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CHAPTER 10
Condensation
M A KEDZIERSKI Building and Fire Research Laboratory National Institute of Standards and Technology Gaithersburg, Maryland
J C CHATO Department of Mechanical and Industrial Engineering University of Illinois–Urbana-Champaign
Urbana, Illinois
T J RABAS Consultant Downers Grove, Illinois
10.1 Introduction 10.2 Vaporspace film condensation 10.2.1 Nusselt’s analysis of a vertical flat plate 10.3 Film condensation on low fins
10.3.1 Introduction 10.3.2 Surface tension pressure gradient 10.3.3 Specified interfaces
10.3.4 Bond number 10.4 Film condensation on single horizontal finned tubes 10.4.1 Introduction
10.4.2 Trapezoidal fin tubes 10.4.3 Sawtooth fin condensing tubes 10.5 Electrohydrodynamic enhancement 10.5.1 Introduction
10.5.2 Vaporspace EHD condensation 10.5.3 In-tube EHD condensation 10.6 Condensation in smooth tubes 10.6.1 Introduction 10.6.2 Flow regimes in horizontal tubes Flow regimes in horizontal two-phase flow Effects of mass flux and quality
Effects of fluid properties and tube diameter Potential role of surface tension
Flow regime mapping Comparison of flow regime maps
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10.6.3 Heat transfer in horizontal tubes Effects of mass flux and quality Effects of tube diameter Effects of fluid properties Effects of temperature difference Gravity-driven condensation Shear-driven annular flow condensation Comparison of heat transfer correlations 10.6.4 Pressure drop
10.6.5 Effects of oil 10.6.6 Condensation of zeotropes 10.6.7 Inclined and vertical tubes 10.7 Enhanced in-tube condensation 10.7.1 Microfin tubes 10.7.2 Microfin tube pressure drop 10.7.3 Twisted-tape inserts 10.8 Film condensation on tube bundles 10.8.1 X-shell condensers (shell-side condensation) Tube-side flow and temperature maldistribution Condensersizing methods
Noncondensable gas management and proper venting techniques 10.8.2 In-tube condensers
Nonuniform outside inlet flow and temperature distributions Noncondensable gas pockets
10.9 Condensation in plate heat exchangers 10.9.1 Introduction
10.9.2 Steam condensation heat transfer 10.9.3 Effect of inclination on heat transfer performance 10.9.4 Effect of inclination on pressure drop
Appendix A Nomenclature References
Condensation is the process by which a vapor is converted to its liquid state Because
of the large internal energy difference between the liquid and vapor states, a signifi-cant amount of heat can be released during the condensation process For this reason, the condensation process is used in many thermal systems
In general, a vaporwill condense to liquid when it is cooled sufficiently orcomes
in contact with something (e.g., a solid oranotherfluid) that is below its equilibrium temperature This chapter is concerned primarily with convective condensation (con-densation of a flowing vaporin a passage) and vaporspace con(con-densation (condensa-tion of stagnate vapor onto a surface) Film condensa(condensa-tion occurs when the condensate
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completely wets the surface in a continuous liquid film and can be associated with eitherconvective orvaporspace condensation Dropwise condensation, usually as-sociated with vaporspace condensation, occurs when the condensate “beads up” on the surface into drops of liquid as a consequence of the liquid’s lack of affinity for the surface Heat transfer coefficients for dropwise condensation can be one to two or-ders of magnitude greater than those for film condensation Unfortunately, dropwise condensation is not easily sustained in practice
10.2 VAPOR SPACE FILM CONDENSATION 10.2.1 Nusselt’s Analysis of a Vertical Flat Plate
Nusselt (1916) published a solution forsteady-state laminarfilm condensation on a vertical flat plate This pioneering work laid the foundation on which those working in the field of condensation still build their research The cross section of the liquid film
as analyzed by Nusselt is shown in Fig 10.1 The vaporcondenses at its saturation temperature (Tsat) due to a cooler wall temperature (Tw) of the vertical plate The thickness of the condensate film (δ) increases along the length (s) due to mass transfer
to the liquid–vapor interface The film is drained by the influence of gravity alone in the downwards direction Consequently, the film velocities in the y and z directions
can be neglected compared to the velocity in the s direction (u) Moreover, the
dm
i m s s. ⫹ dds dsm i
i m s s .
ds q ⬙w =⫺k wdTdy | (ds)
y = 0
z y s
ds
T(y)
u(y)
du
dy y = ␦= 0
g
Energy balance
Saturated vapor
Liquid-vapor interface
Liquid film Wall
⌫(s)
Figure 10.1 Nusselt condensation on vertical flat plate
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velocity and temperature changes in they direction are much larger than those in
thes and z directions Accordingly, both momentum changes and convection in the
s direction are negligible As a result, if there is no shear stress at the liquid–vapor
interface, thes-moment equation becomes
µl d2u
dy2 =dP
Equation (10.1) shows that the viscous forces are balanced by the sum of the gravity force (ρl g) and pressure gradient in the s direction (dP /ds) The momentum
equation can be integrated while using the no-slip condition at the wall and no shear
at the liquid–vapor interface to yield the velocity profile of the film:
u = µy l
y
2 − δ dP
ds − ρl g
(10.2) The liquid mass flow rate of the film per unit width (Γ) is
Γ = δ 0
ρl u s dy = δ3
3νl
dP
ds − ρl g
(10.3) Differentiating eq (10.3) gives
dΓ
ds =
1
3ν1
d ds
δ3
dP
ds − ρl g
(10.4)
An energy balance on an incremental element of the film is shown in Fig 10.1 If the convection of heat along thes direction is neglected, the heat balance becomes
whereλ, kl, andT are the latent heat, thermal conductivity, and temperature of the
film, respectively
By applying the Nusselt assumptions to the energy equation, it reduces to
dT
dy =
Tsat− Tw
whereTsatis the saturation temperature of the condensate andTwis the local wall temperature
Rearranging the incremental energy balance and substituting the temperature gra-dient gives the gragra-dient of the mass flow rate:
dΓ
ds =
−kl(Tsat− Tw)
Equating the foregoing two expressions for the mass flow rate gradient yields the ordinary differential equation that models Nusselt condensation:
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d ds
δ3
dP
ds − ρl g
= −3νlkl(Tsat− Tw)
By applying the Nusselt assumption to thes-momentum equation forthe vaporphase,
one obtainsdP /ds = ρ g g; and using the expression δ(dδ3/ds) = 3
4(dδ4/ds), the
solution to eq (10.8) becomes
δ =
4klµl (Tsat− Tw)s
ρl g(ρ l− ρg )λ
1/4
(10.9)
Forfilm condensation, the convection of heat along lengths can be neglected
com-pared to conduction across the film For these conditions, the temperature gradient
of the film is approximately linear, and the heat transfer coefficient isk l /δ
Con-sequently, the condensation heat transfercoefficient fora vertical flat plate forno interfacial shear is
h =
ρl g(ρ l− ρg )k3
lλ 4µl (Tsat− T w )s
1/4
(10.10)
The average condensation heat transfer coefficient overs = 0 to s = L is
h = 0.943
ρl g(ρ l− ρg )k3
lλ
µl (Tsat− T w )L
1/4
(10.11)
Equation (10.11) gives the heat transfercoefficient forlaminarfilm condensation on
a vertical flat plate for low pressures(ρ v ρg ) and c p,1 (Tsat− T w )/λ < 1 The
average condensation heat transfer coefficient for a horizontal tube of outer diameter
D ois obtained by replacingg with g sin φ and integrating around the tube with respect
to the cylindrical coordinate angleφ from 0 to 180°:
h = 0.729
ρl g(ρ l− ρg )k3
lλ
µl (Tsat− Tw)Do
1/4
(10.12)
10.3.1 Introduction
The first application of a finned surface to condense vapor was probably done with the intent to enhance the heat transfer via additional surface area for a given projected area Today, low fins (<1.5 mm) are specially designed to enhance condensation significantly by inducing surface tension drainage forces that rid the fins of insulating condensate Gravity forces are always present, but the influence of gravity on the
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condensate drainage from a fin can be minimal for short or low fins As described in the remainder of this section, the minimum fin height to encourage surface tension drainage depends on the surface tension of the fluid and shape of the fin
10.3.2 Surface Tension Pressure Gradient
Laplace (1966) has shown that if a liquid–vapor interface is curved, a pressure dif-ference across the interface must be present to establish mechanical equilibrium of the interface The equilibrium condition is described by the difference between the pressure of the liquid (Pl) and the pressure of the vapor (Pg) by the two radii of cur-vature (r) of the interface and the surface tension of the fluid liquid–vapor interface (σ) This may be represented as
P1 − Pg= σ
1
r1 +
1
r2
(10.13)
The radii of curvature are defined as positive from the liquid side of the interface
Figure 10.2 showsr1 as the curvature in the Y–X plane; r2 is the curvature in the X–Z
plane
Consider, forexample, a still pond Because the surface of a pond is flat orof infinite radius, eq (10.13) predicts that the pressure of the water at the liquid side of the liquid–vapor interface is equal to the pressure of the surrounding air Similarly,
eq (10.13) shows that the pressure of a curved liquid film for positive radiir1 and
r2must be greater than the vapor that surrounds it A greater difference between the pressure of the liquid and vapor occurs for large surface tension and small radii of curvature (large curvature)
The liquid–vapor interface of a condensate film must have a curvature (κi) decrease along the fin length to exhibit surface tension drainage The curvature decrease of
Figure 10.2 Convex fin and condensate film