Abstract

Superheater tubes are critical boiler components that operate at relatively higher temperatures and pressure. Amongst the primary concerns for these tubes is the deposition of ash particles on the tube surface, leading to the reduced thickness of the tube due to material corrosion, consequently causing early creep failure of the component. In this research, a novel tube design has been proposed which resembles a teardrop or ogive shape to reduce the drag and concurrently improve the creep life of the superheater tubes. To administer the practicality of novel tubes, metal additive manufacturing (AM), for instance, laser-powder bed fusion (L-PBF), has been proposed. These unconventional designs were assessed and compared with the baseline circular tube design for mechanical design requirements (hoop stress and creep life) and the particle and flue gas flow characteristics around the differently shaped tubes. A thermomechanical finite element (FE) analysis was performed for hoop stress calculations. This study also emphasizes on effect of circumferential thermal variation on hoop stress distribution in tubes. Therefore, a detailed two-dimensional (2D) thermal simulation has been performed to report the circumferential thermal variation on the tube. A computational fluid dynamics (CFD) analysis coupled with particle tracing was performed for gas flow visualization and particle tracing around the proposed shapes and baseline circular-shaped tube design. The Schlieren optic setup was built and leveraged for qualitative validation of the proposed design. The complete design methodology established in the paper shows teardrop-shaped tubes better in terms of drag and creep life in contrast to the circular-shaped tube.

1 Introduction

Despite the growing demand for renewable energy in the future, fossil-fueled power plants remain a crucial part of the energy sector [1]. With the heightened need to minimize the CO2 emissions from fossil-fueled power plants, coal is cofired with biomass as fuel. Biomass is a cost-effective fuel and is shown to reduce CO2, SOx, and NOx emissions [2]. However, it contains significant amounts of contaminants including chlorine and alkali elements such as potassium and sodium. These contaminants increase the propensity of ash deposition on the boiler tubes that can be detrimental for the component's life [3,4]. One of such boiler tubes is a superheater tube. This consequently stipulates serious considerations for superheater tube redesign for increased life.

The primary function of these tubes is to increase the temperature of the saturated high-pressure steam to superheated steam by increasing the steam temperature by 35–50 °C. Due to the presence of high-pressure steam, these tubes experience hoop stresses on the tube walls. With the advent of new generation fossil power plant systems, these tubes operate under higher temperature and pressure loading conditions compared to traditional boiler tubes (∼610 °C and 25–30 MPa) [5]. To superheat the saturated steam, the external surface of the tube is exposed to hot combustion flue gases with a temperature ranging from 800 °C to 1200 °C [6], which also contain undesirable ash particles. These ash particles deposit onto the tube surface and their shape and thickness gradually change over time. Figure 1(a) shows the schematic representation of the superheater tube degradation mechanisms which consequently leads to local thinning of the tube and eventual failure. The flue gases, containing sulfates and chlorides such as Na2SO4 and NaCl, strike the tube surface and react with the ferric oxide or ferrous oxide layer to form low melting alkali trisulfates in the presence of sulfur trioxide (SO3). The melting temperature of these alkali trisulfates (554677°C) lies in the operating temperature of the superheater tube; therefore, liquid phase corrosion [7,8] is enhanced. The overall chemical reactions are outlined below:
Fe2O3+3Na2SO4+3SO32Na3Fe(SO4)32Fe3O4+9Na2SO4+9SO3+1/2O26Na3Fe(SO4)34Fe+3O22Fe2O33Fe+2O2Fe3O4
Fig. 1
Schematic representation of superheater tube containing high-pressure steam and getting exposed to flue gases and ash contaminants: (a) flow conditions that lead to ash deposition; (b) modes of heat transfer during operation; (c) typical degradation due to corrosion; and (d) mechanisms for preferential high-temperature corrosion due to ash deposition
Fig. 1
Schematic representation of superheater tube containing high-pressure steam and getting exposed to flue gases and ash contaminants: (a) flow conditions that lead to ash deposition; (b) modes of heat transfer during operation; (c) typical degradation due to corrosion; and (d) mechanisms for preferential high-temperature corrosion due to ash deposition
Close modal

Reid [9] suggested the presence of macro galvanic couple across a tubular section due to SO3 concentration gradients, hence setting up a gradient in corrosion potential (Fig. 1(d)). A potential difference of 0.5 V is calculated between metal in contact with solid sulfates at one point (just below ash deposits) and fused sulfates at another (around 10 and 2 o'clock position). This is consistent with the observed “wastage flat” near 2 and 10 o'clock locations beneath the ash deposits (see Fig. 1(c)) rather than around the 12 o'clock position. The location of wastage flats also supports the notion of high temperatures and easy access to SO2 from the ambient [7,9]. Around the 12 o'clock position, the deposit thickness is highest, and since its thermal conductivity is very low compared to the metal, it acts as an insulative layer. These boundary conditions should lead to lower temperatures at the 12 o'clock position as compared to other locations where the ash deposit thickness is lesser [10]. These conditions also imply a circumferential thermal gradient around the tube. In addition to high internal pressure, the circumferential thermal gradient can further aggravate hoop (circumferential) stresses in the tube. Increased hoop stresses induce higher creep strain rates and a reduction in time to rupture. In fossil fuel power plants, failure due to creep or long-term overheating is the highest contributor (∼23.4% of all failures) for forced outages [2]. Interestingly, 90% of the creep failures occur in the superheater and reheater tubes.

In addition to this, the ash deposition and slagging reduce the heat transfer ability by 30–60% (Fig. 1(b)) [11]. Therefore, it is imperative to reduce the ash deposition on the tube surfaces while concurrently meeting mechanical design requirements. This overarching goal forms the motivation for this research. Earlier studies for ash deposition on boiler tubes has primarily emphasized studying the chemical composition of ash, predicting ash growth through computational fluid dynamics (CFD) modeling, benchmarking the simulation model with experimental results, studying particle motion in flow and its interaction with the tube surface, and methods to reduce ash particle through external source. A summary of the literature review is provided as a context to the proposed design solution in this research.

Constitution of ash: Analysis of the chemical composition may help to predict the ash melting temperature. Vassilev et al. [12] showed the relationship of the chemical composition of coal ashes with the ash fusion temperature. Holubcik et al. [13] derived the empirical relationship to relate the chemical composition with the biomass ash melting temperature. Kim et al. [14] used the ASTM D1857 method and the thermo-mechanical analysis to measure the ash fusion temperature to predict ash deposition characteristics.

Ash growth prediction via CFD: Computational fluid dynamics models have been used to predict the flue gases flow and concurrent particle deposition on the tubular surface [1517]. Chapela et al. [18] performed CFD modeling using a full three-dimensional (3D)-transient bed model and improved the accuracy of the prediction model by accounting for the boiler geometry. Riccio et al. [19] performed a CFD analysis of particle deposition and verified the calculations with experimental results with a 100-kW test rig facility.

Further, particle–wall interactions have also been considered [20,21]. In addition, based on the evaluation of ash compositions and validation of predictive models, external devices like soot blowers and monitors had been deployed for timely eroding or removal of the ash particles from the tubular surface [22,23].

Tube geometry modification: Also, few research publications focused on changing the tube shapes and their arrangement within the heat exchanger assembly for assessing fouling, heat transfer, and pressure drop. Bouris et al. [24,25] and Tang et al. [26] proposed deposit determined fouling reducing morphology, elliptical, and circular-shaped tubes with different tubular arrangements for lignite utility power plants. Bouris et al. [25] studied these tubes for fouling, heat transfer rate, and pressure drop comparison employing numerical study and experimental methods like laser sheet visualization and laser Doppler anemometer. Tang et al. performed CFD analysis using ansysfluent software for examining ash fouling and thermal-hydraulic characteristics in stacked elliptical tube arrangement. However, the authors did not study these tubes for their mechanical integrity. Also, the numerical study has been performed at a relatively higher Reynolds number; for instance, Bouris et al. examined for Reynolds number varying from 14,000 to 30,000, while Tang et al. studied for Reynolds number ranging from 9000 to 24,000. Li et al. [27] proposed a novel rhombic configuration for heat recovery steam generators for reduced ash deposition. To our knowledge based on extensive literature review and patent search, limited work has been done on the superheater tube's geometry redesign to reduce ash deposition, without affecting the mechanical hoop stresses' requirements or creep life.

1.1 Scope of the Research Paper.

Herein, we propose to modify the target geometry, i.e., the superheater geometry for streamlining the flow and synchronously improving the creep life of the component. We propose a novel-shaped design that resembles a teardrop or an ogive shape for superheater tubes with circular and noncircular inner diameter. Six variants of the proposed tube geometries have been considered and studied. The substantiation of the proposed solution also leads us to two more significant outcomes: first, calculation of circumferential thermal distribution and its impact on the hoop stresses; second, an overall design methodology to down select a tube design for the superheater tubes.

Manufacturability: To enable the manufacturing of such contemporary tubes, additive manufacturing (AM)-based techniques like laser-powder bed fusion (L-PBF) are proposed. Other manufacturing techniques such as extrusion-based method are another viable option to realize such tubes [28]. However, these superheater tubes are welded to a U-bend region in shell and tube heat exchangers. To avoid any sudden geometric aberration at the interface of teardrop-shaped tube and the U-bend, it is suggested to transition the teardrop-shaped tube to circular cross section toward the end. In such scenarios, additive manufacturing-based techniques are significant in empowering such complicated tubes. Additionally, additive manufacturing methods not only empower such contemporary tubes but also allow further modifications such as any dimples, protrusions, or surface aberrations on the inner or outer surface of the tube [29,30]. Teardrop shape with such surface modifications with techniques other than AM can limit the sizing of these features and will require multiple steps to make these tubes [31]. In this paper, for the evaluation of the design, the L-PBF additive manufacturing and polymer-based additive manufacturing techniques have been leveraged.

AM built material properties: In addition to geometric benefits, it is equally important to consider mechanical properties obtained via AM. In this study, stainless steel (SS) 316 L is the chosen material for superheater tubes. Significant progress has been made for AM build SS316 L to minimize porosity and defects. Past studies have shown an increase in density in AM as-built SS316 L from ∼99.2% [32] to 99.82–99.9% [33] through process parameters optimization. AM SS316 L has a relatively shorter creep life benchmarked to conventionally built type SS316 L [33]. As discussed by Li [33], this might be attributed to the significant difference in carbon concentrations in Type 316 SS and AM SS316 L affecting the creep rupture strength. Additionally, it is also possible to enhance the creep properties of AM SS316 L through the heat treatment process for specific parameters [33]. Finally, AM enables complex geometries and can allow tailoring of microstructure that can result in superior properties. Wang et al. [34] reported high strength–high ductile mechanical properties for AM SS316 L compared to conventionally cast SS316 L. Therefore, heterogeneity in AM properties which is usually considered undesirable can become a distinguishing benefit for next-generation component design.

The structure of this research paper is as follows. In Sec. 2, the ash deposition mechanism, inspiration details for novel-shaped tubes, and proposed geometrical details have been discussed. In Sec. 3, the methodology adopted for design validation has been discussed in detail. Broadly, the method involves both computational and experimental approaches for performing comparative studies on mechanical stresses and flow around differently shaped tubes. In Sec. 4, the results of the simulation studies and experiments have been shown which are discussed later in detail in Sec. 5. The conclusion is presented in Sec. 6.

2 Proposed Design for Superheater Tube

In this section, the process for the development of the proposed idea has been discussed in detail. For this, in Sec. 2.1, the dominant ash deposition mechanism and its relationship with the target geometry have been discussed. In the Sec. 2.2, inspiration for the proposed design has been explained. Finally, in Sec. 2.3, computer-aided design drawings and the manufactured parts of the proposed designs have been shown.

2.1 Pipe Geometry and Ash Deposition Mechanism.

Prior investigations explained many mechanisms for ash deposition including inertial impaction, condensation, eddy impaction, thermophoresis, Brownian and eddy diffusion, and chemical reaction [10,35]. Out of all these mechanisms, the dominant one is inertial impaction. This process causes larger particles (>10 μm) to overcome the drag force of the gas flow and deposit on the upstream of the tube surface. Earlier investigations have developed correlations between factors affecting the total mass deposition rate due to inertial impaction. The factors that can be altered to affect the ash deposition include flue gas velocity, particle size, particle density, gas flow properties, particle viscosity and composition, target geometry, tube surface texture to reduce the stickiness, and vibration of the tube. Since there has not been a comprehensive publication considering the interaction between variables, one can hypothesize that there are many opportunities to modify the ash deposition with new manufacturing technologies (e.g., additive manufacturing and cold spray). For example, tube surface chemistry or texture may be varied to reduce the adherence of ash by cold spray. In another example, one may be able to create unique surface geometries that may interact with flow conditions and induce self-vibration to shed the ash deposits from the surface [23]. Self-vibrating tubes could also be designed and manufactured using additive manufacturing to shake off the ash from the tube surface [36].

However, here we focused on altering the tube geometry as the first candidate. Riccio et al. [19] presented detailed mathematical formulation on the development of the total mass deposition rate
mtot=PtotdmpdAsin(γ)+mc,A
(1)

where mtotis total mass deposition flux (kg/m2s), and Ptotis the total fouling probability. Fouling probability is a function of the stickiness of particles (depends on particle chemical composition) and stickiness of the tube surface (depends on vapor condensation). mp is the mass flow rate (kg/s) impacting the deposition area A (m2), γ is the impaction angle at which the particle strikes the tube surface and mc,Ais the vapor condensation flux (kg/m2s). Due to the condensation of particles on the tube surfaces, it forms an adherent layer on the tube surface, further enhancing the ash deposition. This depends on the characteristic length of the tube. In the circular tube, this length is equal to the outer diameter of the tube. Since mass deposition due to condensation is relatively very small as compared to that due to inertial impaction [19], therefore, mc,A can be dismissed for further investigation of target geometry's role in ash deposition. From a geometrical standpoint, the impaction angle (γ) at which the ash particle strikes the tube could be varied. Since mtot is directly proportional to the sine of γ, hence it is preferable to have a lesser impact angle.

2.2 Inspiration for the Novel-Shaped Tube.

Geometries that can offer a lesser impaction angle for particle striking must result in a streamlined design. Airfoil is one of the best examples for streamlining the flow, which has been heavily studied in literature and implemented for practical purposes. Airfoil profile assists in producing a laminar flow. Based on this, a novel tube design that resembles an ogive shape or a teardrop shape is proposed that reduces the impaction angle assisting in streamlined flow and hence reduced deposition. Throughout the work, for simplicity's sake, we further addressed our proposed tube as a “teardrop-shaped” tube.

2.3 Proposed New Geometrical Design and Details.

Total six variants of teardrop shape with circular and noncircular inner diameters were studied (see Fig. 2). These geometries were built using autodeskinventorprofessional 2020. Geometries were created to reduce the impact angle such that it generates a streamlined flue gas flow and hence reduces the total ash deposition mass flux. While creating geometries that have a circular inner profile, the inner diameter is kept equal to the baseline circular tube design. The thickness on the semicircular side of the asymmetric tube is the same as the baseline circular design. In geometry with a noncircular inner profile, the hydraulic diameter is equal to the circular inner profile. However, the other parameters to design the tube are currently user-defined, which could be optimized later. As an example, the dimension for the convex profile teardrop-shaped tube has been shown in Fig. 2(a). All the proposed tubes were additively manufactured on Concept Laser M2 cusing machine in stainless steel 316 L.

Fig. 2
Teardrop shape: (a) with a convex profile and circular ID; (b) with symmetric convex profile and circular ID; (c) with asymmetric convex profile and noncircular ID; (d) with a concave profile and circular ID; (e) with symmetric concave profile and circular ID; (f) with symmetric concave profile and noncircular ID; and (g) proposed tubes were additively manufactured on Concept Laser M2 cusing machine in SS316 L
Fig. 2
Teardrop shape: (a) with a convex profile and circular ID; (b) with symmetric convex profile and circular ID; (c) with asymmetric convex profile and noncircular ID; (d) with a concave profile and circular ID; (e) with symmetric concave profile and circular ID; (f) with symmetric concave profile and noncircular ID; and (g) proposed tubes were additively manufactured on Concept Laser M2 cusing machine in SS316 L
Close modal

3 Research Methodology

The overall research methodology to validate the proposed shapes is summarized in Fig. 3. As a part of the computational approach, a stress analysis was performed to predict the creep life of the proposed tube as compared to the baseline tube design. The steps in the analysis include a thermal model and a structural model of the tube. The second set of analyses was performed with a computational fluid dynamic model to understand the particle motion around the tubes. A Reynolds-averaged Navier–Stokes (RANS) kω SST (shear stress transport) model was built and coupled with the particle tracing module in comsolmultiphysics. Schlieren imaging was performed for the qualitative experimental validation of the fluid flow behavior around the proposed and baseline design.

Fig. 3
Methodology for the validation of the proposed design
Fig. 3
Methodology for the validation of the proposed design
Close modal

3.1 Numerical Modeling: Thermomechanical Finite Element Analysis.

As mentioned earlier in Sec. 1, these tubes contain high-pressure steam at high pressure (15–30 MPa) and temperatures exceeding 610 °C. Such critical operating condition elevates the hoop stress in the tube and is responsible for the stress rupture failure if extensive material waste occurs due to corrosion.

To study the above, a two-dimensional (2D) plane strain thermomechanical finite element (FE) analysis is performed using ansysmechanicalapdl 2021 r2 (student version) for a typical service condition. Herein, thermal and pressure conditions were considered for loading conditions. Since the boiler tubes are long prismatic bars, and all the loads act in the in-plane direction, therefore, it is valid to assume plane strain analysis [37]. Stainless steel 316 L is used as the material for geometry. The 2D geometries were created in autodeskinventorprofessional 2020 and were meshed in ansysapdl 2021 (student version).

The FE study has been performed in three steps: (1) thermal FE model was created to calculate the circumferential thermal variation across a tube, (2) the average temperature calculated at different angular positions was used to perform the conduction run and calculate the intermediate temperatures, and (3) applying these thermal temperatures as body forces for the structural FE model along with pressure loads, the hoop stresses were obtained.

3.1.1 Determination of Temperature Profile Along Tube Surface.

The localized ash deposition may induce temperature variations along the circumference of the tube surface. Such thermal variations might instigate or exacerbate stress-induced corrosions. Previous investigations have not considered the circumferential thermal variations in depth. For example, former analyses have calculated the temperatures by using a one-dimensional (1D) thermal resistance network between flue gases, ash deposits, metal wall, and steam side of the tube [10,38], calculated the thermal variation for ash deposits by assuming constant outer metal wall temperature [19,39], calculated the temperatures for the stacked superheater tube arrangements [40]. Therefore, we need to calculate circumferentially varying surface temperatures across the tube by considering the nonuniform presence of the ash. In this study, a thermal FE model is developed for determining the temperature profile along the tube surface. During these thermal calculations, surface heat transfer coefficients (HTCs) were analytically estimated and provided as boundary conditions to the FE model.

Estimation of HTC: In superheater tubes, heat transfer on the outer surface of the tube occurs due to the combination of radiation and convection. Therefore, the HTC due to convection and radiation was calculated separately and superimposed for effective HTC. For calculating the HTC due to convection, a flow across the cylindrical tube has been considered. For this, the Nusselt number was calculated using the empirical correlation [38,41] (based on experimental values of the average heat transfer coefficient from a heated cylinder)
Nu=C·RemPr(1/3)
(2)
where Nu is the Nusselt number, and the values of C and m vary as per the Reynolds number range. For flue gases, the density of the fluid (ρf) is 0.345 kg/m3, fluid velocity (vf) is 8 m/s, characteristics length (d) is 0.039 m, and dynamic viscosity (μf) is 4.41 × 10−5 kg/m s. The calculated Reynolds number (ρfvfd/μf) was 2442.48; therefore, the chosen value of m and C was 0.683 and 0.466 [38,41]. Hence, the following relationship was used for Nusselt number outside the tube due to convection:
Nu=0.683·Re(0.466)·Pr(1/3)
(3)
Prandtl number was calculated to be 0.73 via μfCp/k(f), where Cp is the specific heat capacity at constant pressure (1146 J/kg K), and k(f) is the thermal conductivity of the flue gas (0.069 W/m °C). The Nusselt number was calculated using Eq. (3) as 23.33. The effective heat transfer coefficient is calculated as below:
heff=hrad+hconv
(4)
hrad=ϵσ(Tf4Td4)/(TfTd)
(5)
hconv=Nu·kf/d
(6)

where ϵ is the emissivity of the surface, Tf is the flue gas temperature, Td is the deposit temperature, σ is the Stefan–Boltzmann constant (5.67 × 10−8 W m−2 K−4), Nu is the Nusselt number, kfluid is the thermal conductivity of the flue gas, and d is the characteristic length which is outside diameter (d0) in this case.

Using Eq. (6), the average heat transfer coefficient due to convection hconv was calculated to be 41.28 W/m2 °C. For calculating the average heat transfer coefficient due to radiation hrad, the emissivity of the deposit was taken as 0.5 [38]. The whole heat transfer process was initially solved as a 1D heat transfer problem to estimate the hrad (see Appendix  A). The calculated hrad value (159.36 W/m2 °C) was finally used to estimate the heff. From 12 to 2 o'clock, the deposit is modeled in the 2D FE model, and thus heffis applied in that region. In the rest of the region, since the ash is not present in a significant amount, thus it is not modeled, and only hconvis applied. However, in this region, the effect of 1 mm of thick ash deposit (radiation and conductance effect) was considered by reducing the bulk film temperatures by ∼10%, which is equivalent to the presence of 1 mm thick ash. The sensitivity was performed for ∼5–15% for bulk film temperatures that are equivalent to 0.05–1.5 mm of deposit presence. The analysis showed that the impact was within ∼2% on the outer wall temperature. Thus, it is acceptable to assume ∼1 mm of wall deposit. This study was verified by both hand-calculation and FE simulations. Thus, the heat transfer coefficient was calculated for every 5-deg sector of the geometry which ranged from 200.65 W/m2 °C to 41.28 W/m2 °C. For inside the tube, since thermal resistance on the flue gas side was found to be approximately four times the steam side [38], the heat transfer coefficient was assumed to be four times higher than the outside heat transfer coefficient. The calculated heat transfer coefficients were then applied in the thermal finite element model (see Fig. 4(a)).

Fig. 4
(a) Heat transfer coefficients in finite element thermal model; (b) boundary and loading conditions applied in the structural FE modeling of the tube
Fig. 4
(a) Heat transfer coefficients in finite element thermal model; (b) boundary and loading conditions applied in the structural FE modeling of the tube
Close modal

Finite element modeling: For thermal modeling, the ash deposit shape has also been modeled along with the circular tube-shaped. Ash deposit profile has been considered elliptical in shape, and a maximum thickness of 0.005 m is applied at the 12 o'clock position [38]. Ash deposits have been provided with a thermal conductivity of 0.6328 W/m °C [38], while the SS316 L has been provided with a thermal conductivity of 19.9 W/m °C. Plane 55 elements (2D thermal solid) were used to create the FE mesh within ansys. The FE model has been chunked in a sector of 5 deg circumferentially, and averaged HTCs have been applied in each sector's inner and outer boundaries. Through this simulation, the temperatures at the outer and inner walls were calculated. The average of the outer and inner walls has been used in the structural FE model (in Sec. 3.1.2) as the body force.

3.1.2 Structural Analysis Using Thermal and Pressure Loads.

In the structural FE model, 2D four-node solid elements were used for the plane strain analysis. Figure 4(b) shows the boundary and loading conditions applied in the structural FE model of the baseline tube. The calculated temperatures from thermal simulations were exported as body forces in the structural model of the tubes. In addition to temperature loads, pressure loads were also applied as surface loads in the structural model. High pressure of 17 MPa was applied at the inner wall, and an atmospheric pressure of 0.1 MPa was applied at the outer wall. The model is fixed in such a manner that it is allowed to grow radially and constrained in the hoop direction (see Fig. 4(b)). The same boundary conditions have been assumed for the proposed tube designs. The thermal coefficient of expansion for SS316 L applied was 1.8 × 10−5 cm/cm per °C.

3.2 Numerical Modeling: Flow Study Around the Differently Shaped Tubes.

The primary motive of studying flow around differently shaped tubes is two folds: First, to quantify the likelihood of particles striking the tube surface for the proposed tubes and the baseline tube. For this, a CFD study coupled with a particle tracking module is performed. In this, the fluid flow is subsequently coupled with particles that follow the stokes drag law. Second, to visualize and qualitatively study the development of different flow regimes for differently shaped tubes, schlieren imaging has been performed and postprocessed.

3.2.1 Computational Fluid Dynamics Modeling: Reynolds-Averaged Navier–Stokes Flow With Particle Tracing in comsolmultiphysics.

A multiphysics model was prepared to perform the comparative study of particle tracing in the tubes using comsolmultiphysics 5.5. For this, a particle tracing module has been used in integration with a single-phase flow module. For illustration purposes, Fig. 5(a) shows the geometric details for circular and teardrop-shaped tube models. Figures 5(b) and 5(c) illustrate the details of the mesh around the circular-shaped tube and teardrop convex profiled tube. The geometric details were inspired by Riccio et al. [19]. To allow comparison, this study is performed using a single tube for both circular and teardrop-shaped tubes.

Fig. 5
(a) Geometric details for both circular tube and teardrop-shaped tube; (b) and (c) mesh details for RANS k–ω SST analysis for circular and teardrop-shaped tube
Fig. 5
(a) Geometric details for both circular tube and teardrop-shaped tube; (b) and (c) mesh details for RANS k–ω SST analysis for circular and teardrop-shaped tube
Close modal

For studying the single-phase flow, the RANS approach has been used with the SST turbulent model. The SST model is a low Reynolds number model which considers k–ε in the freestream and kω model near the walls. For setting up the model, flue gas characteristics, particle density, particle mean size values, and particle mass flow rate were taken from the paper by Riccio et al. [19].

Table 1 summarizes the values for the geometric details, flue gas properties, and particle description for particle tracing. The grid was generated within the software, and the mesh was refined such that y+ values remained near or below one. Additionally, the model was checked for mesh sensitivity. Since the purpose of particle tracing is to quantify and compare the propensity of particles striking differently shaped tubes, therefore certain assumptions were made in the study. Particle to particle interaction was not considered. Uniform particle size (spherical smooth particle size of 68 μm [19]) is considered. Once the particle impacts the tube surface, the particles are assumed to freeze onto the tube surface. In specific, the particle velocity will come to zero, and the particle position will be fixed the instant it contacts the wall. Since rollover, sliding, and rebounding of particles were not considered in the study, temperatures were neither applied to the fluid flow nor to the tubular surface which might impact the shape of the particles. Stokes law has been used for calculating the drag force acting on the particles. Gravitational force and van der Waals forces were not considered. For postprocessing, drag coefficients and the number of particles encountering the tube surfaces will be compared for the circular and proposed tubes.

Table 1

Geometry details, flue gas properties, and particle description

ParametersValuesUnits
The outside diameter of the baseline tube39mm
L2.45m
H0.30m
Inlet flue gas velocity8m/s
Flue gas density0.345kg/m3
Specific heat capacity (flue gas)1146J/kg K
Thermal conductivity (flue gas)0.069W/m K
Dynamic viscosity (flue gas)44.07 × 10−6kg/m s
Particle specific mass flow rate1 × 10−3kg/s
Particle density2200kg/m3
Particle mean diameter68μm
ParametersValuesUnits
The outside diameter of the baseline tube39mm
L2.45m
H0.30m
Inlet flue gas velocity8m/s
Flue gas density0.345kg/m3
Specific heat capacity (flue gas)1146J/kg K
Thermal conductivity (flue gas)0.069W/m K
Dynamic viscosity (flue gas)44.07 × 10−6kg/m s
Particle specific mass flow rate1 × 10−3kg/s
Particle density2200kg/m3
Particle mean diameter68μm

3.2.2 Experimental Approach: Schlieren Optics for Visualizing the Flow Around the Tube.

In order to visualize the flow around differently shaped tubes, Schlieren optics was used. Schlieren optics allows for the visualization of the flow of gases or change in air refractive index via a concave mirror, point-source light, razor blade (light block), and a camera. It is based on the principle that the speed of the light changes in varying density medium, and the path of the light will deviate from its original path (Fig. 6). This technique has been abundantly used in a range of applications for flow visualizations, thermal variations, acoustic waves visualization, the study of materials, and other related studies (see Appendix  B). However, it has found limited applications in boiler tubes and heat exchanger assessments. In our work, we have used this technique due to its capability to allow the user to directly track fluid flow in contrast to tracking particles (for example, fluorescent particles in particle image velocimetry). Also, this technique is easy to set up and comprehend in modern fields of fluid analysis with emerging computer vision techniques and image processing tools [42].

Fig. 6
Schematic representation of Schlieren optic setup
Fig. 6
Schematic representation of Schlieren optic setup
Close modal

Working and setup: In the Schlieren optics setup, point-source light emitting diode (LED) light is projected in the field of view of the concave mirror. A razor blade (light block) is kept at twice the focal length of the concave mirror. A camera with a zoom lens is aligned behind the light block to capture the image of the fluid flow variation in front of the mirror. If the density of the fluid in the proximity of the concave mirror is varying, then the light that gets reflected from the mirror will deviate from its original path. The razor is vertically aligned such that it barely cuts the converged image. If light rays are deflected toward the knife-edge, image portions created from these deflected rays of light will be darkened as compared to image portions due to constant density. Alternatively, if light rays are deflected away from the knife-edge, image portions due to these deflected rays of light will be brighter than the image portions [43]. These are the mechanisms for the enhanced visualization of the fluid flow that is not visible to human naked eyes. An in-house setup was created for the same and has been described in detail in Appendix  C.

Postprocessing technique: Postprocessing was done using opencv (python library) and the open-source software pivlab (digital particle image velocimetry for matlab toolbox) [44]. opencv is used for image segmentation of the schlieren images. These processed images were then used as inputs for pivlab. pivlab is originally designed for the calculation of displacement of the particles based on the cross correlation of the subimages between two consecutive particle image velocimetry images. However, Gena et al. [45] discussed in detail how pivlab is useful for processing other types of pattern displacements than just particle image velocimetry particle images. This tool can be leveraged for schlieren image postprocessing. In the current paper, pivlab has been used for generating streamlines for the flow around differently shaped tubes.

4 Results

4.1 Finite Element Results

4.1.1 Thermal Simulation Results.

From the FE thermal model, temperature contour plots were extracted for the tube wall (see Fig. 7(a)). From the model, averaged temperature values were fetched at every 5-deg interval on the outer wall as well as on the inner wall of the circular tube. The thermal profile on the outer and inner wall with respect to the angular position of the tube is plotted in the polar plot form (see Fig. 7(b)). This profile suggests higher temperatures around the 10 o'clock (∼60 deg) and 2 o'clock (240 deg) position relative to other locations. At the 12 o'clock (0 deg) position, the temperature obtained is the lowest. This conforms with the hypothesis mentioned above. As mentioned earlier, the average temperatures of the inner and outer walls are used as input to the FE structural model.

Fig. 7
(a) Thermal FE simulation results displaying temperatures (°C) contours for the baseline tube wall. (b) Polar plot for the tube wall temperatures on the outer wall, inner wall, and averaged wall temperature.
Fig. 7
(a) Thermal FE simulation results displaying temperatures (°C) contours for the baseline tube wall. (b) Polar plot for the tube wall temperatures on the outer wall, inner wall, and averaged wall temperature.
Close modal

4.1.2 Structural Simulation Results.

The hoop stresses, i.e., stresses in the ϴ-direction in the cylindrical coordinate system (csys-1 and rsys-1 in ansys 2021 r2 mechanicalapdl), were extracted from the FE simulation results. Through the hoop stress contour plots, overall maximum hoop stress and stresses near the 2 o'clock position have been used for the comparison of the differently shaped tubes. The overall maximum hoop stress might or might not be near the 2 o'clock position. However, it would be interesting to understand the nature of stresses near the 2 o'clock location because these regions are known to be susceptible to material degradation. Tensile stresses around this location might aggravate stress-assisted corrosion and accelerate creep failure due to increased hoop stresses [46] (see Fig. 8(h)). The tensile stresses would allow the liquid phase to penetrate through the grain boundaries and thereby lead to crack extension into the subsurface. Therefore, the design must consider the reduction or removal of tensile stresses around the 2 o'clock position. Figure 8 shows the hoop stress contours for the baseline tube (Fig. 8(a)), proposed tube (Fig. 8(b)), and its variants (Figs. 8(c)8(g)). A common scale is used, and tensile stresses (positive hoop stresses) have been plotted for comparison purposes. The gray portions in Figs. 8(b), 8(c), 8(e), and 8(f) indicate only compressive stresses, while the gray portion in Figs. 8(d) and 8(g) contain both compressive and tensile stresses. The tensile stresses in designs D and G are relatively very high as compared to the rest of the designs. The contour plots for hoop stresses clearly indicate the development of compressive stresses in the proposed tubes (design B, C, E, and F) near the 2 o'clock position. In the region of interest (∼2 o'clock position), average stress is marked in Fig. 8 for all the tubes.

Fig. 8
(a)–(g) Tensile hoop stresses in a cylindrical coordinate system for the baseline design and the proposed designs; (h) schematic for stress-assisted corrosion under tensile stresses at the outer wall; and (i) hoop stresses for each design iteration including baseline tube design. Maximum hoop stress indicates the largest magnitude of hoop stress in the tube. Hoop stress at the 2 o'clock position which is the region of interest.
Fig. 8
(a)–(g) Tensile hoop stresses in a cylindrical coordinate system for the baseline design and the proposed designs; (h) schematic for stress-assisted corrosion under tensile stresses at the outer wall; and (i) hoop stresses for each design iteration including baseline tube design. Maximum hoop stress indicates the largest magnitude of hoop stress in the tube. Hoop stress at the 2 o'clock position which is the region of interest.
Close modal
Fig. 9
Velocity field for the baseline tube and proposed tube designs
Fig. 9
Velocity field for the baseline tube and proposed tube designs
Close modal

For clear representation, overall maximum hoop stress and hoop stress at the 2 o'clock position have been plotted in Fig. 8(i). The overall maximum hoop stresses are highest in design D (892 MPa) followed by design G (408 MPa). Amongst designs A, B, C, E, and F, circular-shaped tube (design A) has the highest overall maximum hoop stress (90.3 MPa). The teardrop convex profile shape with a circular inner diameter (design B) shows 37% lower hoop stress at the 2 o'clock position as compared to that of the baseline tube at the same location. Amongst the other proposed variants, the convex profiled shape provides 18–24% less stresses in the region of the interest as compared to the concave profiled teardrop shape. Out of all proposed and baseline designs with a reduction of mechanical stress, the prominence of designs B, C, E, and F appears to be suitable for our application.

4.2 Flow Study Results

4.2.1 Computational Fluid Dynamics Modeling Results.

In this subsection, velocity fields (refer to Fig. 9), drag coefficients, and the number of particles encountering the tube surface for various geometries have been calculated and compared with a baseline tube design. Total drag force per unit length acting on the tube surface was calculated using the velocity field results. The drag force includes both pressure and skin friction forces. For calculating the drag force, local wall shear stress was integrated on all the boundaries of the tube shape. Further, the drag coefficient (a dimensionless number) for each tube was calculated using the following relationship:
Cd=2Fd/ρu2A

where Cd is the drag coefficient, Fd is the drag force (N), ρ is the density of the fluid (kg/m3), u is the flow velocity of the object relative to the fluid (m/s), and A is the reference area (m2). The drag coefficients have been shown in Fig. 10(a). Design B has the lowest drag coefficient which is 21.6% lower than the circular-shaped tube design. While the other variants of the tube design have higher drag as compared to circular-shaped tubes. In addition to velocity fields, the number of particles striking the tube surface was studied. This study was performed for the different number of particles per release that are 250, 500, 600, 750, 1000, and 2000 particles per release. Figure 10(b) shows that in all six cases the number of particles freezing on the circular-shaped tube was higher as compared to teardrop shape convex profile tube design (design B). On average, teardrop convex profiled shape (design B) has a 13% lesser number of particles striking the tube surface as compared to circular-shaped tubes.

Fig. 10
(a) Drag coefficient for each design iteration arranged in descending order; (b) number of particles striking tube surface for differently shaped tubes
Fig. 10
(a) Drag coefficient for each design iteration arranged in descending order; (b) number of particles striking tube surface for differently shaped tubes
Close modal

4.2.2 Schlieren Experiment Results.

Flow around differently shaped tubes (designs A and B) was visualized using Schlieren optics setup in a .mp4 format. These videos were image processed to get streamlines around the tube. The analyses were performed on extracting 20 frames per second of the movie using pythonapi. Further, postprocessing was performed on images extracted from the measured data from teardrop and circular-shaped tubes. Since Schlieren images contain regions of different intensities, therefore, adaptive thresholding was performed using the opencvpython library. In this case, the different threshold value for different regions is calculated by taking the average of the neighborhood values. Figure 11(a) shows Schlieren images for four different frames and the corresponding images after adaptive thresholding. These preprocessed sets of images were then imported into pivlab for image analysis and image postprocessing. Averaging of velocity vector field was performed on results from 80 images. The averaged information was then used to construct the streamlines (see Figs. 11(b) and 11(c)) which has been discussed in detail in Sec. 5.

Fig. 11
(a) First row: original images for circular-shaped tube; second row: images after adaptive thresholding corresponding to images in the first row; third row: original images for teardrop-shaped tube; and fourth row: image after adaptive thresholding corresponding to the image; (b) and (c): streamlines around differently shaped tubes
Fig. 11
(a) First row: original images for circular-shaped tube; second row: images after adaptive thresholding corresponding to images in the first row; third row: original images for teardrop-shaped tube; and fourth row: image after adaptive thresholding corresponding to the image; (b) and (c): streamlines around differently shaped tubes
Close modal

5 Discussions

5.1 Role of Geometry and Creep Life Calculations.

While past investigations used analytical methods to calculate hoop stress due to internal pressure [2,47], in our work, we used finite element methods to calculate the hoop stresses by also considering varying circumferential temperatures in the tube cross section. Circumferential thermal applications lead to nonuniform development of hoop stresses in the circular-shaped tube. However, in other differently shaped tubes, the nonuniform hoop stress development is the consequence of the shape of the tube as well as the circumferential thermal loads.

From Sec. 4.1.2, it can be inferred that the shape of the tube impacts the location of maximum hoop stress development. Designs D and G have the highest hoop stresses and perform poorly than the baseline tube in terms of the mechanical integrity of the tubes. This could be majorly attributed to the geometrical notch at the 12 o'clock and 6 o'clock positions on the inner side of the tube. However, other designs like B, C, E, and F are competitive designs for the circular-shaped tubes. As opposed to circular-shaped tubes, in all these designs, the maximum hoop stress did not develop in the region of interest (i.e., at 2 o'clock position). In all four proposed designs, the overall maximum hoop stress formed either near the ∼12 o'clock position or ∼6 o'clock position. Additionally, the magnitudes of the maximum hoop stresses are lower than the circular-shaped tube which might lead to increased creep life of the superheater tubes.

To estimate the creep life of the baseline and proposed tubes, additively manufactured stainless steel 316 L tubes have been considered. Li [33] developed an empirical relationship between creep stress and Larson–Miller parameter (LMP) for AM SS316 L (manufactured through L-PBF method on Concept Laser M2 cusing machine) at different temperatures. For our case, we leveraged the same relationship at 600 °C to estimate the LMP for differently shaped tubes at 0 h and 10,000 h of operation. Since designs D and G are incompetent for circular-shaped tubes, these tubes were dismissed for further investigation, including creep life calculations. Therefore, LMP was calculated for designs A, B, C, E, and F without degradation and with degradation. For the degradation study, the material degradation of 0.2 mm after 10,000 h of operation was assumed and applied in the geometry from 55 deg to 65 deg (see Fig. 12(a)).2 The LMP, T(C+log10(tf)), where C is material specific constant, tf is time to failure in hours, and T is temperature in Kelvin, was calculated using C = 17 [33]. Figure 12(b) clearly suggests that proposed tube designs B, C, E, and F have lesser LMP, and consequently improved life as compared to the circular-shaped tube. This study also indicates that for circular-shaped tubes, the rate of decrement of life after 10,000 h is slower as compared to the proposed tubes. The FE models with degradation developed for creep life calculations can be modified and extended to real life conditions.

Fig. 12
(a) Local geometric degradation of 0.2 mm after 10,000 hours of operation; (b) predicted LMP at 0 h and after 10,000 h of operation for each design iteration
Fig. 12
(a) Local geometric degradation of 0.2 mm after 10,000 hours of operation; (b) predicted LMP at 0 h and after 10,000 h of operation for each design iteration
Close modal

5.2 Significance of Fluid Flow Characteristics for Superheater Tubes.

Since the Reynolds number of the flue gas flow in our study is of the order of 3, therefore, the contribution from skin friction drag is minimal. Primarily the drag force acting on the tube surface is due to the pressure drag force. Figures 10(a) and 10(b) show that shapes with lower drag coefficients also have a lesser propensity of particles striking the tubular surface. Figure 10(a) displays higher drag coefficients for the symmetric tube designs. This might be mitigated by changing the geometry by reducing the width of the cross section. However, this might either lead to a reduction in the hydraulic diameter thereby causing the pressure drop inside the tube for the steam flow, or increased hoop stresses due to stress concentration. Asymmetric tubes have lower drag coefficients as compared to symmetric tubes which could be attributed to their lower surface area. Also, Fig. 10(a) implies higher drag coefficients for circular-shaped tubes than design B. This can be reasoned out via Schlieren imaging experimental studies. Lower drag coefficient should allow the streamline to follow the geometric curvature, while in the case of higher drag coefficient, the flow will separate early causing turbulent wakes. Figures 11(b) and 11(c) clearly indicate that the flow around circular-shaped tube is more turbulent, while the flow around the teardrop shape is more streamlined. Also, in the case of a teardrop shape, the flow reattaches at the rear of the tube signifying the lesser propensity of vortex shedding at the lee side of the tubes. Additionally, the streamlined flow must enable lower pressure to drop across the heat exchanger. Figure 11(b) also exhibits a high level of mixing of fluid in the circular-shaped tube in contrast to the teardrop-shaped tube. Reduced mixing of flue gas might lessen the ash deposition in the proposed shape as compared to the current superheater tube design. However, reduced mixing and lower turbulence might impact the heat transfer efficiency at the tubular stack level. For this, further optimization of arrangements of tubes could be studied in the future.

5.3 Feasibility of Using Modified Tubes.

Based on the above stress simulations and computational fluid dynamics performed for the proposed tubes and the baseline tubes, tube shape with streamlined geometry and improved creep life can be leveraged as one tool kit to combat the ash deposition. However, the weight and hydraulic diameter of the tubes must also be considered during the design. Since, during the design phase, the hydraulic diameter is kept the same for all the tubes (30 mm), therefore, it can be ruled out in the current case. Weight has been calculated using SS316 L material for all the tubes per unit length. A metric table has been prepared and presented in Table 2 for all the tubes. Each parameter, maximum hoop stress (MPa), hoop stress at 2 o'clock position (MPa), drag coefficient, and the weight of the tube (kg/m), has been normalized. In Fig. 13, normalized parameters are plotted. For normalization, the weightage factor applied for each parameter is kept one, which means that currently, all the parameters in the overall metric are equally important. The x-axis shows the design type, while the y-axis represents the magnitude of the metric. The lower the sum of the overall metric, the better the design is for the superheater heat exchanger tubes. The metric clearly shows that design B, a teardrop convex profiled tube with circular ID, performs best amongst all, while design D, a teardrop convex profiled tube with noncircular ID (symmetric), performs worst.

Fig. 13
Overall metric for down selecting superheater tube shapes; equal weightage is applied for each parameter; lower overall metric represents better design
Fig. 13
Overall metric for down selecting superheater tube shapes; equal weightage is applied for each parameter; lower overall metric represents better design
Close modal
Table 2

Overall metric for several tube designs for superheater tubes

DesignHydraulic diameter (mm)Maximum tensile hoop stress (MPa)Hoop stress at 2 o'clock position (MPa)Drag coefficientWeight of the tube (kg per meter length)
A: Baseline3090.361.20.4973.92
B: Teardrop convex profiled circular ID—asymmetric3065.438.50.3895.92
C: Teardrop convex profiled circular ID—symmetric3062.735.80.7197.91
D: Teardrop convex profiled noncircular ID—symmetric30.42892203.20.7194.93
E: Teardrop concave profiled circular ID—asymmetric3073.147.10.5424.70
F: Teardrop concave profiled circular ID—symmetric3078.146.60.7315.48
G: Teardrop concave profiled noncircular ID—symmetric29.9940890.10.7314.91
DesignHydraulic diameter (mm)Maximum tensile hoop stress (MPa)Hoop stress at 2 o'clock position (MPa)Drag coefficientWeight of the tube (kg per meter length)
A: Baseline3090.361.20.4973.92
B: Teardrop convex profiled circular ID—asymmetric3065.438.50.3895.92
C: Teardrop convex profiled circular ID—symmetric3062.735.80.7197.91
D: Teardrop convex profiled noncircular ID—symmetric30.42892203.20.7194.93
E: Teardrop concave profiled circular ID—asymmetric3073.147.10.5424.70
F: Teardrop concave profiled circular ID—symmetric3078.146.60.7315.48
G: Teardrop concave profiled noncircular ID—symmetric29.9940890.10.7314.91

5.4 Limitations and Future Work.

This research proposes a novel-shaped tube design that can be realized via additive manufacturing. To implement these designs in real-time application in power plants, our proposed work unfolds areas for interesting research, within the allowable limits defined by ASME codes. Since these designs have oblique surfaces, it might increase the rebounding of the ash particles. These rebounded particles can then hit other tubes and start depositing on them. Hence, it is required to study the tube design with the stack of tubes. Also, in this study, the heat transfer calculations were performed for a constant thickness of ash which might vary with respect to time. Additionally, the ash composition affects the conductivity of the ash and emissivity, and hence it might also alter the magnitude of thermal calculations. In this study, for Schlieren optics experiments, the fluid has been treated as a jet in order to increase the sensitivity near the tube wall for visualization purposes. The jet speed is ∼12 m/s, while the speed of the flow considered in CFD is 8 m/s.

6 Conclusion

This paper focuses on geometric modification of circular tubes and the development of design methodology for improved superheater tube design. A novel teardrop-shaped tube design and its variants are proposed which could be enabled via additive manufacturing techniques like L-PBF. These tubes were assessed along with the baseline tube for hoop stress development, creep life, and flow field characteristics around them, and an overall metric was prepared to compare different shapes. Amongst all, teardrop-shaped convex profiled asymmetric tube performs best.

  1. Heat transfer calculation performed for a 2D model of the tube infers higher temperatures around the 2 o'clock position of the tube. The thermal boundary conditions coupled with the structural model show that teardrop convex profiled asymmetric tube encounters ∼37% lower tensile stresses at critical locations and ∼27.5% lower maximum hoop stress as compared to the baseline design. This leads to improved creep life for the superheater tubes.

  2. The particle tracing analysis coupled with single-phase flow turbulent CFD model showed that the teardrop convex profiled design has ∼21% lesser drag than the baseline design. Also, the number of particles encountering the tube surface was lower in the case of teardrop shape convex profiled tube as compared to circular tube design.

  3. Schlieren optics experiment was leveraged to visualize the flow around the differently shaped tube. The streamlines generated through Schlieren videos clearly indicated that the teardrop shape has a more streamlined flow (laminar flow) and reduced mixing of fluid flow as compared to the circular-shaped tube. Reduced mixing of flue gas might lessen the ash deposition in the proposed shape as compared to current superheater tube design.

Acknowledgment

This paper was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05- 00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan.3

Funding Data

  • U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, Advanced Manufacturing Office, and Office of Fossil Energy (Contract No. DE-AC05-00OR22725 with UT-Battelle, LLC; Funder ID: 10.13039/100000015).

  • U.S. DOE, Office of Fossil Energy (Award No. DE-FE-0031831; Funder ID: 10.13039/100006120).

Nomenclature

A =

reference area for drag calculation (m2)

Cp =

specific heat capacity at constant pressure (J/kg K)

d =

characteristic length (m)

d0 =

outside diameter of the tube (m)

Fd =

drag force (N)

H =

height of the geometrical boundary for CFD (m)

hconv =

heat transfer coefficient due to convection (W/m2 °C)

hrad =

heat transfer coefficient due to radiation (W/m2 °C)

k =

thermal conductivity (W/m °C)

L =

length of the geometrical boundary for CFD (m)

mc,A =

vapor condensation flux (kg/m2s)

mtot =

total mass deposition flux (kg/m2s)

T =

temperature (°C)

tf =

time to failure (h)

vf =

fluid flow velocity of flue gas (m/s)

γ =

impaction angle (deg)

μf =

dynamic viscosity of the flue gas (kg/m s)

σ =

Stefan–Boltzmann constant (W m−2 K−4)

Subscripts or Superscripts
conv =

convection

d =

deposit surface

eff =

effective (convection and radiation)

f =

flue gas

o =

outer diameter

rad =

radiation

s =

steam

Dimensionless
C =

material specific constant

Cd =

drag coefficient (2Fd/ρu2A)

LMP =

Larson–Miller parameter T(C+log10(tf))

Nu =

Nusselt number (hd/k)

Pr =

Prandtl number μfCp/k(f)

Re =

Reynolds number (ρfvfd/μf)

Acronyms and Abbreviations
AM =

additive manufacturing

CFD =

computational fluid dynamics

FE =

finite element

HTC =

heat transfer coefficient

L-PBF =

laser-powder bed fusion

RANS =

Reynolds-averaged Navier–Stokes

SS =

stainless steel

SST =

shear stress transport

Footnotes

2

The local fireside corrosion at 2 o'clock position (from 55 deg to 65 deg angular position) was modeled through finite element analysis and maximum hoop stress was calculated for each case.

Heat Transfer Calculation Assuming One-Dimensional Case

The one-dimensional resistance network in a superheater tube consists of combination of radiation and convection (from flue gas to ash), conduction through ash, conduction through metal tube wall, and forced convection from metal wall to high-pressure steam (see Fig. 14 for the thermal resistance network). The heat flux (W/m) can be calculated as below in Eq. (A1):
Qt=(TfTs)/Rt
(A1)
Fig. 14
One-dimensional resistance network in superheater tubes
Fig. 14
One-dimensional resistance network in superheater tubes
Close modal
The overall resistance can be calculated as below in Eq. (A2):
Rt=R1+R2+R3+R4
(A2)
Rt=1/π(hrad+hconv)dd+ln(dddo)2πkdeposit+ln(dodi)2πkmetal+1/πhsteamdi
(A3)
Since
Qt=(TfTd)/R1=(TdTo)/R2=(ToTi)/R3=(TiTs)/R4
(A4)

where Tf is the flue gas temperature (900 °C), Td is the deposit temperature, To is the outer wall temperature, Ti is the inner wall temperature, and Ts is the steam temperature (550 °C). R1 is the thermal resistance from flue gas to ash, R2 is the thermal resistance of ash, R3 is the thermal resistance of the metal wall, and R4 is the thermal resistance of the metal wall to steam. No inner and outer scale has been assumed. A maximum deposit thickness of 0.005 m has been assumed. Using the iterative method and Eqs. (A1), (A3), and (A4), the individual thermal resistances were calculated as 0.032 °C/W, 0.057 °C/W, 0.002 °C/W, and 0.013 °C/W for R1, R2, R3, and R4, respectively (see Fig. 15). This clearly shows that the presence of ash and heat transfer from flue gas to ash offers the highest thermal resistances and leads to decreased heat flux. Additionally, the hrad was calculated as 159.36 W/m2 °C which is further used for the 2D thermal simulation and circumferential thermal profile calculations. Before performing the 2D FE simulations for circumferential thermal gradient calculations, the 2D FE model was created with uniform ash deposition of 0.005 m thickness and was benchmarked with the above 1D hand calculations. FE results were within 2–5% of the hand-calculated values. This indicates the goodness of the FE model for further analysis of thermal gradient with nonuniform ash distribution.

Fig. 15
Calculated thermal resistances for superheater tube for 0.005 m thick ash
Fig. 15
Calculated thermal resistances for superheater tube for 0.005 m thick ash
Close modal

Schlieren Analysis Leveraged in Tube Study

Refer Fig. 16 for the network analysis performed for Schlieren optics used in different domains.

Fig. 16
Network analyses for Schlieren system used in various applications (created with vosviewer)
Fig. 16
Network analyses for Schlieren system used in various applications (created with vosviewer)
Close modal

Schlieren Optics Experimental Setup

For building up the Schlieren optics, the following components were used and aligned to achieve the overall setup:

A concave spherical mirror with aluminum protection of 60 in. focal length and 6 in. of diameter (procured from Edmund Optics, Barrington, NJ) (Fig. 17(a)). A mount for holding the mirror was designed in autodeskinventorprofessional 2020 and was 3D printed using Fortus 400mc (acrylonitrile butadiene styrene material—gray color) (Fig. 17(a)). A razor blade is kept at a distance of ∼120 in. from the mirror center, where the point LED image converges (Fig. 17(c)). A DSLR Canon Camera with a zoom lens (focal length of 75–300 mm and the corresponding f-number is 1:4.5–5.6) (Fig. 17(b)) is kept just behind the light block to capture the videos. For the point LED source, a WOWTAC A1S LED Flashlight (1150 Lumens) covered with an aluminum foil with a point hole (Fig. 17(b)) is used. Camera and point-source LED alignment with mirror is performed using a laser pointer (Fig. 17(b)). Differently shaped tube samples and mounts were designed in autodeskinventorprofessional 2020. They were 3D printed using polylactic acid material (for tubes) and carbon fiber nylon (for mounts) on Makergear M3 at Manufacturing Demonstration Facility, Knoxville, TN. These samples are placed in front of the mirror in the test region (Figs. 17(d)17(g)). For generating the gas flow around the tube surface, an electronic duster—HFC 152a has been used. The refrigerant HFC 152a has a relative vapor density of ∼2.3 (air = 1) and hence provides the visible variation in flow around differently shaped geometry via schlieren imaging. The recorded video can be accessed.4

Fig. 17
Schlieren imaging experimental setup
Fig. 17
Schlieren imaging experimental setup
Close modal

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