化工傳遞過程基礎(chǔ)of equations motion including homeworks

1、Transport PhenomenaDepartment of Chemical EngineeringTianjin University Jingtao Wang, Ph.D The equation of continuity The equation of motionContent in the Previous Classes Outline The materials covered in todays class: Chapter 3:The Equations of Change for Isothermal Systems Section 3.5: THE EQUATIO

2、NS OF CHANGE IN TERMS OF THE SUBSTANTIAL DERIVATIVE Several time derivativesThe simplification of equation of motion Transport PhenomenaIn vector notation, Navier-Stokes eqns may be written down to Transport PhenomenaWhen the Reynolds number Re is very small, the inertial terms in Navier-Stokes equa

3、tions are negligible compared with the viscous terms. Thus, the Navier-Stokes equations reduce to Stokes equations.The elimination of the time-dependent term in the Navier-Stokes equations makes the Stokes equations pseudosteady. Therefore, the velocity and pressure will respond instantaneously to a

4、ny change in the applied pressure or the movement of surfaces.Think： Transport PhenomenaDeformation and retraction of a drop subjected to uniaxial extensional flow. Deformation and Spreading of Liquid Droplets Transport PhenomenaBefore proceeding, we study several different time derivatives that wil

5、l be encountered in transport phenomena. We illustrate these by a example: Observing the concentration of fish in Mississippi River. Because fish swim around, the fish concentration will in general be a function of position (x, y, z) and time (t). Transport PhenomenaSection 3.5 (a):Several time deri

6、vatives Transport PhenomenaThe Partial Time Derivative We can then record the rate of change of the fish concentration at a fixed location. The result is the partial derivative of c with respect to t, at constant x, y, and z. Suppose we stand on a bridge and observe the concentration of fish just be

7、low us as a function of time. Transport PhenomenaThe Total Time DerivativeNow suppose that we jump into a motor boat and speed around on the river, sometimes going upstream, sometimes downstream, and sometimes across the current. All the time we are observing fish concentration. Transport PhenomenaA

8、t any instant, the time rate of change of the observed fish concentration is in which dx/dt, dy/dt, and dz/dt are the components of the velocity of the boat. Transport PhenomenaThe Substantial Time Derivative Next we climb into a canoe and float along with the current, observing the fish concentrati

9、on. In this situation the velocity of the observer is the same as the velocity v of the stream, which has components vx, vy, and vz. Transport PhenomenaIf at any instant we report the time rate of change of fish concentration, we haveThe special operator is called the substantial derivative (隨體導(dǎo)數(shù)，me

10、aning that the time rate of change is reported as one moves with the substance). Transport PhenomenaWe convert equations expressed in terms of into equations written with D/Dt. For any scalar function f (x, y, z, t) we can do the following manipulations: Transport PhenomenaThe quantity in the second

11、 parentheses in the second line is zero according to the equation of continuity. Consequently Eq. 3.5-3 can be written in vector form as Transport PhenomenaSimilarly, for any vector function f(x, y, z, t), When vector function f(x, y, z, t) = v(x, y, z, t) , Transport PhenomenaSection 3.5 (b):The si

12、mplification of equation of motion Transport PhenomenaWe now discuss briefly the three most common simplifications of the equation of motion. (i) For constant and ; (ii) When the acceleration terms in the Navier-Stokes equation are neglected; (iii) When viscous forces are neglected Transport Phenome

13、na(i) For constant and Navier-Stokes equation, first developed from molecular arguments by Navier and from continuum arguments by Stokes: In the second form we have used the modified pressure where h is the elevation in the gravitational field and gh is the gravitational potential energy per unit ma

14、ss. Transport Phenomena(ii) When the acceleration terms in the Navier-Stokes equation are neglected That is, we get which is called the Stokes equation. It is sometimes called the creeping flow equation. The Stokes equation is important in lubrication theory, the study of particle motions in suspens

15、ion, flow through porous media, and swimming of microbes. There is a vast literature on this subject. Transport Phenomena(iii) When viscous forces are neglected That is, the equation of motion eswhich is known as the Euler equation for inviscid fluids. There are no truly inviscid fluids, but there a

16、re many flows in which the viscous forces are relatively unimportant. For examples, the flow around airplane wings (except near the solid boundary), flow of rivers around the upstream surfaces of bridge abutments (橋墩). variables：ux，uy，uz，p；number of equations：4Analysis of continuity and motion equat

17、ions The characteristics of continuity and motion equations :（1）Non-linear partial differential equations；The objective of these equationsto get the distribution of velocity and pressure.Other physical quantities etc.（2）The force balance on a mass point （質(zhì)點(diǎn)）applies only to the analysis of regular la

18、minar flow.Analysis of continuity and motion equations The classification for equations solving ： (1) Very simple laminar flows: the equation will be very simple after simplification. It can be directly worked out by integrationanalytical solution; (2) Some simple laminar flows: the equation can be

19、simplified based on the characteristics of the flow. Then it can be solved by integrationphysical approximate solution; (3) Complex laminar flow: the equation can be solved by using numerical methods; discretizing the equation and find the difference solution (差分解); (4) Turbulent flow: transform the

20、 equations aptly, then based on the problem features and the experiment, find the semi-theoretical solution.Analysis of continuity and motion equations 3.1 Steady laminar flow between two plates1)Simplification of the equations2)Equations solving3)Average flow rate and pressure dropChapter 3 Several

21、 Solutions of Continuity and Motion Equations The fluid flows between two plates, e.g. plate heat exchangers、plate membrane separators.yflow directionxzy0oy0Assuming =constant; steady-state; far away from the inlet and outlet; the fluid flows only along the x direction .1) Simplification of the equations （1）The simplification of equation of continuity（2）The simplification of equation of motionx component：1) Simplification of the equations z component:y component:1) Simpli