### Specific Speed

Enver Karakas, Ebara International Corp., USA, compares gas and hydraulic turbine designs based on specific speed and efficiency requirements.

#### 2.0 Dimensionless parameters used in turbine selection and design

Specific speed is the most common dimensionless (unit-less) parameter used in the determination of the turbine type and achievable efficiency.  Formulation for specific speed can be obtained through utilisation of Buckingham π theorem per the physical variables of the application.  For pump applications, differential head (H), volumetric flowrate (Q), and rotational speed (n) are important variables to define pump characteristics and are used to obtain specific speed.  Pump specific speed is determined by the following equation:

It is important to pay attention to the units when calculating pump specific speed.  Between metric and US unit systems, specific speed values vary significantly.  This is because the form of the equation is not truly dimensionless, unless it is reduced by including gravitational acceleration.  Pump specific speed is a useful guideline to determine the type of turbine.  However, for some turbine applications, due to the importance of power generation, power (P) is taken into account in determination of specific speed and replaces the flowrate (Q) in the equation.  Accordingly, turbine specific speed can be calculated with the following equation:

Over the years, based on proved turbine applications and their operating conditions along with achieved efficiencies, turbine manufacturers established various specific speed maps top assist engineers in their selection and design of the hydraulic components.  Since specific speed is a dimensionless parameter, these specific maps can be applied to any turbine application, regardless of the physical size of the unit and the process.  Besides the specific speed, other dimensionless numbers, such as Reynolds Number, Mach Number, etc., shall be considered in the design and scaling of the turbomachinery.  Maintaining these dimensionless parameters is key in scaling applications.

Specific diameter is also used in selecting the turbine type and/or scaling an existing turbine application.  Specific diameter is calculated based on the actual physical diameter of the turbine wheel (runner) and the process parameters, which are head per stage and volumetric flowrate.

As in the specific speed formula, the specific diameter is not truly dimensionless since the gravitational acceleration is not included in the equation, and it is assumed to be constant.  As the actual value of the gravitational acceleration differs depending on which unit system is used, one should pay attention to the units of each parameter when determining specific speed and specific diameter.

#### 3.0 Specific speed maps, turbine type and achievable efficiency

Specific speed of an application can often be plotted against a parameter that is crucial for that application, or, if the comparison is to be based on a certain parameter, specific speed vs important parameter shall be plotted to determine the characteristics of that application.  There are general specific speed maps that have the specific diameter with a range of efficiencies based on the type of application.  Figure 1 is the specific speed vs specific diameter map, which covers the application region and achievable efficiencies for each type of turbine.¹

It should be noted that Figure 1 is established for a Reynolds number greater than 10^6 – a rotor inlet Mach number of 1.0.  the type of turbine based on the hydraulics varies according to the specific speed and the diameter.  It should also be noted that the turbine efficiencies can be greater than 80% for specific speeds greater than 40 US units.

Depending on the flow direction and path across the turbine wheel, each turbine type is specified as a radial, partial axial (mixed flow), or full axial turbine.  Turbines with a specific speed of 10 or less (US units) are often called impulse turbines and have a relatively higher head with a lower flowrate.  Drag turbines shown in Figure 1 fall under this category.  Piston expanders (displacement and rotary) are often used for high head application with a relatively low volumetric flowrate.  Piston expanders are not centrifugal type turbines, and so do not have a rotating wheel.

Figure 2 covers the centrifugal type turbines and estimates efficiency of those turbines based on a specific speed. ²  All of the published efficiencies at each figure are hydraulic efficiencies that do not include any of the losses due to internal leakage, generator, etc.

According to Figure 2, the turbines with a turbine specific speed of 15 or greater are reaction turbines, which have relatively higher flowrate, but also have lower head compared to impulse expanders.  Reaction turbines utilise nozzle vanes to increase the velocity of the incoming flow to the turbine, which impulse turbines have jets to carry out the same function.  Francis and Kaplan turbines are the most common reaction turbines used in turbomachinery applications.  Francis turbines have a wider range of application region and they are the type of turbine that EIC Cryo offers.  The company’s turbines are mostly used for LNG liquefaction processes, and have proven to be a suitable alternative to Joule Thompson valves to recover BOG and generate electricity.  These expanders are capable of 90% hydraulic efficiency, with a resulting total efficiency of approximately 80%.  Specific speed range from 50 – 80 in US units, with specific diameters ranging from 1.6 to 2.5.  EIC Cryo’s cryogenic turbines are vertically suspended multistage units with two or more turbine wheels (stages) per unit, and have a submerged generator design to enhance generator cooling.  A typical cryogenic turbine of EIC is shown in Figure 3.

#### 4.0 Design and process considerations in cryogenic turbine application

Specific speed is a guideline for design and process engineers to determine the turbine type and estimate the expected hydraulic efficiency of a turbine.  The design of turbomachinery components, which is not limited to the design of hydraulic components, always plays an important role in achieving that efficiency.  Design and process engineers often consider varying the rotational speed, increasing the number of stages, and even increasing the number of turbine units for a given total head and flow requirement of a process.

Rotational speed changes the specific speed linearly and is directly proportional to it.  Speed also varies the head and flowrate of a turbine.  The relationship between rotational speed and process parameters (head and flowrate) is defined by affinity laws for centrifugal turbine applications.  It is often possible to vary the turbine speed by implementing variable frequency drives, or by changing the actual physical diameter of the turbine well.  The impact of the rotordynamics, reliability of the turbine, and cost should be considered when examining a change to the rotational speed.

The most common way to adjust the specific speed is by increasing or decreasing the number of stages of the turbomachinery.  This can be costly and there is always a limit to the number of stages that a turbine can handle.  This limitation is set by rotordynamic behaviour of the turbomachinery, and the maximum allowable pressure of the casings and related components.

Another method of adjusting the specific speed is to increase the number of turbines used in an application and the installation configuration of those turbines in that application.  For relatively high head (pressure) applications, two or more turbines can be installed in series to increase the specific speed to a desired value, while dropping the total pressure to the process needs.  For applications with relatively high flowrates, turbines can be operated in parallel to reduce the specific speed of the turbine.  It is often preferred to have a single turbine handle the process due to limitations of the footprint.  Maintenance requirements and complexity of the system must also be taken into account if multiple turbines are to be operated in parallel or in series, due to the cost impact.  Regardless, it may not be feasible to utilise a single turbine to meet the process needs.  Forcing a single turbine into the process can greatly harm efficiency and result in a product that does not satisfy expectations.

#### 5.0 Conclusion

Selection of turbine type based on specific speed and estimating attainable hydraulic efficiencies for each turbine type have been discussed.  The fundamental differences and similarities of each hydraulic turbine type have been discussed.  The fundamental differences and similarities of each hydraulic turbine type have been shared without getting into finite design details.  Specific speed charts (maps) have proven to be useful for selection and estimation, and are widely used in the turbine design industry.  It is good practice to review and follow the specific speed charts prior to a decision in feasibility of a turbine application.  In conclusion, some adjustments are advantageous and necessary to maintain a turbine’s specific speed within a certain range to ensure high reliability and performance.

#### 6.0 References

1. NICHOLS, K. E., ‘How to Select Turbomachinery for Your Application’, http://www.barber-nichols.com/sites/default/files/wysiwyg/images/how_to_select_turbomachinery_for_your_application.pdf (Accessed on 11/09/15)
2. YOUNG, D. F., MUNSON, B. R., and OKIISHI, T. H., ‘A Brief Introduction to Fluid Mechanics’, 4th edition (2001)