Flow curves

The predicted cumulative cash-flow curve for a project throughout its life forms the basis for more detailed evaluation. Many quantitative measures or indices have been proposed. In each case, important features of the cumulative cash-flow curve are identified and transformed into a single numerical measure as an index. 

Pseudoplastic fluids have no yield stress threshold and in these fluids the ratio of shear stress to the rate of shear generally falls continuously and rapidly with increase in the shear rate. Very low and very high shear regions are the exceptions, where the flow curve is almost horizontal (Figure 1.1). 

Fig. 2. Flow curves (shear stress vs shear rate) for different types of flow behavior.

Viscosity is equal to the slope of the flow curve, Tf = dr/dj. The quantity r/y is the viscosity Tj for a Newtonian Hquid and the apparent viscosity Tj for a non-Newtonian Hquid. The kinematic viscosity is the viscosity coefficient divided by the density, ly = tj/p. The fluidity is the reciprocal of the viscosity, (j) = 1/rj. The common units for viscosity, dyne seconds per square centimeter ((dyn-s)/cm ) or grams per centimeter second ((g/(cm-s)), called poise, which is usually expressed as centipoise (cP), have been replaced by the SI units of pascal seconds, ie, Pa-s and mPa-s, where 1 mPa-s = 1 cP. In the same manner the shear stress units of dynes per square centimeter, dyn/cmhave been replaced by Pascals, where 10 dyn/cm = 1 Pa, and newtons per square meter, where 1 N/m = 1 Pa. Shear rate is AH/AX, or length /time/length, so that values are given as per second (s ) in both systems. The SI units for kinematic viscosity are square centimeters per second, cm /s, ie, Stokes (St), and square millimeters per second, mm /s, ie, centistokes (cSt). Information is available for the official Society of Rheology nomenclature and units for a wide range of rheological parameters (11). 

Thixotropy and Other Time Effects. In addition to the nonideal behavior described, many fluids exhibit time-dependent effects. Some fluids increase in viscosity (rheopexy) or decrease in viscosity (thixotropy) with time when sheared at a constant shear rate. These effects can occur in fluids with or without yield values. Rheopexy is a rare phenomenon, but thixotropic fluids are common. Examples of thixotropic materials are starch pastes, gelatin, mayoimaise, drilling muds, and latex paints. The thixotropic effect is shown in Figure 5, where the curves are for a specimen exposed first to increasing and then to decreasing shear rates. Because of the decrease in viscosity with time as weU as shear rate, the up-and-down flow curves do not superimpose. Instead, they form a hysteresis loop, often called a thixotropic loop. Because flow curves for thixotropic or rheopectic Hquids depend on the shear history of the sample, different curves for the same material can be obtained, depending on the experimental procedure. 

Fig. 1. ADA flow curves of dental waxes. A, impression waxes B, sheet wax C, indirect-inlay wax D, direct-inlay wax E, base-plate wax. The dashed...

Cash-Flow Curves Figure 9-9 shows the cash-flow stages in a project together with their discounted-cash-flow values for the data given in Table 9-4. In addition to cash-flow and discounted-cash-flow curves, it is also instructive to plot cumulative-cash-flow and cumula-tive-discounted-cash-flowcurves. These are shown in Fig. 9-10 for the data in Table 9-4. 

The cost of capital may also be considered as the interest rate at which money can be invested instead of putting it at risk in a manufacturing process. Let us consider the process data listed in Table 9-4 and plotted in Fig. 9-10. If the cost oi capital is 10 percent, then the appropriate discounted-cash-flow curve in Fig. 9-10 is abcdef. Up to point e, or 8.49 years, the capital is at risk. Point e is the discounted breakeven point (DEEP). At this point, the manufacturing process... 

It is not normally possible to make a comprehensive assessment of profitabihty with a single number. The shape of the cumulative-cashflow and cumulative-discounted-cash-flow curves both before and after the breakeven point is an impoiTant factor. 

D. H. Allen [Chem. Eng., 74, 75-78 (July 3, 1967)] accounted for the shape of the cumulative-undiscounted-cash-flow curve up to the... 

Allen accounted for the shape of the cumulative-cash-flow curve... 

C. G. Sinclair [Chem. Process. E/ig., 47, 147 (1966)] has considered similar parameters to the (EMIP) and (IRP) based on a cumulative-discounted-cash-flow curve. 

No single value for a profitability estimate should be accepted without further consideration. An inteUigent consideration of the cumula-tive-cash-flow and cumulative-discounted-cash-flow curves such as those shown in Fig. 9-10, together with experience and good judgment, is the best way of assessing the financial merit of aprojec t. 

Certain pump designs use an internal bypass orifice port to alter head-flow curve. High liquid velocities often erode the orifice, causing the pump to go farther out on the pump curve. The system head curve increase corrects the flow back up the curve. 

Caution should be taken to prevent excessive acceleration. One solution is to provide for opening of the compressor discharge vent valve to increase the compressor flow and increase blower horsepower. This is effective with a centrifugal compressor, however, not with an axial compressor unless the compressor is provided with adjustable stator vanes that are reliable. This can be seen from the performance curve. The head versus flow curve for a given vane setting is extremely steep and opening the vent valve is ineffective. However, if the vanes operate fully open and the vent valve opens, the combined effect is satisfactory. The vent valve must not be oversized. 

At the same time it is not surprising that polymer melts are non-Newtonian and do not obey such simple rules. Fortunately, if we make certain assumptions, it is possible to analyse flow in certain viscometer geometries to provide measurements of both shear stress (t) and shear rate (7) so that curves relating the two (flow curves) may be drawn. 

In fact with a Newtonian liquid y = 4Q/ nR. This latter expression, viz. 4Q/ uR, is obviously much easier to calculate than the true wall shear rate and, since they are uniquely related and the simple expression is just as useful, in design practice it is very common when plotting flow curves to plot against... 

For any particular material the ratio (m/(m + 2)) may be determined from the flow curves and it will be temperature and (to some extent) pressure dependent. 

A plot of apparent viscosity against shear rate produces a unique flow curve for the melt is shown in Fig. 5.3. Occasionally this information may be based on the true shear rate. As shown in Section 5.4(a) this is given by... 

Note that rotational viscometers give true shear rates and if this is to be used with Newtonian based flow curves then, from above, a correction factor of (4n/3 + 1) needs to be applied to the true shear rate. 

Other factors such as the use of additives also have an effect on the shape of the flow curves. Flame retardants, if used, tend to decrease viscosity whereas pigments tend to increase viscosity. Fig. 5.17 shows flow curves for a range of plastics. 

Example S3 The output of polythene from an extruder is 30 x 10 m /s. If the breaker plate in this extruder has 80 holes, each being 4 mm diameter and 12 mm long, estimate the pressure drop across the plate assuming the material temperature is 170°C at this point. The flow curves in Fig. 5.3 should be used. 

From the flow curves for polypropylene at 210 C, at this shear rate, T) = 130 Ns/m, so the shear stress, t, is given by... 

Example 5.9 Calculate the time taken to inflate the bottle in the previous example if the inflation pressure is 50 kN/m. The flow curves given in Fig. 5.3 may be used. 

The viscosity flow curves for these materials are shown in Fig. 5.17. To obtain similar data at other temperatures then a shift factor of the type given in equation (5.27) would have to be used. The temperature effect for polypropylene is shown in Fig. 5.2. 

Polyethylene is extruded through a cylindrical die of radius 3 mm and length 37.5 mm at a rate of 2.12 x 10 m /s. Using the flow curves supplied, calculate the natural time of the process and corrunent on the meaning of the value obtain. ... 

The exit region of a die used to extrude a plastic section is 10 mm long and has the cross-sectional dimensions shown below. If the channel is being extruded at the rate of 3 m/min calculate the power absorbed in the die exit and the melt temperature rise in the die. Flow curves for the polymer melt are given in Fig. 5.3. The product pCp for the melt is 3.3 x 10. ... 

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