System curve operating line

There are two components to the pressure head that has to be supplied by the pump in a piping system  

The static pressure, to overcome the differences in head (height) and pressure. 

The dynamic loss due to friction in the pipe, the miscellaneous losses, and the pressure loss through equipment. 

The static pressure difference will be independent of the fluid flow-rate. The dynamic loss will increase as the flow-rate is increased. It will be roughly proportional to the flow-rate squared, see equation 5.3. The system curve, or operating line, is a plot of the total pressure head versus the liquid flow-rate. The operating point of a centrifugal pump can be found by plotting the system curve on the pump s characteristic curve, see Example 5.3. 

When selecting a centrifugal pump for a given duty, it is important to match the pump characteristic with system curve. The operating point should be as close as is practical to the point of maximum pump efficiency, allowing for the range of flow-rate over which the pump may be required to operate. 

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A large majority of the systems have operating lines and equilibrium curves which can be assumed as straight over the range covered by the design problem. For the conditions of a straight line equilibrium curve, y = mx, Colburn [10, 11] has integrated the relation above to obtain ... 

With this coordinate system the operating line is straight. The equilihiinm curve may be based on actual data for the specific system at column operating conditions or nuy be approximated on the basis of related data. Rousseau and Staton outline steps fix estimating equilibrium curves based on the Hairy s law constant fix unreacted componoit A in the liquid and the equilibrium constant for the chonical reaction of A with the reactive con onoit. 

Fig. 4. Selection of fan size where the soHd line represents a typical setting and the dashed lines the operating extremes, (a) Desirable sizing. The system resistance curve intersects the fan curve near its maximum efficiency. Changes in system resistance from a flow-control element also intersect the fan curve at desirable points for good flow control. The dashed curves also intersect system resistance curves at desirable locations, (b) A fan essentially too large for the system. The intersection of the system curve near the peak of the fan curve results in poor system flow control and perhaps surging.

Figure 14-6 illustrates the graphical method for a three-theoretical-plate system. Note that in gas absorption the operating line is above the equihbrium curve, whereas in distillation this does not happen. In gas stripping, the operating line will be below the equihbrium curve. 

The reason for this simple relationship is that the concept of minimum reflux implies an infinite number of stages and thus no change in composition from stage to stage for an infinite number of stages each way from the pinch point (the point where the McCabe-Thiele operating lines intersect at the vapor curve for a well-behaved system, this is the feed zone). The liquid refluxed to the feed tray from the tray above is thus the same composition as the flash liquid. 

Figure 12-84A illustrates a system with all line friction. Here, the operations would follow the system curve and operate at the intersection with the speed curve. For example, if the speed is cut 10%, the flow decreases 8%. Figure 12-84B shows a system with essentially constant back pressure. Following the operating system curve shows that a small speed cut back of say 10% results in a flow drop of 40%. 

The point where the flow rate of 275 gpm intersects the system curve in Fig. 8-2 (at 219 ft of head) falls between impeller diameters of l and 7 in., as indicated by the O on the line. Thus, the P in. diameter would be too small, so we would need the 7 in. diameter impeller. However, if the pump with this impeller is installed in the system, the operating point would move to the point indicated by the X in Fig. 8-2. This corresponds to a head of almost 250 ft and a flow rate of about 290 gpm (i.e., the excess head provided by the larger impeller results in a higher flow rate than desired, all other things being equal). 

The operating point is located where the centrifugal compressor curves cross the system curve of the process. The system curve can be a constant pressure one (horizontal line), a mostly friction one, or any other. 

Figure 2.120 shows a three-dimensional plot—(a) pressure, (b) flow, and (c) speed—where the system curves form one surface (surface A) and the pump curves form another surface (surface B). The intersection of surfaces A and B is the operating line of the variable-speed pump. 

The variable speed pump operates on the line where the surface formed by the system curves intersects with the surface formed by the pump curves. 

Since the column consists solely of a rectifying section, there is a limit to how many trays can be profitably installed. The system will pinch regardless of stages once the low-boiler concentration in the reboiler approaches the intersection of the operating line with the equilibrium curve. 

As shown in Figure 27, plots of the equilibrium data, the y — x line, and the operating line allow a procedure to calculate the unknown variables for a system, typically, the outlet liquid and vapor compositions. The y = x line simplifies the graphical solution method and intersects the operating line at the feed composition, z. Thus, at this point, y = x = z. The unknown compositions, in the vapor and liquid product streams, are determined by the intersection of the operating line and the equilibrium curve. 

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