Question

could you help me with numbers 1-3

Question 1: Use the Repeating Method Only! Water sloshes back and forth in a tank as shown in the figure below. The frequency of sloshing. 6, is assumed to be a function of the acceleration of gravity. 8. the average depth of the water, h, and the length of the tank, C. Develop a suitable set of dimensionless parameters for this problem using g and l as repeating variables. Question 2, Flow with a Free Surface GIVEN The spillway for the dam shown in Fig. E7.8a is 20 m wide and is designed to carry 125 m/s at Nood stage. A 1:15 model is constructed to study the flow characteristics through the spillway. The effects of surface tension and viscosity are to be neglected. FIND (a) Determine the required model width and Dowrate. (b) What operating time for the model corresponds to a 24-hr period in the prototype? Figure E7.8a (Photo by Marty Melchior. or Flow with a free surface Dependent pi term- (4 pve v pre T o "uestion 3: Pipe Flow Problems (Single Pipe, Type III) GIVEN Water at 60F(y = 1.21 X 10-1/s, see Table 1.5) is to flow from reservoir A to reservoir B through a pipe of length 1700 ft and roughness 0.0005 ft at a rate of Q = 26 ft/s as shown in Fig. E8.13a. The system contains a sharp-edged entrance and four flanged 45° elbows. Elevation , 441 Total length 1700 Elevation 0 FIND Determine the pipe diameter needed. Figure E8.13a What are the applications of Euler number and Froude number and Reynolds number? Vhat does head loss mean? What are the types of losses in pipe flow? Briefly explain about each one in one sentence.

Answer 1

1a) Applications Of Euler Number:-

Euler's number has many practical uses, particularly in higher level mathematics such as calculus, differential equations, discrete mathematics, trigonometry, complex analysis, statistics, among others.

1b) Application Of Froude Number:-

1) Ship hydrodynamics

${\displaystyle \mathrm {Fn} _{V}={\frac {u}{\sqrt {g{\sqrt[{3}]{V}}}}}.}$

2) Shallow water waves:-

For shallow water waves, like for instance tidal waves and the hydraulic jump, the characteristic velocity U is the average flow velocity, averaged over the cross-section perpendicular to the flow direction. The wave velocity, c, is equal to the square root of gravitational acceleration g, times cross-sectional area A, divided by free-surface width B:

{\displaystyle c={\sqrt {g{\frac {A}{B}}}},} $c={\sqrt {g{\frac {A}{B}}}},$

so the Froude number in shallow water is:

{\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {g{\dfrac {A}{B}}}}}.} ${\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {g{\dfrac {A}{B}}}}}.}$

For rectangular cross-sections with uniform depth d, the Froude number can be simplified to:

{\displaystyle \mathrm {Fr} ={\frac {U}{\sqrt {gd}}}.} ${\mathrm {Fr}}={\frac {U}{{\sqrt {gd}}}}.$

For Fr < 1 the flow is called a subcritical flow, further for Fr > 1 the flow is characterised as supercritical flow. When Fr ≈ 1 the flow is denoted as critical flow.

Wind engineering[edit]

When considering wind effects on dynamically sensitive structures such as suspension bridges it is sometimes necessary to simulate the combined effect of the vibrating mass of the structure with the fluctuating force of the wind. In such cases, the Froude number should be respected. Similarly, when simulating hot smoke plumes combined with natural wind, Froude number scaling is necessary to maintain the correct balance between buoyancy forces and the momentum of the wind.

Extended Froude number[edit]

Geophysical mass flows such as avalanches and debris flows take place on inclined slopes which then merge into gentle and flat run-out zones.^[8]

So, these flows are associated with the elevation of the topographic slopes that induce the gravity potential energy together with the pressure potential energy during the flow. Therefore, the classical Froude number should include this additional effect. For such a situation, Froude number needs to be re-defined. The extended Froude number is defined as the ratio between the kinetic and the potential energy:

{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {\beta h+s_{g}\left(x_{d}-x\right)}}},} ${\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {\beta h+s_{g}\left(x_{d}-x\right)}}},}$

where u is the mean flow velocity, β = gK cos ζ, (K is the earth pressure coefficient, ζ is the slope), s_g = g sin ζ, x is the channel downslope position and {\displaystyle x_{d}} $x_{d}$ is the distance from the point of the mass release along the channel to the point where the flow hits the horizontal reference datum; E^p
_pot = βh and E^g
_pot = s_g(x_d − x) are the pressure potential and gravity potential energies, respectively. In the classical definition of the shallow-water or granular flow Froude number, the potential energy associated with the surface elevation, E^g
_pot, is not considered. The extended Froude number differs substantially from the classical Froude number for higher surface elevations. The term βh emerges from the change of the geometry of the moving mass along the slope. Dimensional analysis suggests that for shallow flows βh ≪ 1, while u and s_g(x_d − x) are both of order unity. If the mass is shallow with a virtually bed-parallel free-surface, then βh can be disregarded. In this situation, if the gravity potential is not taken into account, then Fr is unbounded even though the kinetic energy is bounded. So, formally considering the additional contribution due to the gravitational potential energy, the singularity in Fr is removed.

Stirred tanks[edit]

In the study of stirred tanks, the Froude number governs the formation of surface vortices. Since the impeller tip velocity is ωr (circular motion), where ω is the impeller frequency (usually in rpm) and r is the impeller radius (in engineering the diameter is much more frequently employed), the Froude number then takes the following form:

{\displaystyle \mathrm {Fr} =\omega {\sqrt {\frac {r}{g}}}.} ${\mathrm {Fr}}=\omega {\sqrt {\frac {r}{g}}}.$

It must be noted that the Froude number finds also a similar application in powder mixers. It will indeed be used to determine in which mixing regime the blender is working. If Fr<1, the particles are just stirred, but if Fr>1, centrifugal forces applied to the powder overcome gravity and the bed of particles becomes fluidized, at least in some part of the blender, promoting mixing^[9]

Densimetric Froude number[edit]

When used in the context of the Boussinesq approximation the densimetric Froude number is defined as

{\displaystyle \mathrm {Fr} ={\frac {u}{\sqrt {g'h}}}} ${\mathrm {Fr}}={\frac {u}{{\sqrt {g'h}}}}$

where g′ is the reduced gravity:

{\displaystyle g'=g{\frac {\rho _{1}-\rho _{2}}{\rho _{1}}}} ${\displaystyle g'=g{\frac {\rho _{1}-\rho _{2}}{\rho _{1}}}}$

The densimetric Froude number is usually preferred by modellers who wish to nondimensionalize a speed preference to the Richardson number which is more commonly encountered when considering stratified shear layers. For example, the leading edge of a gravity current moves with a front Froude number of about unity.

Walking Froude number[edit]

The Froude number may be used to study trends in animal gait patterns. In analyses of the dynamics of legged locomotion, a walking limb is often modeled as an inverted pendulum, where the center of mass goes through a circular arc centered at the foot.^[10] The Froude number is the ratio of the centripetal force around the center of motion, the foot, and the weight of the animal walking:

{\displaystyle \mathrm {Fr} ={\frac {\text{centripetal force}}{\text{gravitational force}}}={\frac {\;{\frac {mv^{2}}{l}}\;}{mg}}={\frac {v^{2}}{gl}}} ${\displaystyle \mathrm {Fr} ={\frac {\text{centripetal force}}{\text{gravitational force}}}={\frac {\;{\frac {mv^{2}}{l}}\;}{mg}}={\frac {v^{2}}{gl}}}$

where m is the mass, l is the characteristic length, g is the acceleration due to gravity and v is the velocity. The characteristic length l may be chosen to suit the study at hand. For instance, some studies have used the vertical distance of the hip joint from the ground,^[11] while others have used total leg length.^[10][12]

The Froude number may also be calculated from the stride frequency f as follows:^[11]

{\displaystyle \mathrm {Fr} ={\frac {v^{2}}{gl}}={\frac {(lf)^{2}}{gl}}={\frac {lf^{2}}{g}}.} ${\mathrm {Fr}}={\frac {v^{2}}{gl}}={\frac {(lf)^{2}}{gl}}={\frac {lf^{2}}{g}}.$

If total leg length is used as the characteristic length, then the theoretical maximum speed of walking has a Froude number of 1.0 since any higher value would result in takeoff and the foot missing the ground. The typical transition speed from bipedal walking to running occurs with Fr ≈ 0.5.^[13] R. M. Alexander found that animals of different sizes and masses travelling at different speeds, but with the same Froude number, consistently exhibit similar gaits. This study found that animals typically switch from an amble to a symmetric running gait (e.g., a trot or pace) around a Froude number of 1.0. A preference for asymmetric gaits (e.g., a canter, transverse gallop, rotary gallop, bound, or pronk) was observed at Froude numbers between 2.0 and 3.0

1c) Applications Of Reynolds Number:-

1) Reynolds number plays an important part in the calculation of the friction factor in a few of the equations of fluid mechanics, including the Darcy-Weisbach equation.

2) It is used when modeling the movement of organisms swimming through water.

3) Atmospheric air is considered to be a fluid. Hence, the Reynolds number can be calculated for it. This makes it possible to apply it in wind tunnel testing to study the aerodynamic properties of various surfaces.

4) It plays an important part in the testing of wind lift on aircraft, especially in cases of supersonic flights where the high speed causes a localized increase in the density of air surrounding the aircraft.

2) what does head loss mean?

Head loss refers to the measurement of energy dissipated in a system due to friction. It accounts for the totality of energy losses due to the length of a pipe and those due to the function of fittings, valves and other system structures.

3) what are the types of losses in pipe flow? Briefly explain about each one in one sentences.

Major and minor loss in pipe, tubes and duct systems

• Major Head Loss - head loss or pressure loss - due to friction in pipes and ducts.
• Minor Head Loss - head loss or pressure loss - due to components as valves, bends, tees and the like in the pipe or duct system.