Diffuser Design

A vaneless diffuser, or empty space, between the leading edges of diffuser vanes and the impeller tip allows some equalization of velocity and a reduction of the exit Mach number. The vaneless portion, which may have a width as large as 6 percent of the impeller diameter, also effects a rise in static pressure. As with the pump, angular momentum rV_u is conserved, and the fluid path is approximately a logarithmic spiral. Diffuser vanes are set with the diffuser axes tangent to the spiral paths and with an angle of divergence between them not exceeding 12°.

Figure 5.4 Arrangement of diffusers and impeller.

The wedge shaped diffuser vanes are depicted in Figure 5.4.

Since the addition of a vaned portion in the diffusion system results in a small-diameter casing, vanes are preferred in instances where size limitations are imposed. On the other hand, a completely vaneless diffuser is more efficient. If vanes are used, then their number should generally be less than the number of impeller vanes to ensure uniformness of flow and high diffuser efficiency in the range of flow coefficient Vm2/U2 recommended in the previous section.

The vaneless diffuser is situated between circles of radii r2 and r3. At any radial position r the gas velocity V will have both tangential and radial components. The radial component V_r is the same as the meridional component V_m. The mass flow rate at any r is given by

(5.26)

For constant diffuser width b, the product ρrV_m is constant, and the continuity equation becomes

(5.27)

Since angular momentum is conserved in the vaneless space, we can write

(5.28)

where the primed subscript is used to indicate the actual value of tangential velocity component at the impeller exit; however, in the vaneless space, the actual velocity is unprimed.

Typically, the flow leaving the impeller is supersonic, i.e., M_2’ > 1, and flow leaving the vaneless diffuser is subsonic, i.e., M₃ < 1. The radial position at which M = 1 is denoted by r^*; similarly, all other properties at the plane of sonic flow are denoted with a starred superscript, e.g., ρ^*, Vm, a^*, T^*, and α^*. The absolute gas angle α is the angle between V and V_r, i.e., between the direction of the absolute velocity and the radial direction.

Since the radial velocity component can be written as

(5.29)

the continuity equation becomes

(5.30)

Similarly, the angular momentum equation is expressed as

(5.31)

Dividing (5.31) by (5.30), we obtain

(5.32)

Assuming an isentropic flow in the vaneless region, we find

(5.33)

(5.34)

For M=1 , equation (5.34) becomes

(5.35) Substituting (5.34) and (5.35) into (5.33) yields

(5.36) Substituting (5.36) into (5.32) gives

(5.37) The angle α^* is evaluated by substituting α = α_2' and M = M_2' into (5.37). Equation (5.31) can be rewritten as

(5.38)

(5.39) The radial position r* can be found from (5.39) by substituting r = r2

and M = M2', which are known from impeller calculations. Finally, (5.37) can be used to determine α3 from a known M3, and (5.39) can be used to calculate r3 for known values of M3 and α3.

A volute is designed by the same methods outlined in Chapter 4.

The volute functions to collect the diffuser's discharge around the 360°

periphery and deliver it through a single nozzle to the connecting gas-piping system or to the inlet of the next compressor stage.

5.4 Performance

Typical compressor characteristics are shown in Figure 5.5.

Qualitatively, their shape is similar to those of the centrifugal pump, but the sharp fall of the constant-speed curves at higher mass flows is due to choking in some component of the machine. At low flows operation is limited by the phenomenon of surge. Thus, smooth operation occurs on

the compressor map at some point between the surge line and the choke line.

The phenomenon of choking is that associated with the attainment of a Mach number of unity. In the stationary passages of the inlet or diffuser, the Mach number is based on the absolute velocity V. Thus for a Mach number of unity, the absolute velocity equals the acoustic speed a, calculated from

(5.40)

The temperature at this point is calculated from the total temperature T_o using the relation

Figure 5.5 Compressor map.

(5.41)

and setting M = 1. Thus

(5.42)

This Mach number is found near the cross section of minimum area, or throat (At), so that we can estimate the choking, or maximum, mass flow rate from

(5.43)

The pressure P_t at the throat area may be estimated by assuming an isentropic process from the inlet of the stationary component to the throat area. Thus

(5.44)

The process of estimating choked flow rate in the impeller is the same except that relative velocity is substituted for absolute velocity.

When the relative Mach number W/a is set equal to unity in the energy equation of the rotor, namely,

(5.45) we obtain

(5.46) Using the isentropic relation between pressure and temperature and substituting into the continuity relation, the mass flow rate at the throat section of the impeller is given by

(5.47) Thus, it is clear that mass flow for choking in stationary components, given by (5.43), is independent of impeller speed, but that mass flow for choking in the impeller, given by (5.47), actually increases with impeller speed. This is indicated schematically in Figure 5.5.

Referring to Figure 5.5 the point A represents a point of normal operation. An increase in flow resistance in the connected external flow system results in a decrease in Vm2 at the impeller exit and a corresponding increase in Vu2, which results in an increased head or pressure increase. However, the surge phenomenon results when a further increase in external resistance produces a decrease in impeller flow that tends to move the point beyond C, where stall at some point in the impeller leads to change of direction of W2 and an accompanying decrease in the head (or pressure rise) in the impeller. A temporary flow reversal in the impeller and the ensuing buildup to the original flow condition is known as surging. Surging continues cyclically until the external resistance is removed. It is an unstable and dangerous condition and must be avoided by careful operational planning and system design.

5.5 Examples