## P 1

Where C is the tracer concentration at time t recorded by the ith detector. A typical variance curve is shown in Figure 8.19. Figure 8.19 Reduction in variance of concentration of tracer with time Figure 8.19 Reduction in variance of concentration of tracer with time Several experimental techniques may be used, such as acid base titration, electrical conductivity or temperature measurement, measurement of refractive index, light absorption, and so on. In each case, it is necessary to specify...

## 64 Fullydeveloped heat transfer to powerlaw fluids in laminar flow

The heating of a viscous fluid in laminar flow in a tube of radius R (diameter, D) will now be considered. Prior to the entry plane (z < 0), the fluid temperature is uniform at T for z > 0, the temperature of the fluid will vary in both radial and axial directions as a result of heat transfer at the tube wall. A thermal energy balance will first be made on a differential fluid element to derive the basic governing equation for heat transfer. The solution of this equation for the power-law...

## 3 0 02

Where 0 rB rw, the ratio of the yield stress to the wall shear stress. Indeed, Pigford 1955 has asserted that equation (6.33) is applicable to any type of time-independent fluid provided that n0 the apparent flow behaviour index replaces n in equation (6.33). For power-law shear-thinning fluids, equations (6.32) and (6.33) seem to be valid for Gz > 100. 6.5.2 Experimental results and correlations The experimental studies on heat transfer to from purely viscous fluids in laminar flow in...

## 37 Laminar flow of inelastic fluids in noncircular ducts

Analytical solutions for the laminar flow of time-independent fluids in non-axisymmetric conduits are not possible. Numerous workers have obtained approximate numerical solutions for specific flow geometries including rectangular and triangular pipes Schechter, 1961 Wheeler and Wissler, 1965 Miller, 1972 Mitsuishi and Aoyagi, 1973 . On the other hand, semi-empirical attempts have also been made to develop methods for predicting pressure drop for time-independent fluids in ducts of non-circular...

## Vertical upward flow

The interpretation of results for vertical flow is more complicated since they are strongly dependent on the in-situ liquid holdup which , in turn, determines the hydrostatic component of the pressure gradient. Khatib and Richardson 1984 reported measured values of the two-phase pressure drop and liquid holdup for the vertical upward co-current flow of air and aqueous china clay suspensions in a 38 mm diameter pipe. Representative results are shown in Figures 4.17 and 4.18 for air-water and...

## 76 Heat transfer in boundary layers

When the fluid and the immersed surface are at different temperatures, heat transfer will take place. If the heat transfer rate is small in relation to the thermal capacity of the flowing stream, its temperature will remain substantially constant. The surface may be maintained at a constant temperature, or the heat flux at the surface may be maintained constant or surface conditions may be intermediate between these two limits. Because the temperature gradient will be highest in the vicinity of...

## 28 High frequency techniques

In many cases, a comprehensive characterization of the rheological properties of systems, such as concentrated colloidal dispersions, can require measurements of dynamic mechanical behaviour at frequencies outside the range of conventional, commercially available, rheometers (typically 10 3Hz to 102Hz). In particular, consideration of the relative time scales of particle-fluid displacement and interfacial polarization mechanisms in such systems reveals the need for enhanced high frequency...

## Example

The rheological properties of a china clay suspension can be approximated by either a power-law or a Bingham plastic model over the shear rate range 10 to 100s-1. If the yield stress is 15 Pa and the plastic viscosity is 150 mPa-s, what will be the approximate values of the power-law consistency coefficient and flow behaviour index Estimate the pressure drop when this suspension is flowing under laminar conditions in a pipe of 40 mm diameter and 200 m long, when the centre-line velocity is 0.6...

## 86 Mixing in continuous systems

The mixing problems considered so far have related to batch systems in which two or more materials are mixed together and uniformity is maintained by continued operation of the agitator. Consideration will now be given to some of the equipment used for continuous mixing duties. Mixing duties in the plastics industry (and to a lesser extent in food industry) are often carried out in either single or twin screw extruders. The feed to such units usually contains the base polymer in either granular...

## Greek letters

Parameter in Ellis fluid model or kinetic energy correction radial coordinate corresponding to zero shear stress in boundaries of plug of Bingham plastic fluid in annular flow Bingham plastic viscosity (Pa s) non-dimensional radial coordinate (-) non-dimensional inner radius of annulus (-) yield stress in Bingham plastic model (Pa)

## 29 The relaxation time spectrum

The determination of the relaxation spectrum of a visco-elastic fluid from various dynamic shear measurements has been discussed by many workers (e.g. see, Orbey and Dealy, 1991 Baumgaertel and Winter, 1989 Sullivan etal., 1994) and, in the case of a visco-elastic fluid, the problem of determining the relaxation spectrum from oscillatory shear measurements involves the inversion of the following pair of integral equations G Gc C dlnX and (2.35a) where H(X) denotes the continuous relaxation...

## 271 Fourier transform mechanical spectroscopy FTMS

The evolution of visco-elastic properties in non-Newtonian fluids exhibiting time-dependent rheological changes is a matter of wide scientific interest, particularly so in systems undergoing gelation. The gel-point, where a three-dimensional network structure is established, may be identified rheologically by the establishment of a characteristic frequency dependence of the dynamic moduli, and an associated frequency independent loss tangent Winter and Chambon, 1986 . This criterion for...

## 43 Twophase liquidsolid flow hydraulic transport

Hydraulic transport is the conveyance of particulate matter in liquids. Although most of the earlier applications of the technique used water as the carrier medium (and hence the term hydraulic), there are now many industrial plants, particularly in the minerals, mining and power generation industries, where particles are transported in a variety of liquids which may exhibit either Newtonian or non-Newtonian flow behaviour. Transport may be in vertical or horizontal or inclined pipes, but in...

## 22Capillary viscometers

Capillary viscometers are the most commonly used instruments for the measurement of viscosity due, in part, to their relative simplicity, low cost and (in the case of long capillaries) accuracy. However, when pressure drives a fluid through a pipe, the velocity is a maximum at the centre the velocity gradient or shear rate y are a maximum at the wall and zero in the centre of the flow. The flow is therefore non-homogeneous and capillary viscometers are restricted to measuring steady shear...

## 25 Yield stress measurements

Notwithstanding the continuing debate over the very existence of a 'true' yield stress, the concept of an apparent yield stress has been found to be an extremely useful empiricism in many areas of science and engineering Hartnett and Hu, 1989 (see also Chapter 1). A recent comprehensive review Barnes, 1999 has critically assessed the various issues raised in the definition, measurement and application of apparent yield stress behaviour. Any operational definition of apparent yield stress should...

## A

Figure 8.14 Schematic double-celled secondary flow pattern liquid in contact with the tank bottom is essentially stationary while that at higher levels is rotating and will experience centrifugal forces. Consequently, the prevailing unbalanced forces within the fluid lead to the formation of a toroidal vortex. Depending upon the viscosity and type of fluid, the secondary flow pattern may be single-celled as in Figure 8.13 or double-celled, as shown schematically in Figure 8.14. Indeed, these...

## 67 Effect of temperaturedependent physical properties on heat transfer

The theoretical treatments considered so far have been based on the assumption that the thermo-physical properties are constant (i.e. independent of temperature and therefore the velocity profiles do not change over the heat transfer section of the tube. Christiansen and Craig 1962 investigated the effect of temperature-dependent power-law viscosity on the mean values of Nusselt number for streamline flow in tubes with constant wall temperature. They postulated that the flow behaviour index, n...

## 68 Effect of viscous energy dissipation

In the flow of all fluids, mechanical energy is degraded into heat and this process is called viscous dissipation. The effect may be incorporated into the thermal energy balance by adding a source term, SV, (per unit volume of fluid) to the right hand side of equation (6.10). Its magnitude depends upon the local velocity gradient and the apparent viscosity of the fluid. Although, in general, the viscous dissipation includes contributions from both shearing and normal stresses, but under most...

## Problems

The level of difficulty of problems has been graded (a) straightforward, (b) somewhat more complex, and (c) most difficult. In any given Chapter the readers are recommended to tackle problems in increasing order of difficulty. 1.1 The following rheological data have been obtained for a liquid at (a) 295.5 K. By plotting these data on linear and logarithmic scales, ascertain the type of fluid behaviour, e.g. Newtonian, or shear-thinning, or shear-thickening, etc. Also, if the liquid is taken to...

## 74 Laminar boundary layer flow of Bingham plastic fluids over a plate

Outside the boundary layer region, the velocity gradient is zero, and thus the shearing forces must also be zero in the case of Newtonian and power-law fluids, as seen in the preceding section. In contrast, in the case of Bingham plastic fluids, the shear stress approaches the yield stress of the fluid at the outer edge of the boundary layer which must eventually decay to zero over a relatively short distance and thus, once again, there is no shearing force present in the bulk of the fluid...

## 54 Motion of bubbles and drops

The drag force acting on a gas bubble or liquid droplet will not, in general, be the same as that acting on a rigid particle of the same shape and size because circulating patterns are set up inside bubbles and drops. While the radial velocity at the interface is zero, the angular velocity, shear, and normal stresses are continuous across the interface for fluid particles and the velocity gradient in the continuous phase (hence shear stress and drag force) is therefore less than that for a...

## 33 Criteria for transition from laminar to turbulent flow

For all fluids, the nature of the flow is governed by the relative importance of the viscous and the inertial forces. For Newtonian fluids, the balance between these forces is characterised by the value of the Reynolds number. The generally accepted value of the Reynolds number above which stable laminar flow no longer occurs is 2100 for Newtonian fluids. For time-independent fluids, the critical value of the Reynolds number depends upon the type and the degree of non-Newtonian behaviour. For...

## 34 Friction factors for transitional and turbulent conditions

Though turbulent flow conditions are encountered less frequently with polymeric non-Newtonian substances, sewage sludges, coal and china clay suspensions are usually all transported in the turbulent flow regime in large diameter pipes. Therefore, considerable research efforts have been directed at developing a generalised approach for the prediction of the frictional pressure drop for turbulent flow in pipes, especially for purely viscous (power-law, Bingham plastic and Herschel-Bulkley models)...

## 15 Viscoelastic fluid behaviour

In the classical theory of elasticity, the stress in a sheared body is directly proportional to the strain. For tension, Hooke's law applies and the coefficient of proportionality is known as Young's modulus, G, where dx is the shear displacement of two elements separated by a distance dy. When a perfect solid is deformed elastically, it regains its original form on removal of the stress. However, if the applied stress exceeds the characteristic yield stress of the material, complete recovery...

## 32 Laminar flow in circular tubes

Consider the laminar, steady, incompressible fully-developed flow of a time-independent fluid in a circular tube of radius, R, as shown in Figure 3.1. Since there is no angular velocity, the force balance on a fluid element situated at distance r, can be written as p nr2 - p Ap nr2 trz 2nrL 3.1 Figure 3.1 Flow through a horizontal pipe Figure 3.1 Flow through a horizontal pipe This shows the familiar linear shear stress distribution across the pipe cross-section, the shear stress being zero at...

## 88 References

Abid, M., Xuereb C. and Bertrand, J., Chem. Eng. Res. Des. 70 1992 377. Ayazi Shamlou, P. and Edwards, M.F., Chem. Eng. Sci. 41 1986 1957. Bakker, A. and Gates, L.E., Chem. Eng. Prog. 91 Dec, 1995 25. Bakker, A., Morton J.R. and Berg, G.M., Chem. Eng. 101 Mar, 1994 120. Bates, R.L., Fondy P.L. and Corpstein, R.R., Ind. Eng. Chem. Proc. Des. Dev. 2 1963 310. Beckner, J.L. and Smith, J.M., Trans. Inst. Chem. Eng. 44 1966 224. Bourne, J.R. and Butler, H., Trans Inst. Chem. Eng. 47 1969 11....

## 42 Twophase gasnonNewtonian liquid flow

This section deals with the most important characteristics of the flow of a mixture of a gas or vapour and a Newtonian or non-Newtonian liquids in a round pipe. Despite large differences in rheology, two-phase flow of gas-liquid mixtures exhibits many common features whether the liquid is Newtonian or shows inelastic pseudoplastic behaviour. Applications in the chemical, food and processing industries range from the flow of mixtures of crude oil which may exhibit non-Newtonian characteristics...

## 57 Liquidsolid fluidisation

5.7.1 Effect of liquid velocity on pressure gradient As shown schematically in Figure 5.7 for the upward flow of a liquid through a bed of particles, a linear relation is obtained between the pressure gradient and the superficial velocity on logarithmic coordinates up to the point where the bed is fluidised and where expansion of the bed starts to occur A , but the slope of the curve then gradually diminishes as the bed expands. As the liquid velocity is gradually increased, the pressure drop...

## 72 Integral momentum equation

Schilichting 1968 points out that the differential equations for flow in boundary layers require numerical solutions even when the flow is laminar and fluid behaviour Newtonian. However, reasonably good estimates of drag on a plane surface can be obtained by using the integral momentum balance approach due to von Karman, as illustrated in this section. Consider the steady flow of an incompressible liquid of density p over an immersed plane surface. Remote from the surface, the free stream...

## O

Figure 8.25 Variation in turbine impeller designs Figure 8.26 Specially designed impellers a HE-3 b CD-6 c Maxflo 'T' impeller courtesy Chemineer, Inc, Dayton, Ohio Figure 8.26 Specially designed impellers a HE-3 b CD-6 c Maxflo 'T' impeller courtesy Chemineer, Inc, Dayton, Ohio see Figure 8.26 and so on. For dispersion of gases in liquids, turbines and modified turbines are usually employed Figure 8.20 . Commonly two or more disc turbine impellers DT 2 distance apart are mounted on the same...

## 35 Laminar flow between two infinite parallel plates

The steady flow of an incompressible power-law fluid between two parallel plates extending to infinity in x- and z-directions, as shown schematically in Figure 3.15 will now be considered. The mid-plane between the plates will be taken as the origin with the flow domain extending from y b to y b. The force balance on the fluid element ABCD situated at distance y from the mid-plane, can be set up in a similar manner to that for flow through pipes. p 2Wy p Ap 2Wy ryz 2Wdz. 3.61 i.e. ryz 1 y 3.62...

## Gas Newtonian liquid systems

The most widely used method for estimating the pressure drop due to friction is that proposed by Lockhart and Martinelli 1949 and subsequently improved by Chisholm 1967 . It is based on a physical model of separated flow in which each phase is considered separately and then the interaction effect is introduced. In this method, the two phase pressure drop due to friction ApTP , is expressed in terms of dimensionless drag ratios, 0L or 0G defined by the following equations These equations, in...

## Norwood And Metzner

For heat and mass transfer in stirred vessels, additional dimensionless groups which are important include the Nusselt, Sherwood, Prandtl, Schmidt and Grashof numbers. Likewise, in the case of non-Newtonian fluids, an C Height of agitator from base of tank W Width of blades of agitator or paddle Figure 8.5 Typical configuration and dimensions of an agitated vessel appropriate value of the apparent viscosity must be identified for use in equation 8.1 . Furthermore, it may also be necessary to...

## 27 Oscillatory shear measurements

Of the techniques used to characterise the linear visco-elastic behaviour displayed by many non-Newtonian fluids, the oscillatory shear technique which involves either an applied stress or shear rate which varies harmonically with time, is perhaps the most convenient and widely used. The definition of linear visco-elasticity may be expressed in the following form the ratio of the applied stress to strain for any shear history is a function of time alone, and independent of stress magnitude each...

## 321 Powerlaw fluids

For a power-law fluid in a pipe, the shear stress is related to the shear rate by Coulson and Richardson, 1999 Figure 3.2 Schematic representation of shear stress and velocity distribution in fully developed laminar flow in a pipe Figure 3.2 Schematic representation of shear stress and velocity distribution in fully developed laminar flow in a pipe where Vz is the velocity in the axial direction at radius r. Now combining equations 3.2 and 3.3 followed by integration yields the following...

## 36 Laminar flow in a concentric annulus

The flow of non-Newtonian fluids through concentric and eccentric annuli represents an idealisation of several industrially important processes. One important example is in oil well drilling where a heavy drilling mud is circulated through the annular space around the drill pipe in order to carry the drilling debris to the surface. These drilling muds are typically either Bingham plastic or power-law type fluids. Other examples include the extrusion of plastic tubes and pipes in which the...

## 39 Selection of pumps

Non-Newtonian characteristics, notably shear-dependent viscosity and yield stress, strongly influence the choice of a suitable pump and its performance. While no definite quantitative information is available on this subject, general features of a range of pumps commonly used in industry are briefly described here. In particular, consideration is given to positive-displacement, centrifugal, and screw pumps. Difficulties experienced in initiating the flow of pseudoplastic materials owing to...

## 38 Miscellaneous frictional losses

In the analysis of pipe networks, one is usually concerned either with how much power is required to deliver a set flow rate through an existing flow system or with the optimum pipe diameter for a given pump and duty. All such calculations involve determining the frictional pressure losses in the systems, both in the region of fully established flow as has been assumed so far , and in the associated sudden changes in cross-section expansions and contractions and other fittings such as bends,...

## 341 Powerlaw fluids

In a comprehensive study, Dodge and Metzner 1959 carried out a semi-empirical analysis of the fully developed turbulent flow of power-law fluids in smooth pipes. They used the same dimensional considerations for such fluids, as Millikan 1939 for incompressible Newtonian fluids, and obtained an expression which can be re-arranged in terms of the apparent power law index, n', equation 3.26 as follows where A n' and C n' are two unknown functions of n '. Based on extensive experimental results in...

## 24 The controlled stress rheometer

Since the mid 1980s and the advent of reliable 'second generation' controlled-stress rheometers, the controlled-stress technique has become widely established. The facility which most of this type of instrument offers, i.e. of performing three different types of test steady shear, oscillation and creep , makes them particularly cost effective. The instrument referred to here for illustration is a TA Instruments CSL 100 controlled-stress rheometer TA Instruments, UK . The rheometer typically...

## Drag force

The main difficulty in making theoretical estimates of the drag force on a sphere moving in a viscoplastic medium has been the lack of quantitative information about the shape of the sheared cavity. Both Beris et al. 1985 and Blackery and Mitsoulis 1997 have used the finite element method to evaluate the total drag on a sphere moving slowly creeping regime in a Bingham plastic medium and have reported their predictions in terms of the correction factor, X, CDReB 24 which now becomes a function...

## 322 Bingham plastic and yieldpseudoplastic fluids

A fluid with a yield stress will flow only if the applied stress proportional to pressure gradient exceeds the yield stress. There will be a solid plug-like core flowing in the middle of the pipe where Trz is less than the yield stress, as shown schematically in Figure 3.4. Its radius, Rp, will depend upon the magnitude of the yield stress and on the wall shear stress. From equation 3.2 , where Tw is the shear stress at the wall of the pipe. Velocity distribution Shear stress distribution...

## 84 Heat transfer

The rate of heat transfer to process materials may be enhanced by externally applied motion both within the bulk of the material and in the proximity of heat transfer surfaces. In most applications, fluid motion is promoted either by pumping through tubes Chapter 6 or by mechanical agitation in stirred vessels. A simple jacketed vessel is very commonly used in chemical, food, biotechnological and pharmaceutical processing applications to carry out a range of operations. In many cases, heat has...

## 325 Generalised Reynolds number for the flow of timeindependent fluids

It is useful to define an appropriate Reynolds number which will result in a unique friction factor-Reynolds number curve for all time-independent fluids in laminar flow in circular pipes. Metzner and Reed 1955 outlined a generalised approach obviating this difficulty. The starting point is equation 3.21 Equation 3.21 embodies a definite integral, the value of which depends only on the values of the integral function at the limits, and not on the nature of the continuous function that is...

## 83 Gasliquid mixing

Many gas-liquid reactions of industrial significance are carried out in agitated tank reactors, and the design requirements vary from one application to another. For instance, in effluent aeration and in some fermentation reactions, the systems are dilute and reactions are slow so that mass transfer is not likely to be a limiting factor. Energy efficiency is then the most important consideration, and large tanks giving long hold-up times are used. Chlorination and sulphonation reactions, on the...

## 521 Drag on a sphere in a powerlaw fluid

A simple dimensional analysis see example 5.1 of this flow situation shows that the drag coefficient can be expressed in terms of the Reynolds number and the power-law index, i.e. Often for the creeping flow region Re 1 , the numerical results may be expressed as a deviation factor, X n , in the relation between drag coefficient and Reynolds number obtained from Stokes law where Re pV2 ndn m, d being the sphere diameter. The numerical values of X n for both shear-thinning and shear-thickening...

## 235 Moisture loss prevention the vapour hood

When dealing with high concentration samples of low volume, even low moisture loss can have a critical effect on measured rheological properties Barnes et al., 1989 . During prolonged experiments, moisture loss may be minimised by employing a vapour hood incorporating a solvent trap, as shown in Figure 2.6. As noted above, edge effects can be encountered with each of the geometries considered here. They become of particular importance when dealing with samples which form a surface 'skin' in...

## 14 Timedependent fluid behaviour

The flow behaviour of many industrially important materials cannot be described by a simple rheological equation like 1.12 or 1.13 . In practice, apparent viscosities may depend not only on the rate of shear but also on the time for which the fluid has been subjected to shearing. For instance, when materials such as bentonite-water suspensions, red mud suspensions waste stream from aluminium industry , crude oils and certain foodstuffs are sheared at a constant rate following a long period of...

## 234 The parallel plate geometry

In this measuring geometry the sample is contained between an upper rotating or oscillating flat stainless steel plate and a lower stationary plate see Figure 2.5, lower . The upper plate in the example shown is 40mm in diameter. In contrast to the cone-and-plate geometry, the shear strain is proportional to the gap height, h, and may be varied to adjust the sensitivity of shear rate, a feature which readily facilitates testing for wall slip effects Yoshimura and Prud'homme, 1988 . The large...

## 23 Rotational viscometers

Due to their relative importance as tools for the rheological characterisation of non-Newtonian fluid behaviour, we concentrate on this class of rheometers by considering the two main types, namely the controlled shear rate instruments also known as controlled rate devices and controlled stress instruments. Both types are usually supplied with the same range of measuring geometries, principally the concentric cylinder, cone-and-plate and parallel plate systems. The relative merits, potential...

## 12Classification of fluid behaviour

1.2.1 Definition of a Newtonian fluid Consider a thin layer of a fluid contained between two parallel planes a distance dy apart, as shown in Figure 1.1. Now, if under steady state conditions, the fluid is subjected to a shear by the application of a force F as shown, this will be balanced by an equal and opposite internal frictional force in the fluid. For an incompressible Newtonian fluid in laminar flow, the resulting shear stress is equal to the product of the shear rate and the viscosity...