## 12 Dimensions units and physical quantities

A variety of physical quantities of both the fluids and the equipment will be used throughout this book. These quantities must be described quantitatively, for which sets of dimensions and units are required. For example, the density of a fluid is an important quantity that will influence the behavior of the fluid in most situations. The dimensions of density are mass per unit volume M L3. To give the density a numerical value, a set of units must be selected for all the dimensions that are to...

## 1200295 x12 x 103205 x00208

Friction factors for Power-Law fluids Power-law nde n 0.65 Dlrnsrsionless tlowc te 0 ' 2 l-4.tL> Specify the independent variable C Reynolds number Repli Dimensionless pipe diameter D (* Dirrtensionless fkiwrate Cr C Dimensionless vefocity V* F7 Show calculated value on graph Friction factor jo. 00462 Reynolds number 620S5.& Figure 5.19 Friction factor for a power-law fluid calculated using the FLUIDS toolbox Use Figure 5.18 or the FLUIDS toolbox to get the friction factor pl 0.00482

## 266 Pump performance and the system curve

The pump characteristic curve can be used effectively to select a suitable pump for the duty that is required and it can also be used to establish the capacity of a pipeline system as a function of the speed of a pump that already exists in the pipeline. The physical details and frictional characteristics of the pipeline and all fittings and included equipment must be known. The energy balance for the system is given by equation 2.66 W J2 Fi + vAP + gAz + i AV2 (2.66) where Fi is the frictional...

## 67 Continuous thickening of compressible pulps

A particularly important solution of equation 6.77 is the steady condition that describes the behavior of operating industrial thickeners. which can be integrated to give the flux of solid at any level The constant of integration can be evaluated by noting that at the bottom of the thickener the total flux must be equal to the convective outflow qCD. Thus the concentration profile in the compression zone of the thickener is described by the differential equation Hydrodynamic stability requires...

## 265 Net positive suction head

The pumping action of the impeller creates a reduction in pressure in the interior of the pump. This reduction in pressure creates a potential problem that must be considered when evaluating the duty of the pump. The lowest pressure in the pump chamber is usually considerably below the pressure at the inlet flange and it is important that the lowest pressure anywhere in the pump chamber should never fall below the vapor pressure of the liquid that is being pumped. If the pressure does fall...

## Dr m

5.2.1.1 Velocity profile for Newtonian fluids in laminar motion When any fluid flows inside a pipe the axial velocity varies strongly with radial position. The shearing action of the fluid against the pipe wall causes the velocity to be low close to the wall and considerably higher close to axis of the pipe. The radial profile in the axial velocity is easy to calculate whenever the fluid is flowing slowly enough to be in laminar motion. The velocity profile is calculated by integration of the...

## 244 The overall energy balance

When a fluid is moved from one location to another, such as when it is pumped through a piping system, there is usually a redistribution of energy. For example when a fluid flows downward in the gravitational field of the earth, potential energy can be converted to kinetic energy and the fluid can gain energy from a pump. Electrical energy consumed by a pump motor can be converted to kinetic or potential energy of the fluid. As the fluid moves the changes in each type of energy must be balanced...

## 52 Newtonian and nonNewtonian fluids in pipes with circular crosssection

Because of the inherent structure that is exhibited by non-Newtonian slurries, laminar motion of these fluids is encountered far more commonly than with Newtonian fluids. It is therefore profitable to examine some simple cases of laminar flow which can be analyzed completely by theoretical methods. Consider a viscous fluid (Newtonian or non-Newtonian) flowing through a round tube as illustrated in Figure 5.4. Figure 5.4 A force balance over an imaginary cylinder can be used to calculate the...

## 69 Practice problems

The data shown in Table 6.1 were reported by a laboratory that specializes in dewatering technology. The data shows the interface height as a function of time for the standard batch settling test. The slurries were carefully prepared by diluting the original concentrated slurry, adjusting the pH and adding flocculant at 0.02 lb short ton calculated on a dry solids basis. Use the Kynch construction to determine the settling velocity as a function of concentration. Do the data support the Kynch...

## V2

If there is a change in velocity through the fitting, it is the upstream velocity that is used to calculate the value of the velocity head. The frictional dissipation for a variety of common fittings is given in terms of the equivalent additional length of pipe and in terms of the number of lost velocity heads in Table 2.2. A more accurate method is based on experimental data that was obtained in a variety of fittings and is correlated in terms of a loss coefficient Kf. The loss coefficient is...

## 33 Isolated isometric particles of arbitrary shape

In practical engineering situations it is more common to work with particles of irregular shape than it is to work with spherical particles. The shape of a particle is determined by its method of manufacture and these vary widely in the process industries. Common synthesis processes include precipitation, crystallization, deposition, comminution and agglomeration. These processes invariably produce non-spherical particles unless they are designed expressly to produce spheres. Deviations from...

## 262 The generalized pump characteristic curve

The centrifugal pump derives its pumping action from the centrifugal acceleration that is generated when the fluid rotates inside the pump chamber driven by the impeller. The centrifugal acceleration generates an increasing pressure from the center to the outer edge of the impeller. This pressure change is reduced by the frictional drag experienced by the fluid as it moves outward between the blades of the impeller shearing against the surface of the blades and against the inner surfaces of the...

## 422 Velocity at minimum pressure drop

One important consequence of the additional momentum transfer path is that the pressure drop in a pipe carrying slurry does not increase monotonically with the slurry velocity. There is a distinct velocity at which the pressure drop is a minimum. This can be seen in Figure 4.6 where the pressure gradient due to friction, calculated from equation 4.8, is plotted against the velocity for Figure 4.6 Frictional pressure gradient in a 10-cm pipe carrying a slurry of 1-mm silica particles. The volume...

## 133 Vapor pressure

All liquids show a greater or lesser tendency to vaporize and, if allowed to come to equilibrium with its surroundings, a liquid will establish an equilibrium across the liquid-vapor interface. The pressure exerted by the molecules of the fluid in the vapor phase is specific to each liquid, and the equilibrium pressure is called the vapor pressure of the liquid. The vapor pressure is a function of the temperature. If the vapor pressure of a liquid exceeds the prevailing total pressure, the...

## 34 Symbols used in this chapter

Ac Cross-sectional area of particles in plane perpendicular to direction of ap Surface area of particle m2. de Volume equivalent particle diameter m. dp Dimensionless particle size. V Relative velocity between particle and fluid m s. V Dimensionless terminal settling velocity. ps Density of solid kg m3. Si Cd Rep. 2 Rep CD. Sphericity. Superscripts * Indicates that variable is evaluated at the terminal settling velocity. Subscripts M Indicates modified value to take account of non-spherical...

## Vt PT8P7 dp327

This is known as Stokes' law of settling. Stokes law is particularly useful for the analysis of the motion of very small particles. It should not be used if the particle Reynolds number is greater than 0.01. Use Stokes' Law to calculate the terminal settling velocity of a glass sphere of diameter 0.1 mm in a fluid having density 982 kg m3 and viscosity 0.0013 Pa s. Density of glass is 2820 kg m3. Calculate the Reynolds number at terminal settling velocity to establish whether Stokes' Law can be...

## V0 VCq VQj Co VCq6rn

(0> C0) --0-c0 --or V(C0) (6.21) Thus the mudline discontinuity moves downward at the local solids settling velocity - an obvious fact that was used in equation 6.12. An additional condition must be satisfied to ensure that a discontinuity can exist in practice. It must be hydrodynamically stable. To ensure this the density of the slurry above the discontinuity must be less than the density below the discontinuity. If this condition is not satisfied, convection currents would quickly develop...

## 441 Fully stratified flow

When the particles in the suspension are comparatively large and the velocity not too large, most of the particles settle to the bottom of the pipe and are transported as a sliding bed. Most of the solids are supported on the bottom of the pipe as contact load. The movement of this bed is resisted by the mechanical friction between the particles in the bed that are up against the pipe wall and the pipe wall. The resisting force can be calculated as where Ms is the coefficient of friction...

## 522 Bingham plastics in laminar flow in pipes

Substituting the rheological model for a Bingham plastic into equation 5.18 8mb 3 rw 3 rw This is called the Buckingham equation and is often written in the form 8vmb ry i 4 ry+1 (ry)4 (5.30) ReB - is the Reynolds number for the Bingham plastic. The Hedstrom number He (D2pslry Mb) is also useful for the characterization of the rheological properties of the Bingham plastic and equation 5.31 can be written in a number alternative ways which are useful for the solution of particular problems....

## 31 Basic concepts

When a solid particle moves through a fluid it experiences a drag force that resists its motion. This drag force has its origin in two phenomena namely, the frictional drag on the surface and the increase in pressure that is generated in front of the particle as it moves through the fluid. The frictional drag is caused by the shearing action of the fluid as it flows over the surface of the particle. This component is called viscous drag. A region of high pressure P1 is formed immediately in...

## 331 Empirical modifications to drag coefficient correlations

One approach to dealing with non-spherical particles is to find empirical relationships for the coefficients in the drag coefficient vs Rep correlation (see Figure 3.7). A Clift-Gauvin type equation can be used and the parameters A, B, C and D are related to the sphericity as shown in Table 3.2. Haider-Levenspiel equations used for the drag coefficient - Experimental data 1.000 o Experimental data 0.906 - a Experimental data 0.846 v Experimental data 0.806 o Experimental data 0.670 -...

## Mm 7 Ks 7n27Mm nKs 7n1d7587

After expansion of the integrand this yields D (1 + X )3 4 n + 3 2n + 2 3n + 1 where G(n, X) -3 - +-- X + --- X2 + --TX3 (5.90) both of which are dimensionless. Equation 5.88 defines an implicit relationship between 7W and 8V D which cannot be solved in closed form. It is convenient to introduce the dimensionless number H XG(n, X )n1 (5.92) The shear stress at the pipe wall is given by This leads to a relationship between the friction factor and the Reynolds number 2 Tw _ 16 (1 + X) psl V2 Re ,...

## 28 Practice problems

Water is transferred between two reservoirs that are situated 3 km apart. The flow is by gravity and the difference in water level between the two reservoirs is 107.8 m. The pipeline diameter is 15 cm and the surface roughness is 0.085 mm. (a) Calculate the rate at which water is transferred between the reservoirs. (b) What pipe diameter should be used to increase the flow to 215 m3 hr. Mw 0.001 kg ms, pw 1000 kg m3, g 9.81 m s2. 2. An experiment is conducted to estimate the inside surface...

## 43 Head loss correlations for separate flow regimes

While the Durand-Candolios-Worster correlation is useful in the heterogeneous suspension flow regime, it deviates more and more from actual conditions in the other regimes of flow. Experimental observations have shown that different correlations should be used in each of the identifiable flow regimes. Although this is a logical approach it is not straightforward to apply. The main difficulty arises because it is not easy to define the boundaries between the flow regimes. These boundaries are...

## 511 Bingham plastic fluids

Bingham plastics exhibit a linear relationship between shear stress and rate of strain but, unlike Newtonian fluids, this relationship does not exhibit a zero intercept at zero strain rate. The intercept at zero strain rate is called the yield Rate of strain in the fluid d Figure 5.1 Typical stress-rate-of-strain relationship for non-Newtonian fluids Rate of strain in the fluid d Figure 5.1 Typical stress-rate-of-strain relationship for non-Newtonian fluids Figure 5.2 Viscometer data for two...

## 63 Useful models for the sedimentation velocity

A number of models have been proposed to relate the sedimentation velocity to the solids concentration and these have been used with varying success to describe the sedimentation of a variety of slurries that are important in industrial practice. It is usual to express these model equations in terms of the volumetric concentration of solid v expressed as m3 solid m3 slurry. The two representations of the solid concentration are related by where ps is the density of the solid. The most commonly...

## 68 Batch thickening of compressible pulps

The analysis of the behavior of a compressible sediment during batch settling is somewhat more complex than the behavior in the steady-state continuous thickener because batch settling is a dynamic process and the partial differential equation 6.77 must be used to describe the behavior rather than the ordinary differential equation 6.82. The solution to this equation is available in the FLUIDS toolbox. The user can choose between the Richardson-Zaki and extended Wilhelm-Naide models for the...

## 442 Heterogenous flow of settling slurries

Heterogenous flow conditions exist when most of the particles are supported by the fluid and the contact load between settled solids on the pipe wall is negligible. The particles are suspended only when the average velocity is high. At high fluid velocities the particles can be distributed more or less uniformly across the pipe cross-section and the slurry behaves as a homogenous fluid. Between the condition of fully stratified flow that was described in the previous section and homogeneous...

## 431 Flow regime boundaries

The boundaries of the flow regimes are defined in a self-consistent manner by noting that any two regimes are contiguous at their common boundary and therefore each of the two correlation equations must be satisfied simultaneously. For example, the boundary between the sliding bed regime Regime 0 and the saltation regime Regime 1 must lie along the solution locus of the equation 1213 c0'7389 w7717 cQ-0'4054 Fr-1096 107'1 C1'018 f 046 CQ-0'4213 Fr-1'354 4'21 Fr _ S _ 1 4679 C1'083 W'064 CO0'0616...

## 263 Derating of pumps when handling slurries

The simple theory for the pump characteristic curve that is described in the previous section shows that the head developed by a centrifugal pump, measured as head of the fluid being pumped, is independent of the fluid density. This is convenient in practice because the same characteristic curve can be used for fluids of different densities. However, when the pump must transport a slurry, the presence of the solid particles has a significant effect on the performance of the pump. The...

## 514 Shearthinning fluids with Newtonian limit

Some shear-thinning fluids exhibit Newtonian behavior at high strain rates. The rheological properties of these fluids can be modeled using provided that n lt 1 which shows that the fluid exhibits Newtonian behavior at very high strain rates. This is commonly referred to as the Sisko model. Typical data measured using slurries of fine Ti02 are shown in Figure 5.3. The shear stress calculated using equation 5.10 is shown as a series of solid Figure 5.3 Measured shear stress-rate of strain...

## 321 Settling velocity of an isolated spherical particle

When the particle is spherical, the geometrical terms in equation 3.10 can be written in terms of the particle diameter and the drag coefficient of a spherical particle at terminal settling velocity is given by The particle Reynolds number at terminal settling velocity is given by It is not possible to solve equation 3.11 directly because Cd is a iunction oi both and the particle size dp through the relationship shown in Figure 3.2 or that given by either oi the Abraham or Turton-Levenspiel...

## Re3 D3

The velocity, V, is related to the flow rate and the pipe diameter by and substituting this in equation 2.24 Q can be calculated without requiring the pipe diameter or the average velocity to be known. The use of Q for practical problem solving is described in the following illustrative example. Calculate the diameter of a smooth pipe that would transport 0.01178 m3 s of water under a pressure gradient of 180.0 Pa m see Figure 2.8 . Figure 2.8 Data input screen to calculate friction factor for...

## 32 Terminal settling velocity

If a particle falls under gravity through a viscous fluid it will accelerate for a short while but as the particle moves faster the drag force exerted by the fluid increases until the drag force is just equal to the net gravitational force less the buoyancy that arises from the immersion of the particle in the fluid. When these forces are in balance the particle does not accelerate any further and it continues to fall at a constant velocity. This condition is known as terminal settling. The...