Practice Problems on the Dimensional Analysis

Practice Problems on the Dimensional Analysis dim_anal_01

The power, P , to drive an axial flow pump depends on the following variables: density of the fluid,

 angular speed of the rotor,

 diameter of the rotor, D head rise across the pump,



H ( =

 p /

 g ) volumetric flow through the pump, Q a. Rewrite the functional relationship in dimensionless form. b. A model scaled to one-third the size of the prototype has the following characteristics:

 m

= 900 rpm

D m

= 5 in



H

Q m m

= 10 ft

= 3 ft

3

/s

P m

= 2 hp

If the full-size pump is to run at 300 rpm, what is the power required for this pump? What head will the pump maintain? What will the volumetric flow rate be in the prototype?

Answer(s) :

P

 

3 D 5

 f

2







H Q

D

,



D 3





P

P



18 hp



H

P



30 ft Q

P



27 ft /s

3

C. Wassgren, Purdue University Page 1 of 25 Last Updated: 2010 Oct 18


An open cylindrical tank having a diameter D is supported around its bottom circumference and is filled to a depth h with a liquid having a specific weight



. The vertical deflection,



, of the center of the bottom is a function of D , h , d ,



, and E where d is the thickness of the bottom and E is the modulus of elasticity of the bottom material. Form the dimensionless groups describing this relationship.

Answer(s) :



D

 f

2



 h d E

, ,

D D D









The drag characteristics of a blimp 5 m in diameter and 60 m long are to be studied in a wind tunnel. If the speed of the blimp through still air is 10 m/s, and if a 1/10 scale model is to be tested, what airspeed in the wind tunnel is needed for dynamic similarity? Assume the same air temperature and pressure for both the prototype and model.

Answer(s):



V

M



100 m s



A viscous fluid is poured onto a horizontal plate as shown in the figure. Assume that the time, t , required for the fluid to flow a certain distance, d , along the plate is a function of the volume of fluid poured, V , acceleration due to gravity, g , fluid density,



, and fluid dynamic viscosity,



. Determine an appropriate set of dimensionless terms to describe this process. volume, V

Answer(s) : t g d



 f

2



V d

3

,



 d gd



 d



The height of the free surface, h , in a tank of diameter, D , that is draining fluid through a small hole at the bottom with diameter, d , decreases with time, t . This change in free surface height is studied experimentally with a halfscale model. For the prototype tank:

H

D

= 16 in. (the initial height of the free surface)

= 4.0 in. d = 0.25 in.

Experimental data is obtained from the prototype and half-scale model and is given below: h [in.] t [s] h [in.] t [s]

1. Plot, on the same graph, the height data as a function of time for both the model and the prototype.

2. Develop a set of dimensionless parameters for this problem assuming that: h = f( H , D , d , g , t )

3. Replot, on the same graph, the height data as a function of time in non-dimensional form for both the model and prototype.

D g

H h

hole with diameter,

d


Practice Problems on the Dimensional Analysis

Answer(s):

18

16

14

12

10

8

6

4

2

0

0 20 40 t [s] h

H

 f

2



D d

, , g t

H H H





1.2

1.0

0.8

0.6

0.4

0.2

0.0

0 100 200 t*sqrt(g/H)

60

model prototype model prototype

300

80

400



It is desired to determine the wave height when wind blows across a lake. The wave height, H , is assumed to be a function of the wind speed, V , the water density,



, the air density,

 a

, the water depth, d , the distance from the shore, L , and the acceleration of gravity, g . Use d , V , and



as repeating variables to determine a suitable set of pi terms that could be used to describe this problem.

Answer(s):

H



 f

2

 d



 a

L V

, , d gd







Small droplets of liquid are formed when a liquid jet breaks up in spray and fuel injection processes. The resulting droplet diameter, d , is thought to depend on liquid density,



, viscosity,



, and surface tension,



, as well as jet speed, V , and diameter, D . How many dimensionless ratios are required to characterize this process? Determine these ratios.

Answer(s) :

( #

 terms) = (# of variables) – (# of reference dimensions) = 6 – 3 = 3 d

D

 f

2













VD



Reynolds #

,



V D



Weber #











Spin plays an important role in the flight trajectory of golf, Ping-Pong, and tennis balls. Therefore, it is important to know the rate at which spin decreases for a ball in flight. The aerodynamic torque, T , acting on a ball in flight, is thought to depend on flight speed, V , air density,



, air viscosity,



, ball diameter, D , spin rate (angular speed),



, and diameter of the dimples on the ball, d . Determine the dimensionless parameters that result.

Answer(s) :

T



V D

3

 f

2













VD D d



Reynolds #

,



,

V D











We wish to understand what parameters affect the drag force acting on a ship due to the surrounding water. a. What parameters do you expect will be significant in this problem? b. Form dimensionless ratios from these parameters.

Answer(s) :

F

D

 fcn

  

, , ,





F

D  fcn







VL H D W



, , ,

L L L







The pressure drop,

 p , for airflow through a filter depends upon the volume flow rate, Q , density,



, filter thickness,

H , and porosity,



(the percent volume of filter material which is not occupied by filter fibers). The following are pressure-drop data in an air flow at 20



C and 1 atm through a filter 3 cm thick with 45% porosity:

 p [in of H

Q [ft

3

2

0] 0.31 1.24 2.79 4.96 7.75 11.2

Use these data to determine the pressure drop of the same filter material if the thickness is increased to 6 cm and the flow rate is 750 ft

3

/min.



Q p +

 p filter p porosity,



Answer(s) :

 p = 67.9 Pa = 0.273 in. H

2

O

H



A model test of a tractor-trailer rig is performed in a wind tunnel. The drag force, F

D

, is found to depend on the frontal area, A , wind speed, V , air density,



, and air viscosity,



. The model scale is 1:4 ( e.g.

1 m in the model is equivalent to 4 m in the prototype), frontal area of the model is A = 0.625 m

2

. Obtain a set of dimensionless parameters suitable to characterize the model test results. State the conditions required to obtain dynamic similarity between model and prototype flows. When tested at wind speed V =89.6 m/sec, in standard air, the measured drag force on the model was F

D

= 2.46 kN. Estimate the aerodynamic drag force on the full-scale vehicle at V = 22.4 m/sec. Calculate the power needed to overcome this drag force.

Answer(s) :



F

D

2

V A

 f

2









V



A







F

D P



2.46 kN

55.1 kW



The differential equation for small-amplitude vibrations of a simple beam is given by:



A



2

 t 2 y



EI



4 y

 x 4



0 where y



vertical displacement of beam x



horizontal t



time

 

beam density

A



cross-sectional area

I



area moment of inertia

E



Young’s

Rewrite the differential equation in dimensionless form. Discuss the physical significance of any dimensionless terms in the resulting equation.

Answer(s) :

 2 y *

 t *

2



 4 y

 x *

4

*



0



Wind blowing past a flag causes it to flutter. The frequency of this fluttering,



, is assumed to be a function of the wind speed, V , the air density,



, the air dynamic viscosity,



, the acceleration of gravity, g , the length of the flag, l , and the area density of the flag material,



A

. It is desired to predict the flutter frequency of a large flag ( l = 40 ft,



A

= 0.006 slug/ft

2

) in a V = 30 ft/s wind using a smaller ( l = 4 ft) flag. a. Determine the dimensionless terms that characterize this phenomenon. b. What specific conditions must hold in order to maintain full similarity between the prototype and the model? c. Can full similarity be achieved using the same atmospheric air in the prototype and model flows? Explain your answer.

For the remainder of the problem, assume that the same atmospheric air for the prototype is also used in wind tunnel testing of the model. d. What wind tunnel velocity should be used for testing the model? e. Determine the required area density of the model flag material. f. If the model flag flutters at 6 Hz, predict the frequency of the large flag flutter.

Answer(s) :



V l

 f

2









Vl gl

, ,

V 2





A l











V l





M









V l





P









Vl





M











Vl





P



 gl

V 2





M





 gl

V 2





P





 l

A 

M

 





A l



P it is not possible to maintain full similarity using identical prototype and model air

V

M

 

A



P

)

P

= 0.0006

2



A cylinder with a diameter, D , floats upright in a liquid as shown in the figure. When the cylinder is displaced slightly along its vertical axis it will oscillate about its equilibrium position with a frequency,



. Assume that this frequency is a function of the diameter, D , the mass of the cylinder, m , the liquid density,



, and the acceleration due to gravity, g .

If the mass of the cylinder were doubled (assuming the same cylinder material density), by how much would the oscillation frequency change?



D m g

Answer(s) :

Hence, doubling the mass ( i.e.

m

2

= 2 m

1

) will result in a smaller frequency with



2

= 2

-1/6



1



Hoppers are a commonly used device in the handling and storage of particulate materials. A hopper design typically consists of a bin section located above a converging section with a hole located in the bottom through which the particulate material flows (refer to the figures below).

One interesting observation with hopper flows is that the mass flow rate from the hopper exit is independent of the height of the material above the exit and the bin diameter (except when the hopper is nearly empty). The parameters that do affect the discharge rate (assuming cohesionless particles) include the hopper exit diameter, the acceleration due to gravity, the angle of the hopper walls, the friction coefficient between the particulate material and the walls and between the particles themselves, and the bulk density of the material at the discharge plane. a. Perform a dimensional analysis to determine the dimensionless quantities that govern flow from a hopper. b. If the same hopper and particulate material are used ( i.e.

the wall angle and friction properties remain the same), how will the mass flow rate from the hopper change if the hopper exit diameter is doubled? c. Compare the discharge rate found in part (a) with the mass discharge rate expected for a liquid.

Answer(s) :

 b

1 g D

5

E

2

 f

2

 pw



If the wall angle and frictional properties remain constant, then doubling the exit diameter increases the mass flow rate by a factor of 2

5/2 

5.66.



A 1/16 th

-scale model of a weir has a measured flow rate of Q = 2.1 ft

3

/s when the upstream water height is h = 6.3 in. The flow rate is known to be a function of the acceleration due to gravity, g , the weir width (into the page), b , and the upstream water height, h . Furthermore, the flow rate is found to be directly proportional to the weir width, b . What is the flow rate over the prototype weir when the upstream water height is h = 3.2 ft. h

Q g

Answer(s) :

Q gh 5

 f

2 b

 

Q prototype

= 506 ft

3

/s



Velocity measurements in gases and liquids are often accomplished by adding (solid) marker particles to the flow and then measuring the particles’ velocities. To accurately track the gas or liquid, the solid particles must rapidly respond to changes in the flow. This response is a function of the aerodynamic drag exerted on the particle and also the particle buoyancy. The drag and buoyancy are, in turn, functions of the particle size, D , density,

 solid

, fluid density,

 fluid

, fluid dynamic viscosity,

 fluid

, fluid velocity relative to the particle, U , and the acceleration due to gravity, g .

The drag and buoyancy are functions of what three dimensionless (



) groups?

Answer(s) :

 solids

 fluid

 fluid

UD

 fluid

 

3

U gD



Many pharmaceutical processes involve spraying a liquid onto a powder bed surface. The rate at which a liquid drop fully penetrates into the powder bed,

 p

, is of particular interest. Experience has shown that the parameters governing this process include the drop diameter, d , the drop dynamic viscosity,



, the drop surface tension,



, the contact angle between the liquid and the powder bed material,



, the bed porosity,



, and the bed pore radius, R .

Perform a dimensional analysis to express the drop penetration process in dimensionless terms.

Reference: Hapgood, K.P., Litster, J.D., Biggs, S.R., and Howes, T., 2002, “Drop penetration into porous powder beds,” Journal of Colloid and Interface Science , Vol. 253, pp. 353 – 366.

Answer(s) :

 

 p d

 f

2





 

R d







Wet granulation is a common particulate processing operation used to increase particle size. Granulation is often used to improve powder flow properties, reduce dust hazards, and increase blend uniformity by combining various powder components into individual granules. A high shear wet granulator consists of a bowl in which the dry powder ingredients are contained (Fig. 1). Within the bowl is an impeller that is used to blend the components as well as a “chopper” that is used to break apart larger granules. A binding liquid is sprayed onto the powder’s surface to cause individual particles to bind and form larger granules. The impeller and chopper facilitate uniform distribution of the binding liquid and homogeneity of the granulated bed. spray chopper bowl impeller

Figure 1. A schematic of a bottom-driven high shear wet granulator.

Consider the coalescence process whereby two particles with binding liquid on their surfaces collide to form a larger granule. When the particles first approach each other, the viscous binding liquid is squeezed out from between the particles resulting in a dissipation of energy. If the initial kinetic energy of the particles is not dissipated in this viscous event, then the particle surfaces contact and elastic and plastic deformation of the particles occurs. Upon rebound the viscous binding liquid may be entrained back into the gap between the particles resulting in additional energy dissipation. If at any point during these viscous and plastic processes the cumulative energy loss exceeds the initial kinetic energy, the particles will coalesce to form a granule. Details of this coalescence process may be found in the work by Liu et al . (2000).

Assume that two particles collide with an impact velocity U . Each particle has a diameter, d , density,



, elasticity,

E , Poisson’s ratio,



, and yield strength, Y . In addition, assume that the particles have an asperity height, h a

. The binding liquid, with dynamic viscosity,



L

, covers the particle surfaces to a thickness, h

L

. Note that the binding liquid density is assumed to be negligible here since the squeezing of thin viscous films has been shown to be independent of the fluid density.

Perform a dimensional analysis to determine the relevant dimensionless groups govern the coalescence process.

Reference: Liu, L.X., Litster, J.D., Iveson, S.M., and Ennis, B.J., 2000, “Coalescence of deformable granules in wet granulation processes,” AIChE Journal , Vol. 43, No. 3, pp. 529 – 539.

Answer(s) : f

2







Y

E

 d U 2

Yd 3

, h a , h

0





L d 2

, h

0 d









Many particulate processing operations involve the tumbling of particles in rotating drums. Examples of such operations include tumbling blenders, low shear tumble granulators, and pan coaters. An important part of designing these operations is determining how to properly scale the systems from small, lab-scale devices to much larger, industrial-scale devices.

Consider a horizontal cylindrical drum of diameter, D , and length, L , rotating with rotation rate,



. Contained within the drum is a particulate material with diameter, d , and density,



. Determine the dimensionless parameters important to scaling the particles’ speed down the free surface of the tumbling particle bed, V (refer to the figure).

Discuss significant issues regarding the scale-up process.

V

 g

Related Reference : Alexander, A., Shinbrot, T., and Muzzio, F.J., 2002, “Scaling surface velocities in rotating cylinders as a function of vessel radius, rotation rate, and particle size,” Powder Technology , Vol. 126, pp. 174 –

190.

Answer(s) :



V

D

 f

2









H

,



2

D d

,

D



D a Froude #











The instantaneous diameter of a spreading liquid drop on a surface (refer to the figure shown below) is known to be a function of the drop initial diameter, impact speed, liquid density, dynamic viscosity and surface tension, contact angle between the liquid and surface, and time.

Determine the dimensionless parameters important in determining the diameter of the spreading drop.

(The figure is from Roisman et al . (2002).)

Related Reference : Roisman, I.V., Rioboo, R., and Tropea, C., 2002, “Normal impact of a liquid drop on a dry surface: Model for spreading and receding,” Proc. R. Soc. Lond. A , Vol. 458, pp. 1411 – 1430.

Answer(s) :

D

D

0

 f

2













U D

0 ,



2

U D

0 tU

  D

0

0

Weber #

Reynolds #











Reverse-flow cyclone separators (shown below) are commonly used to separate particles from a gas stream.

Particle-laden gas enters tangentially at the top of a conical section, then swirls downward toward the bottom of the cone. Under the action of gravity, particles exit the gas stream at the bottom of the device while the gas stream is redirected toward a low pressure exit pipe at the top of the device.

The parameters that are most important in the separation efficiency,



, of a cyclone separator are the cyclone barrel diameter, D , and other geometric dimensions, L i

, the particle diameter, d

P

, the particle density,



P loading by mass, C (mass of particles to the mass of gass), the gas density,



G

, the gas viscosity,



, the particle

G

( e.g.

with units of Pa

 s) the volumetric flow rate of gas through the cyclone, Q ( e.g.

with units of cubic feet per minute), and the pressure difference between the cyclone inlet and outlet,

 p .

Determine the dimensionless parameters that are most important in determining the efficiency of a cyclone.

Answer(s) :

  f

2











L

D i



P P





St

,



G

Q

,

 pD

D Q

4





G

Re



G



Eu

2

 











In the late 1940s, much of the science concerning nuclear bombs was highly classified. In particular, information regarding the energy released in a nuclear explosion, e.g.

the number of equivalent kilotons of TNT (nowadays the energy is measured in megatons), was top secret. G.I. Taylor, a famous fluid mechanics professor, was asked in

1941 by the British Civil Defence Research Committee of the Ministry of Home Security to predict the dynamics of a blast caused by a nuclear explosion. In his analysis, Taylor assumed that a finite amount of energy, E , is suddenly released in an infinitely concentrated form. The resulting blast wave, with a radius R , then propagates into the surrounding atmosphere, with density



0

and specific heat ratio



= c p

/ c v



, as a function of time, t . Taylor’s analysis resulted in a simple relationship between the blast radius as a function of the time, air density, blast energy, and specific heat ratio. Using declassified photographs of the first nuclear explosion, which occurred at the Trinity test site in New Mexico in 1945, Taylor was able to estimate the energy release to within remarkable accuracy.

Perform a dimensional analysis to determine an expression involving the blast radius as a function of the other significant parameters in the problem.

References : Taylor, G., 1950, “The formation of a blast wave by a very intense explosion. I. Theoretical analyses,”

Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences , Vol. 201, No. 1065, pp.

159 – 174. Taylor, G., 1950, “The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences , Vol. 201, No.

1065, pp. 175 – 186.

Answer(s) :

R



0

1

5

2 t E

1

5

 f

2



The sound power, W , from a turbulent jet through a nozzle is believed to depend on the jet centerline velocity, U , the nozzle diameter, D , the speed of sound, c , and the fluid density,



. The temperature and composition of the fluid in the jet is the same as the ambient fluid. a. Using the Buckingham Pi theorem, find a relation between the dimensionless sound power and the other dimensionless parameter(s) of the system. b. How would the sound power vary if the nozzle diameter is doubled with all other factors staying the same?

Answer(s) :



W

U D 2

 f

2



 U





Mach # increase by a factor of four


Practice Problems on the Dimensional Analysis

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model prototype model prototype

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