## Compression Cycle

Figure 5-17 is a section of a typical multistage compressor, which should aid the reader in following the flow path through the machine.

SHAFT t

Figure 5-17. Flow path through typical stages on a multistage unit. (Courtesy of Elliott Company

SHAFT t

Figure 5-17. Flow path through typical stages on a multistage unit. (Courtesy of Elliott Company

Gas enters the impeller from one of several sources. In the case of the first impeller of a multistage, the flow has moved through an inlet nozzle and is collected in a plenum from which it is then directed into the first impeller. Another possible path occurs when the flow has passed through one or more stages and approaches the impeller through a channel referred to as a return passage. In the return passage, the flow stream passes through a set of vanes. The vanes are called straightener vanes, if the flow is directed axially at the impeller entrance (eye), or guide vanes, if the flow is modified by the addition of prerotation. The final possible path occurs when the flow comes into the compressor from a sidestream nozzle. This stream is directed into the flow stream to mix and be directed into the impeller eye using one of two alternative methods as shown on Figure 5-18. One method is by way of a blank section between the stages where the stream mixing point is immediately ahead of the impeller inlet. This method is used if the sidestream flow is large in comparison to the through flow. The alternative is used when the flow is small compared to the through flow, and consists of injecting the flow into the return passage from the previous stage. The latter has better mixing, and takes less axial space, but has a higher pressure drop. For the former, the opposite is true. It has a lower pressure drop, but exhibits somewhat poorer mixing and uses more axial space, normally at least a

full-stage pitch in length. A stage pitch is defined as the axial distance measured from the entrance of one impeller to the same location on the following impeller. Stage pitch may be a constant, as on low-volume ratio staging, or variable, as may be found in higher-volume ratio stages. The variable stage pitch is commonly used on higher flow coefficient stages using the 3D impeller designs. The importance of physical length will become apparent as the entire compressor is explored, but at this point, it will suffice to say that there never seems to be enough.

Generally, there are no vanes in the inlet of an axial entry compressor (see Figure 5-19). Normally there is no more than the plenum divider vane in the inlet section of the typical multistage compressor, although there are designs that use vanes in this area. These are externally movable and are used to provide flow control for constant speed machines. The use of these vanes will be explored further in the section on capacity control.

After the flow has been introduced into the compressor and has been acted on by one or more stages, it must be extracted. Because there is a relatively large amount of velocity head available in the stream, care must be used when designing the discharge section to keep the head loss low and maintain overall efficiency. The flow from the last stage is gath-

ered in some form of collector, normally a scroll, in an effort to convert as much of the remaining velocity head as possible into pressure. With intermediate extraction, or for some of the in-out designs, a compromise must be made, reducing large passages to preserve axial length.

Having gotten the flow in and out of the machine, a closer examination of just how the compression takes place is needed. An important concept to maintain throughout the following discussion is that all work done to the gas must be done by the active element, the impeller. The stationary element is passive, that is, it cannot contribute any additional energy to the stage. It can only convert the energy and unfortunately contribute to the losses. Figure 5-20 is a schematic diagram of an impeller and the basic inlet and outlet flow vector triangles.

The impeller will be covered in detail in the following sections; therefore. a brief review of the various impeller components is in order. The

impeller consists of a set of vanes radially oriented on a hub. The vanes are enclosed either by a rotating or stationary front and rear shroud. If both front and rear shroud are stationary, the impeller is referred to as an open impeller. If the rear shroud is attached to the vanes and rotates as a part of the impeller assembly, it is referred to as semi-open. If the front shroud is also attached to the vanes and rotates with the assembly, it is referred to as a closed impeller. The vanes may be forward curved, radial, or backward curved, as shown diagramatically in Figure 5-21. Forward curved vanes are normally only used in fans or blowers, and rarely, if ever, used in centrifugal compressors.

Figure 5-21 includes an outlet velocity vector triangle for the various vane shapes. Figure 5-20 shows a backward curved impeller that includes the inlet and outlet velocity vector triangle. Because most of the compressors used in process applications are either backward curved or radial, only these two types will be covered in detail.

Flow Increasing

Figure 5-21. Diagram depicting backwards, radial and forward curved blades. Vector Triangles

Figure 5-21. Diagram depicting backwards, radial and forward curved blades. Vector Triangles

Gas enters the impeller vanes at the diameter dj. The absolute gas velocity approaching the vanes is Vj. As shown in Figure 5-20, the gas approaches the vane in a radial direction after entering the impeller in an axial direction and makes the turn to a radial direction inside the impeller. The vane leading edge velocity is represented by the velocity vector u^ The net velocity is the relative velocity Vrl. It should be noted for this basic example that the relative velocity vector aligns itself with the vane angle pb resulting in zero incidence. In this idealized case, the meridional flow vector Vml is aligned with and equal to the absolute velocity. After passing between the vanes, the gas exits the impeller at the diameter d2. The velocity of the gas just prior to leaving the impeller is the relative velocity Vr2 and leaves at the vane angle (32 in the idealized example. By the addition of the impeller tip velocity vector u2, the absolute leaving velocity V2 is generated. The angle of the absolute flow vector is a2. This is the velocity and direction which the gas assumes as

it leaves the impeller and enters the diffuser. The meridional velocity Vm2 is shown by the radial vector passing through the apex of the outlet velocity triangle. If the vane was radial, rather than backward leaning, |32 = 90°, the relative velocity and the meridional velocity would be equal and aligned.

Slip

In real world application, the gas leaving the impeller will not follow the vane exit angle. The deviation from the geometric angle is referred to as slip. The leaving angle will be referred to as the gas angle fi2. Figure 5-22 shows the discharge velocity vector triangle, including the effect of slip. The terms on the ideal triangle are the same as those used in Figure 5-20. Superimposed over the ideal triangle is the velocity triangle, including the effect of slip. Note that the terms are indicated with the prime (') symbol. While there are numerous papers written on the subject of slip, none seem to present a complete answer. One of the better papers, which summarizes the field and brings the subject into focus, is the one by Wiesner [7]. In this book, for the purpose of understanding the workings of the centrifugal compressor, the Stodola slip equation will be used. It is probably one of the oldest and has been used in practical design prior to the advent of some of the more sophisticated methods available now. Returning to the triangle under discussion, the gas angle, (32, is always less than the geometric angle, p2. In Figure 5-22, projections are

vm, | ||

* Vui * | ||

V | ||

U2 | ||

Figure 5-22. Discharge velocity vector triangle showing the effect of slip

Figure 5-22. Discharge velocity vector triangle showing the effect of slip made onto the tip velocity vector from the absolute gas vectors, V2 and V2. These are labeled as Vu2 and V^2, respectively, and have the designation of tangential component of the absolute velocity. From these vectors, some simple relationships can be presented that will give a reasonable explanation of how the centrifugal compressor geometry relates to its ability to compress gas. The ideal work input coefficient, Ç„ is given by the following expression:

ill)

Ui where

Vu2 = tangential component of the absolute velocity u2 = impeller tip velocity

The ideal head input to the stage is given by

The Stodola slip factor is defined as

C1. 7rsin(32

where

(32 = geometric vane exit angle nv = number of vanes in the impeller

The slip factor SF follows.

Reference is made to Figure 5-22, where

Substituting into Equation 5.4 yields the following slip factor equation:

where

H = stage efficiency, then

For adiabatic head, the head coefficient is defined as |ia and Equation 2.70 is recalled. The geometric and the thermodynamic head relationships for a stage may be equated.

Similarly, for polytropic head, the head coefficient is defined as jip and Equation 2.73 is recalled, the geometric and thermodynamic head relationships, on a per-stage basis, may be equated as above.

In the previous paragraphs, the term specific speed has been used. This is a generalized turbomachinery term used quite successfully with pumps and to some extent with turbines. It can be used with turbocompressors to help delimit the various kinds of machines. It is also used as a general term to describe the need for a correction on multistage machines when the wheel geometry at the current speed will no longer support a reasonable efficiency. For compressors, specific speed is paired with specific diameter to include the geometric factors. In centrifugal compressors, attempts have been made to correlate efficiency directly to these parameters. Most designers feel the relationships, while satisfactory to set bounds, are not adequate for describing impeller efficiency with good resolution. Definitions for specific speed, Ns, and specific diameter, Ds, are

ciency is the numeric average of the impeller efficiency and the diffuser efficiency. Figure 5-25 shows that as the vane angle decreases, the reaction increases. If the efficiency is evaluated for the lower angle, the net stage efficiency is now the weighted average of the two component individual efficiencies, with the higher impeller efficiency contributing a greater influence. A numeric example may help to illustrate the idea.

Example 5-1

Assume

Impeller efficiency = .90 Diffuser efficiency = .60

Calculate an ideal stage efficiency for a radial and a 45° backward leaning impeller.

For the radial impeller, using Equation 5.15,

ANGLE. 32

Figure 5-25. Theoretical reaction without slip,

For a 45° backward leaning impeller,

The diffuser then converts 1 - Rj or .25

The example indicates that an improvement of seven percentage points was achieved by backward leaning the vanes 45°. The obvious question arises. Why not make all impellers high reaction? Maybe this can be put into a good/bad analogy. The good is better efficiency. The bad is a lower head produced by the stage. To see why the head is less, review Figure 5-24. It can be seen that as the outlet angle, |32, is decreased, the tangential component of the absolute velocity, Vu2, is decreased. If Equation 5.1 is recalled, it should be noted that a decrease in Vu2 will decrease the value of the head input coefficient, By carrying a lower value of ^ into Equation 5.9, the head coefficient, (I, is decreased. In Equation 5.10, it is obvious that for a lower (i the output head is decreased. There is some relief in that in Equation 5.9, the stage efficiency r| increases to offset the lowered

However, in real life, this is not enough to make up the difference and the output head of a higher reaction stage is indeed lower. There are several effects that influence a commercial design and, again, the designer is faced with trade-offs. Equation 5.10 indicates that increase in the tip velocity u2 would offset the loss in (I. Impeller stresses and rotor dynamics must also be considered and may act to limit the amount of correction that can be made. Another possibility is using additional stages. A well-proportioned stage is assumed, which brings to light the fact that the high reaction stage tends to use more axial length. This tends to counter the addition of extra stages, especially where the length of the rotor is beginning to cause critical speed problems. Despite the conflicts, changing reaction can sometimes aid the designer in achieving a higher efficiency. Another benefit is a steeper head-capacity curve. Also in some cases, the higher reaction stage seems to perform better where fouling is evident.

### Sizing

Many of the steps used in sizing estimates are also useful for checking bids or evaluating existing equipment. In the latter two endeavors, there is one advantage: someone else has established the initial evaluation criteria. When working from a material balance flow sheet as a starting point, it is sometimes difficult to envision what the compressor should look like. Except for the addition of a few rules of thumb, most of the tools needed have already been established. The method outlined is based on the more conventional multistage compressors used in process service. Earlier, integrally geared, as well as direct expander driven compressors, were briefly described. These compressors may also be sized by the method outlined, but because they are tailored for higher head service, modifications to the method regarding the head per stage and ihe head coefficient are necessary.

To start, convert the flow to values estimated to be the compressor inlet conditions. Initially, the polytropic head equation (Equation 2.73) will be used with n as the polytropic compression exponent. If prior knowledge of the gas indicates a substantial nonlinear tendency, the real gas compression exponent (Equation 2.76) should be substituted. As discussed in Chapter 2, an approximation may be made by using the linear average of the inlet and outlet k values as the exponent or for the determination of the polytropic exponent. If only the inlet value of k is known, don't be too concerned. The calculations will be repeated several times as knowledge of the process for the compression cycle is developed. After selecting the k value, use Equation 2.71 and an estimated stage efficiency of 7.5% to develop the polytropic compression exponent n.

The molecular weight, inlet temperature, and inlet pressure are combined with the compressibility and discharge pressure in Equation 2,73 to estimate the polytropic head. The average of inlet and outlet compressibility should be used, using the polytropic discharge temperature calculated by the following equation to evaluate the discharge compressibility.

where

T2 = absolute discharge temperature of the uncooled section

Tj = absolute inlet temperature of the uncooled section

To determine the number of stages, using the impeller and diffuser defined as the stage, assume 10,000 ft-lb/lb of head per stage. This value can be used if the molecular weight is in the range of 28 to 30. For other molecular weights, this initial value must be modified. As a rule of thumb, lower the head per stage by 100 ft-lb/lb for each unit increase in'molecular weight. Conversely, raise the allowable head per stage 200 ft lb/lb for a unit decrease in molecular weight. The rule of thumb gives the best results for a molecular weight range of 2 through 70. Because this sizing proce dure is being used only to establish the rough size of the compressor, the upper range may be extended with some loss in accuracy.

Once the head per stage has been established, the number of stages can be estimated by taking the total head, as calculated by the head equation, and dividing by the head per stage value. A fraction is usually rounded to the next whole number. However, if the fraction is less than .2, it may be dropped. The stage number should be used to calculate a new head value per stage. This method assumes an uncooled or no sidestream compressor. If either of the two are involved, the uncooled sections can be csti mated, taken one at a time. Assumptions for between-section pressure drop or sidestream mixing can be added to the calculation as appropriate to account for all facets of the process. When all calculations are com pieted, the compressor sections can be arranged to form a complete win.

Before proceeding, a few limits need to be considered. The temperature, if not limited by any other consideration, should not exceed 475 1 This limitation is arbitrary, as centrifugals may be built to higher limits, but the estimator is cautioned not to venture too far into this region without additional considerations. The number of stages per casing should not exceed 8 for rotor dynamics considerations. Also, knowledge of auxiliary nozzle stage pitch would be needed to evaluate exactly how far to venture in this direction. Vendor literature advertises the availability of as many as ten stages; however, an estimate should never go to the edge without a background of considerable experience. These limits can also be used to evaluate proposals and help to determine a series of questions for the vendor skirting the upper limits.

The next step begins by assuming a head coefficient equal to .48. Equation 5.12 can be used to calculate the tip speed, u2. Figure 5-26 can be used to get an impeller diameter estimate from the inlet volume calculated earlier. The diagonal line on the diagram marks the right extremity of each impeller's flow range to guide the user in making the first selection. The tip speed and diameter can be used to calculate an approximate speed, N, by

Ttd i

where d2 = impeller outside diameter

To summarize the sizing to this point: the inlet volume, an overall head, number of stages, head per stage, impeller tip speed, and impeller diameter have been established. The one parameter of interest still missing is the efficiency. To obtain an estimate of efficiency without empirical data, a generalized form may be used. As in the previous chapters, where estimates were involved, the data presented is just one way to approach the problem, and any other reasonable source such as specific vendor data may be used. To use the generalized curve, Figure 5-26, the volume for the first and last stage must be developed. The volume for the first stage is the inlet volume. The volume for the last stage, Q]s, can be estimated by

where

Qin = inlet volume fp = pressure ratio for an uncooled section 7. - number of stages in the uncooled section

Use the inlet and the last stage volume for the uncooled section and use the following equation to calculate the inlet flow coefficient 8.

Nd32

where

Qi = volumetric flow, ft3/min

N = rotational speed, rpm d2 = impeller diameter, in.

Note: This equation is not in the primitive form. While 8 is basically dimensionless, the constant 700 is not easily derived; therefore, units were assigned.

The value for the first-stage flow coefficient should not exceed .1 for a 2D type impeller and for a 3D design, the upper value can be as high as .15. The value for the last stage should be no less than .01. If the flow coefficients should fall outside these limits, another impeller diameter should be selected. It may be necessary to interpolate to obtain a reasonable diameter from Figure 5-26. This can be done because this is an estimate and not bound to an arbitrary line of compressor frames. The diagram was set up to give the user an idea of how a compressor line might be organized. A vendor may quote values outside the guidelines due to the constraints of his available frame sizes. For estimates, values as close as possible to the given guidelines are recommended. At the time of a proposal, the benefit of stages beyond either extreme value of flow coefficient can be evaluated. It should be noted at this point that not all vendors report their flow coefficients on the same basis. If necessary, the parameters for flow coefficient should be obtained to permit evaluation with Equation 5.19. An average of efficiency can be calculated from two efficiencies selected from Figure 5-27. The figure includes efficiency values for 2D and 3D impeller designs. While it would appear obvious that only 3D impellers should be used, there is a caveat. Generally, 3D impellers require more space, that is, the axial stage spacing (stage pitch)

3D |
-----„ | ||||||

2D | |||||||

y | |||||||

___L |
Flow Coefficient Figure 5-27. Centrifugal stage efficiency vs. flow coefficienct for 2D and 3D blading. is longer. This will result in a longer compressor, which makes for possible rotor dynamic problems and does also increase cost. Also, it should be pointed out that the increase in efficiency begins above flow coefficients of .04. The increase in stage pitch can vary from approximately 1.1 to 1.3 times a 2D stage pitch with the values increasing with increased flow coefficient. For the 2D impeller, it should be noted that the peak efficiency occurs at a flow coefficient of approximately .07. The 3D impeller peak efficiency value curve is broader and occurs in the range of from .07 to values as high as 1.3. At this point, after a first pass through the calculation, a new polytropic exponent should be calculated. All values calculated to this point should be rechecked to see if original estimates were reasonable. If the deviation appears significant, a second pass should be made to improve the accuracy. Equation 2.78 can be used to calculate the power for the uncooled section. For an estimate, use a value of 1% for the mechanical losses. If time permits and a more accurate estimate is desired, particularly if the compressor is intercooled or has sidestreams, the velocity head losses through the nozzles can be estimated using the values from Table 2-2. This is possible where the nozzle sizes are available or can readily be estimated. When coolers are involved, the drop through the cooler should be included. Subtract the pressure drop from the inlet pressure (of the stage following the element) and recalculate a modified pressure ratio for the section. The cooler pressure drop can be approximated by using 2% of the absolute pressure at the entrance to the cooler. Because the percentage gives unrealistic values at the lower pressures, a lower limit of 2 psi should be used. Compressors with in-out nozzles used to take gas from the compressor for external cooling and return to the compressor can experience some temperature crossover in the internal sections of the machine. Unless the design has specifically provided for a heat barrier, heating of the return gas can be expected. For a first estimate, a 10°F rise should be used. Balance pistons will be described in the mechanical section of this chapter. Briefly, the balance piston contributes a parasitic loss to the compressor not accounted for in the stage efficiency. The weight flow passing from the balance piston area, normally the discharge, and entering the suction must be added to the flow entering the first stage or the stage receiving the balance piston flow. Unfortunately, the flow is not the only problem, as the return flow also acts to heat the inlet gas. For discharge pressures of 150 psia or less, a value of 1% can be used. For pressures higher than 150 psia but under 1,000 psia, a value of 2Cik is a reasonable starting point. An equation for the heating is tj + tdBP 1 + BP where "W impeller inlet temperature tj = nozzle inlet temperature td = temperature at the balance piston BP = balance piston leakage fraction The relationships are given to help the user size a compressor from scratch. The same relationships can be used in the bid evaluation process. The vendor-provided geometry and performance values can be compared to the original sizing, which should have been performed prior to going out for the bid. The vendor's results can be evaluated using some of the rules of thumb or guidelines provided. Any deviations can be used as a focus for additional discussions. Also, some insight can be gained into the vendor's sizing techniques, particularly the way the vendor trims out a selection. Incremental wheel sizing is fairly universal. Some vendors also offer fixed guide vane sections as part of a stage to aid in the achievement of a particular performance specification. Example 5-2 Using the results of Example 2-2, size a centrifugal air compressor using the sizing procedure. A summary of the results is: Q1 =6,171 cfrn inlet volume wm = 437.5 lbs/min mw = 28.46 molecular weight Pj = 14.7 psia inlet pressure t, = 90.0°F inlet temperature T, = 550°R absolute inlet temperature Rm = 54.29 specific gas constant Add the following conditions to complete the application: k = 1.395 isentropic exponent for air p2 = 40 psia discharge pressure Assumed polytropic efficiency ilp = .75 Step 1. Calculate the polytropic exponent using Equation 2.71. rp = 2.721 pressure ratio Step 2. Calculate the total required polytropic head using Equation 2.73, assuming a value for Zavg = 1: Hp = 36,338.4 ft-lb/lb overall polytropic head Step 3. Determine the number of stages, z, required using the recommended 10,000 ft-lb/lb head per stage. Calculate a new head per stage using four stages: Step 4. Use the geometric form of Equation 5.12 to calculate a tip speed to produce the head per stage just calculated. Also, use the recommended head coefficient p. = .48 in the equation. u2 = 780.7 fps impeller tip speed From Figure 5-26 and the inlet volume, select an initial impeller diameter. d2 =17.3 inches impeller diameter Use Equation 5.17 to calculate the initial speed, N, and use the conversion factors of 12 in/ft and 60 secs/min. 60x12x780.7 71 x 17.3 = 10,342 rpm shaft speed Step 5. The volume into the last impeller, in this example stage 4 inlet, is calculated using the Equation 5.18. 6,171 P 7211~1/4)1/3-608 Qls = 3,869 cfm volume at last stage To obtain an efficiency for the geometry selected, the value of the flow coefficient must be calculated using Equation 5.19 for the first inlet and the last stage flow. 8 = .081 first stage flow coefficient 8 = .051 last stage flow coefficient Using the flow coefficients just calculated and Figure 5-26, the corresponding efficiencies may be looked up: The average is rather easy to calculate. rip = .79 the average efficiency Step 6. Recalculate the polytropic exponent using Equation 2.71 and the new efficiency. Using the new polytropic exponent, calculate the discharge temperature using Equation 5.16. t2 = 327.8°F discharge temperature Calculate the power required using Equation 2.78 and the recommended 1% for mechanical losses. 437.5X36,338.4 p 33,000 x.79 p Wn = 615.9 hp total for the compressor Note, the polytropic head was not recalculated as the change in efficiency only made an approximate 1% difference in original value and is well within the accuracy of an estimate. |

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