17
A vertical surface:
3 2g
1/ 3
Deviation of predicted values from that of measurements For Re ≈ 1000, predicted value is approximately correct For Re < 1000, predicted value is larger For Re > 1000, predicted value is smaller
18
• Alternate form of Reynolds number
Re DeV

De : equivalent diameter当量直径
19
hydraulic radius水力半径: rH
S rH Lp
where S = cross section area of channel流通截面 Lp = perimeter of channel in contact with liquid 润湿周边
D / Dt 0
u w 0 x y z
2
Integrated form of continuity equation
aVa Sa bVb Sb V S m
The flow in channels of circular cross section
g cos 2 2 u ( r ) 2
u

rdr
(4.47)
r
velocity distribution is parabolic
14
• The total mass flow rate of the fluid then is
0 ubdr m
g cos 2 2 u ( r ) 2
16
• Discussion of the assumptions: • 1. No shear stress acting on the interface of liquid and gas(air) • Shear force will increase if the vapor(or gas) shear exist. • 2. Laminar flow • A critical Re number of 2100 has often been used for layer flow, but film thickness measurements indicated a transition at Re around 1200.
equivalent diameter当量直径: De
S De 4rH 4 Lp
20
• For circular pipe
D / 4 D D 4 r D e H rH D 4 For layer flow with free surface of a cylinder or a flat plate L p D rH S D Cylinder:
aVa Db bVb Da
2
2
• If density is taken as constant, thus
Va Db Vb Da
3
The equations of motion
u u u u t u x y w z
4
Euler equation : constant density and zero viscosity
u u u u p u w gx t x y z x
p g y t u x y w z y
2 u 2 u 2 u p x 2 y 2 z 2 x g x
2 2 2 p x 2 y 2 z 2 y g y t u x y w z
(3)the appropriate component of the force of gravity.
8
• One-dimensional flow in the x direction, where
F pa Sa pb Sb Fw Fg
(4.43)
Fw = net force of wall of channel on fluid Fg = component of force of gravity (written for flow in upward direction)
13
Fg cos A 0
A = bL
(4.44)
Fg mg rdu / dr
g cos
Rearranging eq. (4.44) and integrating between limits give
0 du
22
3. Angular-momentum equation(角动量方程)
• Analysis of the performance of rotating fluidhandling machinery such as pumps, turbines, and agitators is facilitated by the use of force moments and angular momentum. force moment 力矩
Review
( u ) ( ) ( w) ( V ) t y z x u w u w t x y z x y z
u w D ( V ) Dt x y z
------------Continuity equation
连续性方程
1
At steady state / t 0 ( u ) ( ) ( w) ( V ) 0 x y z For incompressible fluid，density is a constant.
angular momentum 角动量
23
Impeller(叶轮) of a centrifugal pump (离心泵) or turbine
24
25
(r2u 2 r1u 1 ) T F r2 m
(4.53)
• Equation (4.53) is the angular momentum equation for steady two-dimensional flow. • Applications of Eq. (4.53) are given in Chaps. 8 and 9.
Control volumn L, b, r
film-wise condensation
12
the shear on the upper surface of the element is neglected.
Fg cos A 0
(4.44)
• where Fg = gravity force • = shear stress on lower surface • of control volume • A = area of lower surface of control volume
w w w w t u x y w z
2 w 2 w 2 w p x 2 y 2 z 2 z g z
------------ Navier-Stokes equations
(2) shear stress at the boundary between the fluid stream and the conduit or (if the conduit itself is considered to be part of the system) external forces acting on the solid wall.
2
Flat plate:
S L p
rH
21
• The Reynolds number for flow down a flat plate is defined by the equation.
4 m Re 4rH V 4 L p
（4.51）
p w w w w t u x y w z z g z
5
4.3 MACROSCOPIC MOMENTUM BALANCES
• 1. Momentum of total stream; momentum correction factor • 2. Layer flow with free surface • 3. Angular-momentum equation
g cos m b 3
3 2

(4.48) (4.47)
1/ 3
3 2 g cos
15
3 2 g cos m b 3
(4.49)
/b m
• where Γ is called the liquid loading(凝液负荷). • The units of Γ are kg/m.s or lb/ft.s. • b is called perimeter of channel in contact with liquid 润湿周边
9
( bVb aVa ) F m
F pa Sa pb Sb Fw Fg
(4.43)
Macroscopic momentum balance equation