# An Isothermal Steam Expander for an Industrial Steam Supplying System

Authors: Chen-Kuang Lin; Guang-Jer Lai; Yoshiyuki Kobayashi; Masahiro Matsuo; Min-Chie Chiu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Steam is an essential medium used in the industrial process. To ensure steam quality, small and middle scale boilers are often adopted. However, because a higher steam pressure (compared to the necessary steam pressure) is generated, the boiler’s steam pressure will be reduced via a pressure regulator before the steam is directed through the process. Unfortunately, pressure is somewhat wasted during the reducing process. Therefore, in order to promote energy efficiency, a pressure regulator is replaced by a steam expander. With this steam expander, the pressure will be transformed into mechanical energy and extracted during the expansion process. A new type of isothermal steam expander for an industrial steam supplying system will be presented in the paper. The isothermal steam expander will improve the energy efficiency of a traditional steam expander by replacing the isentropic process with an isothermal expansion process. With this, steam condensation will decrease, energy will increase, and steam quality will be improved. Moreover, the mathematical model of the isothermal steam expander will be established by using the Schmidt theory, the same principle used to analyze Stirling engines. Consequently, by verifying the correctness of the theoretical model for the isothermal steam expander using experimental data, a prototype of 100 c.c. isothermal steam expander is constructed.


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## Body

## 1. Introduction

Because of the continuing increase in the cost of electricity, reducing manufacturing costs using energy-saving processes is essential. To ensure steam quality, the real pressure is often designated at a higher pressure than its targeted pressure; therefore, the steam is passed through a pressure regulator reducing the pressure before steam is introduced into the system. In this reduction process, energy will be wasted. This drawback can be improved if the pressure regulator is replaced by the steam expander [1–6].The isothermal expander was inspired by the Rankine Stirling cycle proposed by Isshiki et al. [7]. In that system, steam was guided into the Stirling cycle to improve output efficiency. In 2009, Yoshiyuki [8] developed an α type steam Stirling engine with a higher output efficiency [9]. In line with the design concept of the Rankine Stirling cycle and an α type steam Stirling engine, a new type of isothermal steam expander used to increase output work is proposed in this paper. Here, an isentropic process is replaced by the isothermal expansion process. The isothermal steam expander can reduce steam condensation, increase output efficiency, and improve steam quality. In order to enhance the isothermal expansion process, the heat regenerator for the traditional steam expander is improved. Also, based on the Schmidt theory, the theoreticalP-V diagram and related output efficiency will be established. Consequently, a prototype 100 c.c. isothermal steam expander is built and used to verify the correctness of the mathematical model.


## 2. Design Concept of the Isothermal Steam Expander

A cogeneration system is very efficient for energy transformation. Current cogeneration systems are often linked to large scale steam turbines. However, the development of a small scale steam supplying system used in the cogeneration system is rare. In order to develop cogeneration systems that extract more output work from the steam’s pressure, research on a small scale cogeneration system linked with a small boiler and a steam expander becomes crucial. In this study, the traditional steam expander is improved. In order to decrease the steam’s condensation, the isentropic process is replaced by an isothermal expansion process using a thermal expansion type steam expander.As indicated in Figure1, the isothermal expansion cycle of the steam expander is composed of two isothermal processes and two isovolumetric processes. High pressure steam will be guided into the isothermal expansion cycle of the steam expander. The cycle will produce heat and then do mechanical work. Subsequently, steam will be discharged and introduced into the manufacturing process. The steam pressure will then be decreased and extra heat will be produced. The resulting quality of the steam released from the isothermal expansion cycle will be improved. Moreover, the output efficiency produced by the isothermal expansion cycle is increased beyond that of the traditional expander.Figure 1
The ideal isothermal expansion cycle for a steam expander.As indicated in Figure2, the isothermal steam expander is similar to the β type Stirling cycle. The phase angle between the piston and the regenerator (displacer) is set at 90°. Because the intake valve and the exhaust valve are set in the isothermal steam expander, high pressure gas will be introduced into the cycle when the piston approaches top dead center. Additionally, low pressure gas will be discharged when the piston approaches bottom dead center. However, the 90° phase angle is not the best value for producing more mechanical work. Therefore, based on the Schmidt theory, an analysis of the best phase angle for the isothermal steam expander is assessed in the following section.Figure 2
The isothermal steam expander.


## 3. Mathematical Model of the Isothermal Steam Expander

The isothermal expander is similar to theβ type Stirling engine cycle. Based on the Schmidt theory [10, 11], the relationship in theP-V diagram between the instant pressure and the volume inside the isothermal expander will be predicted and plotted. Also, based on the β type Stirling cycle’s Schmidt theory, the relationship between the instant pressure and the axial output phase angle can be predicted. The related temperature ratio for the hot end and the cold end, the stroke volume ratio, and the dead space ratio are
(1)τ=VcVh,κ=VspVsd,X=Vdh+Vr+VdcVh
assuming that(2a)ψ=tan-1⁡κsinατ+κcos⁡α+1,(2b)XB=VbVsd,(2c)S=τ+4τX1+τ+τ+1-2XB,(2d)B=τ2+2κτ+1cos⁡α+κ2+2τ+1,(2e)δ=BS.According to the Schmidt theory, the relationship between the instant pressure and the output phase angle is
(3)P=Pmean1-δ21-δ·cos⁡⁡θ-ϕ.For the isothermal expander, the instant pressure will be the same asPh (the pressure of the pressure source) when the valve opens. The average effective Pmean-e can be obtained by including the θhvc and Ph (occurring at the valve closing) in the calculation. Here, P will be the same as Ph at θhvo>θ>θhvc. The instant pressure during the expansion process is
(4)Pe=Pmean-e1-δ21-δ·cos⁡⁡θ-ψwhereθhvc>θ-α>θlvo,Ph=Pmean-e1-δ21-δ·cos⁡⁡θhvc-ψ,Pmean-e=Ph1-δ·cos⁡θhvc-ψ1-δ2,Pe=Ph1-δ·cos⁡θhvc-ψ1-δ·cos⁡⁡θ-ψwhereθhvc>θ-α>θlvo.
Similarly, the instant pressure of the isothermal compression process is
(5)Pc=Pl1-δ·cos⁡⁡θlvc-ψ1-δ·cos⁡⁡θ-ψwhereθlvc>θ-α>θhvo.
Because the instant pressure will be the same as Ph (the pressure of the pressure source) when the valve opens, the relationship between the instant pressure and the axial output’s phase angle for the isothermal expander is
(6)P=Pe=Ph1-δ·cos⁡⁡θhvc-ψ1-δ·cos⁡⁡θ-ψwhereθlvc>θ-α>θlvo,P=Pc=Pl1-δ·cos⁡⁡θlvc-ψ1-δ·cos⁡⁡θ-ψwhereθlvc>θ-α>θhvo,P=Phwhereθhvo>(θ-α)>θhvc,P=Plwhereθlvo>θ-α>θlvc.
Inputting the design parameters and the related operating conditions of the 100 c.c. isothermal expander into the above equations, the resultingP-V diagram is plotted and shown in Figure 3. Figure 3 indicates that the black line profile is the 90° phase angle. The deep gray line represents the 45° phase angle. Finally, the shallow gray line is the 0° phase angle.Figure 3
The influence of theP-V diagram with respect to various phase angles.The output work with respect to the phase angle is depicted in Figure4. As indicated in Figure 4, the maximal output work will occur when the phase angle is 19.6°.Figure 4
Output work with respect to the phase angle (between the piston and the displacer of the isothermal expander).


## 4. Experimental Work of the Isothermal Steam Expander

In order to verify the correctness of the mathematical model of the isothermal expander, a prototype of the 100 c.c. isothermal expansion expander is constructed and shown in Figure5. The surface of the piston and the cylinder is coated with a thin ceramic film. The hot end is made of stainless steel. The total height of the cylinder is 410 mm and the diameter is 50 mm. The piston stroke and the displacer are 50 mm. The venting rate per stroke is 98.2 c.c. In order to conveniently operate and adjust the isothermal expander, the piston is linked to the displacer with a gear. The related phase angle is adjustable.Figure 5
The prototype of the isothermal expander with a capacity of 100 c.c.In order to acknowledge the relationship between the mechanical output work and the phase angle, the internal instant pressure of the isothermal expander is detected by a pressure sensor IC at various phase angles. TheP-V diagram is then obtained. Furthermore, the area of theP-V diagram which shows mechanical output work will be calculated.


## 5. Results and Discussion

In the experimental work, the high pressure steam of the intake pressure source will be replaced by compressed air [12, 13]. The intake pressure will be set at 450 kPa, 400 kPa, and 350 kPa. Because of the retarding force, the air pressure will decrease when the air flows through the channel. The real intake air pressures become 250 kPa, 240 kPa, and 230 kPa. Inputting the data into the formula, we have found that a 19.6° phase angle will produce a maximal output of work. Because the hardware’s resolution of the phase angle is 1°, the setting of the phase angle is 15°, 20°, 25°, 30°, 40°, 65°, and 90°. The related output work can also be measured. Consequently, a comparison at various intake pressure conditions between the theoretical output work and the experimental data is shown in Figure 6. As indicated in Figure 6, the black line represents the theoretical profile of the output work with respect to the phase angle. Figure 6 indicates that the maximal output work will occur when the phase angle is 20°. The output work will decrease when the phase angle is increased or decreased from 20°. Here, it is understood that the output work will be at a minimum when the phase angle is 90°. As indicated in Figure 7, the black line represents the theoreticalP-V diagram. The gray line represents the experimental data. It is obvious that the theoreticalP-V diagram is similar to that of the experimental data.Figure 6
The comparison at various intake pressure conditions between the theoretical output work and the experimental data.Figure 7
The comparison of theoreticalP-V diagram with the experimental data at the same intake pressure.In the experimental work, because of the insufficient cooling rate of the working fluid during the isothermal compression process, the real temperature of the gas will be greater than what the theoretical data shows. Therefore, a rapid pressure increase will occur resulting in output work appearing in a small area of the experimentalP-V diagram. This is why the theoretical area of theP-V diagram is greater than the experimental diagram.


## 6. Conclusion

To improve the output efficiency of the expander, an isothermal steam expander is proposed. The isothermal expander will be similar to theβ type Stirling engine. Because the high pressure gas intake and gas discharge are added inside the isothermal expander, the setting of the phase angle between the piston and the displacer needs to be appropriately adjusted.The intent of this paper has been to provide a mathematical model of an isothermal expander using a modified Schmidt theory. Additionally, a prototype of 100 c.c. isothermal expander was constructed and used to verify the correctness of the isothermal expander’s mathematical model. Results reveal that the theoreticalP-V diagram is similar to that of the experimental data. As can be seen, the maximal theoretical output work of the 100 c.c. isothermal expander will occur at the phase angle of 19.6°. Additionally, the maximal experimental output work occurs when the phase angle is at 20°. Consequently, the approach used for the optimal phase angle of the isothermal steam expander of various sizes proposed in this study efficiently produces maximal output work.



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*Source: 794210-2015-10-12.xml*