8) and (9) express the equality of the interfacial velocity and temperature, respectively, while Eqs. (10)–(12) represent the interfacial shear stress, mass flux and energy flux balances, respectively. The stream functions W and W* are introduced to satisfy the continuity equation, i.e. 1 oðrwÞ ow ; v¼ r or ox 1 oðrw Þ ow ; v ¼ u ¼ r or ox
ð13Þ
u¼
ð14Þ
The conservation equations (Eqs. (1)–(5)) and the boundary and liquid–vapor interfacial conditions (Eqs. (6)–(12)) can be transformed from the original (x, r) coordinates to dimensionless coordinates for the liquid and vapor layers, i.e. [n(x), g(x, r)] and [n(x), g*(x, r)], respectively, by introducing the following transformation variables: g ¼ ðr=xÞðgx3 =4m2 Þ
ð5Þ
549
1=4
n ¼ nðxÞ ¼ u1 =ðgxÞ
¼ ðr=xÞðGrx =4Þ
1=2
1=4
ð15Þ
¼ Rex =Grx1=2 3 2 1=4
ð16Þ
wðx;rÞ ¼ f ðn;gÞ½4mðgx =4m Þ ¼ f ðn; gÞ½4mðGrx =4Þ hðn;gÞ ¼ ðT T sat Þ=ðT w T sat Þ g ¼ ðr=xÞðgx3 =4m2 Þ
1=4
¼ ðr=xÞðGrx =4Þ 3
2 1=4
w ðx;rÞ ¼ F ðn; g Þ½4m ðgx =4m Þ
1=4
1=4
ð17Þ ð18Þ ð19Þ
¼ F ðn; g Þ½4m
ðGrx =4Þ1=4 ð20Þ
where n(x) is the forced-flow parameter, and indicates the effect of the forced vapor flow on free convection. Specifically, the case of n(x) = 0 corresponds to pure free convection, i.e. u1 = 0, while the limiting case of n(x) ! 1 corresponds to pure forced convection. Meanwhile, Rex and Grx are the local Reynolds and Grashof numbers, respectively, and have the forms: Rex ¼ u1 x=m;
Grx ¼ gx3 =m2
ð21Þ
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T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
Substituting Eqs. (15)–(20) into Eqs. (2), (3), (5) and (6)– (12) yields the following system of dimensionless equations: 2 00 f 1 f 1 0 0 f þ 3f þ f f 2f þ þ 2 þ f 5 2 þ 3 þ 1 g g g g g 0 0 f f of f of 1 of f0 þ þ ð22Þ ¼ 2n f 00 þ 2 g on g on g g on h00 1 of f oh h0 ¼ 2n h0 f0 þ þ 4f þ ð23Þ Prg on g on Pr 2 F 1 F 1 F 000 þ 3F þ F 00 F 0 2F 0 þ þ 2 þ F 5 2 þ 3 þ 1 g g g g g F0 F oF F oF 0 1 oF 00 0 þ ð24Þ ¼ 2n F þ 2 F þ g on g on g on g 00
The corresponding boundary conditions are specified as follows: At the tube surface (g = g0): f ðn; g0 Þ ¼ 0;
f 0 ðn; g0 Þ ¼ 0;
hðn; g0 Þ ¼ 1
ð25Þ
In the vapor bulk at the free stream position (g* = 1): F 0 ðn; 1Þ ¼
ðRex =Grx 1=2 Þ n ¼ ; 2 2
F 00 ðn; 1Þ ¼ 0
At the liquid–vapor interface (g = gi, g ¼ f 0 ðn; gi Þ þ
ð26Þ
ð27Þ
hðn; gi Þ ¼ 0 ð28Þ 0 0 f ðn; g Þ f ðn; g Þ F ðn; g Þ F ðn; gi Þ i i 00 i ¼ F R f 00 ðn; gi Þ þ þ ðn; g Þ þ þ i gi g2i gi gi 2
ð29Þ R½4f ðn; gi Þ 2n of ðn; gi Þ=on ¼ 4F ðn; gi Þ 2n oF ðn; gi Þ=on _ h0 ðn; gi Þ ¼ PrM=H ¼ ðPr=H Þ½4f ðn; gi Þ 2n of ðn; gi Þ=on
ð30Þ ð31Þ
In Eqs. (22)–(31), the prime symbol denotes partial differentiation with respect to g (for the f and h functions) or with respect to g* (for the F function). Furthermore, g0 = (r0/x)(Grx/4)1/4 is the value of g at the tube surface (r = r0) and gi = ((r0 + d)/x)(Grx/4)1/4 and gi ¼ ððr0 þ 1=4 dÞ=xÞðGrx =4Þ are the values of g and g*, respectively, at _ are the dimensionless r = r0 + d. Finally, Pr, R, H and M Prandtl number, viscosity ratio, condensation number and condensate mass flux parameter, respectively, and are defined as 1=2 m ql u1 Rex Pr ¼ ; R ¼ ; nðxÞ ¼ ¼ 1=2 ð32Þ 1=2 a q l Grx ðgxÞ H ¼ C p ðT sat T w Þ=hfg 1=4 of ðn; gi Þ _ ¼ m_ x x Grx M ¼ 4f ðn; gi Þ 2n l on 4
1 oðrwÞ ðGrx =4Þ1=2 0 f ðn; gÞ f ðn; gÞ þ ð35Þ ¼ 4u1 Rex r or g 1=4 ow Grx u1 of ðn;gÞ 3f ðn;gÞ ð36Þ gf 0 ðn; gÞ þ 2n v¼ ¼ ox on 4 Rex 1=2 1 oðrw Þ ðGrx =4Þ F ðn; g Þ 0 ¼ 4u1 u ¼ F ðn;g Þ þ ð37Þ r or g Rex 1=4 ow Grx u1 oF ðn;g Þ 0 ¼ g v ¼ F ðn; g Þ þ 2n Þ 3F ðn;g ox 4 Rex on u¼
ð38Þ (
s¼l
3=4 ou lm Grx ¼ 4 2 4 or x
)
f 00 ðn; gÞ þ
f 0 ðn; gÞ f ðn; gÞ g g2
ð39aÞ
( 3=4 ) ou l m Grx F 0 ðn;g Þ F ðn; g Þ 00 ¼ 4 2 F ðn;g Þ þ s ¼l g g2 or x 4
ð39bÞ 1=4 l Grx of ðn; gi Þ 4f ðn; gi Þ 2n ð40Þ x 4 on oT 1=4 ¼ ðk=xÞðT w T sat ÞðGrx =4Þ h0 ðn; g0 Þ ð41Þ qw ¼ k or r¼r0 m_ x ¼
gi ):
f ðn; gi Þ F ðn; gi Þ ¼ F 0 ðn; gi Þ þ gi gi
coefficient h = qw/(Tw Tsat), where qw ¼ kðoT =orÞjr¼r0 is the local surface heat flux; and the local Nusselt number Nux = hx/k. These quantities are defined respectively as
ð33Þ ð34Þ
In the current analysis, the physical quantities of interest include the velocity components u (u*) and v (v*) in the x- and r-directions, respectively; the shear stress s; the local condensate mass flux m_ x ; the local surface heat transfer
h ¼ qw =ðT w T sat Þ ¼ ðk=xÞðGrx =4Þ 0
Nux;w ¼ hx=k ¼ h ðn; g0 ÞðGrx =4Þ
1=4 0
h ðn; g0 Þ
1=4
ð42Þ ð43Þ
Eqs. (22)–(30) collectively describe the mixed convection of a condensate layer flowing along a vertical tube in a forced saturated vapor flow. Meanwhile, Eqs. (12) and (31) denote an energy balance at the liquid–vapor interface and are not required when deriving the numerical solutions of functions f(n, g), h(n, g) and F(n, g*). 3. Numerical solution method Under the assumption that the values of the Prandtl number, Pr, viscosity ratio, R, tube radius, g0, liquid condensate thickness, gi, and forced-flow parameter, n(x), are all known, Eqs. (22)–(31) can be solved using the local nonsimilarity method with truncation at the second level [14] and the finite difference method. In applying the local nonsimilarity method, the parameters of , oh and oF in Eqs. (22)– on on on (31) are defined as g¼
of ; on
/¼
oh ; on
G¼
oF on
ð44Þ
Accordingly, Eqs. (22)–(31) are rewritten as 2 f 1 f 1 f 000 þ 3f þ f 00 f 0 2f 0 þ þ 2 þ f 5 2 þ 3 þ 1 g g g g g 0 f f f g g0 þ ¼ 2n f 00 þ 2 g f 0 þ g g g g
ð45Þ
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
h00 1 f þ 4f þ ð46Þ h0 ¼ 2n h0 g f 0 þ / Pr Prg g 2 F 1 F 1 F 000 þ 3F þ F 00 F 0 2F 0 þ þ 2 þ F 5 2 þ 3 þ 1 g g g g g F0 F F G ð47Þ ¼ 2n F 00 þ 2 G F 0 þ G0 þ g g g g f ðn; g0 Þ ¼ 0; f 0 ðn; g0 Þ ¼ 0; hðn; g0 Þ ¼ 1 ðRex =Gr1=2 x Þ
n ¼ ; F 00 ðn;1Þ ¼ 0 2 2 f ðn; gi Þ F ðn; gi Þ 0 0 f ðn;gi Þ þ ¼ F ðn;gi Þ þ gi gi F 0 ðn;1Þ ¼
ð48Þ ð49Þ ð50Þ
hðn; gi Þ ¼ 0 ð51Þ 0 0 f ðn; gi Þ f ðn;gi Þ F ðn;gi Þ F ðn;gi Þ ¼ F 00 ðn; gi Þ þ R f 00 ðn; gi Þ þ þ þ gi g2i gi gi 2 R½4f ðn; gi Þ 2ngðn; gi Þ ¼ 4F ðn;gi Þ 2nGðn; gi Þ h0 ðn;gi Þ ¼ ðPr=HÞ½4f ðn; gi Þ 2ngðn; gi Þ
ð52Þ ð53Þ ð54Þ
When using the local nonsimilarity method with truncation at 2the second level, the terms involving n og , n o/ and on on o f oG o2 h o2 F n on (n on2 ; n on2 and n on2 Þ are neglected. Consequently, the new system of governing equations and boundary conditions for g, / and G becomes: 2 f 1 f0 f 1 g000 þ 3f þ g00 g0 2f 0 þ 2 þ g f 00 2 7 2 þ 3 g g g g g g 2 ! 0 g g g ¼ 2n g00 þ 2 g g0 þ ð55Þ g g g /00 1 f g þ 4f þ /0 þ 2 f 0 þ / þ 2gh0 ¼ 2n g/0 g0 þ / Pr Prg g g
ð56Þ
2 F 1 F0 F 1 G000 þ 3F þ G00 G0 2F 0 þ 2 þ G F 00 2 7 2 þ 3 g g g g g g 2 ! 0 G G G ¼ 2n G00 þ 2 G G0 þ ð57Þ g g g gðn; g0 Þ ¼ 0; g0 ðn;g0 Þ ¼ 0; /ðn; g0 Þ ¼ 0 1 G0 ðn; 1Þ ¼ ; G00 ðn; 1Þ ¼ 0 2 gðn; gi Þ Gðn; gi Þ 0 g ðn; gi Þ þ ¼ G0 ðn; gi Þ þ gi gi
ð58Þ ð59Þ ð60Þ
/ðn; gi Þ ¼ 0 ð61Þ 0 0 g ðn; g Þ gðn; g Þ G ðn; g Þ Gðn; g Þ i i i i R g00 ðn; gi Þ þ þ þ ¼ G00 ðn; gi Þ þ gi gi g2i gi 2 Rgðn; gi Þ ¼ Gðn; gi Þ /0 ðn; gi Þ ¼ 2
Pr gðn;gi Þ H
ð62Þ ð63Þ ð64Þ
For given values of the Prandtl number and viscosity ratio, Eqs. (45)–(47) and (55)–(57) form a system of coupled partial differential equations for functions f, h, F, g, / and G which is dependent on parameter n and has a total order of 16. In general, these partial differential equations and corresponding boundary conditions, i.e. Eqs. (48)– (54) and (58)–(64), respectively, can be solved using a
551
forward Runge–Kutta method starting at g = g0 and proceeding toward a suitable value of g1 chosen as an approximation to g* = 1. The solution procedure commences by specifying appropriate values for parameters Pr, R, n, g0 (i.e. the dimensionless tube radius) and gi g0 (i.e. the dimensionless liquid film thickness). Initial estimates are then made of f00 , g00 , h 0 and / 0 at the tube surface (i.e. f00 (n, g0), g00 (n, g0), h 0 (n, g0) and / 0 (n, g0)) and these estimates are then substituted into Eqs. (45), (46), (55) and (56). Applying the boundary conditions given in Eqs. (48) and (58), the values of f00 , g00 , f 0 , g 0 , f, g, h 0 , / 0 , h and / are computed at each grid point between the tube surface, i.e. g = g0, and the liquid–vapor interface, i.e. g = gi. Subsequently, the boundary condition of the dimensionless temperature at the liquid–vapor interface is checked using Eqs. (51) and (61), and the initial estimates of h 0 (n, g0) and / 0 (n, g0) are modified accordingly. This process is repeated iteratively until the solutions of Eqs. (51) and (61) satisfy the following convergence criterion: jhðn; gi Þj 6 e;
j/ðn; gi Þj 6 e
where e ¼ 105
ð65Þ
Once the convergence criterion in Eq. (65) has been satisfied, the initial values of F00 , G00 , F 0 , G 0 , F and G at the liquid–vapor interface (i.e. F 00 ðn; gi Þ; G00 ðn; gi Þ; F 0 ðn; gi Þ; G0 ðn; gi Þ; F ðn; gi Þ and Gðn; gi Þ) are derived using the boundary conditions given in Eqs. (50), (52), (53), (60), (62) and (63). Substituting these values into Eqs. (47) and (57), the values of F00 , G00 , F 0 , G 0 , F and G are obtained at all grid points between the liquid–vapor interface, i.e. g ¼ gi , and g* = 1 (approximated by g* = 10 in the present analysis). These values are then checked using the boundary conditions given in Eqs. (49) and (59), and the initial estimates of f00 (n, g0), g00 (n, g0), h 0 (n, g0) and / 0 (n, g0) are modified accordingly. The computational process then returns to the beginning of the solution procedure. The steps described above are repeated iteratively until the solutions of Eqs. (49) and (59) satisfy the following convergence criterion: 0 F ðn; 1Þ n 6 e; j F 00 ðn; 1Þ j6 e; 2 0 G ðn; 1Þ 1 6 e; jG00 ðn; 1Þj 6 e ð66Þ 2 4. Extreme case for comparison purposes In order to validate the current numerical model and to determine the respective effects of the inertia and convection terms on the heat transfer coefficient, a comparison is made between the results obtained for mixed-convection film condensation along the outside surface of a vertical tube with an infinite radius and those presented in the literature for a condensation layer on a vertical isothermal plate under mixed-convection conditions. In considering the case of a tube with an infinite radius, the system of
552
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
governing equations given in Eqs. (45)–(64) is modified by setting parameters g, gi and g* to infinity, and assigning parameter g at the plate surface, i.e. g0, a value of zero. Eqs. (45)–(64) can then be rewritten as f 000 þ 3ff 00 2f 02 þ 1 ¼ 2nðf 00 g f 0 g0 Þ
ð67Þ
00
h þ 4f h0 ¼ 2nðh0 g f 0 /Þ Pr F 000 þ 3FF 00 2F 02 þ 1 ¼ 2nðF 00 G F 0 G0 Þ
ð69Þ
f 0 ðn; 0Þ ¼ 0;
ð70Þ
f ðn; 0Þ ¼ 0;
ðRex =Grx1=2 Þ
F 0 ðn; 1Þ ¼
2 f 0 ðn; gi Þ ¼ F 0 ðn; gi Þ hðn; gi Þ ¼ 0
hðn; 0Þ ¼ 1
n ¼ ; 2
F 00 ðn; 1Þ ¼ 0
ð68Þ
ð71Þ ð72Þ ð73Þ
1=4 Nux 1 Z pffiffiffiffiffiffiffi ¼ K 1 þ 4 Rex 4K H =Pr where K ¼ 0:45 1:2 þ
1 RH =Pr
1=3
Nux ðGrx =4Þ
1=4
¼ 41=4 n0:5 K 1 þ
ð74Þ ð75Þ ð76Þ
5. Results and discussion
g000 þ 3fg00 2f 0 g0 þ gf 00 ¼ 2nðg00 g g02 Þ
ð77Þ
R½f ðn; gi Þ ¼ F
ðn; gi Þ
/00 þ 4f /0 þ 2f 0 / þ 2gh0 ¼ 2n½g/0 g0 / Pr G000 þ 3FG00 2F 0 G0 þ F 00 G ¼ 2nðG00 G G02 Þ gðn; 0Þ ¼ 0;
g0 ðn; 0Þ ¼ 0;
/ðn; 0Þ ¼ 0
ð78Þ ð79Þ ð80Þ
1 G0 ðn; 1Þ ¼ ; G00 ðn; 1Þ ¼ 0 2 g0 ðn; gi Þ ¼ G0 ðn; gi Þ
ð82Þ
/ðn; gi Þ ¼ 0 Rg00 ðn; gi Þ ¼ G00 ðn; gi Þ
ð83Þ ð84Þ
Rgðn; gi Þ ¼ Gðn; gi Þ Pr /0 ðn; gi Þ ¼ 2 gðn; gi Þ H
ð85Þ
ð81Þ
ð86Þ
The system of governing equations given in Eqs. (67)– (86) is solved using the same procedure as that outlined above for the original set of governing equations. Substituting Eq. (33) into Eq. (76) yields: Nux ðGrx =4Þ
1=4
C ðT
¼
Pr ð4f ðn; gi Þ 2ngðn; gi ÞÞ Ja
T Þ
ð87Þ
C DT
where Ja ¼ p hsatfg w ¼ hpfg . For the problem of laminar film condensation on a vertical plate under pure free convection, Rohsenow [2] proposed the following local Nusselt number formulation: 1=4 qgx3 hfg Nux ¼ 0:707 ð88Þ kmðT sat T w Þ Applying the definition for Grx given in Eq. (21), Eq. (88) becomes Nux ðGrx =4Þ
1=4
¼ ðPr=JaÞ
1=4
ð89Þ
Fuji and Uehara [8] proposed the following formulation for the local Nusselt number in laminar film condensation on a vertical plate under mixed convection with negligible inertia effects in the liquid and vapor phases:
gx : U 21
Z¼
Pr 4K 4 n2 Ja
1=4 ð91Þ
where
R½4f ðn; gi Þ 2ngðn; gi Þ ¼ 4F ðn; gi Þ 2nGðn; gi Þ h0 ðn; gi Þ ¼ ðPr=H Þ½4f ðn; gi Þ 2ngðn; gi Þ
00
and
Applying the definitions given in Eqs. (16) and (21), Eq. (90) can be rewritten as
K ¼ 0:45 1:2 þ
00
ð90Þ
Pr R Ja
1=3
In the present analysis, the values of Pr and R used in the governing equations and corresponding boundary conditions (Eqs. (45)–(64) and (67)–(87)) are interpolated from the data presented in [15] for saturated water-vapor at 100 C and 1 atm. The values of the various dimensional and dimensionless physical parameters are summarized in Table 1. Fig. 2 compares the results obtained from the present numerical method for the particular case of a tube with an infinite radius with those presented by Rohsenow [2] and Fujii and Uehara [8] for a vertical plate under pure natural convection condensation (n = 0) and mixed convection (n = 5). In general, the results show that Nux decreases as Ja increases. A larger value of Ja indicates a greater temperature difference between the plate and the vapor. Under such conditions, a thicker condensate layer is formed on the plate, and hence the value of the Nusselt number decreases. The results obtained by Rohsenow [2] and Fujii and Uehara [8] for pure natural convection condensation are virtually identical. Furthermore, the error between these results and those obtained using the current numerical method is less than 1%. Sparrow and Gregg [3] reported that neglecting inertia terms in pure natural convection film condensation induces an error of less than 2%. The good agreement between the current results and Table 1 Values of physical parameters assumed for current saturated water-vapor system Water at 100 C Cp hfg Pr q, q* l, l* k 1=2 R ¼ qql l
4210 J kg1 K1
Vapor at 100 C
– 2257 kJ kg1 1.76 – 957.9 kg m3 0.5956 kg m3 279 · 106 kg m1 s1 12.02 · 106 kg m1 s1 3 1 1 680 · 10 W m K – 193.2
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555 3.2
: Fujji ξ = 5
2.8
: present ξ = 5
2.6
2.4
: Fujji ξ = 0 : Rohsenow ξ = 0 : present ξ = 0
2.2 0.02
0.03
0.04
0.05
0.06
0.07
Ja
Fig. 2. Comparison of solutions presented by Rohsenow [2], Fujii and Uehara [8] and present study for problem of film condensation on vertical plate at n = 0 and n = 5.
those presented in [2] and [8] exists since both previous studies neglected the effects of inertia in the liquid and vapor phases. For the case of mixed-convection condensation (n = 5), it can be seen that the difference between the current results and those presented in [8] increases with increasing Ja, and reaches a value of approximately 3% at Ja = 0.6. As discussed above, a larger value of Ja implies a thicker condensate layer, and therefore, an increased inertia effect. Overall, the results presented in Fig. 2 indicate that the effects of inertia should be taken into account in condensation problems characterized by higher values of n and Ja. Þ In Eqs. (35) and (37), f 0 ðn; gÞ þ f ðn;gÞ and F 0 ðn;g Þ þ F ðn;g g g represent the dimensionless velocity profiles within the liquid film and the vapor phase, respectively. Fig. 3 illus-
trates the variations of these two velocity profiles with the forced-flow parameter, n, for a tube radius of g0 = 1 and a condensate layer thickness of gi g0 = 0.2 (or gi = 1.2). It is observed that both f 0 ðn;gÞ þ f ðn;gÞ and g Þ F 0 ðn; g Þ þ F ðn;g increase as the value of n increases. As a g consequence, it can be inferred that the interfacial velocity increases with increasing n. In other words, the results show that the condensate flow rate can be enhanced by increasing the forced-flow intensity of the vapor bulk. f 0 ðn;gÞ f ðn;gÞ 00 In Eqs. 0(39a) and (39b), f ðn; gÞ þ and g g2 Þ Þ F ðn;g F ðn;g F 00 ðn; g Þ þ g g2 represent the dimensionless shear stress profiles within the liquid film and the vapor phase, respectively. Fig. 4 shows the variation of these two shear stress profiles with the forced-flow parameter, n, for a tube radius and condensate layer thickness of g0 = 1 and gi g0 = 0.2, respectively. It is seen that the shear stress in both the liquid film and the vapor phase increases at higher values of n. As a consequence, it can be inferred that the shear stress at the liquid–vapor interface, i.e. g = gi, increases significantly as the value of n is increased. Thus, it is apparent that the general assumption of a zero interfacial shear stress is invalid for the case where the vapor is subject to a forced flow. The shear stress acting at the wall is an important parameter in many engineering applications. Fig. 5 plots the dimensionless wall shear stress, sw(x2/4lt)(Grx/4)3/4 0 ðn;g Þ f 0Þ (i.e. equal to f 00 ðn; g0 Þ þ g0 0 f ðn;g Þ against the forcedg20 flow parameter, n, as a function of the condensate film thickness for a constant tube radius of g0 = 1. It can be seen that the thickness of the condensate layer has a significant effect on the magnitude of the wall shear stress. The physical reason for this is that higher values of the condensate film thickness, gi g0, imply the farther from the leading edge, which results in a higher velocity and therefore a higher wall shear stress. In other words, the greater the distance from the leading edge, the greater the thickness of the condensate film and hence the higher the gravitationally induced velocity.
3
0.04
0.03
f'(ξ,η)+f (ξ,η)/η
η o = 1.0 η i = 1.2
ξ= 5 2 1 0.5 0
2
ξ= 5 =2 =1 = 0.5 =0
0.02
1 0.01
F'(ξ,η*)+F(ξ,η*)/η*
Nux(Grx/4)-1/4
3
553
η o = 1.0 η i = 1.2 0
0 1
1.04
1.08
1.12
η
1.16
1.2
0
0.4
0.8
1.2
1.6
2
η*-η*ι
Fig. 3. Dimensionless velocity profiles within liquid film and saturated vapor region for n = 0, 0.5, 1, 2 and 5 at gi = 1.2 (dimensionless film thickness = 0.2).
554
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
8
0.2 6 ξ= 5 2 1 0.5 0
0.1
ξ=5
4
=2 =1 = 0.5 =0
2
F''(ξ,η*)+F'(ξ,η*)/η*-F(ξ,η*)/η*
2
f''(ξ,η)+f'(ξ,η)/η-f (ξ,η)/η
η o = 1.0 η i = 1.2
η o = 1.0 η i = 1.2
2
10
0.3
0 0 1
1.04
1.08
1.12
1.16
1.2
0
0.4
0.8
1.2
1.6
2
η*-η*ι
η
Fig. 4. Dimensionless shear stress profiles within liquid film and saturated vapor region for n = 0, 0.5, 1, 2 and 5 at gi = 1.2 (dimensionless film thickness = 0.2).
1.2
1
: ηi -η0 = 0.5 : ηi -η0 = 0.4 : ηi -η0 = 0.3
ηi -η0 = 0.4 0.3 0.2
0.8
η0 = 1.0 0.8
0.6
η0 = 1.0 ξ = 1.0, 3.0, 5.0
θ
τ w(x2/4μυ)(Grx /4)-3/4
1
0.6
0.4
0.4
0.2
0.2
0 0
1
2
3
4
5
ξ Fig. 5. Variation of local wall shear stress with n.
Fig. 6 presents the temperature profiles in the condensate layer, h 0 (n, g), for various values of the condensate film thickness and forced-flow parameter, respectively. (Note that a constant tube radius of g0 = 1 is assumed.) It can be seen that the temperature profile is independent on n and linearly varies from h = 1 at the tube surface to h = 0 at the liquid–vapor interface. In problems involving film condensation, one of the most important parameters is the Nusselt number. Eq. (43) shows that the value of the Nusselt number varies as a function of h 0 (n, g0). Fig. 7 plots the local Nusselt number against the forced-flow parameter, n, for various values of the condensate film thickness and a constant tube radius of g0 = 1. From a close inspection, it is found that the local Nusselt number is independent on n. The results also show that the Nusselt number decreases as the condensate film
1
1.1
1.2
1.3
1.4
1.5
η Fig. 6. Dimensionless temperature profiles within liquid film.
thickness increases. The physical reason for this is that a higher film thickness implies a greater thermal resistance, and hence a lower Nusselt number. Equivalently, the thinner the condensate layer, the steeper the temperature gradient (see Fig. 6), and thus the greater the heat transfer rate. Fig. 8 illustrates the variation of the dimensionless condensate mass flux with the forced-flow parameter as a function of the condensate film thickness and a constant tube radius of g0 = 1. In general, the results show that the condensate mass flux increases with increasing n. This result is physically reasonable since a higher value of the forcedflow parameter implies an increased vapor velocity at the liquid–vapor interface, and then induce an increase in the condensate mass flux. Fig. 8 also shows that the effect of n on the condensate mass flux becomes more pronounced
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
tical tube in dry saturated vapor with a forced flow. A dimensionless parameter, n, has been introduced to represent the forced-flow intensity of the vapor bulk. The results have shown that a higher value of n enhances the condensate flow rate and induces a higher shear stress at the wall. Furthermore, it has been shown that the common assumption of zero interfacial shear stress is invalid for the case of a forced flow of the vapor bulk. The results of this study provide a useful source of reference for designers aiming to improve the thermal performance of shell-and-tube heat exchangers and distillation processes.
5.5
5
Nux(Grx /4)-1/4 = θ'(ξ,η0)
η0 = 1.0 : ηi -η0 = 0.2
4.5
: ηi -η0 = 0.3 : ηi -η0 = 0.4
4
555
3.5
Acknowledgements 3
This work is partially supported by the National Science Council of Taiwan under contract NSC94-2212-E-218-017.
2.5 0
1
2
3
4
5
References
ξ Fig. 7. Variation of local Nusselt number with n.
0.4
: ηi -η0 = 0.5 : ηi -η0 = 0.4 : ηi -η0 = 0.3
mx(Grx /4)-1/4(x/μ)
0.3
η 0 = 1.0 0.2
.
0.1
0 0
1
2
3
4
5
ξ Fig. 8. Variation of dimensionless condensation mass flux with n.
as the thickness of the condensate layer increases. This is to be expected since, as mentioned previously, a higher value of the condensate film thickness implies the farther from the leading edge, and hence the effect of n on the liquid velocity, and thus on the condensate mass flux, increases. 6. Conclusion This study has analyzed the steady-state mixed convection of a condensate film flowing along an isothermal ver-
[1] W. Nusselt, Die oberflachen Kondensation des Wasserdampes, Z. Ver. Deut. Ing. 60 (2) (1916) 541–546. [2] W.M. Rohsenow, Heat transfer and temperature distribution in laminar film condensation, Trans. ASME, J. Heat Transfer 78 (1956) 1645–1648. [3] E.M. Sparrow, J.L. Gregg, Laminar condensation heat transfer on a horizontal cylinder, Trans. ASME, J. Heat Transfer 81 (1959) 291–296. [4] W.J. Minkowycz, E.M. Sparrow, Condensation heat transfer in the presence of noncondensables, interfacial resistance, superheating, variable properties, and diffusion, Int. J. Heat Mass Transfer 6 (6) (1966) 1125–1144. [5] W.J. Minkowycz, E.M. Sparrow, The effect of superheating on condensation heat transfer in a forced convection boundary layer flow, Int. J. Heat Mass Transfer 12 (1) (1969) 147–154. [6] T. Fujii, Theory of Laminar Film Condensation, Springer-Verlag, New York, 1991 (Chapters 1–5). [7] H.R. Jacobs, An integral treatment of combined body force and forced convection in laminar film condensation, Int. J. Heat Mass Transfer 9 (4) (1966) 637–648. [8] T. Fujii, H. Uehara, Laminar filmwise condensation on a vertical surface, Int. J. Heat Mass Transfer 15 (2) (1972) 217–233. [9] K. Lucas, Combined body force and forced convection in laminar film condensation of mixed vapours – integral and finite difference treatment, Int. J. Heat Mass Transfer 19 (7) (1976) 1273–1280. [10] J. Shu, G. Wilks, Mixed-convection laminar film condensation on a semi-infinite vertical plate, J. Fluid Mech. 300 (1995) 207–229. [11] J. Shu, G. Wilks, An accurate numerical method for systems of convection differential–integral equations associated with multiphase flow, Comput. Fluids 24 (6) (1995) 625–652. [12] C.M. Winkler, T.S. Chen, W.J. Minkowycz, Film condensation of saturated and superheated vapors along isothermal vertical surfaces in mixed convection, Numer. Heat Transfer, Part A 36 (2) (1999) 375– 393. [13] C.M. Winkler, T.S. Chen, Mixed convection in film condensation from isothermal vertical surfaces—the entire regime, Int. J. Heat Mass Transfer 43 (2000) 3245–3251. [14] T.S. Chen, Parabolic systems: local nonsimilarity method, in: W.J. Minkowycz (Ed.), Handbook of Numerical Heat Transfer, John Wiley and Sons, New York, 1988, pp. 183–214. [15] J.P. Holman, Heat Transfer, McGraw-Hill, New York, 2002, p. 606.
Applied Thermal Engineering 28 (2008) 547–555 www.elsevier.com/locate/apthermeng
Mixed-convection film condensation along outside surface of vertical tube in saturated vapor with forced flow Tong-Bou Chang
*
Department of Mechanical Engineering, Southern Taiwan University of Technology, Tainan County, Taiwan, ROC Received 12 January 2007; accepted 25 April 2007 Available online 5 May 2007
Abstract This study employs the nonsimilar transformation method to perform a comprehensive investigation into the steady-state mixed convection of a condensate film on an isothermal vertical tube in dry saturated vapor with a forced flow. The analysis takes account of the inertia and convection effects in the condensate layer and the resistance at the liquid–vapor interface. The numerical results indicate that the effect of the forced-flow parameter increases as the thickness of the liquid condensate layer increases. Moreover, it is found that a higher forced-flow intensity increases the condensate flow rate, but has no more than a marginal effect on the temperature profile in the condensate layer. In the extreme case of a tube with an infinite radius, the results show that neglecting the inertia term induces a significant error. Accordingly, inertia effects must be taken into account in condensation problems characterized by higher values of the forced-flow parameter (n) and Jakob number (Ja). 2007 Elsevier Ltd. All rights reserved. Keywords: Film condensation; Vertical tube; Mixed convection
1. Introduction The problem of heat transfer with phase change has received extensive attention in the literature due to its wide range of applications in such diverse fields as heat exchange systems, chemical processing plants, evaporative condensers, distillation facilities, chemical vapor deposition processes, and many other industrial applications. The heat transfer rates of laminar film condensation on vertical or nearly vertical surfaces were first predicted by Nusselt [1] in 1916. In Nusselt’s analysis, the condensate film was assumed to be thin and convective and to have negligible inertial effects. Furthermore, it was assumed that the shear stress at the liquid–vapor interface was zero and that the temperature profile within the condensate film was linear. Following the publication of Nusselt’s findings, many researchers attempted to improve the accuracy of *
Tel.: +886 6 2533131x3533; fax: +886 6 2425092. E-mail address: [email protected]
1359-4311/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.04.012
the original analysis by implementing more realistic assumptions [2–5]. Fujii [6] provides an excellent review of these studies in his investigation into pure forced and pure free condensation. Prior to 1980, very few analytical studies were presented relating to the problem of mixed-convection film condensation [7–9]. However, in those studies which were presented, the authors generally neglected the effects of inertia and convection in the condensate liquid phase and assumed a zero resistance at the liquid–vapor interface. In the 1990s, Shu and Wilks [10,11] studied the mixed convection of saturated vapor along a vertical plate using a perturbation series method and a modified Cebeci box method to solve the corresponding set of coupled differential equations. Winkler and Chen [12,13] investigated the condensation of saturated and superheated vapors along an isothermal vertical plate in forced-convection-dominated and freeconvection-dominated regimes. In their studies, the governing equations were transformed into dimensionless form by means of a group of transformation variables.
548
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
Nomenclature Cp f F g G Grx h hfg H Ja k m_ Nu Pr qw R Re r, x T Tsat
specific heat dimensionless stream function for condensate liquid layer (defined in Eq. (17)) dimensionless stream function for vapor layer (defined in Eq. (20)) gravitational acceleration or nonsimilarity parameter (defined in Eq. (44)) nonsimilarity parameter (defined in Eq. (44)) local Grashof number local heat transfer coefficient latent heat condensation number (defined in Eq. (33)) Jakob number (defined in Eq. (87)) thermal conductivity condensate mass flux Nusselt number Prandtl number surface heat flux viscosity ratio (defined in Eq. (32)) Reynolds number radial and vertical coordinates temperature saturation temperature
The present study performs a theoretical investigation into the problem of heat transfer in a condensate layer on a vertical isothermal tube under mixed-convection conditions. Importantly, the analysis takes account of the inertia and convection effects in the condensate layer and the resistance at the liquid–vapor interface. The complex system of governing equations is transformed into dimensionless form using a dimensionless transformation method, and the resulting equations are then solved using the local nonsimilarity method [14]. When the tube has an infinite radius, the problem becomes one of heat transfer in a condensate layer on a vertical isothermal plate. The results obtained for this extreme case are compared with those presented by Rohsenow [2] and Fujii and Uehara [8] for a vertical plate in order to determine the respective effects of the inertia and convection terms on the heat transfer coefficient.
Tw u, v
wall temperature vertical and radial velocity components
Greek symbols d liquid film thickness g dimensionless variable (defined in Eqs. (15) and (19)) h dimensionless temperature (defined in Eq. (18)) l dynamic viscosity n forced-flow parameter (defined in Eq. (32)) q density s shear stress m kinematic viscosity / nonsimilarity parameter (defined in Eq. (44)) w stream function (defined in Eqs. (17) and (20)) Superscript quantity in vapor layer * Subscripts 1 free stream condition i liquid–vapor interface w wall condition
Under steady-state conditions, the thickness of the liquid film boundary layer, d, is zero at the leading edge of the tube and increases gradually as the liquid flows downward along the tube surface. Furthermore, a vapor
r = ro
r
o
X
X
liquid vapor interface
δ
u∞ , Tsat
2. Analysis This study investigates the problem of condensation heat transfer for an isothermal vertical tube with wall temperature Tw positioned in a forced flow of saturated vapor. As shown in Fig. 1, the vapor has a downward velocity of u1 and a uniform temperature Tsat. If Tw is lower than Tsat, and the liquid wets the surface of the tube ideally, a thin film of condensate is formed on the tube surface.
Fig. 1. Schematic illustration of film condensation along isothermal vertical cylinder in mixed-convection conditions.
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
velocity boundary layer with a thickness of D is formed between the condensate film and the vapor bulk. In analyzing the current steady-state heat transfer problem, the following assumptions are made: 1. The condensate flow is steady and laminar. 2. The wall temperature and vapor temperature are both uniform and remain constant. 3. The properties of the dry vapor and the condensate remain constant. Applying these assumptions, and taking account of the effects of inertia and convection in the condensate layer and the resistance at the liquid–vapor interface, respectively, the equations of continuity, momentum and energy conservation can be expressed as follows: For the condensate film: continuity ou 1 oðrvÞ þ ¼0 ox r or
ð1Þ
momentum
ou ou q1 1 o ou u þv ¼g 1 r þm ox or r or or q
ð2Þ
energy
oT oT k 1 o oT þv ¼ r u ox or qC p r or or
ð3Þ
For the vapor boundary layer: continuity
ou 1 oðrv Þ ¼0 þ ox r or
ð4Þ
momentum ou ou q 1 o ou r þ v ¼ g 1 1 þ m u r or ox or q or
The boundary conditions for the conservation equations given in Eqs. (1)–(5) are as follows: At the tube surface (r = r0): u ¼ 0;
v ¼ 0;
T ¼ Tw
Note that the energy conservation equation is not required for the vapor boundary layer since the vapor phase is in a saturated condition. In the equations above, the x-coordinate is measured from the leading edge of the tube and the r-coordinate is measured from the center line of the tube. Meanwhile, u and v are the velocity components in the x- and r-directions, respectively, T is the temperature, g is the gravitational acceleration, m is the kinematic viscosity, k is the thermal conductivity, and q is the density. Finally, the asterisk symbol (*) indicates that the corresponding quantity relates to the vapor phase, while the subscripts w, i and 1 denote the cooling surface, the liquid–vapor interface and the free stream in the bulk vapor, respectively.
ð6Þ
In the vapor bulk at the free stream position (r ! 1): ou u ! u1 ; ¼0 ð7Þ or At the liquid–vapor interface (r = r0 + d): u ¼ ui ¼ ui T ¼ T i ¼ T sat ou ou l ¼ l or i or i m_ x ¼ ðqu dd=dx qvÞi ¼ ðq u dd=dx q v Þi oT k ¼ m_ x hfg or i
ð8Þ ð9Þ ð10Þ ð11Þ ð12Þ
where l is the dynamic viscosity, hfg is the latent heat of condensation and m_ x is the local condensation mass flux. Eqs. (8) and (9) express the equality of the interfacial velocity and temperature, respectively, while Eqs. (10)–(12) represent the interfacial shear stress, mass flux and energy flux balances, respectively. The stream functions W and W* are introduced to satisfy the continuity equation, i.e. 1 oðrwÞ ow ; v¼ r or ox 1 oðrw Þ ow ; v ¼ u ¼ r or ox
ð13Þ
u¼
ð14Þ
The conservation equations (Eqs. (1)–(5)) and the boundary and liquid–vapor interfacial conditions (Eqs. (6)–(12)) can be transformed from the original (x, r) coordinates to dimensionless coordinates for the liquid and vapor layers, i.e. [n(x), g(x, r)] and [n(x), g*(x, r)], respectively, by introducing the following transformation variables: g ¼ ðr=xÞðgx3 =4m2 Þ
ð5Þ
549
1=4
n ¼ nðxÞ ¼ u1 =ðgxÞ
¼ ðr=xÞðGrx =4Þ
1=2
1=4
ð15Þ
¼ Rex =Grx1=2 3 2 1=4
ð16Þ
wðx;rÞ ¼ f ðn;gÞ½4mðgx =4m Þ ¼ f ðn; gÞ½4mðGrx =4Þ hðn;gÞ ¼ ðT T sat Þ=ðT w T sat Þ g ¼ ðr=xÞðgx3 =4m2 Þ
1=4
¼ ðr=xÞðGrx =4Þ 3
2 1=4
w ðx;rÞ ¼ F ðn; g Þ½4m ðgx =4m Þ
1=4
1=4
ð17Þ ð18Þ ð19Þ
¼ F ðn; g Þ½4m
ðGrx =4Þ1=4 ð20Þ
where n(x) is the forced-flow parameter, and indicates the effect of the forced vapor flow on free convection. Specifically, the case of n(x) = 0 corresponds to pure free convection, i.e. u1 = 0, while the limiting case of n(x) ! 1 corresponds to pure forced convection. Meanwhile, Rex and Grx are the local Reynolds and Grashof numbers, respectively, and have the forms: Rex ¼ u1 x=m;
Grx ¼ gx3 =m2
ð21Þ
550
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
Substituting Eqs. (15)–(20) into Eqs. (2), (3), (5) and (6)– (12) yields the following system of dimensionless equations: 2 00 f 1 f 1 0 0 f þ 3f þ f f 2f þ þ 2 þ f 5 2 þ 3 þ 1 g g g g g 0 0 f f of f of 1 of f0 þ þ ð22Þ ¼ 2n f 00 þ 2 g on g on g g on h00 1 of f oh h0 ¼ 2n h0 f0 þ þ 4f þ ð23Þ Prg on g on Pr 2 F 1 F 1 F 000 þ 3F þ F 00 F 0 2F 0 þ þ 2 þ F 5 2 þ 3 þ 1 g g g g g F0 F oF F oF 0 1 oF 00 0 þ ð24Þ ¼ 2n F þ 2 F þ g on g on g on g 00
The corresponding boundary conditions are specified as follows: At the tube surface (g = g0): f ðn; g0 Þ ¼ 0;
f 0 ðn; g0 Þ ¼ 0;
hðn; g0 Þ ¼ 1
ð25Þ
In the vapor bulk at the free stream position (g* = 1): F 0 ðn; 1Þ ¼
ðRex =Grx 1=2 Þ n ¼ ; 2 2
F 00 ðn; 1Þ ¼ 0
At the liquid–vapor interface (g = gi, g ¼ f 0 ðn; gi Þ þ
ð26Þ
ð27Þ
hðn; gi Þ ¼ 0 ð28Þ 0 0 f ðn; g Þ f ðn; g Þ F ðn; g Þ F ðn; gi Þ i i 00 i ¼ F R f 00 ðn; gi Þ þ þ ðn; g Þ þ þ i gi g2i gi gi 2
ð29Þ R½4f ðn; gi Þ 2n of ðn; gi Þ=on ¼ 4F ðn; gi Þ 2n oF ðn; gi Þ=on _ h0 ðn; gi Þ ¼ PrM=H ¼ ðPr=H Þ½4f ðn; gi Þ 2n of ðn; gi Þ=on
ð30Þ ð31Þ
In Eqs. (22)–(31), the prime symbol denotes partial differentiation with respect to g (for the f and h functions) or with respect to g* (for the F function). Furthermore, g0 = (r0/x)(Grx/4)1/4 is the value of g at the tube surface (r = r0) and gi = ((r0 + d)/x)(Grx/4)1/4 and gi ¼ ððr0 þ 1=4 dÞ=xÞðGrx =4Þ are the values of g and g*, respectively, at _ are the dimensionless r = r0 + d. Finally, Pr, R, H and M Prandtl number, viscosity ratio, condensation number and condensate mass flux parameter, respectively, and are defined as 1=2 m ql u1 Rex Pr ¼ ; R ¼ ; nðxÞ ¼ ¼ 1=2 ð32Þ 1=2 a q l Grx ðgxÞ H ¼ C p ðT sat T w Þ=hfg 1=4 of ðn; gi Þ _ ¼ m_ x x Grx M ¼ 4f ðn; gi Þ 2n l on 4
1 oðrwÞ ðGrx =4Þ1=2 0 f ðn; gÞ f ðn; gÞ þ ð35Þ ¼ 4u1 Rex r or g 1=4 ow Grx u1 of ðn;gÞ 3f ðn;gÞ ð36Þ gf 0 ðn; gÞ þ 2n v¼ ¼ ox on 4 Rex 1=2 1 oðrw Þ ðGrx =4Þ F ðn; g Þ 0 ¼ 4u1 u ¼ F ðn;g Þ þ ð37Þ r or g Rex 1=4 ow Grx u1 oF ðn;g Þ 0 ¼ g v ¼ F ðn; g Þ þ 2n Þ 3F ðn;g ox 4 Rex on u¼
ð38Þ (
s¼l
3=4 ou lm Grx ¼ 4 2 4 or x
)
f 00 ðn; gÞ þ
f 0 ðn; gÞ f ðn; gÞ g g2
ð39aÞ
( 3=4 ) ou l m Grx F 0 ðn;g Þ F ðn; g Þ 00 ¼ 4 2 F ðn;g Þ þ s ¼l g g2 or x 4
ð39bÞ 1=4 l Grx of ðn; gi Þ 4f ðn; gi Þ 2n ð40Þ x 4 on oT 1=4 ¼ ðk=xÞðT w T sat ÞðGrx =4Þ h0 ðn; g0 Þ ð41Þ qw ¼ k or r¼r0 m_ x ¼
gi ):
f ðn; gi Þ F ðn; gi Þ ¼ F 0 ðn; gi Þ þ gi gi
coefficient h = qw/(Tw Tsat), where qw ¼ kðoT =orÞjr¼r0 is the local surface heat flux; and the local Nusselt number Nux = hx/k. These quantities are defined respectively as
ð33Þ ð34Þ
In the current analysis, the physical quantities of interest include the velocity components u (u*) and v (v*) in the x- and r-directions, respectively; the shear stress s; the local condensate mass flux m_ x ; the local surface heat transfer
h ¼ qw =ðT w T sat Þ ¼ ðk=xÞðGrx =4Þ 0
Nux;w ¼ hx=k ¼ h ðn; g0 ÞðGrx =4Þ
1=4 0
h ðn; g0 Þ
1=4
ð42Þ ð43Þ
Eqs. (22)–(30) collectively describe the mixed convection of a condensate layer flowing along a vertical tube in a forced saturated vapor flow. Meanwhile, Eqs. (12) and (31) denote an energy balance at the liquid–vapor interface and are not required when deriving the numerical solutions of functions f(n, g), h(n, g) and F(n, g*). 3. Numerical solution method Under the assumption that the values of the Prandtl number, Pr, viscosity ratio, R, tube radius, g0, liquid condensate thickness, gi, and forced-flow parameter, n(x), are all known, Eqs. (22)–(31) can be solved using the local nonsimilarity method with truncation at the second level [14] and the finite difference method. In applying the local nonsimilarity method, the parameters of , oh and oF in Eqs. (22)– on on on (31) are defined as g¼
of ; on
/¼
oh ; on
G¼
oF on
ð44Þ
Accordingly, Eqs. (22)–(31) are rewritten as 2 f 1 f 1 f 000 þ 3f þ f 00 f 0 2f 0 þ þ 2 þ f 5 2 þ 3 þ 1 g g g g g 0 f f f g g0 þ ¼ 2n f 00 þ 2 g f 0 þ g g g g
ð45Þ
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
h00 1 f þ 4f þ ð46Þ h0 ¼ 2n h0 g f 0 þ / Pr Prg g 2 F 1 F 1 F 000 þ 3F þ F 00 F 0 2F 0 þ þ 2 þ F 5 2 þ 3 þ 1 g g g g g F0 F F G ð47Þ ¼ 2n F 00 þ 2 G F 0 þ G0 þ g g g g f ðn; g0 Þ ¼ 0; f 0 ðn; g0 Þ ¼ 0; hðn; g0 Þ ¼ 1 ðRex =Gr1=2 x Þ
n ¼ ; F 00 ðn;1Þ ¼ 0 2 2 f ðn; gi Þ F ðn; gi Þ 0 0 f ðn;gi Þ þ ¼ F ðn;gi Þ þ gi gi F 0 ðn;1Þ ¼
ð48Þ ð49Þ ð50Þ
hðn; gi Þ ¼ 0 ð51Þ 0 0 f ðn; gi Þ f ðn;gi Þ F ðn;gi Þ F ðn;gi Þ ¼ F 00 ðn; gi Þ þ R f 00 ðn; gi Þ þ þ þ gi g2i gi gi 2 R½4f ðn; gi Þ 2ngðn; gi Þ ¼ 4F ðn;gi Þ 2nGðn; gi Þ h0 ðn;gi Þ ¼ ðPr=HÞ½4f ðn; gi Þ 2ngðn; gi Þ
ð52Þ ð53Þ ð54Þ
When using the local nonsimilarity method with truncation at 2the second level, the terms involving n og , n o/ and on on o f oG o2 h o2 F n on (n on2 ; n on2 and n on2 Þ are neglected. Consequently, the new system of governing equations and boundary conditions for g, / and G becomes: 2 f 1 f0 f 1 g000 þ 3f þ g00 g0 2f 0 þ 2 þ g f 00 2 7 2 þ 3 g g g g g g 2 ! 0 g g g ¼ 2n g00 þ 2 g g0 þ ð55Þ g g g /00 1 f g þ 4f þ /0 þ 2 f 0 þ / þ 2gh0 ¼ 2n g/0 g0 þ / Pr Prg g g
ð56Þ
2 F 1 F0 F 1 G000 þ 3F þ G00 G0 2F 0 þ 2 þ G F 00 2 7 2 þ 3 g g g g g g 2 ! 0 G G G ¼ 2n G00 þ 2 G G0 þ ð57Þ g g g gðn; g0 Þ ¼ 0; g0 ðn;g0 Þ ¼ 0; /ðn; g0 Þ ¼ 0 1 G0 ðn; 1Þ ¼ ; G00 ðn; 1Þ ¼ 0 2 gðn; gi Þ Gðn; gi Þ 0 g ðn; gi Þ þ ¼ G0 ðn; gi Þ þ gi gi
ð58Þ ð59Þ ð60Þ
/ðn; gi Þ ¼ 0 ð61Þ 0 0 g ðn; g Þ gðn; g Þ G ðn; g Þ Gðn; g Þ i i i i R g00 ðn; gi Þ þ þ þ ¼ G00 ðn; gi Þ þ gi gi g2i gi 2 Rgðn; gi Þ ¼ Gðn; gi Þ /0 ðn; gi Þ ¼ 2
Pr gðn;gi Þ H
ð62Þ ð63Þ ð64Þ
For given values of the Prandtl number and viscosity ratio, Eqs. (45)–(47) and (55)–(57) form a system of coupled partial differential equations for functions f, h, F, g, / and G which is dependent on parameter n and has a total order of 16. In general, these partial differential equations and corresponding boundary conditions, i.e. Eqs. (48)– (54) and (58)–(64), respectively, can be solved using a
551
forward Runge–Kutta method starting at g = g0 and proceeding toward a suitable value of g1 chosen as an approximation to g* = 1. The solution procedure commences by specifying appropriate values for parameters Pr, R, n, g0 (i.e. the dimensionless tube radius) and gi g0 (i.e. the dimensionless liquid film thickness). Initial estimates are then made of f00 , g00 , h 0 and / 0 at the tube surface (i.e. f00 (n, g0), g00 (n, g0), h 0 (n, g0) and / 0 (n, g0)) and these estimates are then substituted into Eqs. (45), (46), (55) and (56). Applying the boundary conditions given in Eqs. (48) and (58), the values of f00 , g00 , f 0 , g 0 , f, g, h 0 , / 0 , h and / are computed at each grid point between the tube surface, i.e. g = g0, and the liquid–vapor interface, i.e. g = gi. Subsequently, the boundary condition of the dimensionless temperature at the liquid–vapor interface is checked using Eqs. (51) and (61), and the initial estimates of h 0 (n, g0) and / 0 (n, g0) are modified accordingly. This process is repeated iteratively until the solutions of Eqs. (51) and (61) satisfy the following convergence criterion: jhðn; gi Þj 6 e;
j/ðn; gi Þj 6 e
where e ¼ 105
ð65Þ
Once the convergence criterion in Eq. (65) has been satisfied, the initial values of F00 , G00 , F 0 , G 0 , F and G at the liquid–vapor interface (i.e. F 00 ðn; gi Þ; G00 ðn; gi Þ; F 0 ðn; gi Þ; G0 ðn; gi Þ; F ðn; gi Þ and Gðn; gi Þ) are derived using the boundary conditions given in Eqs. (50), (52), (53), (60), (62) and (63). Substituting these values into Eqs. (47) and (57), the values of F00 , G00 , F 0 , G 0 , F and G are obtained at all grid points between the liquid–vapor interface, i.e. g ¼ gi , and g* = 1 (approximated by g* = 10 in the present analysis). These values are then checked using the boundary conditions given in Eqs. (49) and (59), and the initial estimates of f00 (n, g0), g00 (n, g0), h 0 (n, g0) and / 0 (n, g0) are modified accordingly. The computational process then returns to the beginning of the solution procedure. The steps described above are repeated iteratively until the solutions of Eqs. (49) and (59) satisfy the following convergence criterion: 0 F ðn; 1Þ n 6 e; j F 00 ðn; 1Þ j6 e; 2 0 G ðn; 1Þ 1 6 e; jG00 ðn; 1Þj 6 e ð66Þ 2 4. Extreme case for comparison purposes In order to validate the current numerical model and to determine the respective effects of the inertia and convection terms on the heat transfer coefficient, a comparison is made between the results obtained for mixed-convection film condensation along the outside surface of a vertical tube with an infinite radius and those presented in the literature for a condensation layer on a vertical isothermal plate under mixed-convection conditions. In considering the case of a tube with an infinite radius, the system of
552
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
governing equations given in Eqs. (45)–(64) is modified by setting parameters g, gi and g* to infinity, and assigning parameter g at the plate surface, i.e. g0, a value of zero. Eqs. (45)–(64) can then be rewritten as f 000 þ 3ff 00 2f 02 þ 1 ¼ 2nðf 00 g f 0 g0 Þ
ð67Þ
00
h þ 4f h0 ¼ 2nðh0 g f 0 /Þ Pr F 000 þ 3FF 00 2F 02 þ 1 ¼ 2nðF 00 G F 0 G0 Þ
ð69Þ
f 0 ðn; 0Þ ¼ 0;
ð70Þ
f ðn; 0Þ ¼ 0;
ðRex =Grx1=2 Þ
F 0 ðn; 1Þ ¼
2 f 0 ðn; gi Þ ¼ F 0 ðn; gi Þ hðn; gi Þ ¼ 0
hðn; 0Þ ¼ 1
n ¼ ; 2
F 00 ðn; 1Þ ¼ 0
ð68Þ
ð71Þ ð72Þ ð73Þ
1=4 Nux 1 Z pffiffiffiffiffiffiffi ¼ K 1 þ 4 Rex 4K H =Pr where K ¼ 0:45 1:2 þ
1 RH =Pr
1=3
Nux ðGrx =4Þ
1=4
¼ 41=4 n0:5 K 1 þ
ð74Þ ð75Þ ð76Þ
5. Results and discussion
g000 þ 3fg00 2f 0 g0 þ gf 00 ¼ 2nðg00 g g02 Þ
ð77Þ
R½f ðn; gi Þ ¼ F
ðn; gi Þ
/00 þ 4f /0 þ 2f 0 / þ 2gh0 ¼ 2n½g/0 g0 / Pr G000 þ 3FG00 2F 0 G0 þ F 00 G ¼ 2nðG00 G G02 Þ gðn; 0Þ ¼ 0;
g0 ðn; 0Þ ¼ 0;
/ðn; 0Þ ¼ 0
ð78Þ ð79Þ ð80Þ
1 G0 ðn; 1Þ ¼ ; G00 ðn; 1Þ ¼ 0 2 g0 ðn; gi Þ ¼ G0 ðn; gi Þ
ð82Þ
/ðn; gi Þ ¼ 0 Rg00 ðn; gi Þ ¼ G00 ðn; gi Þ
ð83Þ ð84Þ
Rgðn; gi Þ ¼ Gðn; gi Þ Pr /0 ðn; gi Þ ¼ 2 gðn; gi Þ H
ð85Þ
ð81Þ
ð86Þ
The system of governing equations given in Eqs. (67)– (86) is solved using the same procedure as that outlined above for the original set of governing equations. Substituting Eq. (33) into Eq. (76) yields: Nux ðGrx =4Þ
1=4
C ðT
¼
Pr ð4f ðn; gi Þ 2ngðn; gi ÞÞ Ja
T Þ
ð87Þ
C DT
where Ja ¼ p hsatfg w ¼ hpfg . For the problem of laminar film condensation on a vertical plate under pure free convection, Rohsenow [2] proposed the following local Nusselt number formulation: 1=4 qgx3 hfg Nux ¼ 0:707 ð88Þ kmðT sat T w Þ Applying the definition for Grx given in Eq. (21), Eq. (88) becomes Nux ðGrx =4Þ
1=4
¼ ðPr=JaÞ
1=4
ð89Þ
Fuji and Uehara [8] proposed the following formulation for the local Nusselt number in laminar film condensation on a vertical plate under mixed convection with negligible inertia effects in the liquid and vapor phases:
gx : U 21
Z¼
Pr 4K 4 n2 Ja
1=4 ð91Þ
where
R½4f ðn; gi Þ 2ngðn; gi Þ ¼ 4F ðn; gi Þ 2nGðn; gi Þ h0 ðn; gi Þ ¼ ðPr=H Þ½4f ðn; gi Þ 2ngðn; gi Þ
00
and
Applying the definitions given in Eqs. (16) and (21), Eq. (90) can be rewritten as
K ¼ 0:45 1:2 þ
00
ð90Þ
Pr R Ja
1=3
In the present analysis, the values of Pr and R used in the governing equations and corresponding boundary conditions (Eqs. (45)–(64) and (67)–(87)) are interpolated from the data presented in [15] for saturated water-vapor at 100 C and 1 atm. The values of the various dimensional and dimensionless physical parameters are summarized in Table 1. Fig. 2 compares the results obtained from the present numerical method for the particular case of a tube with an infinite radius with those presented by Rohsenow [2] and Fujii and Uehara [8] for a vertical plate under pure natural convection condensation (n = 0) and mixed convection (n = 5). In general, the results show that Nux decreases as Ja increases. A larger value of Ja indicates a greater temperature difference between the plate and the vapor. Under such conditions, a thicker condensate layer is formed on the plate, and hence the value of the Nusselt number decreases. The results obtained by Rohsenow [2] and Fujii and Uehara [8] for pure natural convection condensation are virtually identical. Furthermore, the error between these results and those obtained using the current numerical method is less than 1%. Sparrow and Gregg [3] reported that neglecting inertia terms in pure natural convection film condensation induces an error of less than 2%. The good agreement between the current results and Table 1 Values of physical parameters assumed for current saturated water-vapor system Water at 100 C Cp hfg Pr q, q* l, l* k 1=2 R ¼ qql l
4210 J kg1 K1
Vapor at 100 C
– 2257 kJ kg1 1.76 – 957.9 kg m3 0.5956 kg m3 279 · 106 kg m1 s1 12.02 · 106 kg m1 s1 3 1 1 680 · 10 W m K – 193.2
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555 3.2
: Fujji ξ = 5
2.8
: present ξ = 5
2.6
2.4
: Fujji ξ = 0 : Rohsenow ξ = 0 : present ξ = 0
2.2 0.02
0.03
0.04
0.05
0.06
0.07
Ja
Fig. 2. Comparison of solutions presented by Rohsenow [2], Fujii and Uehara [8] and present study for problem of film condensation on vertical plate at n = 0 and n = 5.
those presented in [2] and [8] exists since both previous studies neglected the effects of inertia in the liquid and vapor phases. For the case of mixed-convection condensation (n = 5), it can be seen that the difference between the current results and those presented in [8] increases with increasing Ja, and reaches a value of approximately 3% at Ja = 0.6. As discussed above, a larger value of Ja implies a thicker condensate layer, and therefore, an increased inertia effect. Overall, the results presented in Fig. 2 indicate that the effects of inertia should be taken into account in condensation problems characterized by higher values of n and Ja. Þ In Eqs. (35) and (37), f 0 ðn; gÞ þ f ðn;gÞ and F 0 ðn;g Þ þ F ðn;g g g represent the dimensionless velocity profiles within the liquid film and the vapor phase, respectively. Fig. 3 illus-
trates the variations of these two velocity profiles with the forced-flow parameter, n, for a tube radius of g0 = 1 and a condensate layer thickness of gi g0 = 0.2 (or gi = 1.2). It is observed that both f 0 ðn;gÞ þ f ðn;gÞ and g Þ F 0 ðn; g Þ þ F ðn;g increase as the value of n increases. As a g consequence, it can be inferred that the interfacial velocity increases with increasing n. In other words, the results show that the condensate flow rate can be enhanced by increasing the forced-flow intensity of the vapor bulk. f 0 ðn;gÞ f ðn;gÞ 00 In Eqs. 0(39a) and (39b), f ðn; gÞ þ and g g2 Þ Þ F ðn;g F ðn;g F 00 ðn; g Þ þ g g2 represent the dimensionless shear stress profiles within the liquid film and the vapor phase, respectively. Fig. 4 shows the variation of these two shear stress profiles with the forced-flow parameter, n, for a tube radius and condensate layer thickness of g0 = 1 and gi g0 = 0.2, respectively. It is seen that the shear stress in both the liquid film and the vapor phase increases at higher values of n. As a consequence, it can be inferred that the shear stress at the liquid–vapor interface, i.e. g = gi, increases significantly as the value of n is increased. Thus, it is apparent that the general assumption of a zero interfacial shear stress is invalid for the case where the vapor is subject to a forced flow. The shear stress acting at the wall is an important parameter in many engineering applications. Fig. 5 plots the dimensionless wall shear stress, sw(x2/4lt)(Grx/4)3/4 0 ðn;g Þ f 0Þ (i.e. equal to f 00 ðn; g0 Þ þ g0 0 f ðn;g Þ against the forcedg20 flow parameter, n, as a function of the condensate film thickness for a constant tube radius of g0 = 1. It can be seen that the thickness of the condensate layer has a significant effect on the magnitude of the wall shear stress. The physical reason for this is that higher values of the condensate film thickness, gi g0, imply the farther from the leading edge, which results in a higher velocity and therefore a higher wall shear stress. In other words, the greater the distance from the leading edge, the greater the thickness of the condensate film and hence the higher the gravitationally induced velocity.
3
0.04
0.03
f'(ξ,η)+f (ξ,η)/η
η o = 1.0 η i = 1.2
ξ= 5 2 1 0.5 0
2
ξ= 5 =2 =1 = 0.5 =0
0.02
1 0.01
F'(ξ,η*)+F(ξ,η*)/η*
Nux(Grx/4)-1/4
3
553
η o = 1.0 η i = 1.2 0
0 1
1.04
1.08
1.12
η
1.16
1.2
0
0.4
0.8
1.2
1.6
2
η*-η*ι
Fig. 3. Dimensionless velocity profiles within liquid film and saturated vapor region for n = 0, 0.5, 1, 2 and 5 at gi = 1.2 (dimensionless film thickness = 0.2).
554
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
8
0.2 6 ξ= 5 2 1 0.5 0
0.1
ξ=5
4
=2 =1 = 0.5 =0
2
F''(ξ,η*)+F'(ξ,η*)/η*-F(ξ,η*)/η*
2
f''(ξ,η)+f'(ξ,η)/η-f (ξ,η)/η
η o = 1.0 η i = 1.2
η o = 1.0 η i = 1.2
2
10
0.3
0 0 1
1.04
1.08
1.12
1.16
1.2
0
0.4
0.8
1.2
1.6
2
η*-η*ι
η
Fig. 4. Dimensionless shear stress profiles within liquid film and saturated vapor region for n = 0, 0.5, 1, 2 and 5 at gi = 1.2 (dimensionless film thickness = 0.2).
1.2
1
: ηi -η0 = 0.5 : ηi -η0 = 0.4 : ηi -η0 = 0.3
ηi -η0 = 0.4 0.3 0.2
0.8
η0 = 1.0 0.8
0.6
η0 = 1.0 ξ = 1.0, 3.0, 5.0
θ
τ w(x2/4μυ)(Grx /4)-3/4
1
0.6
0.4
0.4
0.2
0.2
0 0
1
2
3
4
5
ξ Fig. 5. Variation of local wall shear stress with n.
Fig. 6 presents the temperature profiles in the condensate layer, h 0 (n, g), for various values of the condensate film thickness and forced-flow parameter, respectively. (Note that a constant tube radius of g0 = 1 is assumed.) It can be seen that the temperature profile is independent on n and linearly varies from h = 1 at the tube surface to h = 0 at the liquid–vapor interface. In problems involving film condensation, one of the most important parameters is the Nusselt number. Eq. (43) shows that the value of the Nusselt number varies as a function of h 0 (n, g0). Fig. 7 plots the local Nusselt number against the forced-flow parameter, n, for various values of the condensate film thickness and a constant tube radius of g0 = 1. From a close inspection, it is found that the local Nusselt number is independent on n. The results also show that the Nusselt number decreases as the condensate film
1
1.1
1.2
1.3
1.4
1.5
η Fig. 6. Dimensionless temperature profiles within liquid film.
thickness increases. The physical reason for this is that a higher film thickness implies a greater thermal resistance, and hence a lower Nusselt number. Equivalently, the thinner the condensate layer, the steeper the temperature gradient (see Fig. 6), and thus the greater the heat transfer rate. Fig. 8 illustrates the variation of the dimensionless condensate mass flux with the forced-flow parameter as a function of the condensate film thickness and a constant tube radius of g0 = 1. In general, the results show that the condensate mass flux increases with increasing n. This result is physically reasonable since a higher value of the forcedflow parameter implies an increased vapor velocity at the liquid–vapor interface, and then induce an increase in the condensate mass flux. Fig. 8 also shows that the effect of n on the condensate mass flux becomes more pronounced
T.-B. Chang / Applied Thermal Engineering 28 (2008) 547–555
tical tube in dry saturated vapor with a forced flow. A dimensionless parameter, n, has been introduced to represent the forced-flow intensity of the vapor bulk. The results have shown that a higher value of n enhances the condensate flow rate and induces a higher shear stress at the wall. Furthermore, it has been shown that the common assumption of zero interfacial shear stress is invalid for the case of a forced flow of the vapor bulk. The results of this study provide a useful source of reference for designers aiming to improve the thermal performance of shell-and-tube heat exchangers and distillation processes.
5.5
5
Nux(Grx /4)-1/4 = θ'(ξ,η0)
η0 = 1.0 : ηi -η0 = 0.2
4.5
: ηi -η0 = 0.3 : ηi -η0 = 0.4
4
555
3.5
Acknowledgements 3
This work is partially supported by the National Science Council of Taiwan under contract NSC94-2212-E-218-017.
2.5 0
1
2
3
4
5
References
ξ Fig. 7. Variation of local Nusselt number with n.
0.4
: ηi -η0 = 0.5 : ηi -η0 = 0.4 : ηi -η0 = 0.3
mx(Grx /4)-1/4(x/μ)
0.3
η 0 = 1.0 0.2
.
0.1
0 0
1
2
3
4
5
ξ Fig. 8. Variation of dimensionless condensation mass flux with n.
as the thickness of the condensate layer increases. This is to be expected since, as mentioned previously, a higher value of the condensate film thickness implies the farther from the leading edge, and hence the effect of n on the liquid velocity, and thus on the condensate mass flux, increases. 6. Conclusion This study has analyzed the steady-state mixed convection of a condensate film flowing along an isothermal ver-
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