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...
GasNewtonian 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...
iii 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...
Solution Gel
There are six variables and three fundamental dimensions M, L, T , and therefore there will be three dimensionless groups. Thus, one can write Writing dimensions of each of these variables Fd MLT-2 m ML-1Tn-2 Choosing p, d, V as the recurring set, the fundamental dimensions M, L and T can be expressed as and the three -groups can be formed as CD f Re, n which is the same relationship as given by equation 5.3 .
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...
18 References
Astarita, G., J. Rheol. 34 1990 275 ibid 36 1992 1317. Barnes, H.A., J. Rheol. 33 1989 329. Barnes, H.A., J. Non-Newt. FluidMech. 70 1997 1. Barnes, H.A., J. Non-Newt. Fluid Mech., 81 1999 133. Barnes, H.A., Edwards, M.F. and Woodcock, L.V., Chem. Eng. Sci., 42 1987 591. Barnes, H.A., Hutton, J.F. and Walters, K., An Introduction to Rheology. Elsevier, Amsterdam 1989 . Barnes, H.A. and Walters, K., Rheol. Acta 24 1985 323. Bird, R.B., Annu. Rev. Fluid Mech. 8 1976 13. Bird, R.B., Armstrong,...
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...