Strain distribution functions

If we accept the premise that the total strain is a key variable in the quality of laminar mixing, we are immediately faced with the problem that in most industrial mixers, and in processing equipment in general, different fluid particles experience different strains. This is true for both batch and continuous mixers. In the former, the different strain histories are due to the different paths the fluid particles follow in the mixer, whereas in a continuous mixer, superimposed on the different paths there is also a different residence time for every fluid particle in the mixer. To quantitatively describe the various strain histories, strain distribution functions (SDF) were defined (56), which are similar in concept to the residence time distribution functions discussed earlier. 

Example 7.9 Strain Distribution Function in Drag (Couette) Flow between Concentric... 

Fig. E7.9b Strain distribution function in G(y) of a power law model fluid in Couette flow between concentric cylinders for the case (c) of Fig. E7.9a...

Example 7.10 The Strain Distribution Function in Parallel Plate Drag Flow Two parallel plates in relative motion with each other can be viewed as an idealized continuous mixer. 

TABLE 7.2 The Strain Distributions Function for Some Simple Flow Geometries with Newtonian Fluids... 

Strain Distribution Function in Poiseuille Flow (a) Derive the SDF F(y) for fully developed isothermal laminar flow of a Newtonian fluid in a tube, (b) Calculate the mean strain, (c) If the length of the tube is 1 m and its radius 0.01 m, what fraction of the exiting stream experiences a total strain of less than 100 ... 

Strain Distribution Function in Parallel Plate Flow (a) Derive the SDF F(y) for the parallel-plate flow with a superimposed pressure gradient in the range — 1/3 < qp/qd < 1/3. The velocity profile is given by... 

The Strain Distribution Function of a Power Law Fluid, in Pressure Flow between Parallel Plates Consider two infinitely wide parallel plates of length L gap II. Polymer melt is continuously pumped in the x direction. Assuming isothermal steady, fully developed flow, (a) show that F(c) is given by... 

We now derive the RTD function/(f) dt and the strain distribution functions / ) dy for isothermal Newtonian fluids in shallow screw channels. 

The Strain Distribution Functions In Chapter 7 we established the relation between interfacial area stretching and the total strain imposed on the fluid. The strain is the product of rate of strain and time, and therefore strain is a function of location in the channel. The strain distribution function (SDF) was defined in Chapter 7 as the fraction of exiting flow rate that experienced a given strain in the extruder. Following similar lines to the derivation of the RTD functions we now derive the SDF. 

Fig. 9.14 Values of F(y) for a 6-in-diameter, 20 1 IVD extruder at constant flow rate (500 lb/h) with screw speed as a parameter. Simulation was made for a square pitched screw with a constant channel depth of 0.6 in. [Reprinted by permission from G. Lidor and Z. Tadmor, Theoretical Analysis of Residence Time Distribution Functions and Strain Distribution Functions in Plasticating Extruders, Polym. Eng. Sci., 16, 450-462 (1976).]...

In continuous mixers, different fluid elements will invariably experience different amounts of strain, as discussed earlier for the screw extruder. Tadmor and Lidor [206] proposed the use of strain distribution functions (SDF), similar to residence time distribution functions (RTD). The SDF for a continuous mixer f(Y)dy is defined as the fraction of exiting flow rate that experienced a strain between y and dy. It is also the probability of an entering fluid element to exit with that strain. The cumulative SDF, F(y), is defined by the following expression ... 

FIGURE 6.17 Comparison of the strain distribution function for circular pipe (CPPF) and parallel plate (PPF) Poiseuille flow, and parallel plate Couette flow (PCF) of Newtonian fluids. 

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