化工傳遞過程基礎8flow through a circular and an annulus

1、Transport PhenomenaDepartment of Chemical EngineeringTianjin University Jingtao Wang, Ph.D Shell momentum balances and boundary conditions Flow of a falling film Content in the Previous Class Outline The materials covered in todays class: Chapter 2:Shell Momentum Balances and Velocity Distributions

2、in Laminar Flow Section 2.3: Flow through a circular tube Section 2.4: Flow through an annulus Transport PhenomenaSection 2.3:Flow through a circular tube Transport PhenomenaDescription of the Problem:The flow of fluids in circular tubes is encountered frequently in physics, chemistry, biology, and

3、engineering. Transport PhenomenaWhat are the transport phenomena? Singapore ethylene chemical plantTransport phenomena in chemical engineering Transport PhenomenaDescription of the Problem:The laminar flow of fluids in circular tubes may be analyzed by the momentum balance. (figure）The only new feat

4、ure introduced here is the use of cylindrical coordinates, (figure）which are the natural coordinates for a pipe of circular cross section. Description of the Problem:We consider the steady-state, laminar flow of a fluid of density and viscosity in a vertical tube of length L and radius R. The liquid

5、 flows downward under the influence of a pressure difference and gravity; the coordinate system is that shown in Fig. 2.3-1. Fig. 2.3-1 Transport PhenomenaAssumption to solve the Problem:We specify that the tube length is very large with respect to the tube radius. Thus, we can ignore end effects th

6、at at the tube entrance and exit the flow will not be parallel to the tube wall. We postulate that vz = vz(r), vr = 0, v = 0, and p = p(z). Then the only nonvanishing components of are rz= zr = -(dvz/dr). Construction of the shell:We select a cylindrical shell of thickness r and length L as our syst

7、em. Transport PhenomenaContributions to the momentum balance: Transport PhenomenaConstruction of the momentum balance and simplification: We now add up the contributions to the momentum balance: (2.3-6) When we divide Eq. (2.3-6) by 2Lr and take the limit as r 0, we get (2.3-8) Transport PhenomenaNo

8、w we write out the components of and (ii) because vz = vz(r), the term vzvz will be the same at both ends of the tube; (iii) because vz = vz(r), the term will be the same at both ends of the tube. Then we make the following simplifications:(i) because vr = 0, we can drop the term vrvz in Eq. 2.3-9a;

9、 in which = p - gz is a convenient abbreviation for the sum of the pressure and gravitational terms. Transport PhenomenaHence, Eq. 2.3-8(2.3-10) Equation 2.3-10 may be integrated to give simplifies to Transport PhenomenaThe constant C 1 is evaluated by using the boundary condition Consequently C 1 m

10、ust be zero, for otherwise the momentum flux would be infinite at the axis of the tube. Therefore the momentum flux distribution is Newtons law of viscosity for this situation (2.3-15) Transport Phenomena Transport PhenomenaFig. 2.3-2 The momentum-flux distribution and velocity distribution for the

11、downward flow in a circular tube. Transport Phenomena Transport Phenomena Transport Phenomena Transport PhenomenaSection 2.3:Flow through an annulus Transport PhenomenaDescription of the Problem:We now solve the steady-state flow of an pressible liquid in an annular region between two coaxial cylind

12、ers of radii R and R as shown in Fig. 2.4-1.The fluid is flowing upward in the tube, that is, in the direction opposed to gravity. Transport PhenomenaAssumption to solve the Problem:We make the same postulates as in the previous section vz = vz(r), vr = 0, v = 0, and p = p(z).Then when we make a mom

13、entum balance over a thin cylindrical shell of liquid, we arrive at the following differential equation: Construction of the momentum balance and simplification: (2.4-1) - Transport PhenomenaIntegration of Eq. 2.4-1 gives (2.4-2) The constant C1 cannot be determined immediately, since we have no inf

14、ormation about the momentum flux at the fixed surfaces r = R and r = R.(2.4-3) All we know is that there will be a maximum in the velocity curve at some (as yet unknown) plane r = R at which the momentum flux will be zero. That is, Transport PhenomenaWhen we solve this equation for C1 and substitute

15、 it into Eq. 2.4-2, we (2.4-4) In this equation the constant of integration C1 has been eliminated by introducing a different constant . The advantage of this is that we know the geometrical significance of . Substituting Newtons law of viscosity to obtain a differential equation for vz (2.4-5) Transport Phenomena Transport Phenomena Transport Phenomena Transport PhenomenaHomework: Read Bird book: 58Pretending that you are a teacher and trying to teach some students about shell balances. Please make a English