8	Pumps and Compressors

“There is no right answer, but there is a best answer.”

—Anonymous

I.    PUMPS

There exists a wide variety of pumps that are designed for various specific applications. However, most of them can be broadly classified into two categories: positive displacement and centrifugal. The primary performance variables for any pump are the pump capacity (e.g., gpm, liters/min, m³/h, etc.) and the pressure or “head” that the pump can develop (e.g., ft or m of fluid). Recall that the head is related to the pressure by h = ΔP/ρg, which involves the density of the system fluid, so that a specific head value is unique to a specific fluid. The most significant characteristics of each of these two types of pumps are described in the following. More detailed descriptions can be found in specialized books (e.g., Karassik et al., 1976; Karassik and McGuire, 2012).

A.    POSITIVE DISPLACEMENT PUMPS

The term positive displacement pump is quite descriptive, because such pumps are designed to displace a more-or-less fixed volume of fluid during each cycle of operation. They include piston, diaphragm, screw, gear, progressing cavity, etc. The volumetric flow rate is determined by the displacement per cycle of the moving member (either rotating or reciprocating) times the cycle rate (e.g., rpm). The flow capacity is thus fixed by the design, size, and operating speed of the pump. The pressure (or head) that the pump develops depends upon the flow resistance of the system in which the pump is installed and is limited only by the size of the driving motor and the mechanical strength of the parts. Consequently, the discharge line from the pump should never be closed off without allowing for recycle around the pump or damage to the pump could result.

In general, PD pumps have limited flow capacity but are capable of developing relatively high pressures. These pumps operate at an essentially constant flow rate, with variable head. They are appropriate for high-pressure requirements, very viscous fluids, and applications that require a precisely controlled or metered flow rate.

B.    CENTRIFUGAL PUMPS

The term “centrifugal pump” is a very descriptive term, since these pumps operate by the transfer of energy (or angular momentum) from a rotating impeller to the fluid, which is normally inside a casing. A sectional view of a typical centrifugal pump is shown in Figure 8.1. The fluid enters at the axis or “eye” of the impeller (which may be open or closed and usually contains radial curved vanes) and is discharged from the impeller periphery.

The kinetic energy and momentum of the fluid are increased by the angular momentum imparted by the high-speed impeller. This kinetic energy is then converted to pressure energy (“head”) in a diverging area (the “volute”) between the impeller discharge and the casing before the fluid exits the pump. The head that these pumps can develop depends upon the pump design and the size, shape, and speed of the impeller, and the flow capacity is determined by the fluid resistance of the system in which the pump is installed. Thus, as will be shown, these pumps operate at approximately constant head and variable flow rate, within limits of course, determined by the size and design of the pump and the size of the driving motor.

FIGURE 8.1 Typical centrifugal pump.

Centrifugal pumps can be operated in a “closed off” condition (i.e., closed discharge line), because the liquid will recirculate within the pump without causing damage. However, such conditions should be avoided, because energy dissipation within the pump could result in excessive heating of the fluid and/or the pump, or unstable operation with adverse consequences. Centrifugal pumps are most appropriate for “ordinary” (i.e., low to moderate viscosity) liquids under a wide variety of flow conditions and are thus the most common type of pump. The following discussion applies primarily to centrifugal pumps.

II.    PUMP CHARACTERISTICS

The Bernoulli equation applied between the suction inlet and the discharge of a pump gives

$-w=\frac{\text{Δ}P}{\text{ρ}}=g{H}_p$

(8.1)

That is, the net energy or work put into the fluid by the pump goes to increasing the fluid pressure, or the equivalent pump head, H_p. However, because pumps are not 100% efficient, some of the energy delivered from the motor to the pump is dissipated or “lost” due to friction in the shear stresses around the high-speed impeller. It is very difficult to separately characterize this friction loss, so it is accounted for in the overall pump efficiency, η_e, which is the ratio of the useful work (or hydraulic work), done by the pump on the fluid (−w) to the work put into the pump by the motor (−w_m):

${\text{η}}_e=\frac{-w}{-w_m}$

(8.2)

The efficiency of a pump depends upon the pump and impeller design, the size and speed of the impeller, and the conditions under which it is operating, and is determined by tests carried out by the pump manufacturer. This will be discussed in more detail later.

When selecting a pump for a particular application, it is first necessary to specify the flow capacity and head required of the pump. Although many pumps might be able to meet these specifications, the “best” pump is normally the one that has the highest efficiency at the specified operating conditions. The required operating conditions, along with knowledge of the pump efficiency, then allow us to determine the required size (e.g., brake horsepower, [HP] or [BHP]) of the driving motor for the pump:

$HP=-w_m\dot{m}=\frac{\Delta PQ}{{\text{η}}_e}=\frac{\text{ρ}g{H}_{p}Q}{{\text{η}}_e}$

(8.3)

Now the power delivered from the motor to the pump is also the product of the torque on the shaft driving the pump (Γ) and the angular velocity of the shaft (ω):

$HP=\Gamma \text{ω}=\frac{\text{ρ}g{H}_{p}Q}{{\text{η}}_e}$

(8.4)

If it is assumed that the fluid leaves the impeller tangentially at the same speed as the impeller (a good approximation), then an angular momentum balance on the fluid in contact with the impeller gives

$\Gamma =\dot{m}\text{ω}{R}_i^2=\text{ρ}Q\text{ω}{R}_i^2$

(8.5)

where R_i is the radius of the impeller, and the angular momentum of the fluid entering the eye of the impeller has been neglected (a good assumption). By eliminating Γ from Equations 8.4 and 8.5 and solving for the pump head, H_p, we get

$H_p=\frac{{\text{η}}_e{\text{ω}}^2{R}_i^2}{g}$

(8.6)

This shows that the pump head is determined primarily by the size and speed of the impeller and the pump efficiency, independent of the flow rate of the fluid. This is approximately correct for most centrifugal pumps over a wide range of flow rates. However, there is a limit to the flow that a given pump can handle, and as the flow rate approaches this limit the developed head will start to drop off. The maximum efficiency (i.e. the “best efficiency point” or BEP) for most pumps occurs near the flow rate where the head starts to drop significantly.

Figure 8.2 shows a typical set of pump characteristic curves as determined by the pump manufacturer. “Size 2 × 3” means that the pump has a 2 in. discharge and a 3 in. suction port. “R&C” and “1⅞ pedestal” are the manufacturer’s designations, and 3500 rpm is the speed of the impeller. Performance curves for impellers with diameters from 6¼ to 8¾ in. are shown, and the efficiency is shown as contour lines of constant efficiency. The maximum efficiency for this pump is somewhat above 50%, although some pumps may operate with efficiencies as high as 80% or 90%. Operation at conditions on the right-hand branch of the efficiency contours (i.e., beyond the “maximum normal capacity” line in Figure 8.2) should be avoided, because this could result in unstable operation. The pump with the characteristics in Figure 8.2 is a slurry pump with a semi-open impeller, designed to pump solid suspensions (this pump can pass solid particles as large as 1¼ in. in diameter). Pump characteristic curves for a variety of other pumps are shown in Appendix H.

Such performance curves are normally determined by the manufacturer from operating data using water at 60°F. Note from Equation 8.6 that the head is independent of fluid properties, although from Equation 8.4 the power is proportional to the fluid density (as is the developed pressure). The horsepower curves in Figure 8.2 indicate the motor horsepower required to pump water at 60°F and must be corrected for density when operating with other fluids and/or at other temperatures. Actually, it is better to use Equation 8.4 to calculate the motor power from the values of the head, flow rate, fluid density, and efficiency at the operating point. The curves in Figure 8.2 labeled “minimum NPSH” refer to the cavitation characteristics of the pump, which will be discussed later.

FIGURE 8.2 Typical pump characteristic curves with system operating line superimposed. (From TRW Mission Pump Brochure.)

III.    PUMPING REQUIREMENTS AND PUMP SELECTION

When selecting a pump for a given application (e.g., a required flow capacity and head), we must specify the appropriate pump type, size and type of impeller, and size (horsepower) and speed (rpm) of the motor that will do the “best” job. “Best” normally means operating in the vicinity of the best efficiency point (BEP) on the pump curve (i.e., not lower than about 75% or higher than about 110% of capacity at the BEP). Not only will this condition do the required job at the least cost (i.e., least power requirement), but it also provides the lowest strain on the pump because the pump design is optimum for conditions at the BEP. We will concentrate on these factors and not get involved with the mechanical details of pump design (impeller vane design, casing dimensions, seals, etc.). More details on these topics are given by Karassik et al. (1976) and Karassik and McGuire (2012).

A.    REQUIRED HEAD

A typical piping application starts with a specified flow rate for a given fluid. The piping system is then designed with the necessary valves, fittings, etc. and should be sized for the most economical pipe size, as discussed in Chapter 7. Application of the energy balance (Bernoulli) equation to the entire system, from the upstream end (point 1) to the downstream end (point 2), determines the overall net driving force (DF) in the system required to overcome the frictional resistance:

$DF={\displaystyle \sum e_f}$

(8.7)

where the change in kinetic energy is assumed to be negligible. The total head (DF) is the net sum of the pump head, the total pressure drop, and the elevation drop:

$\frac{DF}{g}=H_p+\frac{\left(P_1-P_2\right)}{\text{ρ}g}+\left(z_1-z_2\right)$

(8.8)

The friction loss (Σe_f) is the sum of all of the losses from point 1 (upstream) to point 2 (downstream):

${\displaystyle \sum e_f}={\displaystyle \sum _i{\left(\frac{V^2}{2}{K}_f\right)}_i=\frac{8Q^2}{{\text{π}}^2}{\displaystyle \sum _i{\left(\frac{K_f}{D^4}\right)}_i}}$

(8.9)

where the loss coefficients (K_f’s) include all pipes, valves, fittings, contractions, expansions, etc. in the system. Eliminating DF and Σe_f from Equations 8.7 through 8.9 and solving for the pump head, H_p, gives

$H_p=\frac{P_2-P_1}{\text{ρ}g}+\left(z_2-z_1\right)+\frac{8Q^2}{g{\text{π}}^2}{\displaystyle \sum _i{\left(\frac{K_f}{D^4}\right)}_i}$

(8.10)

This relates the system pump head requirement to the specified flow rate and the system loss parameters (e.g., the K_f values). Note that H_p is a quadratic function of Q for highly turbulent flow (i.e., constant K_f’s). For laminar flow, the K_f values are inversely proportional to the Reynolds number, which results in a linear relationship between H_p and Q. A plot of H_p versus Q from Equation 8.10, illustrated in Figure 8.2 as line S1, is called the operating line for the system. Thus, the required pump head and flow capacity are determined by the system requirements as indicated by this line, and we must select the best pump to meet these requirements.

B.    COMPOSITE CURVES

Most pump manufacturers provide composite curves, such as those shown in Figure 8.3 that show the operating range of various pumps. Because of the overlap in these curves, more than one pump may be adequate for a given application. For each pump that provides the required flow rate and head, the individual pump characteristics (such as those shown in Figure 8.2 and Appendix H) are then consulted. The intersection of the system curve with the pump characteristic curve for a given impeller determines the pump operating point. The impeller diameter is selected that will produce the required head (or greater). This is repeated for all possible pump, impeller, and speed combinations to determine the combination that results in the highest efficiency (i.e., least power requirement). Note that if the operating point (H_p, Q) does not fall exactly on one of the (impeller) curves, then the actual impeller diameter that produces the higher head at the required flow rate Q is chosen. However, when this pump is installed in the system, the actual operating point will correspond to the intersection of the system curve (Equation 8.10) and the actual pump impeller curve at this point, as indicated by the X in Figure 8.2.

FIGURE 8.3 Typical pump composite curve. (From TRW Mission Pump Brochure [manufacturer’s catalog].)

IV.    CAVITATION AND NPSH

A.    VAPOR LOCK AND CAVITATION

As previously mentioned, a centrifugal pump increases the fluid pressure by first imparting angular momentum (or kinetic energy) to the fluid, which is converted to pressure in the diffuser or volute section. Hence, the fluid velocity in and around the impeller is much higher than that either entering or leaving the pump, and the pressure is the lowest where the velocity is highest. The minimum pressure at which a pump will operate properly must be above the vapor pressure of the fluid; otherwise the fluid will vaporize (or “boil”), a condition known as cavitation. Obviously, the higher the temperature, the higher the vapor pressure and the more likely that this condition will occur. When a centrifugal pump contains a gas or vapor, it will still develop the same head, but because the pressure is proportional to the fluid density the developed pressure will be several orders of magnitude lower than the pressure for a liquid at the same head. This condition (when the pump is filled with a gas or vapor) is known as vapor lock, and the pump will not function when this occurs.

However, cavitation may result in an even more serious condition than vapor lock. When the pressure at any point within the pump drops below the vapor pressure of the liquid, vapor bubbles will form at that point (this generally occurs on the impeller). These bubbles will then be transported to another region in the fluid where the pressure is greater than the vapor pressure, at which point they will collapse. This formation and collapse of bubbles occurs very rapidly and can create local “shock waves,” which can cause erosion and serious damage to the impeller or pump. (It is often obvious when a pump is cavitating, because it may sound as though there are rocks in the pump!)

B.    NET POSITIVE SUCTION HEAD

To prevent cavitation, it is necessary that the pressure at the pump suction (inlet) be sufficiently high that the minimum pressure anywhere in the pump will always be above the vapor pressure. This required minimum suction pressure (in excess of the vapor pressure) depends upon the pump design, impeller size and speed, and flow rate and is called the minimum required net positive suction head (NPSH or NPSHR). Values of the minimum required NPSH for the pump in Figure 8.2 are shown as dashed lines. The NPSH is almost independent of impeller diameter at low flow rates and increases with flow rate as well as with impeller diameter at higher flow rates. A distinction is sometimes made between the minimum NPSH “required” to prevent cavitation (i.e., the NPSHR) and the actual head (e.g., pressure) “available” at the pump suction (NPSHA). A pump will not cavitate if (NPSHA > NPSHR + vapor pressure head).

The NPSH at the operating point for the pump determines where the pump can be installed in a piping system to ensure that cavitation will not occur. The criterion is that the pressure head at the suction (entrance) of the pump (e.g., the NPSHA) must exceed the vapor pressure head by at least the value of the NPSH (or NPSHR) to avoid cavitation. Thus, if the pressure at the pump suction is P_s and the fluid vapor pressure is P_v at the operating temperature, cavitation will be prevented if

$NPSHA=\frac{P_s}{\text{ρ}g}\ge NPSH+\frac{P_{\text{v}}}{\text{ρ}g}$

(8.11)

The suction pressure P_s is determined by applying the Bernoulli equation to the suction line upstream of the pump. For example, if the pressure at the entrance to the upstream suction line is P₁, the maximum distance above this point that the pump can be located without cavitating (i.e., the maximum suction lift) is determined by applying the Bernoulli equation from P₁ to P_s:

$\begin{array}{|c|}\hline h_{\max }=\frac{P_1-P_{\text{v}}}{\text{ρ}g}-NPSH+\frac{V_1^2-V_s^2}{2g}-\frac{{\displaystyle \sum {\left(e_f\right)}_s}}{g}\\ \hline\end{array}$

(8.12)

where Equation 8.11 has been used for P_s. V₁ is the velocity entering the suction line, V_s is the velocity at the pump inlet (suction), and Σ(e_f)_s is the total friction loss in the suction line from the upstream entrance (point 1) to the pump inlet, including all pipes, fittings, etc. The diameter of the pump suction port is usually bigger than the discharge or exit diameter in order to minimize the kinetic energy head entering the pump, because this kinetic energy decreases the maximum suction lift and enhances the chance of cavitation. Note that if the maximum suction lift (h_max) is negative, the pump must be located below the upstream entrance to the suction line to prevent cavitation.

It is best to be conservative when interpreting the NPSH requirements to prevent cavitation. The minimum required NPSH on the pump curves is normally determined using water at 60°F with the discharge line fully open. However, even though a pump will run with a closed discharge line with no bypass, there will be much more recirculation within the pump if this occurs, which increases local turbulence and corresponding dissipative heating, both of which increase the minimum required NPSH. This is especially true with high-efficiency pumps, which have close clearances between the impeller and pump casing.

C.    SPECIFIC SPEED

The flow rate, head, and impeller speed at the maximum or “best efficiency point” of the pump characteristic can be used to define a dimensionless group called the specific speed:

$N_s=\frac{N\sqrt{Q}}{H^{3/4}}\left(\frac{\text{rpm}\sqrt{\text{gpm}}}{{\text{ft}}^{3/4}}\right)$

(8.13)

Although this group is dimensionless, it is common practice to use selected mixed (inconsistent) units when quoting the value of N_s, that is, N in rpm, Q in gpm, and H in feet. The value of the specific speed represents the ratio of the pump flow rate to the head at the speed corresponding to the maximum efficiency point (BEP) and depends primarily on the design of the pump and impeller. As previously stated, most centrifugal pumps operate at relatively low heads and high flow rates, i.e., high values of N_s. However, this value depends strongly on the impeller design, which can vary widely from almost pure radial flow to almost pure axial flow (like a fan). Some examples of various types of impeller designs are shown in Figures 8.4 and 8.5. Radial flow impellers have the highest head and lowest flow capacity (low N_s), whereas axial flow impellers have a high flow rate and low head characteristic (high N_s). Thus, the magnitude of the specific speed is a direct indication of the impeller design and performance, as shown in Figure 8.5. Figure 8.5 also indicates the range of flow rates and efficiencies of the various impeller designs, as a function of the specific speed. As seen in Figure 8.5, the maximum efficiency corresponds roughly to a specific speed of about 3000.

D.    SUCTION SPECIFIC SPEED

Another “dimensionless” group, analogous to the specific speed that relates directly to the cavitation characteristics of the pump, is the suction specific speed, N_ss:

$N_{ss}=\frac{N{Q}^{1/2}}{{\left(NPSH\right)}^{3/4}}$

(8.14)

The units used in this group are also rpm, gpm, and ft. This identifies the inlet conditions that produce similar flow behavior in the inlet for geometrically similar pump inlet passages. Note that the suction specific speed (N_ss) relates only to the pump cavitation characteristics as related to the inlet conditions, whereas the specific speed (N_s) relates to the entire pump at the BEP. The suction specific speed can be used, for example, to characterize the conditions under which excessive recirculation may occur at the inlet to the impeller vanes. Recirculation involves flow reversal and reentry resulting from undesirable pressure gradients at the inlet or discharge of the impeller vanes, and its occurrence generally defines the stable operating limits of the pump. For example, Figure 8.6 shows the effect of the suction specific speed on the stable “recirculation-free” operating window, expressed as NPSH versus percent of capacity at BEP, for various values of N_ss.

It should be noted that there are conflicting parameters in the proper design of a centrifugal pump. For example, Equation 8.12 shows that the smaller the suction velocity (V_s), the less the tendency to cavitate, that is, the less severe the NPSH requirement. This would dictate that the eye of the impeller should be as large as practical in order to minimize V_s. However, a large impeller eye means a high vane tip speed at the impeller inlet, which is destabilizing with respect to recirculation. Hence, it is advisable to design the impeller with the smallest eye diameter that is practicable.

It should be emphasized that the pump characteristics and performance curves (such as shown in Figure 8.2 and Appendix H) were obtained by the manufacturer using water as the test fluid. Although the pump head (H) values are independent of fluid density, the horsepower curves (HP) are not and must be adjusted for the fluid density. Also, the pump performance will be derated if the fluid viscosity is significantly greater than that of water, often by 10% or more (see, e.g., Wilson et al., 2008; Kalombo et al., 2014).

FIGURE 8.4 Impeller designs and specific speed characteristics. (a) Variation in impeller profiles with specific speed and approximate range of specific speed for the various types. (b) Straight-vane, singlesuction closed impeller. (c) Open mixed-flow impeller. (d) Axial-flow impeller. (e) Semiopen impeller. (f) Open impellers. Notice that the impellers at left and right are strengthened by a partial shroud. (g) Open impeller with a partial shroud. (h) Phantom view of radialvane nonclogging impeller. (Worthington Pump, Inc.) (i) Paper-pulp impeller. (b–f, h, i: Worthington Pump, Inc.) (From Karassik, I.J. and McGuire, J.T., Centrifugal Pumps, 2nd edn., Springer, 2012.)

V.    COMPRESSORS

A compressor may be thought of as a high-pressure pump for a compressible fluid. By “high pressure,” is meant conditions under which the compressible properties of the fluid (gas) must be considered. This normally occurs when the pressure changes by as much as 30% or more. For “low pressures” (i.e., smaller pressure changes), a fan or blower may be an appropriate “pump” for a gas. Fan operation can be analyzed by using the incompressible flow equations, because the relative pressure difference and hence the relative density change are normally small. As with pumps, compressors may be either PD or centrifugal. The former are suitable for relatively high pressures and low flow rates, whereas the latter are designed for higher flow rates but lower pressures. The major distinction in the governing equations, however, depends upon the conditions of operation, that is, whether the system is isothermal or adiabatic. The following analyses assume that the gas is adequately described by the ideal gas law. This assumption can be modified, however, by an appropriate compressibility correction factor, as necessary. For an ideal (frictionless) compression, the work of compression is given by the Bernoulli equation, which reduces to

FIGURE 8.5 Correlation between impeller shape, specific speed, and efficiency. (From Karassik, I.J. et al., Pump Handbook, McGraw-Hill, New York, 1976.)

FIGURE 8.6 Effect of suction speed on stable operating window due to recirculation. Numbers on the curves are the values of the suction specific speed, N_ss. (From Raymer, R.E., Chem. Eng. Progr., 89(3), 79, March 1993.)

$-w={\displaystyle \underset{P_1}{\overset{P_2}{\int }}\frac{dP}{\text{ρ}}}$

(8.15)

The energy balance equation for the gas can be written as

$\Delta h=q+e_f+{\displaystyle \underset{P_1}{\overset{P_2}{\int }}\frac{dP}{\text{ρ}}}$

(8.16)

which says that the work of compression plus the energy dissipated due to friction and any heat transferred into the gas during compression all go to increasing the enthalpy of the gas. Assuming ideal gas properties, the density is given by

$\text{ρ}=\frac{PM}{RT}$

(8.17)

The compression work cannot be evaluated from Equation 8.15 using Equation 8.17 unless the operating condition or temperature is specified. We will consider two cases: isothermal compression and adiabatic compression.

A.    ISOTHERMAL COMPRESSION

If the temperature is constant, eliminating ρ from Equations 8.17 and 8.15 and evaluating the integral gives

$-w=\frac{RT}{M}\ln \frac{P_2}{P_1}$

(8.18)

where the ratio P₂/P₁ is the compression ratio (r).*

B.    ISENTROPIC COMPRESSION

For an ideal gas under adiabatic frictionless (i.e., isentropic) conditions:

$\frac{P}{{\text{ρ}}^k}=\text{constant},\text{where}k=\frac{c_p}{c_{\text{v}}}\text{and}{c}_p=c_{\text{v}}+\frac{R}{M}$

(8.19)

The specific heat ratio k is approximately 1.4 for diatomic gases (O₂, N₂, etc.) and 1.3 for triatomic and higher gases (NH₃, H₂O, CO₂, etc.). The corresponding expression for isothermal conditions follows from Equation 8.17:

$\frac{P}{\text{ρ}}=\text{constant}$

(8.20)

Note that the isothermal condition can be considered a special case of the isentropic condition by setting k = 1. The “constant” in Equation 8.19 or 8.20 can be evaluated from the known conditions at some point in the system (e.g., P₁ and T₁). Using Equation 8.19 to eliminate the density from Equation 8.15 and evaluating the integral leads to

$-w=\frac{R{T}_{1}k}{M\left(k-1\right)}\left[{\left(\frac{P_2}{P_1}\right)}^{\left(k-1\right)/k}-1\right]$

(8.21)

Although it is not obvious by inspection, setting k = 1 in Equation 8.21 reduces the equation to Equation 8.18 (this can be confirmed by application of l’Hospital’s rule).

If we compare the work required to compress a given gas to a given compression ratio by isothermal and isentropic processes, we see that the isothermal work is always less than the isentropic work. That is, less energy would be required if compressors could be made to operate under isothermal conditions. However, in most cases, a compressor operates under more nearly adiabatic conditions (isentropic, if frictionless) because of the relatively short residence time of the gas in the compressor, which allows very little time for heat generated by compression to be transferred away. However, many compressors are fitted with a cooling jacket to remove the heat of compression and to more closely approach isothermal operation. The temperature rise during an isentropic compression is determined by eliminating ρ from Equations 8.17 and 8.19 to give

$\frac{T_2}{T_1}={\left(\frac{P_2}{P_1}\right)}^{\left(k-1\right)/k}=r^{\left(k-1\right)/k}$

(8.22)

In reality, most compressor conditions are neither purely isothermal nor purely isentropic but somewhere in between these limits. This can be accounted for in calculating the compression work by using the isentropic equation (Equation 8.21) but replacing the specific heat ratio k by a “polytropic” constant, γ, where 1 < γ < k. The value of γ is a function of the compressor design as well as the properties of the gas.

C.    STAGED OPERATION

It is often impossible to reach a desired compression ratio with a single centrifugal compressor. In such cases, multiple compressor “stages” can be arranged in series to increase the overall compression ratio. Furthermore, to increase the overall efficiency, it is common to cool the gas between stages by using “interstage cooling.” With interstage cooling to the initial temperature (T₁), it can be shown that as the number of stages increases, the total compression work for isentropic compression approaches that of isothermal compression at T₁.

For multistage operation, there will be an optimum compression ratio for each stage that will minimize the total compression work. This can be easily seen by considering a two-stage compressor with interstage cooling. The gas enters stage 1 at (P₁, T₁), leaves stage 1 at (P₂, T₂), and is then cooled to T₁. It then enters stage 2 at (P₂, T₁) and leaves at P₃. By computing the total isentropic work for both stages (using Equation 8.21), and setting the derivative of this with respect to the interstage pressure (P₂) equal to zero, the value of P₂ that results in the minimum total work can be found. The result is that the optimum interstage pressure that minimizes the total work for a two-stage compression with intercooling to T₁ is

$P_2={\left(P_1{P}_3\right)}^{1/2},\text{or}\frac{P_2}{P_1}=\frac{P_3}{P_2}=r={\left(\frac{P_3}{P_1}\right)}^{1/2}$

(8.23)

That is, the total work is minimized if the compression ratio is the same for each stage. This result can easily be generalized to any number (n) of stages (with interstage cooling to the initial temperature), as follows:

$r=\frac{P_2}{P_1}=\frac{P_3}{P_2}=\cdots =\frac{P_{n+1}}{P_n}={\left(\frac{P_{n+1}}{P_1}\right)}^{1/n}$

(8.24)

If there is no interstage cooling or if there is interstage cooling to a temperature other than T₁, it can be shown that the optimum compression ratio for each stage (i) is related to the temperature entering that stage (T_i) by

$T_i{\left(\frac{P_{i+1}}{P_i}\right)}^{\left(k-1\right)/k}=T_i{r}_i^{\left(k-1\right)/k}=\text{constant}$

(8.25)

D.    EFFICIENCY

The foregoing equations apply to ideal (frictionless) compressors. To account for friction losses, the ideal computed work is divided by the compressor efficiency, η_e, to get the total work that must be supplied to the compressor:

${\left(-w\right)}_{total}=\frac{{\left(-w\right)}_{ideal}}{{\text{η}}_e}$

(8.26)

The energy “lost” due to friction is actually dissipated into thermal energy, which raises the temperature of the gas. This temperature rise is in addition to that due to the isentropic compression, so that the total temperature rise across an adiabatic compressor stage is given by

$T_2=T_1{r}^{\left(k-1\right)/k}+\frac{1-{\text{η}}_e}{{\text{η}}_e}\left(\frac{-w_{ideal}}{c_{\text{v}}}\right)$

(8.27)

SUMMARY

The main points that are important to retain from this chapter include the following:

•  Understand the operation of centrifugal pumps.

•  Understand the information presented on the pump characteristic curves and how they interact with the system flow curve.

•  Understand NPSH and the distinction between NPSHA and NPSHR, and the conditions for cavitation.

•  Understand the use of composite curves.

•  Understand the operation of compressors and staged operation.

PROBLEMS

PUMPS

1.  The pressure developed by a centrifugal pump for Newtonian liquids that are not highly viscous depends upon the liquid density, the impeller diameter, the rotational speed, and the volumetric flow rate.

(a)  Determine a suitable set of dimensionless groups that should be adequate to relate all of these variables.

(b)  You want to know what pressure a pump will develop with a liquid having an SG of 1.4 at a flow rate of 300 gpm using an impeller with a diameter of 12 in. driven by a motor running at 1100 rpm. You have a similar test pump in the lab with a 6 in. impeller driven by an 1800 rpm motor. You want to run a test with the lab pump under conditions that will allow you to determine the pressure developed by the larger pump.

(c)  Should you use the same liquid in the lab as in the larger pump, or can you use a different liquid? Why?

(d)  If you use the same liquid, at what flow rate will the operation of the lab pump simulate that of the larger pump?

(e)  If the lab pump develops a pressure of 150 psi at the proper flow rate, what pressure will the field pump develop at 300 gpm?

(f)  What pressure will the field pump develop with water at 300 gpm?

2.  The propeller of a speed boat is 1 ft in diameter and is 1 ft below the surface of the water. At what speed (rpm) will cavitation occur at the propeller? Water density = 64 lb_m/ft³, vapor pressure of water, P_v = 18.65 mmHg.

3.  You must specify a pump to be used to transport water at a rate of 5000 gpm through 10 miles of 18 in. sch 40 pipe. The friction loss in valves and fittings is equivalent to 10% of the pipe length, and the pump is 70% efficient. If a 1200 rpm motor is used to drive the pump, determine

(a)  The required horsepower and torque rating of the motor

(b)  The diameter of the impeller that should be used in the pump

4.  You must select a centrifugal pump that will develop a pressure of 40 psi when pumping a liquid with an SG of 0.88 at a rate of 300 gpm. From all the pump characteristic curves in Appendix H, select the best pump for this job. Specify pump head, impeller diameter, motor speed, motor efficiency, and motor horsepower.

5.  An oil with a 32.6° API gravity at 60°F is to be transferred from a storage tank to a process unit, that is 10 ft above the tank, at a rate of 200 gpm. The piping system contains 200 ft of 3 in. sch 40 pipe, 25 90° screwed elbows, 6 stub-in tees used as elbows, 2 lift check valves, and 4 standard globe valves. From the pump performance curves in Appendix H, select the best pump for this application. Specify the pump size, motor speed, impeller diameter, operating head, and the efficiency and horsepower of the motor required to drive the pump.

6.  You must purchase a centrifugal pump to circulate cooling water that will deliver 5000 gpm at a pressure of 150 psi. If the pump is driven by an 1800 rpm motor, what should the horsepower and torque rating of the motor be, and how large (diameter) should the pump impeller be, assuming an efficiency of 60%?

7.  In order to pump a fluid of SG = 0.9 at a rate of 1000 gpm through a piping system, a hydraulic power of 60 hp is required. Determine the required pump head, the torque of the driving motor, and the estimated impeller diameter if an 1800 rpm motor is used.

8.  From your prior analysis of pumping requirements for a water circulating system, you have determined that a pump capable of delivering 500 gpm at a pressure of 60 psi is required. If a motor operating at 1800 rpm is chosen to drive the pump, which is 70% efficient, determine

(a)  The required horsepower rating of the motor

(b)  The required torque rating of the motor

(c)  The diameter of the impeller that should be used in the pump

(d)  What color the pump should be painted

9.  You want to pump water at 70°F from an open well, 200 ft deep, at a rate of 30 gpm through a 1 in. sch 40 pipe, using a centrifugal pump having an NPSH of 8 ft. What is the maximum distance above the water level in the well that the pump can be located without cavitating? (Vapor pressure of water at 60°F = 18.7 mmHg)

10.  Steam condensate at 1 atm and 95°C (P_v = 526 mmHg) is returned to a boiler from the condenser by a centrifugal boiler feed pump. The flow rate is 100 gpm through a 2.5 in. sch 40 pipe. If the equivalent length of the pipe between the condenser and the pump is 50 ft, and the pump has an NPSH of 6 ft, what is the maximum height above the condenser that the pump can be located?

11.  Water at 160°F is to be pumped at a rate of 100 gpm through a 2 in. sch 80 steel pipe from one tank to another located 100 ft directly above the first. The pressure in the lower tank is 1 atm. If the pump to be used has a required NPSH of 6 ft of head, what is the maximum distance above the lower tank that the pump may be located?

12.  A pump with a 1 in. diameter suction line is used to pump water from an open hot water well at a rate of 15 gpm. The water temperature is 90°C, with a vapor pressure of 526 mmHg and density of 60 lb_m/ft³. If the pump NPSH is 4 ft, what is the maximum distance above the level of the water in the well that the pump can be located and still operate properly, i.e., without cavitating?

13.  Hot water is to be pumped out of an underground geothermally heated aquifer located 500 ft below ground level. The temperature and pressure in the aquifer are 325°F and 150 psig. The water is to be pumped out at a rate of 100 gpm through 2.5 in. pipe using a pump that has a required NPSH of 6 ft. The suction line to the pump contains four 90° elbows and one gate valve. How far below ground level must the pump be located in order to operate properly?

14.  You must install a centrifugal pump to transfer a volatile liquid from a remote tank to a point in the plant 500 ft from the tank. To minimize the distance that the power line to the pump must be strung, it is desirable to locate the pump as close to the plant as possible. If the liquid has a vapor pressure of 20 psia, the pressure in the tank is 30 psia, the level in the tank is 30 ft above the pump inlet, and the required pump NPSH is 15 ft, what is the closest that the pump can be located to the plant without the possibility of cavitation? The line is 2 in. sch 40, the flow rate is 100 gpm, and the fluid properties are ρ = 45 lb_m/ft³ and μ = 5 cP.

15.  It is necessary to pump water at 70°F (P_v = 0.35 psia) from a well that is 150 ft deep, at a flow rate of 25 gpm. You do not have a submersible pump, but you do have a centrifugal pump with the required capacity that cannot be submerged. If a 1 in. sch 40 pipe is used, and the NPSH of the pump is 15 ft, how close to the surface of the water must the pump be lowered for it to operate properly?

16.  You must select a pump to transfer an organic liquid with a viscosity of 5 cP and SG of 0.87 at a rate of 1000 gpm through a piping system that contains 1000 ft of 8 in. sch 40 pipe, 4 globe valves, 16 gate valves, and 43 std 90° elbows. The discharge end of the piping system is 30 ft above the entrance, and the pressure at both ends is 10 psia.

(a)  What pump head is required?

(b)  What is the hydraulic horsepower to be delivered to the fluid?

(c)  Which combination of pump size, motor speed, and impeller diameter from the pump charts in Appendix H would you choose for this application?

(d)  For the pump selected, what size motor would you specify to drive it?

(e)  If the vapor pressure of the liquid is 5 psia, how far directly above the liquid level in the upstream tank could the pump be located without cavitating?

17.  You need a pump that will develop at least 40 psi at a flow rate of 300 gpm of water. What combination of pump size, motor speed, and impeller diameter from the pump characteristics in Appendix H would be the best for this application? State your reasons for the choice you make. What are the pump efficiency, motor horsepower and torque requirement, and NPSH for the pump you choose at these operating conditions?

18.  A centrifugal pump takes water from a well at 120°F (vapor pressure P_v = 87.8 mmHg) and delivers it at a rate of 50 gpm through a piping system to a storage tank. The pressure in the storage tank is 20 psig, and the water level is 40 ft above that in the well. The piping system contains 300 ft of 1.5 in. sch 40 pipe, 10 std 90° elbows, six gate valves, and an orifice meter with a diameter of 1 in.

(a)  What are the specifications required for the pump?

(b)  Would any of the pumps represented by the characteristic curves in Appendix H be satisfactory for this application? If more than one of them would work, which would be the best? What would be the pump head, impeller diameter, efficiency, NPSH, and required horsepower for this pump at the operating point?

(c)  If the pump you select is driven by an 1800 rpm motor, what impeller diameter should be used?

(d)  What should be the minimum torque and horsepower rating of the motor if the pump is 50% efficient?

(e)  If the NPSH rating of the pump is 6 ft at the operating conditions, where should it be located in order to prevent cavitation?

(f)  What is the reading of the orifice meter, in psi?

19.  Water at 20°C is pumped at a rate of 300 gpm from an open well in which the water level is 100 ft below ground level into a storage tank that is 80 ft above ground. The piping system contains 700 ft of 3½ in. sch 40 pipe, eight threaded elbows, two globe valves, and two gate valves. The vapor pressure of the water is 17.5 mmHg.

(a)  What pump head and hydraulic horsepower are required?

(b)  Would a pump whose characteristics are similar to those shown in Figure 8.2 be suitable for this job? If so, what impeller diameter, motor speed, and motor horsepower should be used?

(c)  What is the maximum distance above the surface of the water in the well at which the pump can be located and still operate properly?

20.  An organic fluid is to be pumped at a rate of 300 gpm, from a distillation column reboiler to a storage tank. The liquid in the reboiler is 3 ft above ground level, the storage tank is 20 ft above ground, and the pump will be at ground level. The piping system contains 14 std elbows, 4 gate valves, and 500 ft of 3 in. sch 40 pipe. The liquid has an SG of 0.85, a viscosity of 8 cP, and a vapor pressure of 600 mmHg. If the pump to be used has characteristics similar to those given in Appendix H, and the pressure in the reboiler is 5 psig, determine

(a)  The motor speed to be used

(b)  The impeller diameter

(c)  The motor horsepower and required torque

(d)  Where the pump must be located to prevent cavitation

21.  A liquid with a viscosity of 5 cP, density of 45 lb_m/ft³, and vapor pressure of 20 psia is transported from a storage tank in which the pressure is 30 psia to an open tank 500 ft downstream, at a rate of 100 gpm. The liquid level in the storage tank is 30 ft above the pump, and the pipeline is 2 in. sch 40 commercial steel. If the transfer pump has a required NPSH of 15 ft, how far downstream from the storage tank can the pump be located without danger of cavitation?

22.  You must determine the specifications for a pump to transport water at 60°C from one tank to another at a rate of 200 gpm. The pressure in the upstream tank is 1 atm, and the water level in this tank is 2 ft above the level of the pump. The pressure in the downstream tank is 10 psig, and the water level in this tank is 32 ft above the pump. The pipeline contains 250 ft of 2 in. sch 40 pipe, with 10 standard 90° flanged elbows and six gate valves.

(a)  Determine the pump head required for this job.

(b)  Assuming your pump has the same characteristics as the one shown in Figure 8.2, what impeller size should be used and what power would be required to drive the pump with this impeller at the specified flow rate?

(c)  If the water temperature is raised, the vapor pressure will increase accordingly. Determine the maximum water temperature that can be tolerated before the pump will start to cavitate, assuming that it is installed as close to the upstream tank as possible.

23.  A piping system for transporting a liquid (μ = 50 cP, ρ = 0.85 g/cm³ from vessel A to vessel B) consists of 650 ft of 3 in. sch 40 commercial steel pipe containing 4 globe valves and 10 elbows. The pressure is atmospheric in A and 5 psig B, and the liquid level in B is 10 ft higher than that in A. You want to transfer the liquid at a rate of 250 gpm at 80°F using a pump with the characteristics shown in Figure 8.2. Determine

(a)  The diameter of the impeller that you would use with this pump

(b)  The head developed by the pump and the power (in horsepower) required to pump the liquid

(c)  The power of the motor required to drive the pump

(d)  The torque that the motor must develop

(e)  The NPSH of the pump at the operating conditions

24.  You must choose a centrifugal pump to pump a coal slurry. You have determined that the pump must deliver 200 gpm at a pressure of at least 35 psi. Given the pump characteristic curves in Appendix H, tell which pump you would specify (give pump size, speed, and impeller diameter) and why. What is the efficiency of this pump at its operating point, what horsepower motor would be required to drive the pump, and what is the required NPSH of the pump? The specific gravity of the slurry is 1.35.

25.  You must specify a pump to take an organic stream from a distillation reboiler to a storage tank. The liquid has a viscosity of 5 cP, an SG of 0.78, and a vapor pressure of 150 mmHg. The pressure in the storage tank is 35 psig, and the inlet to the tank is located 75 ft above the reboiler, which is at a pressure of 25 psig. The pipeline in which the pump is to be located is 2½ in. sch 40, 175 ft long, and there will be two flanged elbows and a globe valve in each of the pump suction and discharge lines. The pump must deliver a flow rate of 200 gpm. If the pump you use has the same characteristics as that illustrated in Figure 8.2, determine:

(a)  The proper impeller diameter to use with this pump

(b)  The required head that the pump must deliver

(c)  The actual head that the pump will develop

(d)  The horsepower rating of the motor required to drive the pump

(e)  The maximum distance above the reboiler that the pump can be located without cavitating

26.  You have to select a pump to transfer benzene from the reboiler of a distillation column to a storage tank at a rate of 250 gpm. The reboiler pressure is 15 psig and the temperature is 60°C. The tank is 5 ft higher than the reboiler and is at a pressure of 25 psig. The total length of piping is 140 ft of 2 in. sch 40 pipe. The discharge line from the pump contains 3 gate valves and 10 elbows, and the suction line has 2 gate valves and 6 elbows. The vapor pressure of benzene at 60°C is 400 mmHg.

(a)  Using the pump curves shown in Figure 8.2, determine the impeller diameter to use in the pump, the head that the pump would develop, the power of the motor required to drive the pump, and the NPSH required for the pump.

(b)  If the pump is on the same level as the reboiler, how far from the reboiler could it be located without cavitating?

27.  A circulating pump takes hot water at 85°C from a storage tank, circulates it through a piping system at a rate of 150 gpm, and discharges it to the atmosphere. The tank is at atmospheric pressure, and the water level in the tank is 20 ft above the pump. The piping consists of 500 ft of 2 in. sch 40 pipe, with one globe valve upstream of the pump and three globe valves and eight threaded elbows downstream of the pump. If the pump has the characteristics shown in Figure 8.2, determine

(a)  The head that the pump must deliver, the best impeller diameter to use with the pump, the pump efficiency and NPSH at the operating point, and the motor horsepower required to drive the pump

(b)  How far the pump can be located from the tank without cavitating

Properties of water at 85°C: viscosity = 0.334 cP, density = 0.970 g/cm³, vapor pressure = 433.6 mmHg.

28.  A slurry pump must be selected to transport a coal slurry at a rate of 250 gpm from an open storage tank to a rotary drum filter operating at 1 atm. The slurry is 40% solids by volume and has an SG of 1.2. The level in the filter is 10 ft above that in the tank, and the line contains 400 ft of 3 in. sch 40 pipe, two gate valves, and six 90° elbows. A lab test shows that the slurry can be described as a Bingham plastic with μ_∞ = 50 cP and τ_o = 80 dyn/cm².

(a)  What pump head is required?

(b)  Using the pump curves in Appendix H, choose the pump that would be the best for this application. Specify the pump size, motor speed, impeller diameter, efficiency, and NPSH. State what criteria you used to make your decision.

(c)  What horsepower motor would you need to drive the pump?

(d)  Assuming the pump you choose has an NPSH of 6 ft at the operating conditions, what is the maximum elevation above the tank that the pump could be located, if the maximum temperature is 80°C? Vapor pressure (P_v) of water is 0.4736 bar at this temperature.

29.  A red mud slurry residue from a bauxite processing plant is to be pumped from the plant to a disposal pond at a rate of 1000 gpm, through a 6 in. ID pipeline that is 2500 ft long. The pipeline is horizontal, and the inlet and discharge of the line are both at atmospheric pressure. The mud has properties of a Bingham plastic, with a yield stress of 250 dyn/cm², a limiting viscosity of 50 cP, and a density of 1.4 g/cm³. The vapor pressure of the slurry at the operating temperature is 50 mmHg. You have available several pumps with the characteristics given in Appendix H.

(a)  Which pump, impeller diameter, motor speed, and motor horsepower would you use for this application?

(b)  How close to the disposal pond could the pump be located without cavitating.

(c)  It is likely that none of these pumps would be adequate to pump this slurry. Explain why, and explain what type of pump might be better.

30.  A pipeline is installed to transport a red mud slurry from an open tank in an alumina plant to a disposal pond. The line is 5 in. sch 80 commercial steel, 12,000 ft long, and is designed to transport the slurry at a rate of 300 gpm. The slurry properties can be described by the Bingham plastic model, with a yield stress of 15 dyn/cm², a limiting viscosity of 20 cP, and an SG of 1.3. You may neglect any fittings in this pipeline.

(a)  What delivered pump head and hydraulic horsepower would be required to pump this slurry?

(b)  What would be the required pump head and horsepower to pump water at the same rate through the same pipeline?

(c)  If 100 ppm of fresh Separan AP-30 polyacrylamide polymer were added to the water in case (b) above, what would be the required pump head and horsepower?

(d)  If a pump with the same characteristics as those illustrated in Figure 8.2 could be used to pump these fluids, what would be the proper impeller size and motor horsepower to use for each of cases (a), (b), and (c) above? Explain your choices.

COMPRESSORS

31.  Calculate the work per pound of gas required to compress air from 70°F and 1 atm to 2000 psi with an 80% efficient compressor under the following conditions:

(a)  Single-stage isothermal compression.

(b)  Single-stage adiabatic compression.

(c)  Five-stage adiabatic compression with intercooling to 70°F and optimum interstage pressures.

(d)  Three-stage adiabatic compression with interstage cooling to 120°F and optimum interstage pressures.

(e)  Calculate the outlet temperature of the air for cases (b), (c), and (d) above. For air: c_p = 0.24 Btu/(lb_m °F), k = 1.4.

32.  It is desired to compress ethylene gas [MW = 28, k = 1.3, c_p = 0.357 Btu/(lb_m °F)] from 1 atm and 80°F to 10,000 psia. Assuming ideal gas behavior, calculate the compression work required per pound of ethylene under the following conditions:

(a)  A single-stage isothermal compressor

(b)  A four-stage adiabatic compressor with interstage cooling to 80°F and optimum interstage pressures

(c)  A four-stage adiabatic compressor with no intercooling, assuming the same interstage pressures as in (b) and 100% efficiency

33.  You have a requirement to compress natural gas (k = 1.3, MW = 18) from 1 atm and 70°F to 5000 psig. Calculate the work required to do this per pound of gas in a 100% efficient compressor under the following conditions:

(a)  Isothermal single-stage compressor

(b)  Adiabatic three-stage compressor with interstage cooling to 70°F

(c)  Adiabatic two-stage compressor with interstage cooling to 100°F

34.  Air is to be compressed from 1 atm and 70°F to 2000 psia. Calculate the work required to do this per pound of air using the following methods:

(a)  A single-stage 80% efficient isothermal compressor.

(b)  A single-stage 80% efficient adiabatic compressor.

(c)  A five-stage 80%efficient adiabatic compressor with interstage cooling to 70°F.

(d)  A three-stage 80% efficient adiabatic compressor with interstage cooling to 120°F. Determine the expression relating the pressure ratio and inlet temperature for each stage for this case by induction from the corresponding expression for optimum operation of the corresponding two-stage case.

(e)  Calculate the final temperature of the gas for cases (b), (c), and (d).

35.  It is desired to compress 1000 scfm of air from 1 atm and 70°F to 10 atm. Calculate the total horsepower required if the compressor efficiency is 80% for

(a)  Isothermal compression.

(b)  Adiabatic single-stage compression.

(c)  Adiabatic three-stage compression with interstage cooling to 70°F and optimum interstage pressures.

(d)  Calculate the gas exit temperature for cases (b) and (c).

Note: C_P = 7 Btu/(lbmol°F); Assume ideal gas.

36.  You want to compress air from 1 atm, 70°F, to 2000 psig, using a staged compressor with interstage cooling to 70°F. The maximum compression ratio per stage you can use is about 6, and the compressor efficiency is 70%.

(a)  How many stages should you use?

(b)  Determine the corresponding interstage pressures.

(c)  What power would be required to compress the air at a rate of 10⁵ scfm?

(d)  Determine the temperature of the air leaving the last stage.

(e)  How much heat (in Btu/h) must be removed by the interstage coolers?

NOTATION

D	Diameter, [L]
DF	Driving force, Equation 8.8, [L2/t2]
ef	Energy dissipated per unit mass of fluid, [FL/M = L2/t2]
g	Acceleration due to gravity, [L/t2]
Hp	Pump head, [L]
HP	Power, [FL/t = M L2/t3]
hmax	Maximum suction lift, [L]
k	Isentropic exponent, (= cp/cv for ideal gas), [—]
Kf	Loss coefficient, [—]
M	Molecular weight, [M/mol]
Ns	Specific speed, Equation 8.13, [—]
Nss	Suction specific speed, Equation 8.14, [—]
NPSH	Net positive suction head, [L]
ṁ	Mass flow rate, [M/t]
P	Pressure, [F/L2 = M/L t2]
Pv	Vapor pressure, [F/L2 = M/L t2]
Q	Volumetric flow rate, [L3/t]
R	Radius, [L]
r	Compression ratio, [—]
T	Temperature, [T]
w	Work done by fluid system per unit mass of fluid, [F L/M = L2/t2]

GREEK

Γ	Moment or torque, [F L = M L2/t2]
Δ()	()2 – ()1
ηe	Efficiency, [—]
ρ	Density, [M/L3]
ω	Angular velocity, [1/t]

SUBSCRIPTS

1	Reference point 1
2	Reference point 2
i	Impeller, ideal (frictionless)
m	Motor
s	Suction line

REFERENCES

Kalombo, J.J.N., R. Haldenwang, R.P. Chhabra, and V.G. Fester, Centrifugal pump de-rating for non-Newtonian slurries, ASME J. Fluids Eng., 136, 131302-1, 2014.

Karassik, I.J., W.C. Krutzsch, W.H. Fraser, and J.P. Messina, Pump Handbook, McGraw-Hill, New York, 1976.

Karassik, I.J. and J.T. McGuire, Centrifugal Pumps, 2nd edn., Springer, New York, 2012.

Raymer, R.E., Watch suction specific speed, Chem. Eng. Progr., 89(3), 79–84, March 1993.

Wilson, K.C., G.R. Addie, A. Sellgren, and R. Clift, Slurry Transport Using Centrifugal Pumps, 3rd edn., Springer, New York, 2008.

*  Note that for PD compressors, the compression ratio is defined as the ratio of the volume change during a cycle. However, for centrifugal compressors, the pressure ratio is more appropriate. For an isothermal ideal gas, the two are the same.