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PIPE EXPANSIONS AND CONTRACTIONS How do we evaluate the head loss in an expansion or a contraction along the length of a pipe? Like other changes in the geometry of a pipe system, we consider these in a similar manner to the other minor losses – a function of the velocity through the system. V2 hL = K L 2g However, which velocity do we use? The purpose of a contraction or expansion is to change the velocity of within the pipe – do we use the upstream or downstream value of velocity (V)? The answer: it depends. But generally the loss is a function of both the upstream and downstream velocities as well as the geometry of the expansion or contraction. Often a velocity is chosen to represent the velocity head (usually the higher velocity) and the KL parameter is a function of the expansion/contraction angle and the area ratio between upstream and downstream pipes (which is the reciprocal of the velocity ratio). 1 Sudden Contractions Sudden contractions are when the area of the pipe reduces suddenly along the length of the pipe (at a 90 degree angle). The downstream velocity will be higher than the upstream velocity. For sudden contractions we use the following equation. V2 2 hL = K L 2g The value for KL is determined experimentally and is a function of A2/A1. The following figure is from your textbook (Section 8.4). Note that when A1 is much larger than A2 we have a coefficient of 0.5, which is the same as the minor entrance loss from a reservoir with 90 degree edges. 2 Sudden Expansions Sudden expansions are when the area of the pipe increases suddenly along the length of the pipe (at a 90 degree angle). The downstream velocity will be lower than the upstream velocity. For sudden expansions we use the following equation. V12 hL = K L 2g Sudden expansions are analogous to the exit loss coefficients. With exit loss coefficients we know that all the kinetic energy (velocity head) is dissipated because there is no velocity in the reservoir. For sudden expansions a portion of the kinetic energy is dissipated – the difference being the difference between the two velocities. Consequently the value for KL can be determined analytically: (V1 − V2 ) 2 hL = 2g And from continuity: V2 A1 = V1 A2 so: 2 A K L = 1 − 1 A2 3 This relationship can also be displayed graphically. Gradual Contraction (Nozzle) For gradual contractions the angle of contraction is something less than 90 degrees as shown in the figure below. Again we use the high-speed velocity to evaluate our head loss. V2 2 hL = K L 2g 4 The losses in a contraction are due to flow separation similar to what we see in a Venturi meter or entrance losses. Values for KL at various angles and Area ratios are shown below. Gradual Expansion (Diffuser) A diffuser acts to slow down the flow velocity by gradually increasing the area of the pipe. Again, we use the high velocity as the reference velocity for our head loss equation. V12 hL = K L 2g The actual value for KL depends heavily on the area ratio and the angle of the diffuser. The following figure (from your text) shows the relationship in terms of the value of KL as the ratio to the equivalent value of KL for a sudden expansion. It can be seen that 5 for angles greater than about 30 degrees an expansion pipe is less efficient than a sudden expansion. 6 PUMPS REVISITED Pumps were already discussed in the course notes (Section 9.11 “Simple Pump Systems”). However, some further explanation is required on pumps operating with multiple stages and operating in series or parallel. Pumps in Series For pumps in series, a new performance curve must be generated. The two pump curves are added together so that for a given flow rate the head added is the sum of the two original pump curves. This new pump curve can be used with the system curve to determine the operation point. The following figure from your text (Section 12.4) illustrates the relationship. Pumps in Parallel When pumps are positioned in parallel one follows a similar approach as for pumps in series except one sums up the flow rates for the pumps in parallel for a particular head to get the new performance curve. The figure below illustrates this (also from your text.) 7 Single and Multistage Pumps Single stage pumps have a single impeller in the pump housing, a multistage pump has 2 or more impellers. The following figure shows a design of a multistage pump with 6 impellers. When considering how to design the pump one must know if the pump h-Q diagram is representative of a single stage of the pump or the pump as a whole. If the diagram represents only one stage, it means that for a given flow rate one could expect to multiply the amount of head generated for a particular flow rate by the number of stages. That is, consider it a number of pumps in series. 8

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