天大化工原理-英文版课件-CHAPTER 4 Basic Equations of Fluid Flow

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• 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.
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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
(3)the appropriate component of the force of gravity.
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• 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)
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
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• Alternate form of Reynolds number
Re DeV

De : equivalent diameter当量直径
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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
Review
( u ) ( ) ( w) ( V ) t y z x u w u w t x y z x y z
angular momentum 角动量
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Impeller(叶轮) of a centrifugal pump (离心泵) or turbine
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(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.
(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.
g cos 2 2 u ( r ) 2
u

rdr
(4.47)
r
velocity distribution is parabolic
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• The total mass flow rate of the fluid then is
0 ubdr m
g cos 2 2 u ( r ) 2
2
Flat plate:
S L p
rH
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• The Reynolds number for flow down a flat plate is defined by the equation.
4 m Re 4rH V 4 L p
（4.51）
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4.4 MECHANICAL ENERGY EQUATION**
• 1. Energy equation for potential flow; Bernoulli equation without friction • 2. Bernoulli equation(柏努利方程): correction for effects of solid boundaries • 3. Kinetic energy of stream • 4. Correction of Bernoulli equation for fluid friction • 5. Pump work in Bernoulli equation
Байду номын сангаас
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
g cos m b 3
3 2

(4.48) (4.47)
1/ 3
3 2 g cos
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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
Control volumn L, b, r
film-wise condensation
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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
p w w w w t u x y w z z g z
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4.3 MACROSCOPIC MOMENTUM BALANCES
• 1. Momentum of total stream; momentum correction factor • 2. Layer flow with free surface • 3. Angular-momentum equation
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• Thus equation may be written
F M M b a
(4.42)
( bVb aVa ) F m
• The momentum correction factor
1 u S dS S V
2
(4.41)
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• Forces acting on the fluid in the direction of the velocity component in the equation include: (1) pressure change in the direction of flow.
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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
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
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( bVb aVa ) F m
F pa Sa pb Sb Fw Fg
(4.43)
Macroscopic momentum balance equation
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2. Layer flow with free surface
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a newtonian liquid, steady flow, constant rate&thickness,laminar flow
u w D ( V ) Dt x y z
------------Continuity equation
连续性方程
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At steady state / t 0 ( u ) ( ) ( w) ( V ) 0 x y z For incompressible fluid，density is a constant.
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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
equivalent diameter当量直径: De
S De 4rH 4 Lp
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• 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
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The equations of motion
u u u u t u x y w z
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Fg cos A 0
A = bL
(4.44)
Fg mg rLbg
The flow is laminar
du / dr
g cos
Rearranging eq. (4.44) and integrating between limits give
0 du