Concept explainers
Repeal Prob. 7-112, but with the distance r from the sound source as an additional independent parameter.
(a)
A dimensionless relationship for I as a function of the other parameters by using the method of repeating variables in mass-based primary dimensions by using distance r from the sound source as an additional independent parameter.
Answer to Problem 113P
Dimensionless relationship between sound intensity and remaining parameter.
Explanation of Solution
Given Information:
Concept used:
The mass-based primary dimension will be used in this question. In this system all the possible variables are replaced by the mass. Mass, time length dimensions are represented as, [M],[L] and [t].
Concept of Buckingham's Pi method will also be used. It is represented as-
Where,
n= number of physical variables
k =independent physical quantities
n = total number of variable parameters
Calculation:
Primary dimensions of each parameter,
The total mass of parameters, n = 5
No. Of primary dimensions, j = 3
Expected no of
Dependant
Rewriting,
Mass,
Time,
Length,
Putting value in
Dependant Pi using independent variable P.
Mass,
Time,
Length,
Putting values in
Thus, from the equation of
Conclusion:
In this way, we are able to produce a dimensionless relationship for sound intensity using an independent parameter.
(b)
The expression for dimensionless relationship of I by using the force-based system of repeating variables by using distance r from the sound source as an additional independent parameter.
Answer to Problem 113P
Intensity by using force-based primary dimension system
For three repeating variables,
Explanation of Solution
Given:
Concept Used:
The force-based primary dimension will be used in this question. In this system all the possible variables are replaced by the mass. Force, time, length dimensions are represented as, [F], [L] and [t].
Concept of Buckingham's Pi method will also be used. It is represented as,
Where,
n= number of physical variables
k =independent physical quantities
n = total number of variable parameters
Calculation:
Now, the primary dimensions of all parameters are given below:
speed of sound =
density =
pressure level =
Sound Intensity,
The total mass of parameters, n = 5
No. Of primary dimensions, j = 3
Expected no of
Dependent p is calculated by using the I dependent variable.
Therefore,
By using primary dimensions,
For,
From equ(1) and equ (2),
Equating the exponents of both sides,
For force:
For time:
Putting the values of a1 and b1 in equation (1),
Dependent p is calculated by using P independent variable.
Therefore,
By using primary dimensions,
For,
From equ(3) and equ (4),
Equating the exponents of both sides,
For mass:
For time:
Putting the values of a2 and b2 in equation (3),
From equation (a) and (b),
Conclusion:
Intensity by using force-based primary dimension system.
For three repeating variables,
Want to see more full solutions like this?
Chapter 7 Solutions
Fluid Mechanics: Fundamentals and Applications
- Hello sir Muttalibi is a step solution in detailing mathematics the same as an existing step solution EXAMPLE 6-1 Momentum-Flux Correction Factor for Laminar Pipe Flow CV Vavg Consider laminar flow through a very long straight section of round pipe. It is shown in Chap. 8 that the velocity profile through a cross-sectional area of the pipe is parabolic (Fig. 6-15), with the axial velocity component given by r4 V R V = 2V 1 avg R2 (1) where R is the radius of the inner wall of the pipe and Vavg is the average velocity. Calculate the momentum-flux correction factor through a cross sec- tion of the pipe for the case in which the pipe flow represents an outlet of the control volume, as sketched in Fig. 6-15. Assumptions 1 The flow is incompressible and steady. 2 The control volume slices through the pipe normal to the pipe axis, as sketched in Fig. 6-15. Analysis We substitute the given velocity profile for V in Eq. 6-24 and inte- grate, noting that dA, = 2ar dr, FIGURE 6–15 %3D Velocity…arrow_forward06:20 all 29% MENG250 - Chapter.. 4-151. Currently eighty-five percent of all neck injuries are caused by rear-end car collisions. To alleviate this problem, an automobile seat restraint has been developed that provides additional pressure contact with the cranium. During dynamic tests the distribution of load on the cranium has been plotted and shown to be parabolic. Determine the equivalent resultant force and its location, measured from point A. 12 lb/ft 0.5 ft W. w = 12(1 + 2x) lb/ft 18 lb/ftarrow_forwardA source and sink of equal strength, m = 25 m²ls, are near a wall, as in Fig. induced by this pair at point A on the wall. 5- 4. . Find the resultant velocity 4 m 3 m 4 m 3 marrow_forward
- An object of mass m falling vertically in a resisting fluid under linear resistance force F= -ev. Find the velocity as a function of time in term of terminal speed v and characteristic time (t). p 3:11arrow_forwardD--- p, FIGURE P7-62 7–63 Consider laminar flow through a long section of pipe, as in Fig. P7–62 0. For laminar flow it turns out that wall roughness is not a relevant parameter unless e is very large. The volume flow rate b through the pipe is a function of pipe diameter D, fluid viscosity µ, and axial pressure gradient dPldx. If pipe diameter is doubled, all else being equal, by what factor will volume flow rate increase? Use dimensional analysis.arrow_forwardThis is a dynamics question. Answer: w=0.0178 rad/sarrow_forward
- |The System in figure below is at 2éc.If the ectmospnen'e pressure is l01-33kpa and the pressure at the bottem of the tan k is 260hPa What is the specific gracity of fluid X? T ake densities of oila89l kg /m³ mercuryz13,55 =13,550 kg/m3 and water = 99849 Im3 oil im Water FluidX Mercury 0 SMarrow_forward4–109 Consider fully developed axisymmetric Poiseuille flow-flow in a round pipe of radius R (diameter D = 2R), with a forced pressure gradient dPldx driving the flow as illustrated in Fig. P4–109. (dPldx is constant and negative.) The flow is steady, incompressible, and axisymmetric about the x-axis. The velocity components are given by 1 dP (r² – R²) 4µ dx u, = 0 U, = 0 и where u is the fluid's viscosity. Is this flow rotational or irro- tational? If it is rotational, calculate the vorticity component in the circumferential (0) direction and discuss the sign of the rotation.arrow_forwardIn deriving the continuity equation, we assumed, for simplicity,that the mass fl ow per unit area on the left face wasjust ρ u . In fact, ρ u varies also with y and z , and thus it mustbe different on the four corners of the left face. Account forthese variations, average the four corners, and determinehow this might change the inlet mass fl ow from ρ u dy dz .arrow_forward
- Show how you can use an energy method to compute the exact value of theradial displacement at x = L. You do not have to (but are welcome to) evaluate the final expression, if it is too complex to do by hand.arrow_forward3-16. Consider a film of liquid draining at volume flow rate Q down the outside of a verti- cal rod of radius a, as shown in Fig. P3-16. Some distance down the rod, a fully devel- oped region is reached where fluid shear balances gravity and the film thickness remains constant. Assuming incompressible laminar flow and negligible shear inter- action with the atmosphere, find an expression for v₂(r) and a relation between Q and film radius b.arrow_forwardA cylinder with a mass 0.222 kg is sliding downwards through a vertically positioned pipe. A thin oil layer exists between the cylinder and the pipe's internal surface. Centerline of the cylinder and the pipe overlap. (yoil =8044.2 N /m3 ; voil=6-106 m²/s). Find the change in the speed of cylinder in the pipe with respect to its unit displacement and the shear stress that acts upon the oil layer. A A L = 128 mm V W 73.8 mm 74 mm 0.1 mm Figure 1arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY