Compressor and turbine models

Though work has been carried out on the aerodynamic design of the turbomachinery to determine the compressors and turbines design parameters but their performance characteristic curve is yet to be generated. However, for nitrogen Brayton cycle, the industrial sponsor has separately carried out the design and generation of performance maps for the compressors and turbine. This data was utilised for the dynamic performance model of the compressors and turbine. For s-CO2

turbomachinery, there are still uncertainties regarding the prediction of the performance. The unique real gas properties of CO2 around the critical point gives challenges when designing or

simulating turbomachinery performance. Hence existing performance curve obtained from other similar work reported in the literature will be adopted to estimate the performance of the s-CO2

turbomachinery. The performance maps for the s-CO2 cycle MC, RC and turbine were obtained

from the work of Carstens et al. (2006).

In order to use these maps, they were scaled to generate new maps of normalised pressure ratios and normalised efficiencies as functions of normalised mass flow rates for a range of shaft speeds (all the normalisation was carried out with respect to design point). The use of performance maps rather than detailed calculations will reduce significantly the burden of such detailed computation in a dynamic model. Also, changes can be made easily to the turbomachinery performance model by simply switching the performance data. Therefore, as more realistic performance characteristic data becomes available from either experimental results or further aerodynamic design calculations they can be easily introduced into the model.

6.2.3.1 Compressor model

For s-CO2 compressors, the most reliable way suggested for constructing a compressor map is

using methods developed for incompressible turbomachinery (pumps). Hence, the pressure rise will be scaled with 𝑈 𝜌 to give the non-dimensional pressure ratio and the mass flow rate scaled with 𝑈𝜌 to give the non-dimensional flow rate or flow coefficient (Gong et al., 2006). 𝑈 is the impeller tip speed proportional to the rotational speed N, while 𝜌 is the fluid density. Therefore, the compressor map describes the relations between the non-dimensional pressure ratio and the flow coefficient, and between the efficiency and the flow coefficient. The non-dimensional coefficients are also normalised to the reference/design point values.

Thus the flow coefficient is defined as (Trinh, 2009; Carstens, 2007):

𝜙 = 𝑚̇ 𝑚̇ 𝜌 𝜌 N 𝑁 (6-37)

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Ideal gas assumption is satisfactory for the compressors in the nitrogen cycle. Therefore, the mass flow rate can be corrected for changing inlet conditions by using the standard method for ideal gas turbomachinery and then normalised as follows:

𝜙 = 𝑚̇ 𝑚̇ 𝑇 𝑇 𝑃 𝑃 (6-38)

The inputs to the compressor models are the mass flow rate, rotational speed and the fluid conditions (pressure, temperature and density) at the compressor inlet. With these inputs, the flow coefficient or corrected mass flow rate is calculated and the performance map is used to obtain corresponding values of non-dimensional pressure ratio and isentropic efficiency defined as:

𝜓 = π 𝜋 𝜌 𝜌 𝑁 𝑁 = 𝑓 (𝜙) (6-39) 𝜂 = 𝜂 𝜂 = 𝑓 (𝜙) (6-40)

Where 𝜓 is the non-dimensional pressure ratio, 𝜋 is the actual pressure ratio, 𝜂 is the normalised isentropic efficiency and 𝜂 is the actual isentropic efficiency

The performance map is incorporated into the model by direct use of the data in a tabular form and table look-up algorithm as well as curve fitting the data with polynomial equations. Once the map data has been obtained, the compressor outlet conditions is computed as follows:

𝑃 = 𝑃 𝜋 (6-41)

ℎ = ℎ +ℎ − ℎ

𝜂 (6-42)

ℎ the enthalpy that would be at the outlet with an isentropic compression. It can be obtained from the fluid thermodynamic properties relations as a function of 𝑃 and 𝑆 (i.e. inlet entropy). Other discharge fluid properties such as outlet temperature, 𝑇 can also be obtained from the fluid thermodynamic properties relations. That is,

𝑇 = 𝑓 (𝑃 , ℎ ) (6-43)

The power consumption of the compressor, 𝑊 is calculated as the product of the mass flow rate and enthalpy rise between the inlet and outlet section.

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The dynamic model of the turbomachinery assumes that the transport delay in the component are small and negligible.

6.2.3.2 Turbine model

The turbine model, similar to the compressor model, uses mass and energy conservation equation. It uses the equation of state of working fluid and the performance maps to provide constitutive relationships. The turbine produces mechanical energy from the thermal energy of the expanding working fluid. The produced mechanical energy is used to run the compressors and/or the electric generator.

The original turbine performance maps are transformed to provide relationship between non- dimensional pressure ratio and flow coefficient and between efficiency and flow coefficient at constant shaft speed parameter. Under turbine conditions, both the CO2 and nitrogen working

fluids were considered as ideal gas. Hence, the flow coefficient is as defined in equation (6-38). The other non-dimensional parameters normalised to their design point values utilised for map scaling are defined as (Carstens, 2007, Trinh, 2009, Gobran, 2013):

Non-dimensional shaft speed:

𝑁 = 𝑁

𝑁 𝑇

𝑇 (6-45)

Non-dimensional pressure ratio:

𝜓(𝜙, 𝑁 ) = 𝜋 − 1

𝜋 − 1= 𝑓 (𝜙, 𝑁 ) (6-46)

Normalised isentropic efficiency:

𝜂 (𝜙, 𝑁 ) = 𝜂

𝜂 = 𝑓 (𝜙, 𝑁 ) (6-47)

These transformation of the characteristic map parameter will enable the use of the maps for turbine inlet fluid conditions and pressure ratios different from the original machine design values. The inputs to the turbine model are the inlet fluid conditions, mass flow rate and rotational speed. These input values are used to compute the flow coefficient and shaft speed parameters. The turbine characteristic maps are incorporated into the model in tabular form and in the form of equations. By using table look-up algorithm and solving the equations the normalised non- dimensional pressure ratio and isentropic efficiency can be obtained and used to determine the actual values as follows:

146 The actual pressure ratio is:

𝜋 = 𝜓 𝜋 − 1 + 1 (6-48)

Isentropic efficiency is:

𝜂 = 𝜂 𝜂 (6-49)

Thus, the exit pressure of the turbine is:

𝑃 =𝑃

𝜋 (6-50)

To determine other gas properties (e.g. temperature and enthalpy) at the turbine outlet, firstly the expansion process is considered isentropic with outlet enthalpy, ℎ . Then the actual turbine exit enthalpy and temperature are calculated as follows:

Turbine outlet enthalpy is:

ℎ = ℎ − 𝜂(ℎ − ℎ ) (6-51)

Exit temperature is obtained from the fluid properties data:

𝑇 = 𝑓 (𝑃 , ℎ ) (6-52)

The power delivered by the turbine, 𝑊 is calculated as the product of the mass flow rate and enthalpy drop between the inlet and outlet section.

𝑊 = 𝑚̇(ℎ − ℎ ) (6-53)