Fluid Mechanics By Khurmi

[Anjali Reyome]

1Hydrostatics

2. Stationary fluid discharge through hole

3. Stationary discharge - overflows (ISO 1438, Swiss Engineers, Hansen, Bazin, Frese)

4. Stationary flow of viscous fluid - constant pipe cross section with output nozzle / diffuser

5. Stationary flow of fluid - various pipe cross sections

6. Water hammer

[1] Mechanika tekutin - ČVUT (Prof. Ing. Jan Ježek, DrSc, Ing. Blanka Vradiov, CSc., Ing Josef Adamec, CSc)

[2] Mechanika tekutin, Sbrka přkladů - ČVUT (Ing. Milan Peťa)

[3] Strojně technick přručka (Svatopluk Černoch)

[4] Mechanika tekutin, VŠB-TU Ostrava (Janalk J., Šťva P.)

[5] Textbook of Machine Design (R.S. KHURMI, J.K. GUPTA)

[6] A TEXTBOOK OF FLUID MECHANICS AND HYDRAULIC MACHINES (Dr. R.K. Bansal)

[7] Roloff / Matek - Maschinenelemente, Normung, Berechnung, Gestaltung

[8] Fluid mechanics, seventh edition (Frank M. White)

[9] 2500 Solved problems in fluid mechanics and hydraulics (Jack Evett, Cheng Liu)

[10] Handbook of hydraulics (Brater, King, Lindell, Wei)

The actual flow rate of a real fluid differs from the theoretical values for the ideal fluid. When the fluid flows out through an opening (or a short nozzle), the contact with the wall is small; thus, the energy dissipation will be small as well. Therefore, it is possible to consider the fluid as inviscid and correct the theoretical results with the following correction factors.

From the perspective of hydraulics, the spillway is a large discharge opening, with no top wall above the beam (Figure A). The spillway may be ideal if the level behind the spillway is lower than the spillway edge, or imperfect (flooded) if the level behind the spillway is higher (Figure B). Ideal spillways are used to determine the amount of fluid flowing. Depending on their shape, spillway edges may be rectangular, triangular, trapezoidal, or circular.

The level h must be measured at a sufficient distance before the spillway (usually 2h - 4h). Above the spillway, the level is lower because part of the positional energy has already been converted into kinetic energy.

Q = Cd * (2/3) * (2 * g)^0.5 * be * he^(3/2)

Cd ... Discharge coefficient

g ... Acceleration of gravity

be ... Effective width

he ... Effective overflow height

he = h + 0.001

be = b + kb

f ... Flooded coefficient

for h/p=0.5, f =1.007 * ((0.975 - h2/h)^1.45)^0.265

for h/p=1.0, f =1.026 * ((0.960 - h2/h)^1.55)^0.242

for h/p=1.5, f =1.098 * ((0.952 - h2/h)^1.75)^0.22

for h/p=2.0, f =1.155 * ((0.950 - h2/h)^1.85)^0.219

1. In paragraph [1], select the fluid, set its parameters (if required), and define environmental parameters.

2. Select the appropriate task and fill in the known input values.

3. The calculations are in the form of a mathematical model of the given problem. Therefore, if the result of the calculation is to be one of the input parameters, it is necessary to iterate over this parameter.

The actual flow rate of the real fluid differs from the theoretical values for the ideal fluid. When the fluid flows out through an opening or short nozzle, the contact with the wall is small; thus, the energy dissipation will be small as well. Therefore, it is possible to consider the fluid as inviscid and correct the theoretical results with the following correction factors.

For the discharge through a hole without a nozzle (figure A), the discharge coefficient can be determined as a function of the Reynolds number Re. The first value on the left is the calculated Re. On the right you can see the discharge coefficient, which can be used instead of the standard value of 0.62.

For large openings (the upper edge near the surface and the ratio of the hole height to the depth of the center of gravity is close to one), a nonlinear distribution of the discharge velocity must be considered.

From the perspective of hydraulics, the spillway is a large discharge opening with no top wall above the beam (Figure A). The spillway may be ideal if the level behind the spillway is lower than the spillway edge, or imperfect (flooded) if the level behind the spillway is higher (Figure B). Ideal spillways are used to determine the amount of fluid flowing. Depending on their shape, spillway edges may be rectangular, triangular, trapezoidal, or circular.

There are a number of formulas and procedures for the calculation of rectangular spillways (Fig. A), or for the calculation of the flow coefficient Cd, which are reported in the literature. Therefore, we present them as well for comparison.

A frequent task to solve for constant cross section piping, bends, valves, etc. The universal Bernoulli equation with the use of loss coefficients is used for the solution. Calculation of flow rate, velocities, losses, and power in constant cross section pipe.

Fill in all dimensional and pressure parameters. You can use the calculations on the right to estimate the loss coefficients. It is possible to choose a negative pressure (p2>p1) and it is possible to choose a negative height (H

If the pipe is not circular, you can enter the value of the pipe area and wetted perimeter after unchecking the check box on the right. The hydrodynamic diameter dh is then determined from these values using the formula dh = 4 * S / C, which is then used in the calculations.

Select the appropriate pipe material from the list. The default roughness value in [mm/in] on the next line is the average of the range indicated in brackets. After unchecking the chec box on the right, you can enter your own value.

Based on the pipe roughness "k" and the coefficient "Re", a number of formulas for calculating the friction loss coefficient for turbulent flow are given in the technical literature. Select the appropriate calculation from the list.

Estimation based on loss integration is used for nozzle (d1> d2) and diffuser (delta 10, when the fluid flow is separated from the pipe wall, the relation zetaO = ((S2 / S1) - 1)^2 * SIN(delta) is used.

The Bernoulli equation is used for the calculation using loss coefficients to calculate the output velocity vo.

Calculation of flow rates, velocities, losses, power in pipes of various cross sections and number of branches.

The velocity of the fluid (kinetic energy) before entering the pipe. In most solved problems (large vessel relative to the volume of the pipeline), it is possible to ignore the velocity and use a zero value. For example, it will be non-zero in the case of a piston that pushes fluid through a pipe.

The sum of the pressure energy potential (p2-p1), the fluid pressure potential (H), and the kinetic energy of the fluid (v0) at the entrance to the pipe.

Usually, the energy level is expressed in meters of fluid column. Select the units in which the energy level is to be expressed on the right.

Curve 1: Loss height hz (losses in the different parts of the pipeline)

Curve 2: Pressure in the pipe (scale on the right in kPa or psi)

Curve 3: Kinematic height

Curve 4: Height points of the beginning of each section relative to the zero line

Simplified (Fig2. Ver1):

The individual sections are divided according to the pipe diameter and all loss coefficients (inlet, bend, valve ...) in one section are summarized into one coefficient (column H), which is applied at the beginning of the relevant section.

Detailed (Fig2. Ver2):

The pipeline is divided into sections so that the relevant loss coefficient (column H) is always at the beginning of the section. This will give you a more detailed graph [6.19]. The overall results (Q, vo ...) are identical..

Enter the diameter. If the pipe is not circular, enter the hydraulic diameter dh (see the description of Surfaces in paragraph E).

Pressing the "V" key copies the value from the first line to the others.

Based on the area ratio (column E) and the chamfer (column F) between the two sections, a loss coefficient at the beginning of the section is proposed (More details can be found in the theoretical part of the help). The loss coefficient in the case of pipeline branching is proposed in a similar way.

ZetaI .... Input losses, losses by section change ( design column G)

ZetaB ... Bending losses (calculation in previous paragraph [5.0])

ZetaV ... Valve losses (table in previous paragraph [5.0])

Reynolds number Re (See help for details - Theory)

Characterizes the flow of a viscous fluid, which depends not only on the mean velocity but on the characteristic dimension of the fluid flow (diameter d or dh), dynamic viscosity and density.

It is used to calculate the friction loss coefficient Lambda (column K).

Design of Friction loss coefficient Lambda. Depends on Re (column J). However, the calculation of Re is backwards dependent on the flow velocity (for which knowledge of Lambda is required). Therefore, there is a sequential iteration of values.

The proposed value is automatically transferred to the cell on the right (column L), which is used for the calculation.

After unchecking the check box on the right, you can enter your own value of the friction loss coefficient.

When regulating the flow of fluid through the pipe, a non-stationary flow is created. As the flow rate decreases, more pressure is created in front of the valve than behind it. In the case of long pipelines transporting liquid, a rapid (emergency) closure of the pipeline can lead to an increase in the pressure which can damage the pipeline.

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