Probabilistic cost analysis determined the cost benefit for applying a protective coating to the wetted surfaces of stainless steel tank walls for concentrating solar power (CSP) thermal storage applications. The model estimated the total material cost of coated 347 or 310 stainless steel (347/310) and the cost of uncoated Inconel 625, which served as the reference tank wall cost. Model results showed that the cost of the coated 347/310 stainless steel was always statistically less than the cost of the bare Inconel 625 when these materials are used for tank walls at representative tank diameters and temperatures for CSP storage applications.

## Introduction

Next-generation CSP plants will operate with advanced power conversion cycles including those that use supercritical carbon dioxide as the working fluid [1]. These cycles will operate at higher temperatures relative to current power cycles to increase their thermal-to-electric conversion efficiency. In addition, higher temperature drops across the power block will increase the stored energy density of the thermal energy storage (TES) system. These improvements in power plant efficiency and thermal storage capacity will require the development of new materials and system integration strategies that enable operation of CSP plants at higher temperatures.

Advanced power cycles require new heat transfer and thermal storage fluids that are thermally stable in the range of 650 °C–850 °C. Current state-of-the-art power plants use nitrate salts that are limited in temperature to about 575 °C [2]. Candidate heat transfer fluids (HTFs) for higher-temperature operation are molten salts and metal alloys—both of which are potentially corrosive to the tank and piping walls that contain them. Salts that contain chlorides, for example, are extremely corrosive to stainless steel. One option that addresses this challenge is to use high-temperature, chemically resistant nickel-based alloys. However, these materials are considerably more expensive than stainless steels—in some cases by an order of magnitude—and their use may negate the cost benefit that results from increased conversion efficiency and stored thermal energy density [3,4].

A second approach is to apply chemically resistant coatings to the wetted surfaces of tanks and piping to protect a less expensive wall material from the corrosive effects of the fluid. Coatings are designed to limit or eliminate degradation and corrosion of the underlying substrate to increase its service life. Different containment materials and coatings have been evaluated to contain aggressive molten salts [5–10]. The coatings must be very dense with minimal defects for good corrosion resistance [11–12]. The majority of coatings development for minimizing molten-salt corrosion has been for waste incinerator, gas turbine, and electric power generation (steam-generating equipment) applications [13–22]. In these systems, impurities from the fuel form salts during combustion. When power turbines and containment alloys reach temperatures greater than 700 °C, the salts form a thin film of molten fluid that attacks the substrate metal alloy. This type of molten-salt corrosion is known as hot corrosion.

Several research efforts within the Department of Energy (DOE's) Sunshot Initiative [23] are developing new HTFs and determining their high-temperature corrosive effects with likely tank wall materials. These efforts include the development of corrosive-resistant coatings that will allow the use of less-expensive containment and piping materials.

### Protective Coatings Development.

The goal for protective coatings research is to develop coatings that protect the substrate (wall or piping material) for the full 30-yr lifetime of the power plant. This requirement is due to practical restrictions that preclude the re-application of coatings after the plant is commissioned. For example, the HTF storage tanks are brought to their operating temperature during plant commissioning and remain at that temperature for the 30-yr plant lifetime. Re-application of the inside wetted surfaces of the tanks would require cooling the tanks to allow personnel access. Also, during normal operation, the tanks never completely drain so as to maintain a minimal HTF level to keep the pump inlets submerged. These restrictions make coating re-application impractical and require that the initial coating that is applied during plant construction last the full 30-yr lifetime of the plant.

Protective coatings are formulated to corrode at an established rate (*μ*m/yr normal to the surface) when in contact with the HTF. This corrosion rate determines the coating thickness that is required to protect the substrate for 30 yr. To protect the storage tank walls from corrosion and eventual failure, coatings are being developed that have a target corrosion rate of 30 *μ*m/yr. To protect for 30 years, these coatings need to have a minimal thickness of 900 *μ*m.

Coatings development for CSP applications has some aspects in common with turbine-blade coatings development [13]. Turbine blades are temperature-cycled between ambient and high temperature during normal operation. In addition to combustion gas products, the blades are impacted by salt particles that are combustion byproducts. Because of these common conditions, we chose to investigate two-layer coatings that are similar to those that were developed for turbine-blade applications. Two-layer coatings have a topcoat that protects the bottom coat and substrate from corrosion. The bottom coat functions as the bond coat that provides an interface between the substrate and topcoat and has thermophysical properties that are intermediate to the substrate and topcoat. The bond coat improves adhesion of the topcoat to the substrate and allows the coating and substrate to tolerate high temperatures and high temperature gradients during thermal cycling.

Protective coatings for HTF storage tank walls have the additional requirement of needing to protect against the corrosive effects of liquid-phase salts that serve as HTFs. This requirement restricts the number of candidate materials for the protective topcoat and requires evaluation of topcoat corrosion rates under a different set of conditions compared to those for turbine blades. For this cost analysis, we chose to evaluate a two-layer coating that is applicable to liquid-phase HTF applications.

### Objective for Coatings Cost Modeling.

We quantified the cost benefit for using protective coatings as the wetted surfaces of heat transfer and storage fluid containment vessels. Specifically, probabilistic cost modeling was used to compare the cost of a storage tank wall composed of a stainless steel substrate with an applied protective coating to the cost of a container wall composed of a bare, nickel-based alloy. The modeling considered the maximum allowable stress of the wall material as a function of temperature and used it to determine appropriate wall thicknesses for each material for a given tank wall diameter and hoop stress.

### Scope of Work.

We developed a probabilistic cost model that accounts for the uncertainties in the coating cost factors, the stainless steel substrate cost, and the reference nickel-alloy cost. This analysis was part of a larger experimental and modeling effort to identify and develop coating materials and methods that can be used for practical, large-scale applications, and applied to the inner walls of CSP thermal storage tanks and other wetted components.

For this analysis, coating costs were estimated assuming that the plasma spray method was used to apply a two-layer coating to the substrate. The first layer was a nickel/chromium formulation that forms a bond coat with the substrate and helps the top, protective coat to adhere to the substrate material. The second layer was an α-alumina topcoat and protects the substrate material from corrosion due to contact with the fluid. The probabilistic model evaluated the cost of the two-layer coating plus 347 or 310 stainless steel (347/310) to the cost of a bare nickel-alloy (Inconel 625) for a set of representative CSP storage tank temperature and pressure conditions. The properties and costs of 347 and 310 stainless steels are very similar, so the modeling results are applicable to both materials.

The representative conditions for this comparative analysis were specified assuming application of the coating to the inside surface of a storage tank wall at the base of the tank. Wall sections at the tank base are thicker than those in the upper parts of the tank to withstand the greater fluid pressure and resulting hoop stress at the base. For this scenario, the hoop stress is a function of fluid pressure and tank diameter. We chose to evaluate the costs of tank walls at the base of the storage tanks that had diameters from 5 to 15 m and operated at temperatures of 650 °C, 760 °C, and 815 °C.

## Thermal Spray Methods

The plasma spray method is one of several thermal spray methods that are used to apply coatings to large surfaces typically in industrial settings [24]. They are used to apply protective coatings to the wetted surfaces of large storage tanks for the petrochemical industry. Because of their widespread use at large scale, they are considered the most commercially developed and economical methods for applying protective coatings.

All thermal spray methods melt particles of the coating material as they pass through a high-temperature torch at high velocity. The torch may be maneuvered manually or using robotics. The melted particles impinge onto the substrate surface where they form a continuous film that rapidly freezes to form the protective coating. The differences between the methods are the heat source for generating the high-temperature particle melt and the gas flow composition. Heat sources are combustion of acetylene, hydrogen, methane, or other hydrocarbons with oxygen, electric-arc plasma, or radio-frequency plasma.

When a plasma method is used, the particles are entrained in an inert gas such as argon, helium, or nitrogen. Coating parameters include initial particle size, particle flow rate, particle velocity, torch temperature, and gas flow rate. The plasma method that is assumed for this analysis uses electricity as the heat source to melt the particles entrained in nitrogen gas as they pass through the torch. The cost factors that make up the coating cost are labor for setup, surface preparation, and coating application, powders and gas feedstocks, electricity, and equipment use fee.

## Probabilistic Cost Modeling

The probabilistic cost methodology and model included all relevant cost factors for determining the total cost of the coating. Cost-factor categories are coating materials, energy, consumables, and labor. For comparison, an alternative, straight-forward approach to estimate cost is to identify the cost of each of the cost factors and input them into a cost model that generates the estimated total cost of the coating. This approach is deterministic modeling and assumes a single value for each input variable (cost-factor) to generate a single value for the output variable (cost estimate for a particular coating). The limitation of this methodology is that the single value for the estimated total cost does not provide any information regarding the uncertainty of the estimate. With this approach, there is no information available that provides the level of confidence that the coated stainless steel will reach its intended cost goal—i.e., be less than the cost of an uncoated nickel-alloy substrate.

Deterministic modeling does not consider the fact that there are significant uncertainties in the values for most or all of the cost factors of a new technology. Uncertainty is typically most significant in the early stages of the research and development of a new technology when specific materials and processes have not been fully characterized or may not be accurately known. Probabilistic modeling accounts for inherent uncertainty in the input variables.

Probabilistic modeling provides a means to quantify the inherent uncertainty in the input variables to the cost model and provides an estimate of confidence and reliability in the predicted total cost of the candidate coating [25]. Output from the probabilistic model gives the probability that the output variable is above or below a particular threshold value. In addition, sensitivity analysis [26] is used with probabilistic analysis to rank and quantify the input variables that most significantly impact the total cost of the coatings—and cost factors that are most important can be further developed and refined. Use of this modeling approach allows the economics of the technology to be evaluated in a more systematic manner.

### Methodology.

Probabilistic modeling assigns a distribution, instead of a single value, to each input variable [27]. The distribution is a probability function that may be uniform, normal, triangular, or other. Uniform functions have a constant probability within their specified range. Normal distributions are characterized with a mean and standard deviation [28]. Triangular functions are specified with three values—minimum, maximum, and most probable [29].

The choice between using a uniform probability distribution and one in which the probability varies within the input range (normal, triangular) depends on the source and certainty of the cost and performance data. Sources for cost information are vendor quotes, literature values, or communications with experts in the art. Uniform distributions are typically used in cost models for materials, systems, and processes that are in an early stage of development. In these cases, cost and performance factors have a high level of uncertainty and available information may only provide a range of values for a cost or performance factor with no additional information to estimate the most likely value within the range.

For this analysis, we used the cost model to estimate costs of coatings with known compositions. The method for applying the coatings is commercially established and the costs of the coating materials and ancillary supplies were available through Internet surveys. For these reasons, we had good estimates for the range of values for each cost and performance factor and good estimates for the most likely value for each factor. This level of certainty allowed us to use triangular probability distributions for this analysis. Figure 1 shows the probability function and cumulative distribution function (CDF) for a triangular distribution having a minimum = 1, maximum = 5, and most probable = 2.

Each of the input variables is sampled from its distribution using a random sampling method to generate a set of *N* samples. The samples from all the input variables form an input variable matrix that is used to generate a set of cost outcomes. But first, samples from the input variables are paired to generate *N* sets of input variables for the cost model. Sample pairing of the input parameters is random (zero correlation) or correlated based on known correlations between the various cost factors. The *N* paired sets of selected samples are used to generate a set of *N* estimated total cost outcomes or realizations.

Several types of random sampling methods are used to generate the input variable sample sets including Monte Carlo [30] and Latin hypercube sampling (LHS) [31]. The Monte Carlo method selects samples from the full sample range according to its probability distribution. LHS is similar to Monte Carlo but uses a stratified sampling technique that divides the uncertainty distribution into *N* intervals. For any distribution, the CDF is divided into intervals with each interval having equal probability of being sampled. One sample is taken from each of the *N* intervals to ensure that the entire range of values is sampled. The sampling is not done uniformly within a particular interval (unless the distribution is uniform), but is based on the overall probability function within that interval. The benefit of LHS is that the minimum number of model outcomes that are required to generate a stable probability distribution of the total cost is less than the minimum number required for Monte Carlo sampling.

Multiple cases are typically run for several values of *N*, and the outcome distributions of the total coatings costs for all cases are compared. At some value of *N*, the outcome mean, standard deviation, and corresponding CDF stabilize and do not vary as *N* is increased. This value of *N* gives the final outcome distribution. The output variable data are treated statistically like any sampled data. Confidence intervals (i.e., 95%) can be determined using the sample mean, standard deviation, sample size, and corresponding student's *t*-distribution value. Confidence intervals can also be obtained from the CDF.

### Benefits.

Probabilistic modeling has significant benefits compared to deterministic modeling. Probabilistic modeling generates a distribution of model outcomes (estimated coating cost) that indicates the range in which the coating cost will fall. It allows one to estimate the probability that the coating cost will be less than a target cost and gives information regarding the sensitivity of the coating cost to input variables. Overall, it provides more information that allows one to evaluate the economic potential of a candidate coating.

## Coating and Wall Cost Model

Triangular probability functions were generated for the coating and wall material cost factors listed in Table 1. For each cost-factor, Table 1 lists the minimum, maximum, and most-probable values. Generation of the probability function from the three values (minimum, maximum, and most probable) was performed by setting the probability of the minimum and maximum values to 0 and then determining the probability of the most probable input value that generated an integrated probability of 1 for each function. The triangular functions were then integrated stepwise to generate the CDFs. The CDFs were sampled using LHS sampling *N* times and the corresponding values for the input factors were calculated. The sample sets were randomized to generate *N* sets of input factors and *N* outcomes from the cost model. All calculations were performed using a Python script written by the author.

### Coating Cost.

Travel, setup, preparation, and coating application times were obtained from discussions with experts in thermal spray coatings. Electricity and gas usage, and powder utilization factors were obtained from similar discussions and from materials handbooks. Electricity pricing was based on local rates. Gas pricing was obtained from local vendors.

Powder cost quotes were considerably higher than the corresponding on-line bulk prices for these powders. Powders that are used in this method must have very narrow particle-size distributions and must be free flowing. The additives and processing required to make the powders free flowing are proprietary, but they presumably add significant cost to the powders.

The required powder mass per unit coating area was the product of the coating density and coating thickness for the topcoat and similarly for the bond coat. The density of the nickel-based bond coat was 9000 kg/m^{3}, and the thickness ranged from 12.7 to 38.1 *μ*m with a most probable value of 25.4 *μ*m. The density of the alumina topcoat was 4000 kg/m^{3}, and the thickness ranged from 900 to 1350 *μ*m with a most probable value of 1125 *μ*m. The minimum value (900 *μ*m) was based on the target top (protective) coat corrosion rate of 30 *μ*m/yr for 30 years.

### Wall Cost.

To determine the coating cost benefit, wall costs were estimated for storage tank walls at the base of the tanks as a function of tank diameter and fluid temperature. Wall costs were estimated for bare (uncoated) Inconel 625 and coated 347/310 stainless steel. For all cases, the fluid height in the tanks was 9 m, and the fluid density was 2000 kg/m^{3}. These values generate a fluid pressure at the tank base of 25.6 psi. Tank diameters ranged from 5 to 15 m. Current commercial thermal storage tanks are somewhat larger than this range, but they operate at lower temperature drops so their thermal capacities are comparable [32]. Tank wall temperatures were 650 °C, 760 °C, and 815 °C.

The safety factor distribution ranged from 2 to 3 and had a most probable value of 2.5. The allowable stresses for Inconel 625 and 347/310 stainless steels are functions of temperature.

Equation (8) indicates that the wall thickness (and cost per unit area) needs to increase with tank diameter to maintain an appropriate hoop stress. The wall thicknesses for all combinations of tank temperature and diameter were calculated for Inconel 625 and 347/310 stainless steels according to Eqs. (7) and (8). The mass of wall material per unit area was calculated as the product of wall thickness and material density.

Tables 2 and 3 summarize wall properties and thicknesses for 347/310 stainless steels and Inconel 625 assuming a safety factor of 2. In the probabilistic cost model, the safety factor varied from 2 to 3 to generate a distribution of wall thickness outcomes for the two wall materials. In Table 2, some of the wall thicknesses (>0.1 m) may exceed the practical limit for rolling and welding 347/310 stainless steel plate to form the tank wall. In these cases, storage capacities can be met with multiple tanks of smaller diameter.

Wall costs were calculated as the product of wall material cost ($/kg) and wall material mass (kg/m^{2}). Wall material costs for 347/310 stainless steel and Inconel 625 plate were estimated from Internet quotes [34] and used to generate triangular cost distributions for the two wall materials. For all outcomes (*N*), the cost of the coating was added to the cost of the 347/310 stainless steel. The coating cost benefit was determined by comparing the cost of bare Inconel 625 with the cost of coated 347/310 stainless steel.

## Modeling Results

The probability cost model was run to determine mean values, standard deviations, and 95% confidence intervals for the 1) protective coating cost ($/m^{2}), 2) 347/310 stainless steel with protective coating ($/m^{2}), and 3) bare Inconel 625 ($/m^{2}).

The cost factors in Table 1 were sampled 1000 times (*N* = 1000) and the sample sets were randomized to generate 1000 sets of input factors. The cost model used the sample sets to generate 1000 cost outcomes for the coating and wall costs. This process was performed for all tank temperature/diameter combinations. We chose to sample 1000 times because the mean value of the coating cost outcomes did not vary by more than 0.1% for sample sizes greater than 800.

A histogram of the 1000 outcomes for the coating cost is shown in Fig. 2. The mean was $369.83/m^{2}, and standard deviation (σ) was $53.62/m^{2}. The mean values for coated 347/310 stainless steel wall and bare Inconel 625 wall costs ($/m^{2}) for storage tanks operating at 650 °C, 760 °C, and 815 °C are presented in Figs. 3–5. These figures show that in all cases of tank temperature and diameter, the mean cost of the coated 347/310 stainless steel is less than the mean cost of the bare Inconel 625 by a factor of 2.5–4.

To characterize the statistical significance of these results, 95% confidence bars (mean ± *t* × σ/√*N*, *t* = 2, *N* = 1000) were calculated for all results shown in Figs. 3–5 and are listed in Tables 4–6. Values for the two-sided 95% confidence intervals show that cost differences between coated 347/310 stainless steel walls and bare Inconel 625 walls for all cases are statistically significant.

The cost per unit area increases with higher surface areas and tank diameters because the hoop stress increases with tank diameter. This effect causes the wall thickness to increase to meet the safety factor requirement for allowable wall stress and increases the wall cost per unit area. Almost all costs were based on cost per unit area, so economies of scale were not realized. Travel and setup times were fixed costs, but they were small relative to the other costs.

### Sensitivity Rank Ordering of Cost Inputs Using Multiple Regression Analysis.

Multiple regression analysis was used to determine the sensitivity of the coating cost to the cost input factors. This analysis ranks the cost inputs according to the magnitude and sign of their effects on the coating cost. Multiple regression analysis determined the magnitude and sign of standard regression coefficients (SRCs) for the cost inputs [26]. SRCs are also referred to as β coefficients for rank ordering of cost inputs [25,27].

Results of the rank ordering are shown in Fig. 6. The two powder utilization factors correlate negatively with coating cost: coating cost decreases when utilization factors increase, as expected. All other cost factors correlate positively with coating cost. Figure 6 shows that coating cost is most sensitive to the topcoat cost factors, coating time, and bond coat quantity. The remaining factors have less impact on coating cost. Electricity cost, inert gas cost, gas usage, and travel/setup time all generated SRCs less than 0.02 (absolute) and were not considered significant.

## Conclusions

Probabilistic cost analysis was used to estimate likely costs for a two-layer protective coating that can be applied to tank wall materials using the plasma spray method. The first layer was a nickel formulation that forms a bond coat and helps the top, protective coat to adhere to the substrate material. The second coat was an α-alumina topcoat and protects the substrate material from corrosion. The probabilistic model included all cost factors needed to estimate the likely cost of the protective coating. The model was used to estimate the wall cost of the bare Inconel 625 and the wall cost of coated 347/310 stainless steel.

Comparing the resulting wall costs of the bare Inconel 625 to the cost of the coated 347/310 stainless steel substrate showed that the cost of the coated 347/310 stainless steel was always statistically less than the cost of the bare Inconel 625 when these materials are used for the tank wall at likely tank diameters and temperatures for CSP storage applications. In all cases, the cost of the coated 347/310 stainless steel was less than the cost of the bare Inconel 625 by a factor of 2.5–4. Multiple regression analysis determined that the coating cost is most sensitive to the topcoat cost factors, coating time, and bond coat quantity. The remaining factors have less or no significant impact on coating cost. In conclusion, there is a clear cost benefit for using protective coatings to protect the wetted surfaces of tank and other components in CSP plants that operate at temperatures associated with advanced power cycles for next-generation electricity production.

## Acknowledgment

We wish to thank Aaron Hall of Sandia National Laboratories in Albuquerque, NM, for providing us a basic understanding of thermal spray methods and their applications.

This work was supported by the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08-GO28308 with the National Renewable Energy Laboratory. For funding support, the authors thank the DOE Office of Energy Efficiency & Renewable Energy, Solar Energy Technologies Office, SunShot Initiative.

## Nomenclature

### Variables

*c*_{bond}=bond coat cost ($/kg)

*c*_{electricity}=electricity rate ($/kWh)

*c*_{equipment}=equipment use fee ($/hr)

*c*_{gas}=gas unit cost ($/kg)

*c*_{labor}=loaded labor rate ($/hr)

*c*_{top}=topcoat cost ($/kg)

*C*_{equipment}=cost of equipment ($/m

^{2})*C*_{energy}=cost of electricity ($/m

^{2})*C*_{gas}=cost of inert gas ($/m

^{2})*C*_{labor}=cost of labor ($/m

^{2})*C*_{powder}=cost of powder ($/m

^{2})*f*_{safety}=wall thickness safety factor

*m*_{bond}=bond coat areal mass (kg/m

^{2})*m*_{top}=topcoat areal mass (kg/m

^{2})*N*=number of sample realizations or outcomes

*p*_{electricity}=electrical power (kW)

*P*_{fluid}=fluid pressure at tank base (psi)

*r*_{gas}=gas usage (kg/hr)

*r*_{tank}=tank radius (m)

- size =
project coating size (m

^{2}) *t =*student-

*t*factor*t*_{coating}=coating time (hr/m

^{2})*t*_{prep}=coating preparation time (hr/m

^{2})*t*_{travel}=travel time to project site (hr)

*u*_{bond}=bond coat utilization factor

*u*_{top}=topcoat utilization factor

*x*_{wall}=tank wall thickness (m)

*β*=standard regression coefficient

*σ*=standard deviation ($/m

^{2})*σ*_{allowable}=allowable stress (psi)

*σ*_{hoop}=hoop stress (psi)