天津大学化工流体课后题答案.docx

1.3 A differential manometer as shown in Fig. is sometimes used to measure small pressure difference. When the reading is zero, the levels in two reservoirs are equal. Assume that fluid B is methane（甲烷）, that liquid C in the reservoirs is kerosene (specific gravity = 0.815), and that liquid A in the U tube is water. The inside diameters of the reservoirs and U tube are 51mm and 6.5mm , respectively. If the reading of the manometer is145mm., what is the pressure difference over the instrument In meters of water, (a) when the change in the level in the reservoirs is neglected, (b) when the change in the levels in the reservoirs is taken into account? What is the percent error in the answer to the part (a)? Solution：pa=1000kg/m3 pc=815kg/m3 pb=0.77kg/m3 D/d=8 R=0.145mWhen the pressure difference between two reservoirs is increased, the volumetric changes in the reservoirs and U tubes (1)so (2)and hydrostatic equilibrium gives following relationship (3)so (4)substituting the equation (2) for x into equation (4) gives (5)（a）when the change in the level in the reservoirs is neglected, （b）when the change in the levels in the reservoirs is taken into accounterror=1.4 There are two U-tube manometers fixed on the fluid bed reactor, as shown in the figure. The readings of two U-tube manometers are R1=400mm，R2=50mm, respectively. The indicating liquid is mercury. The top of the manometer is filled with the water to prevent from the mercury vapor diffusing into the air, and the height R3=50mm. Try to calculate the pressure at point A and B. Figure for problem 1.4Solution: There is a gaseous mixture in the U-tube manometer meter. The densities of fluids are denoted by , respectively. The pressure at point A is given by hydrostatic equilibrium is small and negligible in comparison withand H2O , equation above can be simplified= =10009.810.05+136009.810.05=7161N/m=7161+136009.810.4=60527N/mDdpapaHhAFigure for problem 1.51.5 Water discharges from the reservoir through the drainpipe, which the throat diameter is d. The ratio of D to d equals 1.25. The vertical distance h between the tank A and axis of the drainpipe is 2m. What height H from the centerline of the drainpipe to the water level in reservoir is required for drawing the water from the tank A to the throat of the pipe? Assume that fluid flow is a potential flow. The reservoir, tank A and the exit of drainpipe are all open to air.Solution:Bernoulli equation is written between stations 1-1 and 2-2, with station 2-2 being reference plane:Where p1=0, p2=0, and u1=0, simplification of the equation 1The relationship between the velocity at outlet and velocity uo at throat can be derived by the continuity equation: 2Bernoulli equation is written between the throat and the station 2-2 3Combining equation 1,2,and 3 givesSolving for HH=1.39m1.6 A liquid with a constant density kg/m3 is flowing at an unknown velocity V1 m/s through a horizontal pipe of cross-sectional area A1 m2 at a pressure p1 N/m2, and then it passes to a section of the pipe in which the area is reduced gradually to A2 m2 and the pressure is p2. Assuming no friction losses, calculate the velocities V1 and V2 if the pressure difference (p1 - p2) is measured.Solution: In Fig1.6, the flow diagram is shown with pressure taps to measure p1 and p2. From the mass-balance continuity equation , for constant where 1 = 2 = ,For the items in the Bernoulli equation , for a horizontal pipe,Then Bernoulli equation becomes, after substituting for 2,Rearranging,Performing the same derivation but in terms of 2,1.7 A liquid whose coefficient of viscosity is flows below the critical velocity for laminar flow in a circular pipe of diameter d and with mean velocity V. Show that the pressure loss in a length of pipe is .Oil of viscosity 0.05 Pas flows through a pipe of diameter 0.1m with a average velocity of 0.6m/s. Calculate the loss of pressure in a length of 120m.Solution: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 area 1From velocity profile equation for laminar flow 2substituting equation 2 for u into equation 1 and integrating 3rearranging equation 3 givesFigure for problem 1.81.8. In a vertical pipe carrying water, pressure gauges are inserted at points A and B where the pipe diameters are 0.15m and 0.075m respectively. The point B is 2.5m below A and when the flow rate down the pipe is 0.02 m3/s, the pressure at B is 14715 N/m2 greater than that at A. Assuming the losses in the pipe between A and B can be expressed as where V is the velocity at A, find the value of k.If the gauges at A and B are replaced by tubes filled with water and connected to a U-tube containing mercury of relative density 13.6, give a sketch showing how the levels in the two limbs of the U-tube differ and calculate the value of this difference in metres. Solution: dA=0.15m; dB=0.075mzA-zB=l=2.5mQ=0.02 m3/s,pB-pA=14715 N/m2When the fluid flows down, writing mechanical balance equation0.295making the static equilibriumFigure for problem 1.91.9The liquid vertically flows down through the tube from the station a to the station b, then horizontally through the tube from the station c to the station d, as shown in figure. Two segments of the tube, both ab and cd，have the same length, the diameter and roughness.Find:（1）the expressions of , hfab, and hfcd, respectively.（2）the relationship between readings R1and R2 in the U tube.Solution:(1) From Fanning equationandsoFluid flows from station a to station b, mechanical energy conservation gives hence 2from station c to station dhence 3From static equationpa-pb=R1（-）g -lg 4pc-pd=R2（-）g 5Substituting equation 4 in equation 2 ，thentherefore 6Substituting equation 5 in equation 3 ，then 7ThusR1=R21.10 Water passes through a pipe of diameter di=0.004 m with the average velocity 0.4 m/s, as shown in Figure. 1) What is the pressure drop DP when water flows through the pipe length L=2 m, in m H2O column?LrFigure for problem 1.102) 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.4）if kerosene flows through this pipe，how do the variables above change？（the viscosity and density of Water are 0.001 Pas and 1000 kg/m3，respectively；and the viscosity and density of kerosene are 0.003 Pas and 800 kg/m3，respectively）solution:1） from Hagen-Poiseuille equation2）maximum velocity occurs at the center of pipe, from equation 1.4-19so umax=0.42=0.8m3）when u=V=0.4m/s Eq. 1.4-17 *4) kerosene:1.12 As shown in the figure, the water level in the reservoir keeps constant. A steel drainpipe (with the inside diameter of 100mm) is connected to the bottom of the reservoir. One arm of the U-tube manometer is connected to the drainpipe at the position 15m away from the bottom of the reservoir, and the other is opened to the air, the U tube is filled with mercury and the left-side arm of the U tube above the mercury is filled with water. The distance between the upstream tap and the outlet of the pipeline is 20m. a) When the gate valve is closed, R=600mm, h=1500mm; when the gate valve is opened partly, R=400mm, h=1400mm. The friction coefficient is 0.025, and the loss coefficient of the entrance is 0.5. Calculate the flow rate of water when the gate valve is opened partly. (in m/h)b) When the gate valve is widely open, calculate the static pressure at the tap (in gauge pressure, N/m). le/d15 when the gate valve is widely open, and the friction coefficient is still 0.025.Figure for problem 1.12Solution：(1) When the gate valve is opened partially, the water discharge isSet up Bernoulli equation between the surface of reservoir 11 and the section of pressure point 22，and take the center of section 22 as the referring plane, then （a）In the equation (the gauge pressure) When the gate valve is fully closed, the height of water level in the reservoir can be related to h (the distance between the center of pipe and the meniscus of left arm of U tube).（b） where h=1.5mR=0.6mSubstitute the known variables into equation bSubstitute the known variables equation a9.816.66=the velocity is V =3.13m/s the flow rate of water is 2） the pressure of the point where pressure is measured when the gate valve is wide-open. Write mechanical energy balance equation between the stations 11 and 3-3，then （c）since input the above data into equation c，9.81the velocity is: V=3.51 m/sWrite mechanical energy balance equation between thestations 11 and 22, for the same situation of water level （d）since input the above data into equation d，9.816.66=the pressure is: Figure for problem 1.171.17.Sulphuric acid of specific gravity 1.3 is flowing through a pipe of 50 mm internal diameter. A thin-lipped orifice, 10mm, is fitted in the pipe and the differential pressure shown by a mercury manometer is 10cm. Assuming that the leads to the manometer are filled with the acid, calculate (a)the weight of acid flowing per second, and (b) the approximate pressure drop caused by the orifice.The coefficient of the orifice may be taken as 0.61, the specific gravity of mercury as 13.6, and the density of water as 1000 kg/m3Solution:a)b) approximate pressure drop588.6Pa *2.1 Water is used to test for the performances of pump. The gauge pressure at the discharge connection is 152 kPa and the reading of vacuum gauge at the suction connection of the pump is 24.7 kPa as the flow rate is 26m3/h. The shaft power is 2.45kw while the centrifugal pump operates at the speed of 2900r/min. If the vertical distance between the suction connection and discharge connection is 0.4m, the diameters of both the suction and discharge line are the same. Calculate the mechanical efficiency of pump and list the performance of the pump under this operating condition.Solution: Write the mechanical energy balance equation between the suction connection and discharge connection wheretotal heads of pump is efficiency of pump is since N=2.45kWThen mechanical efficiency The performance of pump is Flow rate ，m/h26Total heads，m18.41Shaft power ，kW2.45Efficiency ，%53.12.2 Water is transported by a pump from reactor, which has 200 mm Hg vacuum, to the tank, in which the gauge pressure is 0.5 kgf/cm2, as shown in Fig. The total equivalent length of pipe is 200 m including all local frictional loss. The pipeline is f573.5 mm , the orifice coefficient of Co and orifice diameter do are 0.62 and 25 mm, respectively. Frictional coefficient l is 0.025. Calculate: Developed head H of pump, in m (the reading R of U pressure gauge in orifice meter is 168 mm Hg)Solution:Equation(1.6-9)Mass flow rate2) Fluid flow through the pipe from the reactor to tank, the Bernoulli equation is as followsfor V1=V2Dz=10m Dp/rg=7.7mThe relation between the hole velocity and velocity of pipeFriction lossso H=7.7+10+5.1=22.8mHg2.3 . A centrifugal pump is to be used to extract water from a condenser in which the vacuum is 640 mm of mercury, as shown in figure. At the rated discharge, the net positive suction head must be at least 3m above the cavitation vapor pressure of 710mm mercury vacuum. If losses in the suction pipe accounted for a head of 1.5m. What must be the least height of the liquid level in the condenser above the pump inlet?Solution:From an energy balance,Where Po=760-640=120mmHgPv=760-710=50mmHgUse of the equat