Modular design and feasibility analysis of a Rankine Compression GasTurbine system for industrial cogeneration

Master of Science Thesis

KTH School of Industrial Engineering and Management Energy Technology TRITA-ITM-EX 2018:637

Division of HPT SE-100 44 STOCKHOLM

Modular design and feasibility analysis of a Rankine Compression

Gas Turbine system for industrial cogeneration

Daniel Carrion da Fonseca

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Master of Science Thesis TRITA-ITM-EX 2018:637

Modular design and feasibility analysis of a Rankine Compression Gas Turbine system

for industrial cogeneration

Daniel Carrion da Fonseca

Approved Examiner

Reza Fakhrai

Supervisor

Reza Fakhrai

Commissioner Contact person

Abstract

A substantial share of the world’s energy is consumed in the form of heat in many distinct industries, which usually generates it through a boiler and consumes the heat by using steam as an energy carrier. In such factories, a cogeneration installation is economically and environmentally beneficial.

In this thesis, a new type of cogeneration cycle for industrial biomass-fired boilers is studied, the Rankine Compression Gas turbine (RCG) cycle. Consisting in a combined Rankine and Brayton cycle that can be easily integrated in industries using steam boilers, the system presents clear advantages such as a up to 35%

higher power output compared to a simple steam turbine system and a quick demand response in the electricity generation, being able to change its power output within seconds.

Firstly, a time-based model is developed by implementing the differential equations that represent the dynamic behavior of all components, culminating in an easy-to-use Matlab-Simulink model; this model presents a robust modular concept, in which distinct systems and equipment can be studied.

Then, a 40 kW pilot RCG system and a 100 kW commercial-scale RCG system are designed, aiming to investigate the system’s behavior and select its components. It is proven that most of the RCG components are found as off-the-shelf products and, thus, with reduced cost when compared to customized ones. The simulations show the system advantages and establish key design criteria for the RCG.

Finally, an economic feasibility analysis of commercial-scale RCG systems is carried out. Even though the

system’s feasibility is considerably sensitive to the electricity cost, capacity factor and fuel cost, the RCG

system presents a payback time from 2 to 4 years and a levelized cost of electricity between €0.07 and €0.12

per kWh.

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Abstrakt

En väsentlig del av världens energikonsumtion sker i form av värme inom industrier. Dessa industrier genererar oftast värmen med hjälp av en värmepanna där ånga agerar som energibärare. I dessa industrier är en kraftvärmeinstallation både ekonomisk och miljömässigt försvarbar.

I denna rapport studeras en ny teknik för kraftvärmecykler för industriella värmepannor drivna på biomassa, en så kallad Rankine Compression Gas turbine (RCG) cykel. Bestående av en kombinerad Rankine och Brayton cykel som enkelt kan integreras inom industrier som använder ångpanna. Systemet visar på tydliga fördelar, som up till 35% högre kraftutflöde, jämfört med ett system med en enkel ångturbin. Den har även snabb respons gällande el generering, med möjlighet att ändra produktionen inom fåtal sekunder.

Inledningsvis utvecklades en tidsbaserad modell genom att implementera differentialekvationerna som representerar komponenternas dynamiska beteende, vilket resulterade i en användarvänlig Matlab-Simulink modell. En modell som representerar en robust modulärt koncept, där distinkta system och utrustning kan studeras.

Därefter designades ett 40 kW pilot RCG system och ett 100 kW RCG system i kommersielle skala, med förhoppningen att undersöka systemets beteende och välja ut dess komponenter. Det visades att de flesta RCG komponenter är standardprodukter och var därmed billigare än specialbeställda. Simuleringarna visar systemets fördelar samt väsentliga kriterier gällande design och utformning av RCG.

Slutligen utfördes en ekonomisk genomförbarhetsanalys av storskaliga RCG system. Trots att systemets

genomförbarhet är relativt känslig för elkostnader, kapacitetsfaktorn och bränslekostnader, ger RCG

systemet en återbetalningstid från 2 till 4 år samt en stabil elkostnad mellan 0,07 och 0,12 euro/kWh

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Table of Contents

Abstract ... 2

1 Background ... 6

1.1 Industrial steam systems ... 6

1.2 Cogeneration ... 7

2 Rankine compression gas turbine technology ... 8

2.1 Technology explanation ... 8

2.2 Proposed system (s) ... 9

2.3 Thesis motivation and outline ...10

3 System modelling ...12

3.1 Thermodynamic model ...12

3.2 System dynamic model ...15

3.2.1 Compressor ...16

3.2.2 Turbines ...18

3.2.3 Pipelines ...19

3.2.4 Valves ...22

3.2.5 Heat exchanger ...22

3.2.6 Transmission ...25

3.3 Model integration ...26

3.4 Control strategy ...29

4 Technical assessment ...30

4.1 General equipment considerations ...30

4.1.1 Steam turbine ...30

4.1.2 Compressor ...31

4.1.3 Gas turbine ...32

4.1.4 Heat exchanger ...32

4.2 Design of a 40 kW system ...33

4.2.1 Background information ...33

4.2.2 Equipment selection ...33

4.2.3 System design...38

4.2.4 System performance ...41

4.2.5 Control Strategy ...47

4.2.6 Sensitivity analysis ...47

4.3 Upgrade to a 100 kW system ...53

4.3.1 Equipment selection ...53

4.3.2 System design and performance analysis...55

5 Economic assessment of commercial-scale RCG systems ...61

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5.1 Commercial-scale RCG ...61

5.2 Cost and savings ...62

5.3 Feasibility analysis ...66

5.4 Sensitivity analysis ...68

6 Conclusion ...73

Bibliography ...74

Appendix A: Matlab-Simulink model ...76

Appendix B: Turbine map estimation ...87

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1 Background

This thesis is carried out in cooperation with the company Heat Power BV, which commercializes and holds the patent for the Rankine Compression Gas Turbine system (RCG). The company is currently entering the marketing with a commercial product that aims to provide peak-shaving cogeneration for industries using steam boiler systems.

This section is intended to provide some background information about industrial steam systems and a brief explanation about industrial cogeneration before explaining the RCG itself and the thesis objectives.

1.1 Industrial steam systems

The industrial sector is responsible for a large amount of the world’s energy consumption, accounting for roughly 35% of it (Ashish K. Sharma, 2017). This massive quantity of energy is used to meet different energy demands in the sector, from feeding a electric motor to drying bulk material, and, thus, the energy is used in different forms. The largest demands in the sector are electricity and process heat.

Process heat is, basically, applied in different industries for processes such as drying, calcination, distillation and extrusion (Roger Cundapí, 2017), accounting for approximately 45% to 65% of the total industry demand. So, in total, industrial process heat accounts for around 20% of the world’s total energy consumption.

Regarding industrial process heat, it can be divided mostly into direct-heating, which means that the combustion gases are directly used to transfer energy, and heating using another medium. The most common medium is steam for several reasons, e.g. it is non-hazardous, energy can be stored as latent heat, it has a high energy density and it is cheap (Ashish K. Sharma, 2017).

There are distinct ways to produce steam. A recurrent modern technology is the utilization of a furnace to burn fuel and have several water/steam tubes in the hot gases’ way, thus, transferring the heat through convection. Figure 1 presents a typical boiler: water inlet passes through an economizer, then it is transformed into steam in a bank of tubes and, finally, passes through a superheater. This type of system is multi-fuel, being suitable for using waste and biomass as a fuel.

Figure 1 - A typical vertical boiler. Extracted from (Vakkilainen, 2017)

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There are numerous concrete examples of segments that use steam, such as pulp and paper, food processing, textiles, chemical and wood processing. All those segments rely partially or totally in their steam systems to meet their heat demands. Often, the heating requirements are different within the same factory; meaning that distinct steam temperatures and, mainly, pressures are required.

A frequent solution for regulating the steam pressure is using a pressure reducing station. The working principle is basically to reduce the pressure with a valve and, sometimes, decrease the steam temperature with a desuperheater system (U.S. Department of Energy, 2004). On the other hand, this system is wasting potential energy that could be used, for example, in a steam turbine, which could generate power with this pressure difference, by expanding the steam. Such a solution, however, is not usually applied.

One of the issues is the fact that this solution is not usually feasible for low power production. Another possible reason for that is that the steam supply should always be guaranteed and, thus, no disturbance in it could be accepted. In this case, if the steam turbine is used to drive a generator, the electricity production cannot be at part load at any time.

To conclude, there is room, in industrial steam systems, to take advantage of this steam pressure drop for energy generation. Such a solution should be feasible, attend the industry requirements and be seamlessly integrated.

1.2 Cogeneration

As it was stated, a high amount of energy is used in industries worldwide and most of it is used as process heat. However, there is still a considerable quantity of energy that is consumed in the electricity form to power many devices, such as electric motors.

Thus, the concept of combined heat and power/cogeneration gets more and more relevant in the industrial sector. Companies that have both requirements could take advantage of their heat demand, which is usually locally produced, to produce their own electricity. Such systems present not only a financial advantage but also an environmental benefit and better usage of resources, namely the fuel.

However, when it comes to small cogeneration plants, with lower energy demand/production, the cost per

kW of electricity produced increases greatly as the power generation decreases (U.S. Environmental

Protection Agency, 2017). This is one of the main challenges in small-scale cogeneration, as some of the

components still have a high cost per kw of electricity produced.

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2 Rankine compression gas turbine technology

In this section, the Rankine compression gas turbine (RCG) technology is detailed.

2.1 Technology explanation

The RCG technology was first developed as an alternative to conventional gas turbine’s technologies. Its reasoning was based on filling a gap on the current energy generation systems, which would only reach high efficiencies and economic feasibility for high power productions, by creating a technology that would be feasible and efficient for the range between 1 and 10 MW. It was identified, by that time, that small gas turbines featuring a recuperator would fit a range up to 100 kW and that the conventional combined gas turbine and Rankine cycle would fit the range from 10 MW (Ouwerkerk, 2009).

The RCG technology consists in combined Rankine and gas turbine cycle, however with a different configuration of its components. In order to better explain this innovative cycle, it is helpful to first explore the conventional combined Brayton and Rankine cycle.

The main existing combined cycle technology involves the utilization of a regular Brayton cycle, in which the turbine and compressor are coupled, and the utilization of its exhaust gases as an input for an “isolated”

Rankine cycle (Figure 2). This cycle presents high efficiency and it is widely used for high power generation.

Figure 2 - Conventional combined gas turbine and Rankine cycle. (C: compressor; T: turbine; ST: steam turbine; G: generator). (Ouwerkerk & de Lange, 2006).

On the other hand, the RCG offers a distinct combined cycle, by rearranging its components. The main innovation is that the steam turbine shaft power is used as an input power for the compressor; by applying this concept, it can be stated that the compressor is driven by the waste heat generated by the Brayton cycle.

On top of that, as the turbine is not required for driving the compressor, it is decoupled from it; this results

in having a free power turbine (Figure 3).

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Figure 3 - Rankine compression gas turbine cycle diagram. (C: compressor; T: turbine; ST: steam turbine; G: generator). Adapted from (Ouwerkerk &

de Lange, 2006).

One of the advantages of having all output power on a free power turbine is that the shaft can operate easily on changing rotational speeds and loads, being able to be a quick response peak shaving device. Additionally, this characteristic not only enables the RCG to be used as a power generation technology but also enables it to be a mechanical drive, which is useful in other application such as ship propulsion.

As stated before, the application of such system is mainly focused on power outputs up to 10 MW. The thermodynamics theory of the system will be clarified in further sections, providing more information on technical aspects and elucidating the system’s advantages.

2.2 Proposed system (s)

Even though the RCG was developed aiming to be a range specific type of combined cycle using natural gas as a fuel, the technology has been proven to be useful for other applications. Heat Power BV (the company who holds RCG’s patent), targeting to get the system to the market, has encountered many different business cases that led the company to pivot and focus on a particular application.

The targeted application comprises heat recovery in industrial steam systems for small and medium

enterprises, mainly using biomass and waste as a fuel. The company’s proposed system aims electricity

cogeneration in the range between 100 kW and 2 MW. Essentially, the requirements for an industry to fit

as a possible customer for such system is that it uses steam in different pressures in its process and that this

steam is produced by the utilization of an industrial boiler. Many industries fulfil these requirements; some

examples are the food and beverage, pulp and paper and wood processing industries.

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The system consists of an add-on to existing process steam systems. Basically, it takes advantage of the steam expansion required, which is usually done by an expansion valve, to install a steam turbine; this steam turbine drives a compressor that compresses the air through a heat exchanger installed in the furnace. The hot compressed gases are expanded in a gas turbine and, finally, the still hot exhaust gases are injected back into the furnace. A scheme of the system can be viewed in Figure 4.

Figure 4 - Scheme of the RCG add-on to industrial steam systems. Extracted from (Heat Power BV, n.d.).

As this system takes advantage of an existing steam system, its capital expenditure is drastically reduced, making it feasible to have a small-scale electricity generation system on site. This add-on enables a better fuel utilization and a cheaper-than-the-grid electricity. In addition, as the steam turbine can run independently from the power turbine, the heat (steam) and electricity production can be decoupled;

together with the fast response of the free power turbine, it culminates in a peak shaving system.

Furthermore, this system presents a power generation between 15% and 35% higher compared to the alternative of connecting the generator directly in the steam turbine.

Finally, it also has a high energy efficiency as the exhaust gases from the gas turbine are reinjected into the furnace, being used again in the boiler.

It is also worth mentioning that even though the system can handle different types of fuel, the company focuses on the renewable energy sector, mainly in the utilization of biomass and waste.

2.3 Thesis motivation and outline

The company developed a 5 kW pilot system aiming to demonstrate the system’s proof of concept to

possible investors and potential clients. This pilot was installed in a wood processing factory, in the

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Netherlands; this industry uses biomass as its furnace’s fuel and the steam system is used to dry wet wood in the production line.

In order to keep developing the product, the company received investment to develop a closer to commercial scale system, which has been defined to be around 40 kW. This system is considered to be a new milestone for the technology and it aims to prove all the functionalities and advantages of the RCG technology.

In this regard, the first goal of this thesis is to design the 40 kW RCG system concerning the thermodynamic point of view, equipment selection, and main components design. Furthermore, this thesis also targets to develop the project of an upgrade of this 40 kW system to a higher power output system (closer to 100 kW), which will be presented to investors, focusing on guaranteeing a next round of investment.

The methodology applied to carry out this thesis can be divided into few steps. First, a review and thermodynamic analysis of the system is conducted so that the system can be completely understood, and its advantages and drawbacks can be well explored. Then, a model of the system is developed in order to be able to easily simulate the system behavior, output and process data; this model is intended to fulfil two main demands: be a quick tool to estimate new systems power performance and to evaluate the RCG’s transient operation. It is crucial to assess the system’s transient performance as its quick demand response is one of its selling points and this investigation will lead to a better control strategy.

Next, by using the model, the 40 kW system and a 100 kW upgrade of this system are technically assessed.

In this stage, not only the components will be selected and designed but also the system performance will be investigated. This study also includes a technical sensitivity analysis of the system related to its main parameters.

Finally, an economic analysis of commercial-scale systems is performed. The commercial-scale RCG

systems are based on scaling up the 100 kW system studied in the previous section. This investigation aims

to assess the system feasibility concerning its main influencing factors, such as fuel cost and capital

expenditure. The methodology for this section is to evaluate all the relevant factor that implies in the system

economic feasibility and calculate pertinent economic parameters.

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3 System modelling

In this section, the RCG technology is studied from a modelling perspective. The aim of the model is to be able to simulate a transient response of the system. The utilization of a transient model also enables a general steady-state evaluation of the system and, thus, being useful for all required analysis, both in technical and economic field.

3.1 Thermodynamic model

Considering the system as an add-on to an existing steam production system, some of the components can be neglected for this analysis. The main components to be examined are the gas turbine, the steam turbine, the heat exchanger and the air compressor. Figure 5 presents a diagram of the proposed RCG add-on system and its boundaries.

Figure 5 - RCG system diagram.

In the steam turbine, steam enters at a high pressure and leaves at a lower pressure; this expansion extracts energy from the steam stream and transform it into shaft power. Ideally, this process happens as an isentropic expansion, according to equation (1).

𝑃 _𝑠𝑡 = 𝑚 _𝑠𝑡 ̇ ( ℎ ₁ − ℎ _2𝑠 ) ⁽¹⁾

In which 𝑃 _𝑠𝑡 is the steam turbine power in kW, 𝑚 _𝑠𝑡 ̇ is the mass flow of steam in kg/s, ℎ ₁ is the enthalpy in stream 1 and ℎ _2𝑠 is the hypothetical enthalpy in stream 2 considering an isentropic expansion, both in kJ/kg.

However, no turbine is capable of performing an isentropic expansion and, thus, turbines always present an isentropic efficiency. The higher the efficiency, the closer the expanding process is to an isentropic expansion. So, the power output from the steam turbine can be calculate as stated in equation (2).

𝑃 _𝑠𝑡 = 𝜂 _𝑠𝑡 𝑚 _𝑠𝑡 ̇ ( ℎ ₁ − ℎ _2𝑠 ) (2)

-13- In which 𝜂 𝑠𝑡 is the steam turbine’s isentropic efficiency.

As steam properties database is widely available, equation (2) can be easily used to calculate the steam turbine power.

Another key component of the RCG system is the air compressor, which is a component that takes air from the environment, compressing and forcing it to flow. The work required for an ideal compressor to function is expressed in equation (3).

𝑃 _𝑐 = 𝑚 _𝑎 ̇ (ℎ ₃ − ℎ _4𝑠 ) ⁽³⁾

In which 𝑃 𝑐 is the compressor power in kW, 𝑚 𝑎 ̇ is the air mass flow in kg/s, ℎ 3 is the enthalpy in stream 3 and ℎ 4𝑠 is the hypothetical enthalpy in stream 2 considering an isentropic compression, both in kJ/kg.

Again, there is no compressor that can perform an isentropic compression and, therefore, there is an isentropic efficiency related to the process, as shown in equation (4). This equation can be rewritten as it is shown in equation (5).

𝑃 _𝑐 = 𝑚 _𝑎 ̇ (ℎ ₃ − ℎ _4𝑠 )

𝜂 _𝑐 ⁽⁴⁾

𝑃 _𝑐 = 𝑚 _𝑎 ̇ 𝑐 _𝑝,𝑎 (𝑇 ₃ − 𝑇 _4𝑠 )

𝜂 _𝑐 ⁽⁵⁾

In which 𝜂 _𝑐 is the compressor’s isentropic efficiency, 𝑐 _𝑝,𝑎 is the specific heat at constant pressure of the air in kJ/kg.K, 𝑇 3 is stream 3 temperature and 𝑇 4𝑠 is the hypothetical stream 4 temperature considering an isentropic compression, both in K.

Considering that the air behaves as an ideal gas, the isentropic relation expressed in equation (6) is valid.

The combination of equation (6) with equation (5) gives equation (7), which relates the compressor’s power with its air flow and pressure ratio.

𝑇 4𝑠

𝑇 ₃ = ( 𝑃 4

𝑃 ₃ )

𝛾−1

𝛾 (6)

𝑃 _𝑐 =

𝑚 _𝑎 ̇ 𝑐 _𝑝,𝑎 𝑇 ₃ (( 𝑃 ₄ 𝑃 ₃ )

𝛾−1 𝛾 − 1) 𝜂 _𝑐

(7)

In which 𝛾 is the specific heat ratio (the specific heat constant pressure divided by the specific heat at constant volume), 𝑃 ₄ is the air pressure in the stream 4 and 𝑃 ₃ is the air pressure in the stream 3.

The compressor’s outlet temperature is evaluated by equation (8).

𝑇 ₄ 𝑇 ₃ = 1 +

( 𝑃 ₄ 𝑃 ₃ )

𝛾−1 𝛾 − 1 𝜂 _𝑐

(8)

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As it has already been stated, the main difference between the RCG technology and a regular combined Rankine and gas turbine cycle is the fact that the steam turbine is directly driving the air compressor. So, the steam turbine’s power output is the compressor power input (𝑃 _𝑠𝑡 = 𝑃 _𝑐 ).

The gas turbine can be thermodynamically evaluated in a similar way. As equation (2) is also valid for the gas turbine and that air behaves as an ideal gas (equation (6)), an expression for the power output of the turbine is developed and shown in equation (9).

𝑃 _𝑔𝑡 = 𝑚 ̇ 𝑐 _{𝑎 𝑝,𝑎} 𝑇 ₅ 𝜂 _𝑔𝑡 (1− ( 𝑃 ₆ 𝑃 ₅ )

𝛾−1

𝛾 ) ⁽⁹⁾

In which 𝜂 𝑔𝑡 is the gas turbine’s isentropic efficiency, 𝑃 6 is the air pressure in the stream 6 and 𝑃 5 is the air pressure in the stream 5.

The turbine’s outlet temperature is evaluated by equation (10).

𝑇 ₆

𝑇 ₅ = 1 + 𝜂 _𝑡 (( 𝑃 ₆ 𝑃 ₅ )

𝛾−1

𝛾 − 1) ⁽¹⁰⁾

Another key component is heat exchanger. In this equipment, heat is transferred from the furnace’s combustion gases to the pressurized air. The relation between the total heat transferred and the pressurized air temperature is expressed by equation (11).

𝑄 _𝐻𝑋 = 𝑚 ̇ 𝑐 _{𝑎 𝑝,𝑎} (𝑇 ₅ − 𝑇 ₄ ) ⁽¹¹⁾

In which 𝑄 𝐻𝑋 is the heat transferred in the heat exchanger.

Finally, the last main equipment is the generator, which is evaluated in a simple way. The generator uses the shaft power of the turbine and transforms it in electricity; however, not all the shaft power is converted, as there are some losses in this device. In order to evaluate it, a generator efficiency is applied according to equation (12)

𝑃 _𝑔𝑒 = 𝑃 _𝑔𝑡 𝜂 _𝑔𝑒 ⁽¹²⁾

In which 𝑃 𝑔𝑒 is the generator’s power generation and 𝜂 𝑔𝑒 is the generator’s efficiency.

A steady-state simulation of the RCG performance can be carried out by applying the previously mentioned equations and defining some boundary conditions, such as the steam line pressures, air compressor compression ratio, etc. Additionally, together with the equations from next sub-section, a transient model can be created.

There a few relevant parameters that should be examined to assess the RCG’s performance. The first one will be called “power boost”; this parameter relates the gas turbine power with the steam turbine power.

The reasoning underpinning this this parameter is that it presents how much power could be generated in comparison with a hypothetical solution of coupling the generator directly in the steam turbine. Equation (13) presents a formulation for this parameter.

𝑃𝑜𝑤𝑒𝑟 𝑏𝑜𝑜𝑠𝑡 [%] = ( 𝑃 _𝑔𝑡

𝑃 _𝑠𝑡 − 1) × 100% ⁽¹³⁾

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Another appropriate parameter to be evaluate the system is the marginal electrical efficiency. Using the electric efficiency to evaluate such a system would lead to a wrong conclusion, as the main objective of the steam system is to deliver heat. The marginal electrical efficiency can be defined as the relation between the electricity generation and the extra fuel required, compared to the fuel demanded to produce only heat (Harvey, 2010).

There are, basically, two streams of energy that are used by the RCG system: the steam energy, that would be used, otherwise, to meet the heat demand and the energy that is extracted from the furnace through the heat exchanger. Nevertheless, the gas turbine’s outlet is re-injected in the furnace, which means that this energy is still used to generate steam. Hence, the extra fuel energy required to power the RCG system can be calculated through equation (14)

𝑄 _{𝑒𝑥𝑡𝑟𝑎} = 𝑄 _𝑠𝑡

𝜂 _𝑏 + 𝑄 _𝐻𝑋 − 𝑄 _𝑔𝑡𝑜 ⁽¹⁴⁾

In which 𝑄 _{𝑒𝑥𝑡𝑟𝑎} the extra fuel energy required for the RCG system, 𝑄 _𝑠𝑡 is the energy absorbed in the steam turbine, 𝜂 _𝑏 is the boiler’s efficiency and 𝑄 _𝑔𝑡𝑜 is the energy at the gas turbine’s outlet.

As the energy absorbed in the steam turbine can be considered equal to the steam turbine’s power, due to the first law of thermodynamics (energy conservation) and the air re-injected in the furnace will be substituting atmospheric air inlet in the furnace (at same conditions as stream 3 in Figure 5), equation (14) can also be written according to equation (15). If re-arranged and substituted by the previous equations, equation (15) can be expressed also as equation (16).

𝑄 _{𝑒𝑥𝑡𝑟𝑎} = 𝜂 _𝑠𝑡 𝑚 _𝑠𝑡 ̇ (ℎ ₁ − ℎ _2𝑠 )

𝜂 _𝑏 + 𝑚 ̇ 𝑐 _{𝑎 𝑝,𝑎} (𝑇 ₅ − 𝑇 ₄ ) − 𝑚 _𝑎 ̇ 𝑐 _𝑝,𝑎 (𝑇 ₆ − 𝑇 ₃ ) ⁽¹⁵⁾ 𝑄 _{𝑒𝑥𝑡𝑟𝑎} = 𝜂 _𝑠𝑡 𝑚 _𝑠𝑡 ̇ (ℎ ₁ − ℎ _2𝑠 )

𝜂 _𝑏 + 𝑃 _𝑔𝑡 − 𝑚 _𝑎 ̇ 𝑐 _𝑝,𝑎 (𝑇 ₃ − 𝑇 ₄ ) ⁽¹⁶⁾ Therefore, the marginal electric efficiency of the RCG system is evaluated, as previously stated, as the electricity generation divided by the extra energy required by the system, represented in equation (17).

𝜂 _{𝑚𝑎𝑟𝑔} = 𝑃 _𝑔𝑒

𝑄 _{𝑒𝑥𝑡𝑟𝑎} = 𝑃 _𝑔𝑒

𝜂 _𝑠𝑡 𝑚 _𝑠𝑡 ̇ (ℎ ₁ − ℎ _2𝑠 )

𝜂 _𝑏 + 𝑃 _𝑔𝑡 − 𝑚 _𝑎 ̇ 𝑐 _𝑝,𝑎 (𝑇 ₃ − 𝑇 ₄ )

(17)

In which 𝜂 _{𝑚𝑎𝑟𝑔} is the marginal electrical efficiency of the RCG.

3.2 System dynamic model

This section is intended to present the model used in the transient analysis of the system for all components.

The components’ governing equations that are shown below were used to develop a software in the Matlab- Simulink platform that is capable of simulating not only the steady-state but also the transient behavior of the RCG system.

Initially, it is crucial to identify which are the system’s boundaries, so that the relevant equipment can be

modelled. As the primary goal of this model is to evaluate the proposed add-on system, the relevant

equipment are the ones presented in Figure 5, namely the steam turbine, the air compressor, the heat

exchanger, the power turbine and the generator. Additionally, valves and piping are modelled since they

present a relevant behavior in the system performance and operation.

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Consequently, some boundaries must be defined, such as the steam inlet temperature and pressure, the steam outlet pressure, the air room temperature, etc. Those boundaries are elucidated in each component model explanation.

3.2.1 Compressor

The compressor is one of the key components in the RCG system and its proper simulation is crucial. The reason for that is not only to proper predict its performance but also to predict some harmful phenomena that might happen, namely compressor surging. Surge can be defined as “a sudden drop in delivery pressure, and, with violent aerodynamic pulsation which is transmitted throughout the whole machine”

(Saravanamuttoo, 2001). This phenomenon generates a counter flow in the compressor and can severely damage it.

There are distinct ways to predict the behavior of a compressor, differing in the level of complexity and the result’s quality (Schobeiri, 2016). For instance, one way to predict it is by using a compressor map, which presents all the relevant data (flow, pressure ratio and efficiency) in a graph. This method presents a good global performance but does not give many details on the dynamic behavior.

Another option would be carrying out the thermodynamic analysis of the adiabatic compression process, by using constructive parameters of the compressor such as its impeller’s angles. This option would express a close-to-reality result, although at the cost of performing a substantial number of calculations. In addition, a considerable amount of the compressor’s constructive information would be required.

There is a solution to have reliable transient formulation without all this constructive information and without requiring massive computational processing capacity. First developed in order to predict the behavior of axial compressors, this method foresee the transient response of the compression process to a deviation from the steady-state compression process (Greitzer, 1976). As the steady-state performance of a compressor is expressed by its map, this method is convenient to be applied.

The so-called Greitzer compression model considers the compressor as an actuator disk, shown in Figure 6. This model consists of: a compressor, represented by a pipe, with a Lc length, and an actuator disk, which represents the impeller, applying a force into the fluid; the pipe/vessel in which the compressor delivers the fluid, represented by the plenum; and throttle valve that represents the air outlet. This model consists of an

“emptying-and-filing” formulation.

Figure 6 - Greitzer compression model. (Greitzer, 1976).

The application of the one-dimensional linear momentum equation in the compressor results in equation

(18).

-17- 𝑑𝑚̇

𝑑𝑡 = − ∫ 𝐴(𝑥) 𝐿

𝑑𝑃 𝑑𝑥

𝐿𝑐 0

+ 𝐹 _𝑎𝑑 ⁽¹⁸⁾

In which 𝑚̇ is the fluid mass flow in kg/s, 𝐴(𝑥) is the cross-section area of the compressor pipe at the distance x from the starting point in meters, 𝐿 𝑐 is the compressor’s equivalent pipe length in meters, 𝑃 is the fluid pressure in Pa and 𝐹 𝑎𝑑 is the resultant force that the fluid is subjected to in Newtons.

This equation can be rewritten as presented in equation (19).

𝑑𝑚̇

𝑑𝑡 = − 𝐴

𝐿 _𝑐 ∆𝑃 + 𝐹 _𝑎𝑑 ⁽¹⁹⁾

Considering that at steady-state, the mass flow is constant, equation (20) is valid.

𝐴

𝐿 _𝑐 ∆𝑃 _𝑠𝑠 (𝑚̇, 𝑁) = 𝐹 _𝑎𝑑 ⁽²⁰⁾

In which ∆𝑃 𝑠𝑠 (𝑚̇, 𝑁) is the steady state pressure ratio for a given mass flow and rotational speed (N), in Pa.

Finally, equation (21) presents the dynamic behavior of an axial compressor, according to Gretizer’s formulation. It states that the mass flow changes directly proportionally to the difference between the actual pressure ratio and the pressure ratio the compressor would present, in steady-state, at the current rotational speed and mass flow.

𝑑𝑚̇

𝑑𝑡 = 𝐴

𝐿 _𝑐 (∆𝑃 _𝑠𝑠 (𝑚̇, 𝑁) − ∆𝑃) ⁽²¹⁾

Additionally, it shows that the relation between those two quantities is equal to a parameter ( _𝐿 ^𝐴

𝑐 ) that is related to the compressor geometry. This parameter can be evaluated either by the approximate length of the compressor and its average cross-sectional area or by carrying out experiments with the compressor.

The plenum can be easily modelled by the mass-conservation equation, which is represented in equation (22).

𝑉 _𝑝 𝑑𝜌 _𝑝

𝑑𝑡 = 𝑚̇ _𝑐 − 𝑚̇ _𝑣 ⁽²²⁾

In which 𝜌 𝑝 is the plenum density in kg/m ³ , 𝑉 𝑝 is the plenum’s volume in m ³ , 𝑚̇ 𝑐 is the mass flow of the compressor and 𝑚̇ _𝑣 is the mass flow in the throttle valve, both in kg/s.

Using the perfect gas law, considering the process a polytropic process and considering experimental results (Greitzer, 1976), equation (23) presents the plenum’s pressure differential equation.

𝑑𝑃 _𝑝 𝑑𝑡 = 𝑎 ²

𝑉 _𝑝 (𝑚̇ _𝑐 − 𝑚̇ _𝑣 ) ⁽²³⁾

In which 𝑃 _𝑝 is the plenum’s pressure in Pa and 𝑎 is the speed of the sound in this medium in m/s.

In addition, the power related to the compressor is evaluated by equation (7) and the compressor’s outlet temperature is calculated by equation (8).

It is important to mention that, in further studies, this model was proved to have good results for centrifugal

compressors as well (Hansen, et al., 1981). So, this model can be applied no matter the compressor

technology. Additionally, this model assumes that there is a compressor map to check the steady-state

operating conditions.

-18- 3.2.2 Turbines

The gas turbine is the component responsible for delivering all the shaft power outlet of the system and, thus, it is important that the simulation can predict its response to the control mechanisms.

As it is also a turbomachine, just like the compressor, it can also be modelled in a similar way, as an actuator disk. The difference is that the fluid flows from the plenum to the turbine, represented by the actuator disk, as presented in Figure 7.

Figure 7 - Expansion model. Adapated from (Greitzer, 1976).

To avoid being repetitive, the equations are not developed as in the compressor’s case, although the reasoning behind it is the same one. The result is the two equations represented by equations (24) and (25).

𝑑𝑚̇

𝑑𝑡 = 𝐴

𝐿 _𝑡 (∆𝑃 _𝑠𝑠 (𝑚̇, 𝑁) − ∆𝑃) ⁽²⁴⁾

𝑑𝑃 _𝑝 𝑑𝑡 = 𝑎 ²

𝑉 _𝑝 (𝑚̇ _𝑡 − 𝑚̇ _𝑣 ) ⁽²⁵⁾

In which 𝑚̇ 𝑡 is the mass flow through the turbine in kg/s and 𝐿 𝑡 is the equivalent length of the turbine in meters.

It is important to mention that this method also relies on the availability of a turbine’s map.

So, essentially, this formulation can be applied for both the gas turbine and the steam turbine. However, as it is intended to be an add-on system, their integration with the existing system occurs differently.

The gas turbine inlet is the RCG pipeline and its outlet is the furnace pressure, which is defined by the existing process. Hence, both equation (24) and (25) are used in the gas turbine simulation. Additionally, equation (9) is used to evaluate the gas turbine power output and equation (10) is used to evaluate the turbine’s outlet temperature. The gas turbine map shows the pressure ratio/mass flow relation and the turbine efficiency for each rotational speed.

On the other hand, the steam turbine works with pressure difference between the steam inlet pipe and the steam outlet pipe. Those values might change depending on the process requirements at each instant and, therefore, they are considered boundaries. It means that the plenum equation (equation (25)) is not required to simulate its behavior, as it would require simulating the whole steam process, beyond the RCG’s defined boundaries.

As it will be explained in further sections of this thesis, the type of turbine that is more suitable for the RCG

is the impulse turbine. Its basic working principle is a nozzle expansion of steam, generating a high-speed

steam jet that moves a wheel, as shown in Figure 8.

-19-

Figure 8 - Impulse steam turbine principle. Extracted from (Ouwerkerk, 2009).

In this case, the turbine map is different, and it is related to the nozzle. The map relates the tip speed ratio, to the turbine’s isentropic efficiency. The tip speed ratio is the relation between the steam jet and the tip speed of the turbine. Equation (26) presents those relations. Additionally, the mass flow through the nozzle is determined by equation (27).

𝑇𝑖𝑝 𝑠𝑝𝑒𝑒𝑑 𝑟𝑎𝑡𝑖𝑜 = 𝑢

𝑐 ₀ = 2𝜋𝑟 _𝑆𝑇 𝑁 _𝑆𝑇 60

√2(ℎ 𝑆𝑇,𝑖𝑛 − ℎ _{𝑆𝑇,𝑜𝑢𝑡} )

(26)

In which, ℎ _{𝑆𝑇,𝑖𝑛} is the steam inlet enthalpy in kJ/kg, ℎ _{𝑆𝑇,𝑜𝑢𝑡} is the isentropic enthalpy outlet in kJ/kg, 𝑟 _𝑆𝑇 is the steam turbine radius in m and 𝑁 𝑆𝑇 is the steam turbine rotational speed in RPM.

𝑚̇ _𝑆𝑇 = 𝐶 _{𝑛𝑜𝑧𝑧𝑙𝑒} 𝑃 _{𝑆𝑇,𝑖𝑛}

√𝑇 𝑆𝑇,𝑖𝑛

(27)

In which 𝑚̇ _𝑆𝑇 is the steam mass flow in kg/s, 𝐶 _{𝑛𝑜𝑧𝑧𝑙𝑒} is the nozzle’s constant, 𝑃 _{𝑆𝑇,𝑖𝑛} is the inlet steam pressure in Pa and 𝑇 𝑆𝑇,𝑖𝑛 is the steam’s inlet temperature in K.

3.2.3 Pipelines

The reason to model the pipelines is the fact that they some other dynamic behavior to the system; firstly, they generate a pressure drop in the fluid and, secondly, because the pipelines also can present oscillating behaviours.

The pipelines’ dynamics can be modelled by both the mass conservation equation and the linear momentum equation (Matko, et al., 2001). The result of the first equation is the same as the plenum’s pressure equation in the previously mentioned compressor/turbine cases and it is presented in equation (28).

𝑑𝑃 _𝑝𝑙 𝑑𝑡 = 𝑎 ²

𝑉 _𝑝𝑙 (𝑚̇ _𝑖𝑛 − 𝑚̇ _𝑜𝑢𝑡 ) ⁽²⁸⁾

In which 𝑉 𝑝𝑙 is the pipeline volume in m ³ and 𝑃 𝑝𝑙 is the pipeline’s pressure in Pa.

Concerning the linear momentum equation applied to the pipeline and neglecting the gravity forces that the fluid is subjected to, the result is present in equation (29).

𝑑𝑚̇

𝑑𝑡 = ∑ 𝐹 _𝑒𝑥𝑡 = − 𝜕𝑃

𝜕𝑥 𝐴 − 𝐹 _𝑓 ⁽²⁹⁾

-20-

In which P is the pressure in Pa, A is the cross-section area of the pipeline and 𝐹 𝑓 is the friction force in Newtons.

The friction force can be evaluated by using the well-established Darcy–Weisbach equation. This formulation was established through a series of pipe-flow experiments, resulting in a reliable equation that can predict the friction related to a flow for distinct pipe roughness and flow regime (White, 2011). When applied to equation (29), the result is presented in equation (30).

𝑑𝑚̇

𝑑𝑡 = 𝜕𝑃

𝜕𝑥 𝐴 − 𝑓𝑚̇ ² 2𝐷𝐴𝜌̅ = ∆𝑃

𝐿 𝐴 − 𝑓𝑚̇ ²

2𝐷𝐴𝜌̅ ⁽³⁰⁾

In which ∆𝑃 is the pressure difference between the pipeline’s inlet and outlet in Pa, L is the pipeline’s length, D is the pipeline’s diameter in meters, 𝜌̅ is the average density of the fluid in kg/m ³ and 𝑓 is the Darcy friction factor.

The Darcy friction factor can be determined by different methods. One well-established methodology is consulting the Moody chart for pipe friction, which is a good method for a steady-state simulation. However, as this model is intended to predict a transient behaviour, the Colebrook equation (equation (31))is used to evaluate the friction factor.

1

√𝑓 = −2 log ( 𝜖 ⁄ 𝑑

3.7 + 2.51

𝑅𝑒 _𝑑 √𝑓 ) ⁽³¹⁾

Additionally, there are other friction sources rather than the distributed friction in the pipeline’s flow and they are called “minor losses”; some examples of minor losses are: pipe entrance and exit, expansions and contractions, bends, tees and valves (White, 2011). In this pipeline model, the bends are relevant pressure drop sources and, thus, they are also considered in the pipeline dynamics.

When this phenomenon is included in equation (30), the result is equation (32).

𝑑𝑚̇

𝑑𝑡 = ∆𝑃

𝐿 𝐴 − 𝑓𝑚̇ ²

2𝐷𝐴𝜌̅ − 𝐾𝑚̇ ²

2𝐴𝐿𝜌̅ ⁽³²⁾

In which K is the minor loss coefficient, which can be found through experimental results and can be

consulted in graphs such as the one presented in Figure 9.

-21-

Figure 9 - Minor loss coefficient for a 90° bend. Extracted from (White, 2011).

As it has been shown, equation (28) is equivalent to plenum’s equation in the turbine and compressor model, what raises the question on whether the pipeline dynamics should be considered before or after the turbomachinery plenum.

Aiming to tackle this issue, an experimental study has been carried out to compare the results for different model configurations of the combination of a compressor with the pipeline transient equations. This study conclude that the closer-to-reality model is referent to the pipeline dynamics being added between the valve and the plenum (Yoon, et al., 2011), as shown in Figure 10.

Figure 10 - Block diagram of the compressor and pipeline model

Basically, this diagram shows each component’s numerical model inputs and outputs. For example, the plenum’s equation (equation (23)) uses the mass inlet from the compressor (equation (21)) and the mass outlet from the pipeline (equation (32)(30)); while its result, the plenum’s pressure is used in the compressor and pipeline equations.

Unfortunately, this study has not been carried out for turbines, probably because the surge effect is not a recurrent problem in turbines. So, in order to include the pipeline dynamics, it was chosen to include it following the same logic of the compressor’s result, between the throttle valve and the turbine’s plenum, as shown in Figure 11.

Figure 11 - Block diagram of the turbine and pipeline model.

-22- 3.2.4 Valves

The throttling valves were also contemplated by Greitzer in the compression model (Greitzer, 1976). The valve dynamics can be derived from the one-dimensional linear momentum equation, just like the compressor and the turbine, considering it an actuator disk.

To avoid being repetitive, the steps to reach the formulation are to present. However, they are analogue to the equation development from equation (18) to equation (21), resulting in equation (33).

𝑑𝑚̇

𝑑𝑡 = 𝐴

𝐿 _𝑉 (∆𝑃 − ∆𝑃 _𝑠𝑠 (𝑚̇, 𝐶𝑉)) ⁽³³⁾

In which A is the cross-section area of the valve in m ² , 𝐿 𝑉 is the equivalent length of the valve pipe in m,

∆𝑃 _𝑠𝑠 (𝑚̇, 𝐶𝑉) is the steady-state pressure difference, in Pa, in the valve for a given mass flow and flow coefficient (CV), which is explained further in this thesis, and ∆𝑃 is the actual pressure difference in the valve in Pa.

The valve’s performance in steady-state is determined by a flow coefficient, which is usually represented as CV. This coefficient is found out by performing experiments and it describes the relation between pressure drop and flow. The current Heat Power’s supplier provides equation (34) to evaluate the flow when the pressure downstream is more than 50% of the inlet pressure and equation (35) when the downstream pressure is less than 50% of its inlet.

𝐶𝑉 = 𝑄

380 √ 𝑆𝐺 × 𝑇

∆𝑃 × 𝑃 2

(34)

𝐶𝑉 = 𝑄

205 𝑃 ₁ √𝑑 × 𝑇 ⁽³⁵⁾

In which Q is the flow rate in Nm ³ /s and SG is the specific gravity of the fluid referred to air, T is the fluid temperature in K, ∆𝑃 is the pressure drop in kg/cm ² , d is the valve diameter in m, 𝑃 1 is the inlet pressure and 𝑃 2 is the outlet pressure, both in kg/cm ² .

Equation (35) represents the choked flow through the valve, in which the flow is not related to the pressure drop anymore.

It is worth mentioning that control valves usually have a mechanism to actively change its flow coefficient, being able to control a process variable by partially closing and opening itself.

3.2.5 Heat exchanger

The heat exchanger is another crucial component on the RCG system that should be dynamically modelled.

The heat exchanger is the equipment responsible for the heat transfer between the combustion gases, in the kiln, and the RCG’s compressed air. Heat exchangers are usually classified according to their type of construction and their flow arrangement (DeWitt, et al., 2007).

There mainly three possible flow arrangements, which are referred to the relative hot and cold fluid’s flow

direction: parallel flow, counterflow and cross flow (Figure 12). The decision between each of them is usually

-23-

related to physical requirements, as their effectiveness can be easily ranked, being the counterflow the most effective arrangement and the parallel flow the less effective one.

Figure 12 - Flow arrangements for heat exchangers. Extracted and adapted from (DeWitt, et al., 2007).

Regarding the type of construction there many possible options such as concentric tubes, as shown in Figure 12’s two first options, bundle of tubes, as shown in Figure 12’s last option, parallel plates, shell and tube, etc.

In order to focus on the viable options for the RCG system, it is essential to understand that most of the first systems that will be installed, such as the current one, are retrofits from existing steam generation systems. Often it will be difficult to place a counterflow heat exchanger in the combustion gases path, as the furnace has not been projected to accommodate it. Consequently, the most likely heat exchanger to fit RCG systems is a bundle of tubes in a cross-flow arrangement.

Basically, the heat flux mechanism is a forced convection from the hot fluid to the outer fouling layer in the tube, then energy is transferred from the outer fouling to the tube and to the inner fouling layer by conduction and, finally, transferred from the inner fouling layer to the cold fluid, inside the tubes, by convection again. Figure 13 depicts the mentioned components

Figure 13 - Heat exchanger by heat transferring component. Extracted and adapted from (DeWitt, et al., 2007).

-24-

This is a complex methodology to be implemented as there are many components. Although, as the heat exchanger will be operating at forced convection in both sides, the conduction heat fluxes might be neglected. In order to evaluate it, a dimensionless parameter is used, the Biot number, expressed in equation (36).

𝐵𝑖 = ℎ𝐿 _𝑐 𝑘

(36)

In which 𝐵𝑖 is the Biot number, ℎ is the convection heat transfer coefficient in W/m ² K, 𝐿𝑐 is the pipe ratio of volume to surface area in m, and 𝑘 is the tube thermal conductivity in W/mK.

The Biot number presents the ratio of thermal resistivity between the convection and the conduction processes. If Bi < 0.1, it is reasonable to state that the tube presents a uniform distribution of temperature.

As the order of magnitude of Biot number for this type of heat exchanger is around 0.01, this assumption can be applied correctly. Hence, the heat flux mechanism is simplified to a convection flow from the hot fluid to the tube and another convection flow from the tube the cold fluid.

So, the heat exchanger model implemented in this system will divide the heat exchanger pipes into a number N of elements and analyse the heat transfer in each of those sections. For each of them, the energy inflow is represented in equation (37).

𝑄 _𝑖𝑛 − 𝑄 _𝑜𝑢𝑡 = 𝑚 𝑁 𝑐 _𝑝 𝑑𝑇

𝑑𝑡 = 𝐴 _𝑒𝑥𝑡 ℎ _𝑒𝑥𝑡 (𝑇 ∞,𝑒𝑥𝑡 − 𝑇) − 𝐴 _𝑖𝑛𝑡 ℎ _𝑖𝑛𝑡 (𝑇 − 𝑇 ∞,𝑖𝑛𝑡 ) ⁽³⁷⁾ In which m is the total heat exchanger mass in kg, N is the number of discrete elements in it, 𝑐 𝑝 is the heat capacity of the heat exchanger material in W/kgK, 𝐴 𝑒𝑥𝑡 is the external area of the discrete element, ℎ 𝑒𝑥𝑡 is the external convection heat transfer coefficient of the discrete element in W/m ² K, 𝐴 𝑖𝑛𝑡 is the internal area of the discrete element, ℎ 𝑖𝑛𝑡 is the internal convection heat transfer coefficient of the discrete element in W/m ² K, T is the temperature of the heat exchanger discrete element in K, 𝑇 ∞,𝑒𝑥𝑡 and 𝑇 ∞,𝑖𝑛𝑡 are the external and internal fluid temperatures in K.

In the external flow case, 𝑇 ∞,𝑒𝑥𝑡 is to be considered the bulk fluid temperature, defined by the average between the inlet and outlet temperature, which are evaluated for the flow as a whole by equation (38). On the other hand, 𝑇 ∞,𝑖𝑛𝑡 is the fluid temperature in each section of the heat exchanger, which is per discrete element and is represented by equation (39).

𝑄 𝑖𝑛 = ∑ 𝐴 𝑒𝑥𝑡 ℎ 𝑒𝑥𝑡 (𝑇 ∞,𝑒𝑥𝑡 − 𝑇) = 𝑚̇𝑐 𝑝 (𝑇 𝑜𝑢𝑡 − 𝑇 𝑖𝑛 ) ⁽³⁸⁾ 𝑄 _𝑜𝑢𝑡 = 𝐴 _𝑖𝑛𝑡 ℎ _𝑖𝑛𝑡 (𝑇 − 𝑇 _{∞,𝑖𝑛𝑡} ) = 𝑚̇𝑐 _𝑝 (𝑇 _𝑜𝑢𝑡 − 𝑇 _𝑖𝑛 ) ⁽³⁹⁾ In which in the right-hand side of the equation all the properties and temperatures are related to the fluids, external in equation (38) and internal in equation (39).

Additionally, the internal flow conditions are evaluated with the same pipeline equations, presented in equations (28) and (32), as the pressure drop is considered to be one main issue in the heat exchanger.

Finally, aiming to apply the pipeline dynamics in the heat exchanger, the tubes’ curves and the fluid inlet

and outlet must be accounted as a minor pressure loss. Figure 14 presents the loss coefficients for the bundle

of tubes inlet (sudden contraction) and outlet (sudden expansion). As the tubes will have a greatly smaller

diameter than the air pipelines before and after the heat exchanger, _𝐷 ^𝑑 will be considered close to zero. Thus,

the fluid dynamics of the heat exchanger is presented in equations (40) and (41).

-25-

Figure 14 - Sudden expansion and sudden contraction minor pressure loss coefficients. Extracted from (White, 2011).

𝑑𝑃 _𝐻𝑋 𝑑𝑡 = 𝑎 ²

𝑉 _𝐻𝑋 (𝑚̇ _𝑖𝑛 − 𝑚̇ 𝑜𝑢𝑡 ) ⁽⁴⁰⁾

𝑑𝑚̇

𝑑𝑡 = ∆𝑃

𝐿 𝐴 − 𝑓𝑚̇ ²

2𝐷𝐴𝜌̅ − 𝐾 _{𝑐𝑢𝑟𝑣𝑒𝑠} 𝑚̇ ²

2𝐴𝐿𝜌̅ − 0.4𝑚̇ ²

2𝐴𝐿𝜌 _{𝑖𝑛𝑙𝑒𝑡} − 1𝑚̇ ²

2𝐴𝐿𝜌 _{𝑜𝑢𝑡𝑙𝑒𝑡} ⁽⁴¹⁾

3.2.6 Transmission

The steam turbine is coupled with the compressor through a transmission, which is intended to be a belt transmission. Although, the formulation presented in this section can be applied for any type of transmission.

The transmission system, in this case, is modelled with three components: the steam turbine shaft, and

intermediate shaft and the compressor shaft as presented in Figure 15. The stream turbine shaft drives the

intermediate transmission shaft, with a certain transmission ratio, which drives the compressor, with another

transmission ratio.

-26-

Figure 15 - Steam turbine and compressor transmission scheme.

This component is modelled by modelling the intermediate shaft dynamic response, present in equation 𝑑𝑁

𝑑𝑡 =

𝜂 _𝑡𝑟 𝑃 _𝑠𝑡 − 𝑃 _𝑐 𝜂 _𝑡𝑟 𝑁 ∑ 𝐼

(42)

In which N is the intermediate transmission shaft rotational speed in rpm, 𝜂 𝑡𝑟 is the transmission efficiency, 𝑃 _𝑠𝑡 is the steam turbine power output in kW, 𝑃 𝑐 is the compressor power input in kW and ∑ 𝐼 is the sum of all three elements rotational inertia in kgm ² .

The turbomachines’ power output/input are determined by their thermodynamic relations, already presented and their current rotational speed is determined by multiplying their respective transmission ratio to the intermediate shaft by the intermediate’s shaft rotational speed.

It is worth mentioning that a belt transmission, such as the timing belt for example, presents a transmission efficiency between 97% and 99%, being almost neglectable (Nisbett, 2008).

3.3 Model integration

The system governing equations that are expressed in the previous subsections of this chapter must be integrated in a software in order to simulate the transient response of the system. The thermodynamic, fluid dynamic and mechanical equations are mainly differential equations that require an integration process, making Matlab-Simulink a good tool to be used.

This software presents a user-friendly graphic interface with many functionalities embedded. One of the main functionalities is the integration over time block, in which it receives an initial state and the current derivative, giving the output. One example to depict this function is presented in Figure 16, in which the constructive and process data are received in the left-hand side of the block “pipeline dynamics” so that the time derivative of the pressure and mass flow can be evaluated (equations (28) and (32)) and, then, integrated in their integration blocks (represented a 1/s).

The same logic was applied for all the equipment present in the RCG system.

-27-

Figure 16 - Pipeline dynamic response in Simulink.

An overview of a simple RCG system model is shown in Figure 17. The two green blocks present the air and steam conditions, which are boundaries for the RCG system. In this overview, it is possible to analyze each blocks’ process inputs and outputs. Additionally, there are many non-process related inputs, such as diameters and turbomachinery maps, that are include inside each of those blocks.

It is important to point out that even though Figure 17 presents a simple RCG system, the software is carried out in such a way that it is modular, scalable and robust, meaning that many distinct arrangements can be evaluate with the same software; some examples of possibilities are: placing valves between pipelines, using two or more heat exchangers, placing compressors and turbines in parallel and series and create by- pass pipelines.

Appendix A: Matlab-Simulink model presents the system’s model, component by component and shows

how to integrate them.

-28-

Figure 17 - Overview of a simple RCG system model in Matlab-Simulink.

-29-

3.4 Control strategy

Besides the power boost, the RCG also presents, as a selling point, its quick response power generation.

This advantage is because all the power output is on the gas turbine, which acts as a free power turbine and can vary its power output quickly.

To perform this quick adjustment of the power generation, a reliable and robust control strategy should be used. There are three possible strategies that are studied in this thesis:

• By-pass the heat exchanger and, thus, have a lower turbine inlet temperature.

• By-pass the turbine and, thus, reduce the mass flow over the turbine.

• Throttle the air flow and, thus, reduce the turbine inlet pressure.

• By-pass both the turbine and the heat exchanger at once.

Those options are investigated in the next chapter of this thesis, in the 40 kW system design.

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4 Technical assessment

In this section, the RCG systems are designed. Firstly, the main goal is to design the 40 kW system, which still is not expected to be at a commercial scale, but it will be used as a showcase for the product. Then, an upgrade of this system is developed, reaching around 100 kW of electric power output.

The design includes selection and arrangement of the main equipment, control strategy, instrumentation and the performance analysis.

4.1 General equipment considerations

This sub-section aims to point out some important considerations about some of the RCG equipment.

Initially, it is important to understand that one of the main issues for the RCG design and equipment selection is to find feasible solutions for small-scale generation. Products that require a high technological solution are usually expensive for small-scale power generation; for example, in a large-scale biomass power plant, it is feasible to implement a high efficiency steam turbine, while on a small-scale one, the trade-off is not advantageous.

In order to have a feasible final project, it is crucial to seek off-the-shelf products, which do not require extra engineering to be developed and, thus, are cheaper than customized products, which would present a better performance.

Following this reasoning, an evaluation/benchmarking of the main components availability and technologies is carried out to narrow down the equipment options. The findings are presented subsequently by component.

4.1.1 Steam turbine

The selection of the steam turbine must take into consideration from technical to economic aspects.

Concerning the technical aspects, there two main categories to be considered for the selection process, which are the reaction and the impulse turbines.

The impulse turbine, essentially, works by carrying out all the expansion process in a nozzle, generating a high-speed steam jet, which’s energy is transformed into mechanical energy in the turbine blades, which have a “bucket” form. On the other hand, in the reaction turbine, part of the expansion is done in a stator and then the rest occurs in the moving blades of the turbine.

The selection process between one of those two major types can be summarized by a few characteristics.

An impulse turbine presents better efficiency for small steam volumes and higher pressure ratio; an impulse turbine presents a lower number of stages, compared to a reaction turbine; an impulse turbine offers a longer time between maintenance events (Bloch, 2009) (Sarkar, 2015).

To sum up, the impulse turbine presents some of the characteristics desired for this project, which are the larger maintenance interval, lower number of stages, what reduces costs. Also, it presents higher pressure ratios at considerably low steam flows.

Commercially, products in different ranges can be found. For a small-scale turbine, it is usual to find multistage steam turbines with efficiencies up to 65% and single stage turbines with efficiencies in the 40%

range (U.S. Environmental Protection Agency, 2017); those ranges were also validated by a commercial

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benchmarking. It is interesting to point out the difference between those efficiencies and the ones for large- scale power plants, which can get up to 90%, showing one of the already mentioned challenges of the small- scale cogeneration.

4.1.2 Compressor

There are different compressor technologies in the industry used for distinct applications, each of them with their own advantages and drawbacks. The two compressors to be considered for the RCG application are the centrifugal and the axial compressors, both rotodynamic machines. Operationally, in the centrifugal compressor, the air enters the impeller in the axial direction and is forced to leave in the radial direction; on the other hand, the axial compressor does not change the direction of the fluid.

When comparing those two options, it can be generally stated that the centrifugal compressors are more suitable for smaller flows and they have a higher pressure ratio per stage (Kappis, 2013). Additionally, axial compressors present a higher peak-efficiency, however it has a lower off-design efficiency, being less robust (Boyce, 2012). As the RCG is intended for multiple operating points and, generally, small air flow, the centrifugal compressor is more advantageous. Also, Figure 18 gives an overview on different technologies and visually shows the robustness of centrifugal compressors.

Figure 18- Different compressor types ranges regarding mass flow and pressure ratio. Extracted from (Kappis, 2013).

A market benchmarking has reinforced that centrifugal compressors are the most suitable compressors for this application. Two main options were found for this equipment: multistage and single stage. More expensive, the multistage compressors are available for different applications, they present a wide range of operating points, are more expensive and can be found with isentropic efficiencies from 60% to 72% in the off-the-shelf small-scale set of products.

On the other hand, single stage compressors were found to be less costly, have a narrower operating range

and, more important, with isentropic efficiencies up to 80% in the off-the-shelf small-scale category. Thus,

being preferable to be used over the multistage options.