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1、1.3.6 Macroscopic Momentum Balances2. Layer flow with free surfaceIn one form of layer flow the liquid layer has a free surface and flows under the force of gravity over an inclined or vertical surface. If such flow is in steady state, with fully developed velocity gradients, the thickness of the la
2、yer is constant. There is so little drag at the free liquid surface that the shear stress there can be ignored. If the flow is laminar and the liquid surface is flat and free from ripples, the fluid motion can be analyzed mathematically.A layer of liquid flowing in steady flow at constant rate and t
3、hickness over a flat plate. The plate is inclined at an angle with the vertical. The breadth of the layer in the direction perpendicular to the plane of the figure is b, and the thickness of the layer in the direction perpendicular to the plate is The upper surface of the control volume is in contac
4、t with the atmosphere, the two ends are planes perpendicular to the plate at a distance L apart, and the lower surface is the plane parallel with the wall at a distance r from the upper surface of the layer. lThe possible forces acting on the control volume in a direction parallel to the flow are th
5、e pressure forces on the ends, lthe shear forces on the upper and lower faces, land the component of the force of gravity in the direction of flow. lFrom this equation, noting that A=bL and Since the flow is laminar Rearranging and integrating between limits give Equation (1.3-40) shows that in lami
6、nar flow on a plate the velocity distribution is parabolic. (1.3-40) Shell Momentum Balance for Falling Film We now use an approach similar to that used for laminar flow inside a pipe for the case of flow of a fluid as a film in laminar flow down under the force of gravity over a vertical surface. T
7、he control volume for the falling film is shown in Fig. 2.9-3a. Fluid thickness is x and has a length of L in the vertical z direction. This region is sufficiently far from the entrance and exit regions so that the flow is not affected by these regions. This means the velocity u(x) does not depend o
8、n position z. The shear stress can be ignored at the freeliquid surface. If the flow is laminar and the liquid surface is flat and free from ripples.The fluid motion can be analyzed mathematicallyThe possible forces acting on the control volume in a direction parallel to the flow arel the pressure f
9、orces at the endsl the shear forces on the facesl the component of the force of gravity in the direction of flow. Since the pressure on the outer surface is atmospheric, the pressures on the control volume at the ends of the volume are equal and oppositely directed.Also, by assumption, the shear on
10、the free liquid surface of the element is neglected.The two forces remaining are therefore the shear force on the right surface of the control volume and the component of the gravity in the direction of flow.From this equation, noting that A=WL and Fg=gLWxSoSince the flow is laminar, = -du/dx andRea
11、rranging and integrating between limitsgives (the boundary condition that u= 0 at x = )and Where is the total thickness of the liquid layer2.9-25 This means the velocity profile is parabolic, as shown in Fig. 2.9-3b. The maximum velocity occurs at x = 0 in Eq. (2.9-25) and is 2.9-26 The average velo
12、city can be found: 2.9-27 Substituting Eq. (2.9-25) into (2.9-27) and integrating 2.9-28 Industrial processes necessarily require the flow of fluids through pipes, tubes ,and channels with a noncircular cross section.This section deals with the steady flow of incompressible fluids through closed pip
13、es and channels1.4 Incompressible Flow in Pipes and Channels Figure Fluid element in steady flow through pipe.RFlowpdL-(p+dp)rConsider the steady flow of fluid of constant density in fully developed flow through a horizontal pipe. Visualize a disk-shaped element of fluid, concentric with the axis of
14、 the tube, of radius r and length dL, as shown in Fig. forces acting on the element:vForce from the pressure on the upstream faces of the disk: Abpb =r2(p+dP)vForce from the pressure on the downstream faces of the disk: Aapa=r2pvShear force acting on the rim of the element:(2rdL) So the total force
15、acting on the element isSimplifying this equation and dividing by r2dL(1.4-1)In steady flow, the pressure at any given crosssection of a stream tube is constant, so thatdp/dL is independent of r, dp/dL=p/L, Eq. (1.4-1) then becomes1.4-1 lThis means that the momentum flux varies linearly with the rad
16、ius, as shown in Fig., and the maximum value occurs at r = R at the wall. Substituting Newtons law of viscosity, into Eq. we obtain the following differential equation for the velocity: Integrating using the boundary condition that at the wall, u= 0 at r = R, we obtain the equation for the velocity
17、distribution 1.4-15 This result shows that the velocity distribution is parabolic, as shown in Fig The average velocity V for a cross section is found by summing up all the velocities over the cross section and dividing by the cross-sectional areasubstituting u into equation and integrating Equation
18、 called HagenPoiseuille equation, relates the pressure drop and average velocity for laminar flow in a horizontal pipe.lThe maximum velocity for a pipe is found from Eq. (1.4-15) and occurs at r = 0: lwe find that 1.4-16Also, 1.4-17 Velocity Profiles in PipesFigure is a plot of the relative distance
19、 from the center of the pipe versus the ratio of local velocity to maximum velocity u/umax. For laminar flow, the velocity profile is a true parabola. The velocity at the wall is zero. In many engineering applications the average velocity V in a pipe is the most useful. In some cases only the umax a
20、t the center point of the tube is measured. The relationship between umax and V can be used to determine V. experimentally measured values of V/umax are plotted as a function of the Reynolds numbers The average velocity is precisely 0.5 times the maximum velocity at the center for laminar flow. On t
21、he other hand, for turbulent flow, the curve is somewhat flattened in the center and the average velocity is about 0.8 times the maximum. This value of 0.8 varies slightly, depending upon the Reynolds number。lThe liquid layer has a free surface and flows in laminar flow under the force of gravity ov
22、er a vertical surface, the velocity profile is ( )lAverage velocity is ( )Problem 1The average velocity is ( ) times the maximum velocity at the center of tube for laminar flow. The average velocity is ( ) times the maximum velocity at center for turbulent flowThe maximum value of the momentum flux
23、occurs ( ) for a laminar flow.The maximum value of the velocity occurs ( ) for a fluid flow in pipe. In laminar flow the velocity distribution with respect to radius is ( ) with the apex at the centerline of the pipe The pressure drop is ( ) to the average velocity for the laminar flowProblem Water
24、(=0.001 Pas, =1000 kg/m3) passes through a pipe of diameter di=0.001 m with the average velocity 1 m/s, as shown in Figure. 1) What is the pressure drop P when water flows through the pipe length L=2 m, in m H2O column?2) Find the maximum velocity and point r at which it occurs. 3) Find the point r at which the average velocity equals the local velocity.rLsolutionIt is laminarSince the flow is laminar, we haveso p=1.22m H2OAnd average velocityr=0, The velocity is the maximum velocity we have a distribution of velocitylet V =u, so
