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Unformatted text preview: ed above was first developed by Nusselt in 1916
under the following simplifying assumptions:
1. Both the plate and the vapor are maintained at constant temperatures of
Ts and Tsat, respectively, and the temperature across the liquid film varies
linearly.
2. Heat transfer across the liquid film is by pure conduction (no convection
currents in the liquid film).
3. The velocity of the vapor is low (or zero) so that it exerts no drag on the
condensate (no viscous shear on the liquid–vapor interface).
4. The flow of the condensate is laminar and the properties of the liquid
are constant.
5. The acceleration of the condensate layer is negligible.
Then Newton’s second law of motion for the volume element shown in Figure
10–24 in the vertical xdirection can be written as
Fx max 0 since the acceleration of the fluid is zero. Noting that the only force acting
downward is the weight of the liquid element, and the forces acting upward
are the viscous shear (or fluid friction) force at the left and the buoyancy
force, the force balance on the volume element becomes
Weight
l g( Fdownward ↓ Fupward ↑
Viscous shear force Buoyancy force
du
y)(bdx)
(bdx)
g(
y)(bdx)
l
dy Canceling the plate width b and solving for du/dy gives
du
dy g( l )g(
l y) Weight
ρl g(δ – y) (bdx) Buoyancy force
ρv g(δ – y) (bdx) y 0
x δ
dx
=0
at y = 0 y Idealized
velocity
profile
No vapor drag
Idealized
temperature
profile Ts
g
Liquid, l Tsat
Linear FIGURE 10–24
The volume element of condensate
on a vertical plate considered
in Nusselt’s analysis. cen58933_ch10.qxd 9/4/2002 12:38 PM Page 536 536
HEAT TRANSFER Integrating from y 0 where u 0 (because of the noslip boundary condition) to y y where u u(y) gives
g( u(y) )g l y2
2 y l (1012) The mass flow rate of the condensate at a location x, where the boundary layer
thickness is , is determined from
·
m (x) l u(y)dA A l u(y)bdy y0 (1013) Substituting the u(y) relation from Equation 10–12 into Eq. 10–13 gives
gb l( ·
m (x) ) l 3 3 2 d
dx (1014) l whose derivative with respect to x is
·
dm
dx gb l( ) l
l (1015) which represents the rate of condensation of vapor over a vertical distance dx...
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 Spring '10
 Ghaz

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