Dimensional Analysis and ScaleUp

20. In the steady flow of a Newtonian fluid through a long uniform circular tube, if NRe < 2000 the flow is laminar and the fluid elements move in smooth straight parallel lines. Under these conditions, it is known that the relationship between the flow rate and the pressure drop in the pipe does not depend upon the fluid density or the pipe wall material.

(a) Perform a dimensional analysis of this system to determine the dimension-less groups that apply. Express your result in a form in which the Reynolds number can be identified.

(b) If water is flowing at a rate of 0.5 gpm through a pipe with an ID of 1 in., what is the value of the Reynolds number? If the diameter is doubled at the same flow rate, what will be the effect on the Reynolds number and on the pressure drop?

21. Perform a dimensional analysis to determine the groups relating the variables that are important in determining the settling rate of very small solid particles falling in a liquid. Note that the driving force for moving the particles is gravity and the corresponding net weight of the particle. At very slow settling velocities, it is known that the velocity is independent of the fluid density. Show that this also requires that the velocity be inversely proportional to the fluid viscosity.

22. A simple pendulum consists of a small, heavy ball of mass m on the end of a long string of length L. The period of the pendulum should depend on these factors, as well as on gravity, which is the driving force for making it move. What information can you get about the relationship between these variables from a consideration of their dimensions? Suppose you measured the period, T, of a pendulum with mass m1 and length L1. How could you use this to determine the period of a different pendulum with a different mass and length? What would be the ratio of the pendulum period on the moon to that on the earth? How could you use the pendulum to determine the variation of g on the earth's surface?

23. An ethylene storage tank in your plant explodes. The distance that the blast wave travels from the blast site (R) depends upon the energy released in the blast (E), the density of the air (p), and time (t). Use dimensional analysis to determine:

(a) The dimensionless group(s) that can be used to describe the relationship between the variables in the problem

(b) The ratio of the velocity of the blast wave at a distance of 2000 ft from the blast site to the velocity at a distance of 500 ft from the site

The pressure difference across the blast wave (AP) also depends upon the blast energy (E), the air density (p), and time (t). Use this information to determine:

(c) The ratio of the blast pressure at a distance of 500 ft from the blast site to that at a distance of 2000 ft from the site

24. It is known that the power required to drive a fan depends upon the impeller diameter (D), the impeller rotational speed (&>), the fluid density (p), and the volume flow rate (Q). (Note that the fluid viscosity is not important for gases under normal conditions.)

(a) What is the minimum number of fundamental dimensions required to define all of these variables?

(b) How many dimensionless groups are required to determine the relationship between the power and all the other variables? Find these groups by dimensional analysis, and arrange the results so that the power and the flow rate each appear in only one group.

25. A centrifugal pump with an 8 in. diameter impeller operating at a rotational speed of 1150 rpm requires 1.5 hp to deliver water at a rate of 100 gpm and a pressure of 15 psi. Another pump for water, which is geometrically similar but has an impeller diameter of 13 in., operates at a speed of 1750 rpm. Estimate the pump pressure, flow capacity, and power requirements of this second pump. (Under these conditions, the performance of both pumps is independent of the fluid viscosity.)

26. A gas bubble of diameter d rises with velocity V in a liquid of density p and viscosity p.

(a) Determine the dimensionless groups that include the effects of all the significant variables, in such a form that the liquid viscosity appears in only one group. Note that the driving force for the bubble motion is buoyancy, which is equal to the weight of the displaced fluid.

(b) You want to know how fast a 5 mm diameter air bubble will rise in a liquid with a viscosity of 20 cP and a density of 0.85 g/cm3. You want to simulate this system in the laboratory using water (p = 1 cP, p = 1 g/cm3) and air bubbles. What size air bubble should you use?

(c) You perform the experiment, and measure the velocity of the air bubble in water (Vm). What is the ratio of the velocity of the 5 mm bubble in the field liquid (Vf ) to that in the lab (Vm)?

27. You must predict the performance of a large industrial mixer under various operating conditions. To obtain the necessary data, you decide to run a laboratory test on a small-scale model of the unit. You have deduced that the power (P) required to operate the mixer depends upon the following variables:

Tank diameter D Impeller diameter d

Impeller rotational speed N Fluid density p Fluid viscosity p

(a) Determine the minimum number of fundamental dimensions involved in these variables and the number of dimensionless groups that can be defined by them.

(b) Find an appropriate set of dimensionless groups such that D and N each appear in only one group. If possible, identify one or more of the groups with groups commonly encountered in other systems.

(c) You want to know how much power would be required to run a mixer in a large tank 6 ft in diameter, using an impeller with a diameter of 3 ft operating at a speed of 10 rpm, when the tank contains a fluid with a viscosity of 25 cP and a specific gravity of 0.85. To do this, you run a lab test on a model of the system, using a scale model of the impeller that is 10 in. in diameter. The only appropriate fluid you have in the lab has a viscosity of 15 cP and a specific gravity of 0.75. Can this fluid be used for the test? Explain.

(d) If the preceding lab fluid is used, what size tank should be used in the lab, and how fast should the lab impeller be rotated?

(e) With the lab test properly designed and the proper operating conditions chosen, you run the test and find that it takes 150 W to operate the lab test model. How much power would be required to operate the larger field mixer under the plant operating conditions?

28. When an open tank with a free surface is stirred with an impeller, a vortex will form around the shaft. It is important to prevent this vortex from reaching the impeller, because entrainment of air in the liquid tends to cause foaming. The shape of the free surface depends upon (among other things) the fluid properties, the speed and size of the impeller, the size of the tank, and the depth of the impeller below the free surface.

(a) Perform a dimensional analysis of this system to determine an appropriate set of dimensionless groups that can be used to describe the system performance. Arrange the groups so that the impeller speed appears in only one group.

(b) In your plant you have a 10 ft diameter tank containing a liquid that is 8 ft deep. The tank is stirred by an impeller that is 6 ft in diameter and is located 1 ft from the tank bottom. The liquid has a viscosity of 100 cP and a specific gravity of 1.5. You need to know the maximum speed at which the impeller can be rotated without entraining the vortex. To find this out, you design a laboratory test using a scale model of the impeller that is 8 in. in diameter. What, if any, limitations are there on your freedom to select a fluid for use in the lab test?

(c) Select an appropriate fluid for the lab test and determine how large the tank used in the lab should be and where in the tank the impeller should be located.

(d) The lab impeller is run at such a speed that the vortex just reaches the impeller. What is the relation between this speed and that at which entrain-ment would occur in the tank in the plant?

29. The variables involved in the performance of a centrifugal pump include the fluid properties (^ and p), the impeller diameter (d), the casing diameter (D), the impeller rotational speed (N), the volumetric flow rate of the fluid (Q), the head

(H) developed by the pump (AP = pgH), and the power required to drive the pump (HP).

(a) Perform a dimensional analysis of this system to determine an appropriate set of dimensionless groups that would be appropriate to characterize the pump. Arrange the groups so that the fluid viscosity and the pump power each appear in only one group.

(b) You want to know what pressure a pump will develop with a liquid that has a specific gravity of 1.4 and a viscosity of 10 cP, at a flow rate of 300 gpm. The pump has an impeller with a diameter of 12 in., which is driven by a motor running at 1100 rpm. (It is known that the pump performance is independent of fluid viscosity unless the viscosity is greater than about 50 cP.) You want to run a lab test that simulates the operation of the larger field pump using a similar (scaled) pump with an impeller that has a diameter of 6 in. and a 3600 rpm motor,. Should you use the same liquid in the lab as in the field, or can you use a different liquid? Why?

(c) If you use the same liquid, what flow rate should be used in the lab to simulate the operating conditions of the field pump?

(d) If the lab pump develops a pressure of 150 psi at the proper flow rate, what pressure will the field pump develop with the field fluid?

(e) What pressure would the field pump develop with water at a flow rate of 300 gpm?

30. The purpose of a centrifugal pump is to increase the pressure of a liquid in order to move it through a piping system. The pump is driven by a motor, which must provide sufficient power to operate the pump at the desired conditions. You wish to find the pressure developed by a pump operating at a flow rate of 300 gpm with an oil having a specific gravity (SG) of 0.8 and a viscosity of 20 cP, and the required horsepower for the motor to drive the pump. The pump has an impeller diameter of 10 in., and the motor runs at 1200 rpm.

(a) Determine the dimensionless groups that would be needed to completely describe the performance of the pump.

(b) You want to determine the pump pressure and motor horsepower by measuring these quantities in the lab on a smaller scale model of the pump that has a 3 in. diameter impeller and a 1800 rpm motor, using water as the test fluid. Under the operating conditions for both the lab model and the field pump, the value of the Reynolds number is very high, and it is known that the pump performance is independent of the fluid viscosity under these conditions. Determine the proper flow rate at which the lab pump should be tested and the ratio of the pressure developed by the field pump to that of the lab pump operating at this flow rate as well as the ratio of the required motor power in the field to that in the lab.

(c) The pump efficiency (^e) is the ratio of the power delivered by the pump to the fluid (as determined by the pump pressure and flow rate) to the power delivered to the pump by the motor. Because this is a dimensionless number, it should also have the same value for both the lab and field pumps when they are operating under equivalent conditions. Is this condition satisfied?

31. When a ship moves through the water, it causes waves. The energy and momentum in these waves must come from the ship, which is manifested as a ''wave drag'' force on the ship. It is known that this drag force (F) depends upon the ship speed (V), the fluid properties (p, p), the length of the waterline (L), and the beam width (W) as well as the shape of the hull, among other things. (There is at least one important ''other thing'' that relates to the ''wave drag,'' i.e., the energy required to create and sustain the waves from the bow and the wake. What is this additional variable?) Note that ''shape'' is a dimensionless parameter, which is implied by the requirement of geometrical similarity. If two geometries have the same shape, the ratio of each corresponding dimension of the two will also be the same.

(a) Perform a dimensional analysis of this system to determine a suitable set of dimensionless groups that could be used to describe the relationship between all the variables. Arrange the groups such that viscous and gravitational parameters each appear in separate groups.

(b) It is assumed that ''wave drag'' is independent of viscosity and that ''hull drag'' is independent of gravity. You wish to determine the drag on a ship having a 500 ft long waterline moving at 30mph through seawater (SG = 1.1). You can make measurements on a scale model of the ship, 3 ft long; in a towing tank containing fresh water. What speed should be used for the model to simulate the wave drag and the hull drag?

32. You want to find the force exerted on an undersea pipeline by a 10 mph current flowing normal to the axis of the pipe. The pipe is 30 in. in diameter; the density of seawater is 64 lbm/ft3 and its viscosity is 1.5 cP. To determine this, you test a 1l in. diameter model of the pipe in a wind tunnel at 60°F. What velocity should you use in the wind tunnel to scale the measured force to the conditions in the sea? What is the ratio of the force on the pipeline in the sea to that on the model measured in the wind tunnel?

33. You want to determine the thickness of the film when a Newtonian fluid flows uniformly down an inclined plane at an angle 0 with the horizontal at a specified flow rate. To do this, you design a laboratory experiment from which you can scale up measured values to any other Newtonian fluid under corresponding conditions.

(a) List all the independent variables that are important in this problem, with their dimensions. If there are any variables that are not independent but act only in conjunction with one another, list only the net combination that is important.

(b) Determine an appropriate set of dimensionless groups for this system, in such a way that the fluid viscosity and the plate inclination each appear in only one group.

(c) Decide what variables you would choose for convenience, what variables would be specified by the analysis, and what you would measure in the lab.

34. You would like to know the thickness of a syrup film as it drains at a rate of 1 gpm down a flat surface that is 6 in. wide and is inclined at an angle of 30° from the vertical. The syrup has a viscosity of 100 cP and an SG of 0.9. In the laboratory, you have a fluid with a viscosity of 70 cP and an SG of 1.0 and a 1 ft wide plane inclined at an angle of 45° from the vertical.

(a) At what flow rate, in gpm, would the laboratory conditions simulate the specified conditions?

(b) If the thickness of the film in the laboratory is 3 mm at the proper flow rate, what would the thickness of the film be for the 100 cP fluid at the specified conditions?

35. The size of liquid droplets produced by a spray nozzle depends upon the nozzle diameter, the fluid velocity, and the fluid properties (which may, under some circumstances, include surface tension).

(a) Determine an appropriate set of dimensionless groups for this system.

(b) You want to know what size droplets will be generated by a fuel oil nozzle with a diameter of 0.5mm at an oil velocity of 10 m/s. The oil has a viscosity of 10 cP, an SG of 0.82, and a surface tension of 35 dyn/cm. You have a nozzle in the lab with a nozzle diameter of 0.2 mm that you want to use in a lab experiment to find the answer. Can you use the same fuel oil in the lab test as in the field? If not, why not?

(c) If the only fluid you have is water, tell how you would design the lab experiment. Note: Water has a viscosity of 1 cP and an SG of 1, but its surface tension can be varied by adding small amounts of surfactant, which does not affect the viscosity or density.

(d) Determine what conditions you would use in the lab, what you would measure, and the relationship between the measured and the unknown droplet diameters.

36. Small solid particles of diameter d and density ps are carried horizontally by an air stream moving at velocity V. The particles are initially at a distance h above the ground, and you want to know how far they will be carried horizontally before they settle to the ground. To find this out, you decide to conduct a lab experiment using water as the test fluid.

(a) Determine what variables you must set in the lab and how the value of each of these variables is related to the corresponding variable in the air system. You should note that two forces act on the particle: the drag force due to the moving fluid, which depends on the fluid and solid properties, the size of the particle, and the relative velocity; and the gravitational force, which is directly related to the densities of both the solid and the fluid in a known manner.

(b) Is there any reason why this experiment might not be feasible in practice?

37. You want to find the wind drag on a new automobile design at various speeds. To do this, you test a 1/30 scale model of the car in the lab. You must design an experiment whereby the drag force measured in the lab can be scaled up directly to find the force on the full-scale car at a given speed.

(a) What is the minimum number of (dimensionless) variables required to completely define the relationship between all the important variables in the problem? Determine the appropriate variables (e.g., the dimensionless groups).

(b) The only fluids you have available in the lab are air and water. Could you use either one of these, if you wanted to? Why (or why not)?

(c) Tell which of these fluids you would use in the lab, and then determine what the velocity of this fluid past the model car would have to be so that the experiment would simulate the drag on the full-scale car at 40 mph. If you decide that it is possible to use either fluid, determine the answer for each of them.

(d) What is the relationship between the measured drag force on the model and the drag force on the full scale car? If possible, determine this relationship for the other fluid as well. Repeat this for a speed of 70 mph.

(e) It turns out that for very high values of the Reynolds number, the drag force is independent of the fluid viscosity. Under these conditions, if the speed of the car doubles, by what factor does the power required to overcome wind drag change?

38. The power required to drive a centrifugal pump and the pressure that the pump will develop depend upon the size (diameter) and speed (angular velocity) of the impeller, the volumetric flow rate through the pump, and the fluid properties. However, if the fluid is not too viscous (e.g. less than about 100 cP), the pump performance is essentially independent of the fluid viscosity. Under these conditions:

(a) Perform a dimensional analysis to determine the dimensionless groups that would be required to define the pump performance. Arrange the groups so that the power and pump pressure each appear in only one group.

You have a pump with an 8 in. diameter impeller that develops a pressure of 15 psi and requires 1.5 hp to operate when running at 1150 rpm with water at a flow rate of 100 gpm. You also have a similar pump with a 13 in. diameter impeller, driven by a 1750 rpm motor, and you would like to know what pressure this pump would develop with water and what power would be required to drive it.

(b) If the second pump is to be operated under equivalent (similar) conditions to the first one, what should the flow rate be?

(c) If this pump is operated at the proper flow rate, what pressure will it develop, and what power will be required to drive it when pumping water?

39. In a distillation column, vapor is bubbled through the liquid to provide good contact between the two phases. The bubbles are formed when the vapor passes upward through a hole (orifice) in a plate (tray) that is in contact with the liquid. The size of the bubbles depends upon the diameter of the orifice, the velocity of the vapor through the orifice, the viscosity and density of the liquid, and the surface tension between the vapor and the liquid.

(a) Determine the dimensionless groups required to completely describe this system, in such a manner that the bubble diameter and the surface tension do not appear in the same group.

(b) You want to find out what size bubbles would be formed by a hydrocarbon vapor passing through a 1/4 in. orifice at a velocity of 2 ft/s, in contact with a liquid having a viscosity of 4 cP and a density of 0.95 g/cm3 (the surface tension is 30 dyn/cm). To do this, you run a lab experiment using air and water (surface tension 60 dyn/cm). (1) What size orifice should you use, and what should the air velocity through the orifice be? (2) You design and run this experiment and find that the air bubbles are 0.1 in. in diameter. What size would the vapor bubbles be in the organic fluid above the 1/4 in. orifice?

40. A flag will flutter in the wind at a frequency that depends upon the wind speed, the air density, the size of the flag (length and width), gravity, and the ''area density'' of the cloth (i.e. the mass per unit area). You have a very large flag (40 ft long and 30 ft wide) which weighs 240 lb, and you want to find the frequency at which it will flutter in a wind of 20 mph.

(a) Perform a dimensional analysis to determine an appropriate set of dimen-sionless groups that could be used to describe this problem.

(b) To find the flutter frequency you run a test in a wind tunnel (at normal atmospheric temperature and pressure) using a flag made from a cloth that weighs 0.05 lb/ft2. Determine (1) the size of the flag and the wind speed that you should use in the wind tunnel and (2) the ratio of the flutter frequency of the big flag to that which you observe for the model flag in the wind tunnel.

41. If the viscosity of the liquid is not too high (e.g., less than about 100 cP), the performance of many centrifugal pumps is not very sensitive to the fluid viscosity. You have a pump with an 8 in. diameter impeller that develops a pressure of 15 psi and consumes 1.5 hp when running at 1150 rpm pumping water at a rate of 100 gpm. You also have a similar pump with a 13 in. diameter impeller driven by a 1750 rpm motor, and you would like to know what pressure that pump would develop with water and how much power it would take to drive it.

(a) If the second pump is to be operated under conditions similar to that of the first, what should the flow rate be?

(b) When operated at this flow rate with water, (1) what pressure should it develop and (2) what power would be required to drive it?

42. The pressure developed by a centrifugal pump depends on the fluid density, the diameter of the pump impeller, the rotational speed of the impeller, and the volumetric flow rate through the pump (centrifugal pumps are not recommended for highly viscous fluids, so viscosity is not commonly an important variable). Furthermore, the pressure developed by the pump is commonly expressed as the ''pump head,'' which is the height of a column of the fluid in the pump that exerts the same pressure as the pump pressure.

(a) Perform a dimensional analysis to determine the minimum number of variables required to represent the pump performance characteristic in the most general (dimensionless) form.

(b) The power delivered to the fluid by the pump is also important. Should this be included in the list of important variables, or can it be determined from the original set of variables? Explain.

You have a pump in the field that has a 1.5 ft diameter impeller that is driven by a motor operating at 750 rpm. You want to determine what head the pump will develop when pumping a liquid with a density of 50 lbm/ft3 at a rate of 1000 gpm. You do this by running a test in the lab on a scale model of the pump that has a 0.5 ft diameter impeller using water (at 70°F) and a motor that runs at

1200 rpm.

(c) At what flow rate of water (in gpm) should the lab pump be operated?

(d) If the lab pump develops a head of 85 ft at this flow rate, what head would the pump in the field develop with the operating fluid at the specified flow rate?

(e) How much power (in horsepower) is transferred to the fluid in both the lab and the field cases?

(f) The pump efficiency is defined as the ratio of the power delivered to the fluid to the power of the motor that drives the pump. If the lab pump is driven by a 2 hp motor, what is the efficiency of the lab pump? If the efficiency of the field pump is the same as that of the lab pump, what power motor (horsepower) would be required to drive it?

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