## Abstract

Supercritical carbon dioxide (sCO_{2}) power cycles show promising potential of higher plant efficiencies and power densities for a wide range of power generation applications such as fossil fuel power plants, nuclear power production, solar power, and geothermal power generation. sCO_{2} leakage through the turbomachinery has been one of the main concerns in such applications. To offer a potential solution, we propose an elastohydrodynamic (EHD) seal that can work at elevated pressures and temperatures with low leakage and minimal wear. The EHD seal has a very simple, sleeve-like structure, wrapping on the rotor with minimal initial clearance at micron levels. In this work, a proof-of-concept study for the proposed EHD seal was presented by using the simplified Reynolds equation and Lame’s formula for the fluid flow in the clearance and for seal deformation, respectively. The set of nonlinear equations was solved by using both the conventional Prediction–Correction (PC) method and modern Physics-Informed Neural Network (PINN). It was shown that the physics-informed deep learning method provided good computational efficiency in resolving the steep pressure gradient in the clearance with good accuracy. The results showed that the leakage rates increased quadratically with working pressures and reached a steady-state at high-pressure values of 15∼20 MPa, where Q = 300 g/s at 20 MPa for an initial seal clearance of 255 μm. This indicates that the EHD seal could be tailored to become a potential solution to minimize the sCO_{2} discharge in power plants.

## 1 Introduction

The production of efficient, cost-effective, and environmentally friendly power is an issue that affects everyone on the planet. Over the past century, power plants have been the primary source of power production. Although there is an unprecedented effort for greener energy generation—including on-shore and off-shore windmills and fuel cells—power plants continue and will continue to be one of the major energy sources for years to come. Unfortunately, this technology has not developed rapidly enough to keep up with the power demand from society [1]. Thermal efficiency has been a major design concern in power plants, and there are continuous efforts to increase efficiency. However, the traditional open Brayton cycles or indirect-fired, closed Rankine cycle technologies have reached their saturation as far as efficiencies are concerned. Different working fluids, such as supercritical carbon dioxide (sCO_{2}), are being explored to determine their impact on the efficiency of power cycles [2]. The use of sCO_{2} as a working fluid in steam power plants has shown a potential to produce higher plant efficiency and power density, lower water consumption, reduced physical footprint, and overall, more cost-effective power generation [3–11].

If successful, sCO_{2} power cycles have the potential to be used in a wide range of power generation applications, including fossil fuel-based power plants, nuclear power production, solar power, and geothermal power generation. sCO_{2} power cycles can offer several benefits, including higher cycle efficiencies because of sCO_{2}’s unique fluid and thermodynamic properties, less emissions, compact turbomachinery, and reduced plant size, cost-effectiveness, rapid response to load transients, lesser water usage as well as water-free capability in dry-cooling applications, and heat source flexibility. However, technology readiness must be demonstrated for 10–600 MWe power plants and at sCO_{2} temperatures and pressures of 350–700 °C and 20–35 MPa to unlock the full potential of sCO_{2} power cycles [12]. Advanced seal technology plays a vital role in improving the overall efficiency of a power cycle. Seals are mainly used to minimize gas recirculation and gas leakage from the primary path. Gas recirculation and leakage occur in rotor stages, cavities, and stator stages, while gas leakage from the primary path occurs in flanges, vane pivots, and the compressor end [13]. The lack of suitable seals at sCO_{2} operating conditions is one of the main challenges at the component level [14]. As a remedy, there is a worldwide effort to develop effective seal designs for sCO_{2} power technology.

Some of the current sealing solutions that are utilized in this method of power generation are labyrinth seals, dry gas seals, compliant foil seals, and finger seals. Labyrinth seals are considered to be the more conventional seal design. Dry gas seals are also used in larger supercritical CO_{2} power cycles. Other advanced contact seals like finger and brush seals were also developed for this application. These conventional seals are discussed in detail in the following paragraphs.

Labyrinth seals are used in turbomachinery to control leakage between high- and low-pressure zones. These seals can be used in gas, steam, and hydraulic turbines [15]. This seal utilizes a set of teeth that face the spinning rotor. Since the seal never contacts the rotor, this seal is considered a non-contact seal. As the fluid travels through the teeth, pressure continually decreases minimizing the overall leakage rate. Due to the intrinsic design of the seal, leakage rates are heavily dependent on the operating conditions and critical dimensions for each application, which include pressure, shaft speed, and shaft diameter. In general, the leakage rate of labyrinth seals is proportional to the gap area and is inversely proportional to the number of teeth on the seal [16]. In most cases, leakage decreases by almost 20% in high shaft speed conditions [17]. It has been shown that the efficiency losses could be as high as 0.65% for a 500 MWe utility-scale power plant when labyrinth seals are used, which is too large to be neglected [18]. As for environmental concerns, excessive CO_{2} leakages contribute to the greenhouse effect and thus have a negative impact on the environment.

Brush seals are another common sealing solution used in turbomachinery. These seals consist of brushes that are aligned with a rotor in the tangential direction. These seals allow the rotor to move due to vibration or thermal expansion. Due to this property, these seals offer lower leakage rates than labyrinth seals. A backing plate in the axial direction supports the bristles and prevents them from bending in the rotational direction of the rotor. This seal design is considered a contact seal as the brushes contact the spinning rotor [19]. Since the seals contact the spinning rotor, these seals suffer from material wear significantly quicker when compared to non-contact sealing solutions. They also cause wear on the rotor surface. To improve the efficiency of the seal and to retrofit the design to multiple applications, many different parameters of the seal can be adjusted. These changes include bristle thickness, lay angle, bristle length, backing plate gap, and front plate gap. In some applications, multiple brush seals are used to achieve the desired leakage rate.

Compliant foil seals offer a similar sealing solution to both labyrinth seals and brush seals. These seals are mostly used in secondary flow systems of aircraft engines [20]. These seals consist of a seal membrane, a floating seal ring, a seal base, and several elastic supporting sheets. The compliant seal can move and adjust to any changes in rotor diameter caused by thermal expansion and/or vibrations. This movement is due to the elastic supporting sheets. Since the seal never touches the spinning rotor, this seal design is considered a non-contact seal. The life span of the seal is significantly better due to this no-contact operating configuration [21]. This design allows leakage to be minimized in the small gaps of secondary flow systems [22]. There are several reasons why this seal design has not gained traction in the supercritical CO_{2} power cycle and turbomachinery application. First is the perception that this design is only good for lightweight and high-speed rotors. Second is inexperience in foil bearing design leading to the belief that this design is not robust enough for demanding environments like high loads and temperature. Researchers are currently working on developing this design and change the perception of this sealing solution [23].

Finger seals have a lower manufacturing cost when compared to brush seals. This seal design is also able to better adapt to thermal changes than labyrinth seals. Due to these characteristics, finger seals have been adopted for use in various aero-engine applications [24]. Like brush seals, finger seals fall into the same category as contact seals. This is because the fingers contact the rotor. Over time, this results in wearing and malfunctioning in various components of the seal. In most cases, the fingers are made from either metal or carbon/composite. Finger seals are a significant advancement from labyrinth seals as they offer lower leakage rates and higher overall efficiency. It was also found that for many years during the development of this seal, researchers focused on the leakage through the radial clearance of the finger seal. The leakage that made it past the finger seal bundles was not considered when calculating the efficiency of the seal. This resulted in a lower overall efficiency rating for the seal once all leakage points were considered. Due to all the components in this seal, the complexity is a major downfall of this specific seal design.

Dry gas seals have several key benefits over other traditional mechanical sealing systems for turbomachinery [25]. This seal design can withstand significantly high rotational speeds (20,000–30,000 rpm) without the aid of a traditional oiling system. They also offer extremely competitive leakage rates when compared to other high-performance shaft-end seals. Due to the nature of the seal design, dry gas seals are able to work efficiently with a wide variety of gasses and operating conditions [26]. These benefits have enabled this seal design to replace traditional oil ring seals in compressors. Dry gas seals can also be run in dual and triple seal configurations when sealing volatile fluid systems. These configurations help to further reduce the leakage of these harmful chemicals into the environment [25]. The multistage design also allows for each stage to be optimized to handle a certain pressure value resulting in an overall more efficient seal. Due to the complexity of the seal, labyrinth seals are still more likely to be selected for this specific application.

So far, conventional seals all suffer from the incapability of handling sCO_{2} pressure and temperature in one way or another. There is an ongoing worldwide effort to develop effective sealing technologies for sCO_{2} turbomachinery [27–31]. White et al. [2] conducted a much-needed review of sCO_{2} power generation technology recently. According to their comprehensive review, the existing sCO_{2} turbomachinery designs employ mostly labyrinth seals, including the sCO_{2} test loops at Sandia National Laboratories, Southwest Research Institute/General Electric, and Echogen Power Systems—all in the United State—or dry gas seals such as in Supercritical CO_{2} Integral Experiment Loop (SCIEL) at KAIST in South Korea. In terms of power capacity, labyrinth and dry gas seals are used for the power ranges of 0.3–10 MWe and 10–500 MWe, respectively, with an overlap between 3 and 10 MWe. The review points out that the efforts concentrate on the application and development of dry gas seals for utility-scale sCO_{2} power plants. In this line of work, recently, Kim et al. [16] presented both numerical and experimental studies of the sCO_{2} critical flow model for their labyrinth seal design, and the isentropic flow was verified to model the CO_{2} leakage flow mechanism. The gap effect was simulated and reviewed for a single-phase flow. Their simulation model was developed based on choked flow equation, Hodkinson’s equation, and labyrinth seal design equation. Then, they compared their simulation results with the experimental data. Their experimental results included data from both single- and two-phase flows. Overall, their study provided useful information to further understand the performance characteristics of labyrinth seals, leading to an optimal seal design.

Bidkar et al. [18] discussed significant design challenges in the context of a representative sCO_{2} face seal. Their seal has a diameter as big as 24 in. and a pressure differential higher than about 1000 psia. In their study, they indicated the expected operating conditions for shaft-end seals on utility-scale sCO_{2} turbines. Unique characteristics of sCO_{2} were considered and a hydrodynamic face seal concept was proposed for end sealing application. They performed a computational fluid dynamics (CFD) analysis of the face seal film and demonstrated the feasibility of their concept by investigating the thermal analysis. Their study showed promise in replacing existing labyrinth seal and work at high working pressure.

Cao et al. [32] took an approach to increase the overall system stability for sCO_{2}. They established a dynamic three-dimensional numerical model of a staggered labyrinth seal as a potential solution. They utilized a previously proposed CFD model for transient flow to calculate and predict the dynamic force coefficients for the seal. The proposed seal was considered to operate under various rotor axial shifting distances, rotor convex plate heights/widths, seal cavity heights, and clearance conditions. Subsequently, they compared the results with conventional sealing systems and showed that their system is comparatively more stable.

Hylla et al. [33] worked on implementing carbon floating ring seals (CRS) in sCO_{2} systems. They demonstrated the theoretical model of CRS to predict the physical behavior of the leakage flow. They performed experimental investigations using an integrally geared compressor test rig. They used three different seals and found out that the pressure difference along the axial directions significantly changed. Then, they validated their pressure difference and mass flowrate results by comparing them with the numerical results of the model. This is a significant study as it shows promising evidence of a desired decrease for both pressure along the seal and mass flowrate.

Zhu et al. [34] proposed a reasonably simplified computational model for labyrinth seal by neglecting the compressibility of sCO_{2}. They provided literary proof that there is no exact computational model for the internal flow characteristic of sCO_{2} labyrinth seals. Their numerical model is universally applicable to the relevant empirical coefficients, including flow discharge coefficient and kinetic residual coefficient. Additionally, they introduced the homogeneous two-phase flow model and modified some essential computational parameters to compute the two-phase outlet conditions. They presented the latest experimental results of round-hole, see-through, and stepped-staggered sCO_{2} labyrinth seals and verified their 1-D computation method as well as sealing performances with the experimental results. They successfully demonstrated the consistency of the 1-D method with the experimental results. The higher efficiency of their proposed staggered seal over other types was also justified. This study provides a potential way of simplifying numerical methods.

Dry gas seals for supercritical fluid Brayton cycles have no well-established design guidelines. Fairuz and Jahn [35] conducted a study to understand the performance of dry gas seals. It is challenging because of the non-linearity and unpredictable thermodynamic behavior of sCO_{2} as the working fluid. Their study included a CFD investigation of fluid leakage at possible locations of the seal in operating condition. They investigated two inlet conditions with both real fluid and ideal gas. They also implemented a comparison study to understand real gas effects that affect CO_{2} thermal and transport properties near the critical point. Then, they presented their comparison results for the same seal geometry operating with air and CO_{2} at four operating points. Furthermore, they validated their results by comparing their predictions with previously published experimental results. This study is critical because it gives insight into designing dry gas seals for sCO_{2} power cycles.

As one of the key players in the turbomachinery industry, non-contacting dry gas seals can also be a potential solution for sCO_{2} power cycles. Armin et al. [36] proposed a design solution for such a non-contacting dry gas seal for CO_{2} power cycles. They designed, fabricated, and tested their proposed design for liquid, gaseous, liquid–gas, and supercritical phases. Their result shows an isothermal seal expansion due to low leakage while maintaining a non-contacting dynamic operation. Moreover, they ran a deliberate failure test to justify the low leakage demand.

Closed-loop Brayton cycles have the highest efficiency with sCO_{2} as a working fluid. However, conventional shaft-end seal limits extracting its maximum efficiency. Spiral groove seal has shown much potential to replace shaft-end seal as an alternative in recent studies. Yan et al. [37,38] studied the influence of different turbulence effects on the spiral grooves dry gas seal performance. Theoretical and experimental tests were conducted on sCO_{2} dry gas seal prototypes for isothermal or adiabatic flow. For the theoretical study, leakage rates and flow fields under different operating conditions were calculated by coupling the Reynolds and energy equations for the test conditions. They were then compared with the experimental test results. Their study shows that the turbulence effect has a recognizable impact on the leakage rate. In the future, they want to extend their study to investigate structural deformations of the seal.

In a similar effort, Rimpel et al. [39] worked on restricting existing sealing limitations for recompression Brayton cycles. They presented a new test rig design for a conceptual 24-in. film riding face seal applicable in 450 MWe utility-scale sCO_{2} turbines. The test rig was designed for a shaft speed of 3600 rpm, maximum supply temperature of 400 °F, upstream cavity pressure of 75 bar, and maximum downstream cavity pressure of 10 bar. Their analysis shows that excessive thrust load and downstream cavity pressure in the test rig would be mitigated in case there are any seal failure scenarios. This study is a work in progress. This study shows how to imitate the natural environment without any probable accident while running the experimental analysis.

There are still major roadblocks in the full realization of sCO_{2} power technology such as high leakage rates through shaft-end seals [40,41]. High leakage rates are not desired, because of their negative impact not only on efficiency but also on the environment. As far as efficiency is concerned, sCO_{2} power cycles operate on closed-loop cycles such as steam Rankine cycles. Thus, the sCO_{2} past the shaft-end seal cannot be condensed back to the liquid phase to be recovered and fed back to the system. It must be recompressed to the sCO_{2} states, which will require additional power input, i.e., additional compressor power, penalizing the efficiency of the whole cycle. Each seal design has merits in both efficiency and production cost depending on the application. More recently, non-contact seals like film riding seals and hydrodynamic seals are being developed to further decrease leakage past the seal and improve efficiency. Non-contact seals have been found to offer less leakage and longer life since they are not in contact with the rotor-like finger and brush seal designs. Researchers are currently working toward creating a non-contact seal that can be used in supercritical CO_{2} power cycles and other high-pressure turbomachinery applications. The hydrodynamic face seal is one of the best candidates for sCO_{2} operating conditions due to its ability to withstand large pressure differentials. These pressure differentials can be as high as 1000 psia. Currently, large-scale hydrodynamic face seals are not commercially available due to both manufacturing and design challenges.

Despite these efforts, there is still a need for effective sealing technologies for sCO_{2} turbomachinery. To offer a potential solution, we propose an EHD seal that can work at elevated pressures and temperatures with low leakage and minimal wear. To offer a potential solution, we propose an EHD seal that can work at elevated pressures and temperatures with low leakage and minimal wear. The EHD seal has a very simple, sleeve-like structure, wrapping on the rotor with minimal initial clearance. The combined effect of the pressures at the top and bottom of the seal forces the seal to bend downward to provide a throttling effect in the clearance, which minimizes leakage [42–48].

The foundation of the elastohydrodynamic seal concept dates back to 1886 when Osborne Reynolds published a paper discussing the pressure distribution and load-carrying capacity of fluid films [49]. EHD lubrication, specifically, has made major strides in the last few decades. The first major publication involving this topic specifically was published in the late 1930s. Since then, the study of elastohydrodynamic lubrication has played a critical role when designing a variety of machine components. The components that utilize this phenomenon include anything from bearings to gears [29].

Due to the complexity and nature of elastohydrodynamic lubrication, computer modeling and Computational Fluid Dynamics (CFD) are widely used to help understand and apply this lubrication technique to many different applications, including single and multistage seals for turbomachinery [50–52].

Next, the novelty of the paper is discussed. In this paper, we are utilizing the simplified Reynolds equation and Lame’s formula to model the fluid flow and structural deformation of the seal, respectively. This requires the solution of a set of highly nonlinear, coupled differential equations. Solving these nonlinear equations using conventional optimization methods requires careful selection of the flowrates and often results in convergence issues. Physics-informed deep learning methods are becoming popular and have shown the potential to solve complex nonlinear equations [53–74]. It is one of the most advanced tools to solve partial differential equation problems numerically [75,76]. PINN infers unknown solutions for physics of interest by using the limited amount of given data and governing equations of conservation laws [77]. It provides a paradigm shift into data-driven modeling for physical systems. PINN’s framework is relatively simple, meshless, and can be applied successfully for high-speed flow solutions [78]. The novelty of the paper comes from the verification of the proposed seal design by using modern solution tools, such as PINN, for physics-based analytical modeling.

In this work, the design equations are solved using PINN. The results are discussed to verify the working mechanism of the proposed seal design. Also, the advantages of the PINN method over the conventional Prediction–Correction (PC) methods are discussed.

The rest of the article is organized as follows: Sec. 2 introduces the proposed seal design and discusses the design methodology, the governing mathematical equations for the flow inside the clearance and seal deformation along with the details of conventional PC and PINN methods. The results are presented and discussed in Sec. 3. Conclusions and future work are presented in Secs. 4 and 5, respectively.

## 2. Materials and Methods

### 2.1 Proposed Elastohydrodynamic Seal Design.

The proposed EHD seal utilizes the proven electrohydrodynamic lubrication sealing mechanism [79–82], which eliminates wear and reduces both leakage and overall cost. As shown in Fig 1(a), the main EHD seal is attached to a back ring. The back ring sits on a rotor with a cold clearance of *h*, where *P*_{0} = *P _{e}*. Once the rotor reaches operating speed, pressure

*P*

_{0}>>

*P*, which results in a decaying pressure distribution in the clearance (Fig 1(b)). In this condition, the pressure along the top of the seal is equal to the operating pressure

_{e}*P*

_{0}. Since the root of seal is fixed, i.e., it is welded onto the back ring, the pressures at the top and bottom of the seal will cause the seal to deform to create the minimum possible clearance for the given conditions. One conclusion one might reach is that the seal would bend downward to contact the rotor and block the flow. However, when such a contact occurs the pressure differential across the contact region will force the seal to open, allowing flow to continue. Recall that

*P*

_{0}>

*P*always during operation.

_{e}isBased on the discussion earlier, there will always be a viable minimum clearance between the EHD seal and the rotor. This will provide the necessary throttling effect to minimize the leakage. The proposed EHD seal design for sCO_{2} turbomachinery has several benefits which include:

*Low leakage*. The self-regulated minimum clearance throttles the sCO_{2}leaking flow, improving the cycle efficiency.*Minimal wear*. The EHD seal operates in non-contact conditions.*Low cost*. The simple sleeve structure results in low seal cost and minimal wear, saving maintenance costs.*No stress concentration*. The EHD seal design eliminates sharp angles and stress concentration risks.

### 2.2 Design Methodology.

*P*) variation along the clearance and leakage rate (

*Q*) is given by

*h*is the clearance along the sleeve,

_{c}*ρ*is the density, and

*µ*is the viscosity. In this work, the seal fixation at the root of the seal is not considered for simplicity when the seal deformation is evaluated. Using Lame’s formula [9] for a thick-walled cylinder, the clearance can be obtained as

*h*

_{0}is the initial clearance,

*P*

_{0}is the working pressure, and

*k*

_{1}and

*k*

_{2}are the coefficients given by

*E*is Young’s modulus, and

*D*and

*D*

_{0}are the inner and the outer diameters of the sleeve, respectively. The diameters are given by

*D*

_{rotor}is the diameter of the rotor and

*t*

_{seal}is the EHD seal. The boundary conditions on the inlet and outlet boundaries are

*P*

_{0}is the working pressure acting on the seal and

*P*is the pressure at the outlet, which is equal to the atmospheric pressure. The viscosity is a function of pressure and is given by the Barus equation

_{e}*µ*

_{0}denotes the dynamic viscosity of the working fluid at the atmospheric pressure, and

*α*is the pressure–viscosity coefficient. The density is a function of pressure and is given by the Dowson–Higginson formula

*ρ*

_{0}is the density of the working fluid at the atmospheric pressure. Equations (1)–(8) constitute the isothermal EHD governing equations for the high-pressure sleeve. The equations are highly nonlinear, and two different methods to solve these equations are discussed next.

### 2.3 Prediction–Correction Algorithm.

With the fluid properties being a function of pressure, Eq. (1) is highly nonlinear. The importance behind Eq. (1) is that the mass leakage rate, *Q*, is equal to the product of fluid properties and pressure gradient at any position along the length of the clearance. The pressure will vary along the length of the sleeve, but the principle of mass conservation limits the *Q* to be constant. The pressure and mass leakage rate are the variables to be solved for, and the boundary conditions are the constraints on Eq. (1). The boundary condition on the right end is the atmospheric pressure, *P _{e}* and on the left end is the working pressure,

*P*

_{0}. To solve Eq. (1), in the first iteration, a value for

*Q*is assumed and the pressure is solved from the inlet to the outlet using an Ordinary Differential Equation (ODE) solver. The value of

*Q*is adjusted based on the error between the pressures obtained from ODE and the actual pressure at the right end. Careful selection of

*Q*is required to solve the ODE. A large value of

*Q*results in convergence issues in ODE solver due to resulting in negative pressures. Using the Newton-Raphson method, steepest descent, or other optimization methods results in infeasible values for

*Q*, resulting in convergence issues. A PC method is used to find the value of

*Q*for the given boundary conditions.

The method to carefully solve for the value of leakage rate *Q* using the PC method can be seen in Fig. 2. To begin, a starting value for the leakage rate is chosen, *Q ^{i}*, and the flow equation along the clearance, Eq. (1), is solved using an ODE solver with pressure at the inlet,

*P*

_{0}, as the initial condition until

*x*=

*L*. Using the pressure at the outlet

*x*=

*L*and the ambient pressure,

*P*, the error (

_{e}*ɛ*) is calculated.

If the error is below the specified tolerance, the mass leakage rate is decreased; otherwise, the mass leakage rate is increased. The ODE is then solved again with a new leakage rate, *Q ^{i}*

^{+1}. The loop of predicting the pressure field, along the clearance, for a given flowrate,

*Q*, and updating the

^{i}*Q*based on the error in the predicted pressure at the outlet, is repeated until the absolute value of

^{i}*ɛ*is below a threshold (

*ɛ*

_{tol}). This method helps with the selection of a leakage rate,

*Q*, for a given inlet pressure

*P*

_{0}. This method is known to result in convergence issues when the flow equations are being solved for high inlet pressure values,

*P*

_{0}. Because of this, a parametric sweep will be needed to ramp the inlet pressure up from ambient value (

*P*

_{x}_{= L}) to the actual working pressure (

*P*

_{0}).

The parametric sweep needed to solve the flow equations for a final working pressure, *P*_{0}, is outlined in Fig. 3. Using the parametric method, the pressure field (*P*) and leakage rate (*Q*) are solved for using ambient air pressure as the starting point. The working pressure is then increased by ΔP. The resulting flow equation is solved using the PC with the obtained *Q _{j}* as the initial flowrate as shown in Fig. 2. Using a parametric sweep with the PC method, to solve the flow equations Eq. (1), for any given working pressure P

_{0}makes this method computationally expensive. Other methods such as physics-informed deep learning method are proposed as a more efficient solving method.

### 2.4 Physics-Informed Deep Learning.

Physics-informed deep learning can be used to solve the flow equations, Eqs. (1)–(8), for any boundary pressure condition. The architecture used for solving the flow equations using a PINN can be seen in Fig. 4. The network is made of several input and output layers. These layers are connected by a network of hidden layers. Each layer contains artificial neurons. The output of each neuron is passed onto the next layer as an input. As this process happens, the neurons do a weighted sum of the inputs and add a bias to the sum. An activation function is also applied to the output of each neuron before it is passed onto the next layer as an input. Each neuron is associated with a bunch of weights and a bias. The number of weights for each artificial neuron is equal to the number of inputs to the neuron. The neural networks are very robust in generating a complex function between multiple inputs and outputs of the training data by identifying patterns in the data [83–88].

Training the neural network is finding the weights and bias of each neuron using the data of interest. This training requires minimizing the loss function [89], which allows the neural network to predict output values close to the actual values for unseen/new data points. The sum of squares of the difference between the predicted and actual output values is known as the loss function. The methodology of identifying a nonlinear map between high dimensional input–output data seems to be a naïve task [75]. The PINN helps to supplement the neural networks to solve some of the most complex nonlinear mathematical equations of any physical phenomena. It is well known to solve two classes of problems: data-driven solution and data-driven discovery of partial differential equations (PDE). A data-driven solution of a partial differential equation style problem will be solved in this work.

*P*,

*Q*) are of the same scale when solving for both flow and seal deformation within the clearance of the seal. The following parameters are used for scaling:

*D*for diameters,

*L*for axial dimensions,

*h*

_{0}for clearance,

*P*

_{0}for pressure and E Young's modulus, 1/

*P*

_{0}for pressure–viscosity coefficient,

*μ*

_{0}for viscosity,

*ρ*

_{0}for density, and $\pi D\rho 0h03/12\mu 0L$ for mass leakage. The final governing equations in the dimensionless form are given by

*x*is considered as the input and the pressure

*P*and leakage rate

*Q*, as the output. The goal of the PINN is to estimate a value of

*P*,

*Q*at any input

*x*for given boundary conditions by finding the weights and bias of the NNs. First, random points are generated inside the domain, where Eq. (1) is trained. The loss function for the current problem will be the sum of squares errors in the boundary conditions, and the errors in the differential equation as shown below

*N*

_{grid}is the number of random points selected inside the domain.

*P*

_{NN}and

*Q*

_{NN}are the pressure and the leakage rate predicted by the PINN. The second term in Eq. (18) is due to the error in Eq. (17) for the predicted value of leakage rate by the PINN and is given by

The boundary conditions at inlet and outlet are given by Eqs. (15) and (16). The main goal of the PINN is to determine a pressure field and leakage rate which minimizes the total loss function. This means that if the PINN selects the exact value for both pressure field and leakage rate for a given input pressure or boundary condition, the field satisfies the Eqs. (10) and (17), which results in the total loss function equaling zero. The important parameter in the PINN is the choice of the activation function and in this work, *tanh* is used as the activation function. The Adam optimizer is used to train the PINN by minimizing the total loss, i.e., Eq. (18). Then, the weights of the artificial neurons are calculated. The PINN needs to be trained for each pressure boundary condition and the weights of the neural networks (NN) are different for each boundary condition. However, the weights obtained for one boundary condition could be used as initial weights of the PINN for training at different inlet pressures, and it reduces the training time for new inlet pressure conditions.

## 3 Results and Discussions

The accuracy of the PINN is investigated by solving the flow equations along the clearance considering the fluid as liquid and comparing the results with the solution obtained from the PC algorithm. The main fluid properties and parameters used in this investigation can be seen in Table 1. In this work, the python programming language was used to execute the discussed algorithms. The SciPy package [90] is utilized for the ODE solution algorithm, and the TensorFlow package [58] is used for the PINN method. The domain is divided into a uniform grid of 1000 nodes for the ODE, and the nodes are selected randomly for the PINN.

With the outlet pressure set to atmospheric pressure, the flow equations are solved for different working pressure conditions. The PC method is used to solve flow equations. This was done by using 20,000 iterations, and the step size of 1e5 Pa is used for the parametric sweep. For the PINN, the input is the spatial variable, *x*, and the output is the scaled pressure and leakage rate. Four layers with 30 neurons in each layer are used. Figure 5 shows the variation of training loss with the iteration number of the Adams optimizer for a typical case. The training of PINN is stopped when the loss reaches 1e−5.

To solve the flow equations for a larger value of working pressure (*P*_{0}) using the PC method, the parametric sweep with the starting value of *P*_{0} as the ambient pressure (*P _{e}*) is used. Also, the resulting leakage rate (

*Q*) is used as the starting value for the next step as shown in Fig. 3. In each step of the parametric sweep, the pressure is increased by a value of 1e5 Pa, whereas the PINN can solve the flow equations for any arbitrary value of working pressure (

*P*

_{0}) without any parametric sweep. This demonstrates that PINN is more powerful than conventional optimization techniques due to its nature of solving complex nonlinear equations efficiently.

In this study, both the PC and PINN methods were compared using 200 simulation cases. The cases varied in working pressures (*P*_{0}) between 0.2 and 20 MPa. Figure 6 shows the comparison of percentage Root Mean Square (RMS) of $Lossdp\xaf$ using the PC and the PINN method for different working pressure conditions. It was found that the RMS loss was random in the case of PINN. This is due to random initialization [91] of the weights and biases of artificial neurons in NN for every simulation. Using the PC method, the RMS loss continuously increased. Using both methods, the $Lossdp\xaf$ is below 0.6%.

The scatter plot of percentage RMS loss in the pressure boundary condition at the outlet (*Loss*_{BC2}) for different simulation cases with working pressures between 0.2 and 20 MPa is shown in Fig. 7. The *Loss*_{BC2} increases with working pressure using both the PINN and PC methods. The PINN method performed marginally better than the PC method. Using both methods, the loss values are below 1.2%, where ∼1.2% is singled out for PINN in the vicinity of the highest pressure of 20 MPa. All else is below 0.8%. A graphical comparison of the predicted pressure for both the PINN and PC can be seen in Fig. 8. For the PINN, the scaled pressure is obtained, and the actual pressure is calculated by multiplying the scaled value by *P*_{0}. It shows that both methods provide the same results. It was found that the pressure curve decreases along the clearance using both methods. For lower working pressures (*P*_{0} = 0.2 MPa), the pressure decreases almost linearly, and for higher working pressures (*P*_{0}), the decrease is more nonlinear. The variation of clearance (*h _{c}*) under different working pressures (

*P*

_{0}) is calculated using Eq. (11) and is shown in Fig. 9. For lower working pressures, the clearance decreases almost linearly toward the outlet, and for higher working pressures, there is a steeper contraction towards the outlet.

Using both the PINN and PC methods, the leakage rate (*Q*) is calculated using different working pressures (*P*_{0}). A comparison of both methods is shown in Fig. 10. It was found that both methods generated similar results. The leakage rate, *Q*, increases with initial working pressures. Once the pressure reaches higher values, the increase in leakage rate begins to level off and stabilize.

For different boundary conditions, similar pressure fields and leakage rates were obtained using both methods. It was found that a parametric sweep was needed to solve for higher working pressures (*P*_{0}) using the PC method. This was not the case for the PINN method as it can be applied for any condition.

In addition to discussion of the solution methodologies, it is also important to discuss the physical meaning of the obtained solutions. Figure 8 confirms the decaying pressure distribution in the clearance. It was found that the pressure distribution becomes more nonlinear as the working pressures increase, especially after *P*_{0} = 11 MPa. This is also shown by the clearance thickness displayed in Fig. 9. After *P*_{0} = 11 MPa, the clearance decreases more rapidly, particularly after axial length, *x* = 18 mm. This is due to the nonlinear effects of pressure on the density, *ρ*, and viscosity, *μ*, which are felt more strongly at higher pressures.

Recalling that the seal root at *x* = 26.5 mm is fixed to the back ring, the clearance thicknesses shown in Fig. 9 will result in seal deformations similar to those shown in Fig 1(b), which proves the hypothesis discussed in Sec. 2.1 as far as achievable by the simplified Reynolds equation and Lame’s formula. To obtain similar seal deformations depicted in Fig 1(b), one needs to modify the boundary conditions and implement them in the solution procedures discussed in this work. This is left as a part of future work along with others discussed in the next section.

## 4 Conclusions

The main outcomes of this study can be summarized as follows:

A proof-of-concept study for a novel EHD seal was presented by using the simplified Reynolds equation and Lame’s formula for a thick-walled cylinder.

The set of the nonlinear equation was solved by using the conventional PC method and PINN.

The solutions obtained from both methods are closely matched. However, the PC method required the parametric sweep to converge at high working pressures to avoid convergence issues, which resulted in large computational times.

The pressure field decreased linearly from inlet to outlet at lower working pressures, whereas the decrease was nonlinear at higher working pressures.

The clearance decreased linearly from the inlet to outlet at lower working pressures and nonlinearly at higher working pressures.

The clearance thickness results proved that there would be a throat happening closer to the root of the seal.

The leakage rate followed a quadratic trend, making its peak around

*P*_{0}= 20 MPa, where*Q*= 300 g/s.The proposed seal could be tailored to minimize the leakage rates to become a potential candidate for sCO

_{2}power technology.

## 5 Future Work

As for future work, we will attempt to investigate the following:

In the present analysis, the root of the seal is not fixed, and we will implement additional boundary conditions to incorporate that.

We will perform a parametric sweep analysis for different values of rotor diameter, seal thickness, seal length, elastic modulus, and so on.

In the present work, we picked the working fluid to be a liquid for proof-of-concept purposes. However, we will attempt to incorporate the real gas properties of sCO

_{2}, which seems to be rather challenging.In the present study, the effect of temperature on viscosity and density was neglected. In future work, we will attempt to incorporate the temperature effects as well.

We will incorporate the rotational effects in future analyses.

Most importantly, we will design and fabricate a 2″ static test rig to demonstrate the EHD seal experimentally.

## Acknowledgment

This work has been financially supported by the U.S. Department of Energy (DOE) STTR Phase I and II grants under Grant No. DE-SC0020851, in which Dr. Sevki Cesmeci and Dr. Hanping Xu serve as the Principal Investigator and Project Manager, respectively. This paper reflects the views of the author(s) only and does not necessarily reflect the views or opinions of DOE. The intellectual property of the proposed seal concept is protected under Patent No. 63/200,712, “Clearance seal with an externally pressurized sleeve.”

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

- CFD =
computational fluid dynamics

- EHD =
elastohydrodynamic

- FEA =
finite element analysis

- sCO
_{2}= supercritical CO

_{2}- NN =
neural networks

- ODE =
ordinary differential equation

- PC =
prediction–correction

- PDE =
partial differential equation

- PINN =
Physics-informed neural network

- RMS =
root mean square

- SCIEL =
supercritical CO

_{2}integral experiment loop

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