Flow behaviors time-independent

TIME-INDEPENDENT FLOW BEHAVIOR Newtonian Model... 

Figure 1. Time-independent flow behavior of fluid...

Fig. 5 shows the relation of shear stress and shear rate of silver paste with different wt % of thinner. The trend of non-Newtonian behavior is consistent with the results found by Chhabra Richardson, (1999) for the types of time-independent flow behavior. The time-independent non-Newtonian fluid behavior observed is pseudoplasticity or shear-thinning characterized by an apparent viscosity which decreases with increasing shear rate. Evidently, these suspensions exhibit both shear-thinning and shear thickening behavior over different range of shear rate and different wt% of thinner. The viscosity and shear stress relationship with increasing percentage of thinner is plotted in Fig 6. It is clearly observed that both viscosity and shear stress decreases resp>ectively. 

Based on a mechanical model in a time-independent flow, de Gennes derivation tries to extrapolate it to a time-dependent chain behavior. His implicit assumptions have been criticised by Bird et al. [55]. More recent calculations extending the de Gennes dumbbell to the bead-spring situation [56] tend, nevertheless, to confirm the existence of a well-characterized CS transition results with up to 100-bead chains show a critical value of the strain rate at scs = 0.5035/iz which is just 7% higher than the value predicted by de Gennes. 

Figure 1-2 Basic Shear Diagram of Shear Rate versus Shear Stress for Classification of Time-Independent Fiow Behavior of Fiuid Foods Newtonian, Shear-Thinning, and Shear-Thickening. Also, some foods have yield stress that must be exceeded for flow to occur Bingham and Herschel-Bulkley (H-B).

Flow models have been used also to derive expressions for velocity profiles and volumetric flow rates in tube and channel flows, and in the analysis of heat transfer phenomenon. Numerous flow models can be encountered in the rheology literature and some from the food rheology literature are listed in Table 2-1. Also, here those models that have found extensive use in the analysis of the flow behavior of fluid foods are discussed. Models that account for yield stress are known as viscoplastic models (Bird et al., 1982). For convenience, the flow models can be divided in to those for time-independent and for time-dependent flow behavior. 

For the most part, PFDs exhibit shear-thinning (pseudoplastic) rheological behavior that is either time-independent or time-dependent (thixotropic). In addition, many PFDs also exhibit yield stresses. The time-independent flow curves are illustrated in Figure 1. The shear-thinning behavior appears to be the result of breakdown of relatively weak structures and it may have important relationship to mouthfeel of the dispersions. Because the viscosity of non-Newtonian foods is not constant but depends on the shear rate, one must deal with apparent viscosity defined as ... 

This suggests that the dispersed layer may be treated as if it were stagnant. The momentum transfer between the pure bulk oil and water phase will be greatly reduced, implying that they will have independent flow behavior and may be treated separately. Also, the residence time within... 

The Newtonian and non-Newtonian fluids discussed in section 8.2 are time independent that is, the viscosity remains constant as long as the shear rate does not change. However, some fluids exhibit time-dependent flow behavior and their viscosities change with the time of shearing. Two most important time-dependent fluids are thixotropic and rheopectic. At a fixed shear rate, the viscosity of a thixotropic fluid decreases with time, while the viscosity of a rheopectic fluid increases with time (Figure 8.7). 

Since each input of mass to a perfect plug flow unit is independent of what has been input previously, its condition as it moves along the reactor will be determined solely by its initial condition and its residence time, independently of what comes before or after. Practically, of course, some interaction will occur at the boundary between successive inputs of different compositions or temperatures. This is governed by diffusional behaviors which are beyond the scope of the present work. 

Chapter HI relates to measurement of flow properties of foods that are primarily fluid in nature, unithi.i surveys the nature of viscosity and its relationship to foods. An overview of the various flow behaviors found in different fluid foods is presented. The concept of non-Newtonian foods is developed, along with methods for measurement of the complete flow curve. The quantitative or fundamental measurement of apparent shear viscosity of fluid foods with rotational viscometers or rheometers is described, unithi.2 describes two protocols for the measurement of non-Newtonian fluids. The first is for time-independent fluids, and the second is for time-dependent fluids. Both protocols use rotational rheometers, unit hi.3 describes a protocol for simple Newtonian fluids, which include aqueous solutions or oils. As rotational rheometers are new and expensive, many evaluations of fluid foods have been made with empirical methods. Such methods yield data that are not fundamental but are useful in comparing variations in consistency or texture of a food product, unit hi.4 describes a popular empirical method, the Bostwick Consistometer, which has been used to measure the consistency of tomato paste. It is a well-known method in the food industry and has also been used to evaluate other fruit pastes and juices as well. 

If a chemical reaction is operated in a flow reactor under fixed external conditions (temperature, partial pressures, flow rate etc.), usually also a steady-state (i.e., time-independent) rate of reaction will result. Quite frequently, however, a different response may result The rate varies more or less periodically with time. Oscillatory kinetics have been reported for quite different types of reactions, such as with the famous Belousov-Zha-botinsky reaction in homogeneous solutions (/) or with a series of electrochemical reactions (2). In heterogeneous catalysis, phenomena of this type were observed for the first time about 20 years ago by Wicke and coworkers (3, 4) with the oxidation of carbon monoxide at supported platinum catalysts, and have since then been investigated quite extensively with various reactions and catalysts (5-7). Parallel to these experimental studies, a number of mathematical models were also developed these were intended to describe the kinetics of the underlying elementary processes and their solutions revealed indeed quite often oscillatory behavior. In view of the fact that these models usually consist of a set of coupled nonlinear differential equations, this result is, however, by no means surprising, as will become evident later, and in particular it cannot be considered as a proof for the assumed underlying reaction mechanism. 

To fully understand the behavior of biological materials we need to address the issue of viscoelasticity. When a weight is placed on viscoelastic material, there is an instantaneous elastic response and a time-dependent viscous response (see Figure 7.1). For polymers the elastic response reflects the change in macromoleular conformation, which is usually time independent if no bonds are broken. The viscous response is the flow of macromolecules by each other similar to what happens during the flow of fluids in a tube. Fluid flow is a time-dependent process. Polymers exhibit viscoelastic behavior because they have both a time-independent response and a time-dependent response. 

The analysis above refers to time-independent velocity fields and equal diffusion coefficients for all species, but Straube et al. (2004) have shown that similar behavior applies to mixing in time-dependent chaotic flows. It was shown numerically that A is a linear function of the reaction rate and the transition in the stability of the spatially uniform state to non-homogeneous perturbations takes place when the positive Lyapunov exponent of the local dynamics is equal to the exponent describing the decay rate of the dominant eigenmode. 

Proteins in solution are also sensitive to orthokinetic aggregation, i.e., shear-induced aggregation. We have observed this phenemenon at very low shear rates in the case of BSA, ovalbumin, and BLG [28,29] in Fig. 1, one sees that the viscosity of the protein solution increases with time till it reaches a plateau the phenomenon becomes less and less pronounced as the shear rate increases, so much so that the solution displays time-independent and Newtonian flow behavior in the usual shear rate range. 

This section contains several models whose spatiotemporal behavior we analyze later. Nontrivial dynamical behavior requires nonequilibrium conditions. Such conditions can only be sustained in open systems. Experimental studies of nonequilibrium chemical reactions typically use so-called continuous-flow stirred tank reactors (CSTRs). As the name implies, a CSTR consists of a vessel into which fresh reactants are pumped at a constant rate and material is removed at the same rate to maintain a constant volume. The reactor is stirred to achieve a spatially homogeneous system. Most chemical models account for the flow in a simplified way, using the so-called pool chemical assumption. This idealization assumes that the concentrations of the reactants do not change. Strict time independence of the reactant concentrations cannot be achieved in practice, but the pool chemical assumption is a convenient modeling tool. It captures the essential fact that the system is open and maintained at a fixed distance from equilibrium. We will discuss one model that uses CSTR equations. All other models rely on the pool chemical assumption. We will denote pool chemicals using capital letters from the start of the alphabet. A, B, etc. Species whose concentration is allowed to vary are denoted by capital letters... 

By far, the most imderstood behaviors are those of Newtonian fluids and time-independent non-Newtonian suspensions. A series of flow equations and charts have been developed in order to predict their flow characteristics. For other types of non-Newtonians the flow equations, if they can be developed at all, are much more complicated. However, under certain assumptions, for example, steady-state flow without acceleration (flow in straight pipes without nozzles, bends, orifices, etc.), these fluids can often be treated as time independent too. 

In the quantitative analysis of most extrusion problems, the polymer melt generally is considered to be a viscous, time-independent fluid. This assumption is, of course, a simplification, but it usually allows one to find a relatively straightforward solution to the problem. This assumption will be used throughout the rest of this book, unless indicated otherwise. In the analysis of any flow problem, however, it should be remembered that elastic effects may play a role. Also, some flow phenomena, such as extrudate swell, clearly cannot be analyzed unless the elastic behavior of... 

This was conceived as a means of predicting the flow behavior of time-independent fluids, particularly shear-thinning and shearthickening fluids. It is really nothing more than an equation to fit the plot of shear stress versus shear rate for such a fluid when both are plotted on logarithmic scaled paper. 

However, in the real world, circuit elements exhibit much more complex behavior the simple concept of resistance cannot be used and in its place, impedance, a more general circuit parameter, is used. Like resistance, imp ance is the ability of the system to impede the flow of electrical current through it. Though it is similar to resistance, impedance is not time independent it is a time- or frequency-dependent parameter. Similar to resistance, impedance is defined as the ratio of the time-dependent current to the time-dependent potential. 

---