化工四大化学(专业)CHAPTER4BasicEquationsofFluidFlow(课堂讲义)

1、1 Review )( )()()( V z w yx u t )(V z w yx u Dt D -Continuity equation 连续性方程 z w yx u z w yx u t 2 For incompressible fluid，density is a constant. At steady state 0/t 0/DtD ()()() ()0 uw V xyz 0 uw xyz 3 SVSVSVm bbbaaa Integrated form of continuity equation The flow in channels of circular cross sec

2、tion 2 a b bb aa D D V V If density is taken as constant, thus 2 a b b a D D V V 4 z u w y u x u u t u x g x p z u y u x u 2 2 2 2 2 2 z w yx u t y g y p zyx 2 2 2 2 2 2 z w w y w x w u t w z g z p z w y w x w 2 2 2 2 2 2 - Navier-Stokes equations The equations of motion 5 Euler equation : constant

3、density and zero viscosity z u w y u x u u t u x p g x z w yx u t y p g y z w w y w x w u t w z p g z 6 4.3 MACROSCOPIC MOMENTUM BALANCES 1. Momentum of total stream; momentum correction factor 2. Layer flow with free surface 3. Angular-momentum equation 7 Thus equation )( aabb VVmF (4.42) ab MMF ma

4、y be written The momentum correction factor dS V u S S 2 1 (4.41) 8 Forces acting on the fluid in the direction of the velocity component in the equation include: pressure change in the direction of flow. shear stress at the boundary between the fluid stream and the conduit or (if the conduit itself

5、 is considered to be part of the system) external forces acting on the solid wall. the appropriate component of the force of gravity. 9 One-dimensional flow in the x direction, gwbbaa FFSpSpF where Fw = net force of wall of channel on fluid Fg = component of force of gravity (written for flow in upw

6、ard direction) (4.43) 10 )( aabb VVmF gwbbaa FFSpSpF (4.43) Macroscopic momentum balance equation 11 2. Layer flow with free surface 12 a newtonian liquid, steady flow, constant rate Bernoulli equation without friction n2. Bernoulli equation(柏努利方程): correction for effects of solid boundaries n3. Kin

7、etic energy of stream n4. Correction of Bernoulli equation for fluid friction n5. Pump work in Bernoulli equation 28 1. Energy equation for potential flow; Bernoulli equation without friction Potential flow: density is constant and viscosity is zero. Therefore Euler equation is useful 29 The x compo

8、nent of the Euler equation Eq. (4.31) is gp Dt DV (4.31) x g x p z u w y u x u u t u (4.45) 30 ua ub Datum surface 31 nFor this steady unidirectional flow process, Eq.(4.45) 0cos )2/( 2 ug dx dp u dx ud u (4.56) x g x p x u u x g x p z u w y u x u u t u turn to: (4.45) 32 cosxZZ a dxdZcos dxdZ /cos

9、0 1)2/( 2 dx dZ g dx dp dx ud (4.57) Equation (4.57) is the point form of the Bernoulli equation without friction. Eq.(4.56) turn to 33 nIn fps units 22 22 b b ba a a u gZ pu gZ p (4.58a) c b c bb c a c aa g u g gZp g u g gZp 22 22 (4.58b) Equation (4.58) is known as the Bernoulli equation without f

10、riction. It is a particular form of a mechanical energy balance Integrating Eq. (4.57) over the system shown in Fig. 4.8 gives 34 Discussion about Bernoulli Equation 22 22 b b ba a a u gZ pu gZ p Static energy 静压能 压力能 potential energy 势能，位能 kinetic energy 动能 J/kg total mechanical energy per unit mas

11、s , J/kg (a) Each term in Eq. (4.58) is a scalar and has the dimensions of energy per unit mass, 35 (b) Total mechanical energy is constant. Mechanical energy can change from one form to other form. 36 37 (c) for a stationary fluid, density is constant Eq. (4.58a) become )( ba ab ZZg pp (2.4) Equati

12、on of hydrostatic equilibrium is the particular form of Bernoulli equation 38 n(d) to identify the streamline or stream tube, to choose definite upstream and downstream stations, and to choose datum surface. 39 nEXAMPLE 4.4. Brine, specific gravity 60F /60F = 1.15, is draining from the bottom of a l

13、arge open tank through a 50-mm pipe. The drainpipe ends at a point 5 m below the surface of the brine in the tank. Considering a streamline starting at the surface of the brine in the tank and passing through the center of the drain line to the point of discharge, and assuming that friction along th

14、e streamline is negligible, calculate the velocity of flow along the streamline at the point of discharge from the pipe. 40 aa b b 5m Solution. choose station a as upstream station at the brine surface station b as downstream station at the end of the streamline at the point of discharge. The datum

15、for measurement of heights can be taken through station b 41 22 22 b b ba a a u gZ pu gZ p (4.58a) pa = pb = atmospheric pressure ua is negligible, Zb = 0 and Za = 5 m. 2 2 1 5 b ug aa b b 5m 42 Fluid is flowing in a s i p h o n . Friction along the streamline is negligible. nCalculate the pressure

16、at 2-2, 3-3, 4-4, 5-5 . 43 nTake 1-1 as upstream station and 6-6 as downstream station. 6-6 as datum face 22 22 b b ba a a u gZ pu gZ p 2/1 2 6 Vg smV/43. 4 6 44 nUsing Bernoulli equation from 1-1 to 2-2. 2-2 is taken as datum level Z1=3m, p1= 101330Pa, u2=4.43m/s Get p2=120990Pa Same method, we can

17、 get p3=91560Pa, p4=86660Pa, p5=91560Pa 45 p2=120990Pa p3=91560Pa, p4=86660Pa, p5=91560Pa 46 n three modification: nKinetic energy of stream nFluid friction nPump work 2. Bernoulli equation: correction for effects of solid boundaries 47 3. Kinetic energy of stream nConsider an element of cross-secti

18、onal area dS. The mass flow rate through this is udS 2 2 Vm Ek 48 nKinetic energy flow rate through area dS is 22 )( 32 dSuu udSEd k The total rate of flow of kinetic energy through the entire cross section S is, assuming constant density within the area S, dSuE S k 3 2 (4.59) 49 (4.60) SV dSu dSu d

19、Su m E S S S k 33 2 1 2 1 SV dSu m EV Sk 2 2 3 2 2 2 Vm Ek Define 50 n is 2.0 for laminar flow and is about 1.05 for highly turbulent flow. Kinetic energy correction factor 动能校正因子 SV dSu S 3 3 (4.61) 51 52 53 54 4. Correction of Bernoulli equation for fluid friction nFor incompressible fluids, the B

20、ernoulli equation becomes f bb b baa a a h V gZ pV gZ p 22 22 (4.62) 55 hf Friction, friction loss, energy loss 阻力损失 Unit: J/kg energy per unit mass hf, is always positive. It is zero in potential flow hf represents the loss of mechanical energy at all points between stations a and b. hf is not inte

21、rconvertible with the mechanical energy quantities. 56 nTwo kinds of friction loss Friction generated in unseparated boundary layers is called skin friction（表面摩擦阻力）. When boundary layers separate and form wakes, additional energy dissipation appears within the wake, and friction of this type is call

22、ed form friction（形体阻力） since it is a function of the position and shape of the solid. 57 58 nexample 4.5. 59 5. Pump work in Bernoulli equation f bb b baa a a h V gZ pV gZ p 22 22 (4.65)* f bb b b p aa a a h V gZ p W V gZ p 22 22 60 nWp : the work done by the pump per unit mass of fluid. Shaft work(轴功), J/kg Then the net work to the fluid(work