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INTRODUCTION
DIMENSIONS, UNITS, AND PHYSICAL QUANTITIES
Note that if the leading number in an answer is 1, it does not count as a significant figure, e.g. 1248 has three significant figures. Note that a calculator will provide 19.924 N, which contains four significant figures (the leading 1 does not count).
GASES AND LIQUIDS
PRESSURE AND TEMPERATURE
Note that we do not use the degree symbol when expressing temperature in degrees Kelvin, nor do we capitalize the word 'kelvin'. We read '100 K' as 100 Kelvin in the SI system (remember that the SI system is a special metric system).
PROPERTIES OF FLUIDS
The velocity of the inner cylinder is RO and the velocity of the outer cylinder is zero. If the pressure is below the vapor pressure, the molecules will escape the liquid; it is called boiling when water is heated to the temperature at which the vapor pressure equals the atmospheric pressure.
THERMODYNAMIC PROPERTIES AND RELATIONSHIPS
The car is driven in Arizona where the temperature of the tire on the asphalt reaches 65–C. Calculate the shear stress at (a) the wall of the pipe, (b) at a radius where is 0.2 cm, and (c) at the centerline of the pipe.
INTRODUCTION
PRESSURE VARIATION
In a fluid at rest there is no acceleration, so the pressure variation of Eq. If the pressure variation in the atmosphere is desired, then Eq. 2.9) would be used with the ideal gas law p¼rRT to give.
MANOMETERS
In the troposphere (between the earth's surface and an altitude of about 10 km) where the temperature (in kelvin) is T¼28820:0065z, Eq. 2.12) can be integrated to give the pressure change. Since point 4 is shown to be open to the atmosphere, the pressure there is zero pressure: p4 ¼0.
FORCES ON PLANE AND CURVED SURFACES
2.6(b) and 2.6(c), the desired forces can be calculated, provided that the force FW acting through the centroid of the area can be found. The forces FHandFV are the horizontal and vertical components of the force of the water acting on the gate.
ACCELERATING CONTAINERS
What rotational speed is required so that the water just touches point A? Also find the force on the bottom of the cylinder. The force of the water on the gate is given by Eq. Giving moments about the hinge.
INTRODUCTION
FLUID MOTION
The vorticity and angular velocity components are 0 for an irrotational flow; the fluid particles do not rotate, they only deform. It is the deformation of fluid particles that leads to the internal stresses in a flow.
CLASSIFICATION OF FLUID FLOWS
Yet it would be a laminar flow, because there would be no mixing of fluid particles. The airflow can be assumed to be incompressible if the velocity is sufficiently low.
BERNOULLI’S EQUATION
This flow is very turbulent at this Reynolds number, in contrast to our observation of the calm flow. Determine the rate of change of temperature of a liquid particle in the stream at point (2, 1.22) at t¼2 s. 3.19.
INTRODUCTION
SYSTEM-TO-CONTROL-VOLUME TRANSFORMATION
3 Figure 4.1 System and fixed control volume. where we simply added and subtracted E1ðtþDtÞ in the last row. Note that the first relation in the last line above refers to the control volume, so that.
CONSERVATION OF MASS
It leaves the volume through two tubes, one 2 cm in diameter and the other with a mass flow of 10 kg=s. If the velocity out of the 2 cm diameter tube is 15 m=s, determine the rate at which the mass inside the volume is changing.
THE ENERGY EQUATION
Also predict the pressure just upstream of the nozzle (the losses through the nozzle can be neglected). If the correct units are included at the points in our equations, the units will come out as expected, i.e. the units of WW_T must be J=s.
THE MOMENTUM EQUATION
EXAMPLE 4.7 The relatively fast water flow in a horizontal rectangular channel can suddenly 'jump' to a higher level (an obstruction downstream can be the cause). Determine the force components of the water on the nozzle and the magnitude of the resultant force.
INTRODUCTION
It is often easier to solve problems using cylindrical or spherical coordinates; the differential equations using those two coordinate systems are presented in Table 5.1.
THE DIFFERENTIAL CONTINUITY EQUATION
It would be necessary if there are temperature differences at the boundaries, or if viscous effects are so great that temperature gradients develop in the flow. 0:80 ¼215 m=ðs·mÞ The best estimate of the density gradient, using the given information, is then.
THE DIFFERENTIAL MOMENTUM EQUATION
Finally, if an incompressible flow is assumed, then HHHHH·V¼0;the Navier-Stokes equations result rDu. where the z direction is vertical. The differential momentum equations (the Navier-Stokes equations) can be solved with relative ease for some simple geometries.
THE DIFFERENTIAL ENERGY EQUATION
Assume a plane flow in which only the x-andy components are nonzero and viscous and gravity effects are negligible. Experiments show that a stagnant fluid region exists in the face of a sudden increase in the height of the bottom of a channel in a stratified flow.
INTRODUCTION
DIMENSIONAL ANALYSIS
When temperature is required, as in the flow of a compressible gas, an equation of state such as: can be expressed dimensionally as. where the brackets mean 'the dimensions of'. The RT product does not introduce any additional dimensions. EXAMPLE 6.1 It is assumed that the pressure drop over a length of a pipe depends on the average velocity V, the diameter of the pipe D, the average height of the roughness elements of the pipe wall, the fluid density and the fluid viscosity.
SIMILITUDE
Estimate the flow rate and pressure rise that would be expected in the model study. What speed should be chosen for the model study and what drag force would be expected on the prototype if a force of 80 N is measured on the model.
INTRODUCTION
ENTRANCE FLOW
A laminar flow cannot exist for Re.7700; a value of 1500 is used as the limit for a conventional flow. The pressure variation for the laminar flow is higher in the inlet region than in the fully developed region due to the larger wall shear and the increasing momentum flux.
LAMINAR FLOW IN A PIPE
In terms of Reynolds number, the friction factor for laminar flow is (Combine Eqs. EXAMPLE 7.1 The pressure drop across a 30 m length of 1 cm diameter horizontal pipe carrying water at 20–C is measured to be 2 kPa .
LAMINAR FLOW BETWEEN PARALLEL PLATES
If both plates are stationary and the flow is due only to a pressure gradient, it is a Poiseuille flow. The same result can be obtained by solving the appropriate Navier-Stokes equation; if that is not important, go directly to Sec. EXAMPLE 7.2 The thin layer of rain at 20°C flows at a relatively constant depth of 4 mm over a parking lot.
LAMINAR FLOW BETWEEN ROTATING CYLINDERS
Estimate (a) the flow velocity, (b) the surface shear, (c) the Reynolds number, and the surface velocity. The velocity profile can be assumed to be half of the profile shown in Fig.
TURBULENT FLOW IN A PIPE
The outer edge of the wall region can be as low as uty=n¼3000 for a low Reynolds number flow. The viscous wall layer does not play a role for a rough pipe. Solution: First, the average velocity and Reynolds number are V¼Q. a) To find the wall shear stress, let us first find the friction factor. e) The height of the roughness elements is given as 0.0015 mm (drawn pipe), which is less than the thickness of the viscous layer.
OPEN CHANNEL FLOW
Both EGL and HGL are tilted downstream due to tube losses. Determine the flux using (a) Moody's diagram and (b) the alternative equation. and head loss is.
INTRODUCTION
The boundary layer is so thin that it can be ignored when solving for the inviscid flow. The impure flow solution provides the lift, which is not significantly affected by the viscous boundary layer, and the pressure distribution at the surface of the body, as well as the velocity at that surface (since the inviscid solution ignores viscosity effects, the fluid does not stick to the boundary, but slips off the edge).
FLOW AROUND BLUNT BODIES .1 Drag Coefficients
Free-stream innocuous flow is usually non-rotating, although it may be a swirling flow with vorticity, eg, the flow of air near the ground around a tree trunk or water near the ground around a pole in a river; the water digs a hole in the sand in front of the pole, and the air digs a similar hole in the snow in front of the tree, quite an interesting observation. The moment due to the drag force, which acts at the center of the sign, is.
FLOW AROUND AIRFOILS
The lift and drag on aerofoils will not be calculated from the flow conditions, but from graphical values of the lift and drag coefficients. Lift is directly proportional to angle of attack until just before stall is encountered.
POTENTIAL FLOW .1 Basics
For a real flow, there would be a separated region at the rear of the cylinder, but the flow over the front (perhaps over the entire front half, depending on the Reynolds number) can be approximated by the potential flow shown in outline. To create flow around a rotating cylinder, as in Fig. 8.10, add a vortex to the stream function of Eq. recognizing that the radius of the cylinder remains unchanged since a vortex does not affect vr.
BOUNDARY-LAYER FLOW .1 General Information
Estimate the shear velocity, the thickness of the viscous wall layer and the boundary layer thickness at the end of the plate (assume a turbulent layer from the leading edge). At the end of the plate, estimate (a) the wall shear stress, (b) the maximum value of the boundary layer, and (c) the flow rate through the boundary layer.
INTRODUCTION
SPEED OF SOUND
9.2(b) the source is moving at a subsonic speed, which is less than the speed of sound, i.e. the source. High-amplitude waves called shock waves, which emanate from the leading edge of blunt-nosed airfoils, also form zones of calm, but the angles are greater than those created by the Mach waves.
ISENTROPIC NOZZLE FLOW
The case of a convergent-divergent nozzle allows a supersonic flow to occur provided the receiver pressure is sufficiently low. If the receiver pressure is equal to the reservoir pressure, no flow occurs, represented by curve A.
NORMAL SHOCK WAVES
