8
Click here to load reader
Click here to load reader
Microgravity Sci. Technol. (2013) 25:95–102DOI 10.1007/s12217-012-9328-3
ORIGINAL ARTICLE
Analysis of Film Condensation Along a Vertical Flat PlateUnder Sinusoidal G-Jitter
Pradyumna Ghosh · Abhishek Sarkar · Subir Das
Received: 4 September 2010 / Accepted: 15 October 2012 / Published online: 14 November 2012© Springer Science+Business Media Dordrecht 2012
Abstract Transient phenomenon of laminar film con-densation along a vertical flat plate under sinusoidalg-jitter has been analyzed, based on the same assump-tions of Nusselt’s analysis of film condensation wherethe heat transfer within the liquid film is by pure con-duction. The momentum equation retains the transientterm. The perturbed acceleration due to gravity hasbeen assumed to be a sinusoidal function of time andfrequency of oscillation for the simplicity. The resultantequation has been solved analytically and the velocityprofiles and mass flow rate under such g-jitter has beensimulated. It has been observed that at the differentheights the velocity response with time is purely sinu-soidal with different amplitude. Last but not the least;boundary layer thickness is also oscillating with time,hence heat transfer coefficient. However, the entireanalysis is an extension of Nusselt’s analysis of filmcondensation which includes transient response.
P. Ghosh (B) · A. SarkarDepartment of Mechanical Engineering,Indian Institute of Technology,Banaras Hindu University,Varanasi 221005, Indiae-mail: [email protected]
S. DasDepartment of Applied Mathematics,Indian Institute of Technology,Banaras Hindu University,Varanasi 221005, India
Keywords Analytical-solution · Film condensation ·Oscillatory boundary layer thickness and heat transfercoefficient · Sinusoidal g-jitter
Nomeclature
Tsat Saturation temperature of vapour [K]Ts Temperature of the vertical plate [K]t Time [s]x, y x and y coordinates [m]X Body forceg Acceleration due to gravity,
Eign function [m/s2]u x and y components of velocity
respectively [m/s]r Chosen functionv Chosen functionhx Local heat transfer coefficient [W/m2K]hL Average heat transfer coefficient [W/m2K]NuL Average Nusselt number [–]U∞ Free stream velocity of vapour [m/s]m Condensate mass flow rate [kg/s]ρl Density of liquid [kg/m3]ρv Density of vapour [kg/m3]�(x) Condensate mass flow rate per
unit width [kg/m-s]δ(x,t), δ Instantaneous boundary layer
thickness at height x [m]kl Thermal conductivity of liquid [W/m-K]h fg Latent heat of vapourization [kJ/kg]q′′
s Surface heat flux [W/m2]νl Kinematic viscosity of liquid [m2/s]go Constant part of g [m/s2]ω Natural frequency of vibration [rad/s]
96 Microgravity Sci. Technol. (2013) 25:95–102
f Excitation frequency [Hz]τ Time period [s]S Surface area [m2]
Introduction
In a gravity-free environment and in the absence ofradiation, heat transfer in a fluid medium is affected bypure diffusion. However, with a g-jitter, the situationmay change. Such a field produces a fluctuating bodyforce if density gradients are present. This results anoscillating fluid flow with both fluctuating and time-independent components. Many analysis has been car-ried out for laminar condensation along vertical andinclined plate but transient film condensation alongvertical plate has been studied by Reed et al. (1987)which includes inertia as well as interfacial shear stress.Transient film condensation along finite horizontalplate has been carried out by Prasad and Jaluria (1982).Transient film condensation on a vertical plate embed-ded in porous medium has been described by Masoudet al. (2000), using mathematical model analytically todetermine the velocity profiles, the film thickness. Oneof the earliest numerical studies on condensation hasbeen proposed by Faghri and Chow (1991): the filmcondensation process is analyzed for an annulus con-denser configuration by applying the Nusselt theory forlaminar film condensation for co-current vapour flows.The role of surface tension for two-phase heat and masstransfer processes in the absence of gravity has been an-alyzed by Straub (1994). It summarizes of experimentalresults available that time (1994) and the theoreticalanalysis of the surface tension physical phenomenonwhen the buoyancy reduced and transport processesare determined by the properties at the interface alone.Chen et al. (2006) presents a review of the recent in-vestigations concerning the condensation process in theabsence of body forces. With the development of moreadvanced applications, a spacecraft thermal control sys-tem must be able to transport and dissipate large quan-tities of heat over moderate distances. Standard me-chanically pumped single-phase loops, widely used inthe past, are being abandoned due to the large amountof pumping power required to overcome the pressuredrops and meet the high heat removal demands ofadvanced spacecraft thermal control. The combinationof high heat flux capacity, low weight-to-dissipated-power ratio and high degree of temperature uniformityprovided by phase-change condensing radiators offeran effective alternative. The condensation heat transferis used in these applications to transfer heat from thesource to the radiators, where, through phase change,
the heat transfer can be achieved and ultimately dis-sipated from the radiator surface to space. Effectivecondensing radiators are capable of dissipating highheat fluxes with small temperature differences and havea high heat-rejected to radiating-mass ratio. Vapourcondensation on curvilinear disk-shaped fin at resid-ual microgravity has been described by Marchuk andKabov (2008). But there is no reported study so far toanalyze the film condensation under g-jitter. However,in the present investigation, laminar film condensationalong a vertical plate under such reduced oscillatinggravitational field (g-jitter) has been solved analyticallywith the same assumptions of Nusselt’s analysis of filmcondensation where the heat transfer within the liquidfilm is by pure conduction. The Navier–Stoke equationretained the transient term. Also, the acceleration dueto gravity has been modified and has been assumedto be a sinusoidal function of time and frequency ofoscillation for simplicity which has a significant applica-tion in space in the condensing radiator for space craftthermal management system.
Mathematical Modeling
Integral Analysis
Laminar steady film condensation of a pure single-component saturated vapour has been analyzed byNusselt as per Incropera and DeWitt (2006). Figure 1depicts the process of laminar film condensation on avertical plate from a quiescent vapour. The film origi-nates at the top of the plate and flows downward underthe influence of gravity. The instantaneous thicknessδ(x,t) and the condensate mass flow rate m increasewith increasing x because of continuous condensationat the liquid–vapour interface, which is at Tsat. More-over, there exists a finite shear stress at the liquid–vapour interface, contributing to a velocity gradient inthe vapour, as well as in the film.
Where all liquid properties should be evaluatedat the film temperature T f = (Tsat + Ts) /2, and h fg
should be evaluated at Tsat. In the present integralanalysis on the basis of Nusselt integral analysis of filmcondensation (Faghri and Chow 1991), effects of g-jitterhave been introduced. Transient laminar film conden-sation along an isothermal wall has been solved analyt-ically under oscillatory g-jitter as there is no previouswork for transient film condensation under includingthe effect g-jitter. In this analysis g-jitter is modeledby a unidirectional, harmonically oscillating, and small-amplitude gravitational field go sin ωt. Buoyancy-drivenlaminar flow has been considered, of a Boussinesq
Microgravity Sci. Technol. (2013) 25:95–102 97
Fig. 1 Laminar steady filmcondensation on a verticalplate (Incropera and DeWitt2006)
fluid in a fluctuating gravitational field in which theliquid density is assumed to be constant except in thebuoyancy term of the equation of motion.
In the present analysis, it has been considered thata vertical plate maintained at a constant temperatureTs which is kept in an environment filled with vapourmaintained at it’s saturation temperature Tsat movingat a constant velocity U∞.
Governing Differential Equation
The x-momentum equation is given as:
ρl∂u∂t
= −∂p∂x
+ μl∂2u∂y2
+ X (1)
where X = Body force.The body force within the film is equal to ρlg, and
the pressure gradient may be approximated in termsof conditions outside the film. That is, the bound-ary layer approximation
(∂p/∂y) ≈ 0, it follows that(
∂p/∂x) ≈ ρvg.
Substituting these values in Eq. 1, and rearranging,we get,
∂u∂t
= νl∂2u∂y2
+{
g(
1 − ρv
ρl
)}(2)
Now to account for the g-jitter, substituting g =go sin ωt in the above equation, we get,
∂u∂t
= νl∂2u∂y2
+{
go sin ωt(
1 − ρv
ρl
)}(3)
The dimension of each term turns up to be (m/s2).The equation turns up to be Nusselt equation of filmcondensation for the following assumptions
1. Steady state i.e. ∂u∂t = 0
2. sin ωt = 1
Solution Methodology
Let Q(y, t) =
{go sin ωt
(1 − ρv
ρl
)}.
Boundary conditions:
u (0, t) = 0 & u (δ (x, t) , t) = U∞ (4)
where δ(x,t) is the instantaneous boundary layer filmthickness.
Initial condition:
u(y, 0
) = 0 (5)
The mathematical problem defined by Eqs. 3–5 consistsof a non-homogeneous partial differential equationwith non-homogeneous BCs (Chen et al. 2006).
We consider a y reference velocity distribution r(y,t) with only the property that satisfy the given non-homogeneous BCs, it means only that
r (0, t) = 0 & r (δ, t) = U∞
Let r(y, t) = yδ(t)
U∞ (6)
Again the difference between the desired solution u(y,t) and the chosen function r(y, t) is employed.
v(y, t) ≡ u
(y, t)− r
(y, t)
(7)
∵ both u(y, t) and r(y, t) satisfy the same linear BC aty = 0 and y = δ, it follows that v(y, t) satisfies the relatedhomogeneous BCs:
v (0, t) = 0 (8)
v (δ, t) = 0 (9)
The PDE satisfied by v(y,t) is derived by substituting
u(y, t) = v
(y, t)+ r
(y, t)
98 Microgravity Sci. Technol. (2013) 25:95–102
hence
∂r(y, t)∂t
= − yδ′(t)δ2(t)
U∞, where δ′(t) = dδ(t)dt
.
If u(x, t) satisfies Eq. 2 then function v (y, t) = u (y, t) −r (y.t) satisfies
∂v∂t
= vl∂2v∂y2
+ Q(y, t) + yδ′(t)δ2(t)
U∞
= vl∂2v∂y2
+ Q1(y, t) (10)
The initial condition is usually altered:
v(y, 0
) ≡ 0 − r(y, 0
) = − yδ
U∞ ≡ g(y)
It can be seen that in general only the BC have beenmade homogeneous.
∴ The problem transforms to:PDE:
dv∂t
= vl∂2v∂y2
+{
go sin ωt(
1 − ρv
ρl
)}
= vl∂2v∂y2
+ Q(y, t) (11)
BCs:
v (0, t) = 0 (12)
v (δ, t) = 0 (13)
IC:
v(y, 0
) = − yδ
U∞ = g(y) (14)
Solution by the Method of Eigenfunction Expansion
Consider the eigen-functions of the related homoge-neous problem.
The related homogeneous problem is
dv∂t
= vl∂2v∂y2
v (0, t) = 0
v (δ, t) = 0 (15)
The eigenfuntions of the related homogeneous prob-lem is
d2φ
dy2+ λφ = 0
φ(0) = 0
φ(δ) = 0 (16)
We know that the eigenvalues are λn = ( nπδ
)2, n =1, 2, 3, . . .,∞ and the corresponding eigenfunctions areφn(y) = sin nπy
δ. The method of eigenfunction expan-
sion employed to solve the homogeneous problem (11)with homogeneous BCs (12) and (13), consists in ex-panding the unknown solution v(y,t) in a series of therelated homogeneous eigenfunctions:
v(y, t) =∞∑
n=1
an(t)φn(y, t) (17)
For each fixed t, v(y, t) is a function of y, and hence v(y,t) will have generalized Fourier series. The generalizedFourier coefficients will vary as ‘t’ varies. Thus, thegeneralized Fourier coefficients are functions of time,an(t).
Substituting Eq. 17 into Eq. 10 results in,
∞∑
n=0
(dan(t)
dt− an(t)
)φn +
∞∑
n=0
an(t)∂φn(y, t)
∂t
=∞∑
n=0
(dan(t)
dt− an(t)
)φn − δ′(t)
δ2(t)
∞∑
n=0
an(t)nπy cosnπyδ(t)
= Q1(y, t)
However, this analysis is true only when δ′(t)δ2(t) << 1.
So second term of expression becomes negligible incomparison of the first term. Hence the closed formsolution is a good approximation of the generalizedsolution.
Equation 17 automatically satisfies the homoge-neous BCs. We emphasize then by stating that v(y,t) and φn(y) satisfy the homogeneous BC. The initialcondition is satisfied if
g(y) =∞∑
n=1
an(0)φn(y)
Due to orthogonality of the eigenfunctions, we candetermine the initial values of the generalized Fouriercoefficients.
an (0) =
δ∫
0g(y)φn(y)dy
δ∫
0φ2
n(y)dy
(18)
Let I1 =δ∫
0
g(y)φn(y)dy = (−1)nU∞δ
nπand
I2 =δ∫
0
φ2n(y)dy = δ
2
Microgravity Sci. Technol. (2013) 25:95–102 99
∴ an (0) = I1
I2
∴ an (0) = 2(−1)nU∞nπ
(19)
All that remains is to determine an(t) such that Eq. 17solves the Eq. 11. If the v(y, t) solves the same homo-geneous BC as does φn(y), then the necessary term-by-term differentiation can be justified.
Term-by-term differentiating v(y, t),
∂v∂t
=∞∑
n=1
ddt
an(t)φn(y) and
∂2v∂2 y
=∞∑
n=1
an(t)∂2
∂y2φn(y) = −
∞∑
n=1
an(t)λnφn(y)
∵ φn(y) satisfies∂2φn
∂y2+ λnφn = 0
Substituting these results into equation, we get,
∞∑
n=1
[dan
dt+ λnvlan
]φn(y) = Q(y, t) (20)
L.H.S. is the generalized Fourier series for Q(y, t). Dueto orthogonality of φn(y), we obtain the first order DEfor an(t).
dan
dt+ λnvlan =
δ∫
0Q(y, t)φn(y)dy
δ∫
0φ2
n(y)dy
≡ qn(t) (21)
R.H.S. is a known function of time (and n), the Fouriercoefficient of Q(y, t),
Q(y, t) =∞∑
n=1
qn(t)φn(y)
Equation 21 is a non-homogeneous linear first orderequation. Perhaps the easiest method to it is to multiplyit by the integrating factor eλnvl t. Thus,
eλnvl t(
dan
dt+ λnvlan
)= d
dx
(aneλnvl t
) = qneλnvl t
Integrating from 0 to t, we get,
an(t)eλnvl t − an(0) =t∫
0
qneλnvl tdt
∴ an(t) = 2(−1)nU∞e−λnvl t
nπ
+ e−λnvl t
t∫
0
qneλnvl tdt (22)
This, the solution of original PDE is given as
u(y, t) = v
(y, t)+ r
(y, t)
∴ u(y, t) =∞∑
n=1
an(t)φn(y) + yU∞δ
∴ u(y, t) =∞∑
n=1
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(2(−1)nU∞e−λnvl t
nπ
)+ e−λnvl t
t∫
0
⎡
⎢⎢⎢⎣
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
δ∫
0
[{go sin ωt
(1 − ρv
ρl
)}sin nπy
δ
]dy
δ∫
0φ2
ndy
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
eλnvl t
⎤
⎥⎥⎥⎦
dt
⎫⎪⎪⎪⎬
⎪⎪⎪⎭
sinnπyδ
+ yU∞δ
(23)
∴ We get,
u (y, t) =∞∑
n=1
⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
(2 (−1)n U∞e−λnvl t
nπ
)+ e−λnvl t
(2
nπ
[1 − (−1)n]
)
⎧⎪⎨
⎪⎩
(go
(1 − ρv
ρl
))(1
(λnvl)2 + ω2
)∗
[eλnvl t (λnvl sin ωt − ω cos ωt)
]
⎫⎪⎬
⎪⎭
⎫⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎭
sinnπyδ
+ yU∞δ
(24)
100 Microgravity Sci. Technol. (2013) 25:95–102
The dimensionless velocity (u(y,t)/u∞) has been plottedfor different (y/δ) and dimensionless time in Figs. 2and 3.
Now the condensate mass flow rate per unit depth isgiven by,
m(x, t)b
=t∫
0
δ∫
0
ρlu(y, t)dydt = �(x, t) (25)
Using Eq. 24 and solving for �(x, t) we get,
�(x, t) =∞∑
n=1
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
2δ(−1)nU∞ρl [1 − (−1)n](1 − e−λnvl t
)
n2π2λnvl
+(
2ρlδ [1 − (−1)n]2
n2π2
)
⎡
⎢⎢⎢⎢⎣
(go
(1 − ρv
ρl
))(1
(ω2 + λ2
nv2l
)
)
∗{
λnvl (1 − cos ωt)ω
− sin ωt}
⎤
⎥⎥⎥⎥⎦
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
+ ρlU∞δt2
(26)
At a portion of the liquid–vapour interface of unitwidth and length dx, the rate of heat transfer into thefilm, dq, must equal the rate of energy release due tocondensation at the interface. Hence
dq = h fgdm(x, t) (27)
Fig. 2 Transient velocity response at different locations insidethe boundary layer, where ω = π/2, Ui = 0.0001 m/s
Fig. 3 Transient velocity response at different locations insidethe boundary layer, where w = π /4, Ui = 0.0001 m/s
Since advection is neglected, it also follows that the rateof heat transfer across the interface must equal the rateof heat transfer to the surface. Hence
dq = q′′s (b .dx) (28)
Fourier’s law may be used to express the surface heatflux as
q′′s = kl(Tsat − Ts)
δ(x, t)(29)
Equating Eqs. 27, 28 and 29, and solving for δ(x, t), weget,
δ(x, t) = kl(Tsat − Ts)
�(x, t)h fg= kl�T
�(x, t)h fg(30)
The dimensionless form of instantaneous boundarylayer thickness is as follows:
δ(x, t) = JaPr
· μ1
�t
And the heat transfer coefficient is,
hx = kl
δ(x, t)(31)
i.e.
hx = �(x, t)h fg
�T(32)
Parametric Study
It has been observed from the solution of the gov-erning equations that velocity profile and mass flow
Microgravity Sci. Technol. (2013) 25:95–102 101
rate per unit width are oscillatory trigonometric func-tion. In that context, without loss of generality themass flow rate per unit width has been assumed as,� (x, t) = A sin(ωt) + B cos(ωt). To understand the re-sponse, heat transfer coefficient and in velocity profilefor a required amount of condensate mass flow rateper unit width to be � (x, t) = 10−5 sin (ωt − π/10) +10−10 cos (ωt + π/4) , the velocity profiles have beenestimated for Ui = 10−4 m/s. The velocity of each ofthe cases at different dimensionless heights (y/δ) withinthe boundary layer has been plotted in Figs. 2 and 3.For calculation following thermo physical propertiesare used:
�T = 10 ◦C, kl = 0.68 W/m-K,
h fg = 2257 KJ/kg, Ja = 0.01, Pr = 7.1
Boundary layer thickness δ(x,t) and heat transfercoefficient have also been observed oscillating atdifferent heights (x) of the plate where,
� (x, t) = 10−5 sin(ωt − π/10) + 10−10 cos(ωt + π/4)
However, the example has been selected in such waythat the maximum value of variation in the ratio δ′(t)
δ2(t) is0.15 which is <<1 in Fig. 4 and in case of ω = π /4 inFig. 5 is 0.225 << 1
From Fig. 4, we find the instantaneous BL thicknessto be strongly oscillatory. Boundary layer first growsin downard direction that is positive direction but asgravitational field reverses due to sinosodial nature theboundary layers grows in upward direction. This is acharacteristic feature where there is oscillatory pressure
Fig. 4 BL thickness Vs Time for w = π/2, Ja = 0.01, Pr = 7.1
Fig. 5 BL thickness Vs Time for w = π /4, Ja = 0.01, Pr = 7.1
gradient due to sinusoidal g-jitter and thus the flow isreversed. However, an effort has been made to under-stand the effect of frequency on the boundary layerdevelopment.
The oscillation of the heat transfer coefficient is alsoevident from Eq. 31.
Conclusions
Velocity profiles at different height are oscillating withtime due to g jitter effect. But it has been observed,for such a nonlinear system response with an excitationfrequency (ω), the PDE solution indicates that theoscillation will contain other harmonics which has beendemonstrated through FFT (Fast Fourier Transform)and it is quite expected from a non linear system re-sponse. At low free stream velocities, the velocitiesof condensate are almost the same at different valuesof y. On the contrary as the free stream velocity in-creases the velocity of condensate also increases as wego away from the wall. At an excitation frequency off = 0.25 Hz, we have observed frequency clustering inalmost all heights of the plate irrespective of the massflow rate. However, when the excitation frequency isreduced to f = 0.125 Hz, there is no frequency cluster-ing with higher mass flow rate. The instantaneous BLthickness is strongly oscillatory due to g-jitter effect andflow reversal. The maximum value of instantaneous BLthickness also increases with increase in x component(height) of the plate. Transient response of heat trans-fer coefficient shows oscillation and with the decreas-ing boundary layer thickness heat transfer coefficient
102 Microgravity Sci. Technol. (2013) 25:95–102
increases and vice versa. Moreover, the entire analysisis an extension of Nusselt analysis of film condensationwhich includes transient response.
References
Chen, Y., Sobhan, C.B., Peterson, G.P.: Review of condensationheat transfer in microgravity environments. J. Thermophys.Heat Transf. 20, 353–360 (2006)
Faghri, A., Chow, L.C.: Annular condensation heat transfer inmicrogravity environment. Int. Commun. Heat Mass Trans-fer 18, 715–729 (1991)
Haberman, R.: Elementary Partial Differential Equations, 3rdedn. Prentice Hall, Englewood Cliffs (1998)
Incropera, F.P., DeWitt, D.P.: Fundamentals of Heat andMass Transfer, 5th edn, pp. 533–593. Wiley, New York(2006)
Marchuk, I., Kabov, O.: Vapour condensation on curvilineardisk-shaped fin at microgravity. Microgravity Sci. Technol.20, 165–169 (2008)
Masoud, S.A., Al-Nimar, M.A., Alkam, M.K.: Transient film con-densation on a vertical plate imbedded in porous medium.Transp. Porous Media 40, 345–354 (2000)
Prasad, V., Jaluria, Y.: Transient film condensation on horizontalplate. Chem. Eng. Commun. 13, 327–342 (1982)
Reed, J.C., Gerner, F.M., Tien C.L.: Transient condensation ofa laminar film onto a vertical plate. In: AIAA-1987-1534,Thermophysics Conference, 22nd, Honolulu, HI, 8 pp. (1987)
Straub, J.: The role of surface tension for two-phase heat andmass transfer in absence of gravity. Exp. Therm. Fluid Sci.9, 253–273 (1994). ISBN 3-540-42074-6
