Cool-down Process Research Articles

Combined computational-experimental investigation of residual stresses and pre-cracking in mode I behaviour of thick adhesively bonded GFRP composite joints

This paper presents a novel Finite Element (FE) simulation approach to examine the mode I fracture of thick adhesive joints used particularly in the trailing edge of the wind turbine blades. The approach involved FE models of the DCB specimens focusing on aspects overlooked in the existing literature. There has been limited investigation on residual stresses caused by thermal mismatch between composites and adhesives. Similarly, the impact of generating notches/pre-cracks in the adhesive layer during the preparation of Double Cantilever Beam (DCB) specimens on residual stresses has received minimal attention. Additionally, the Cohesive Zone Model, commonly used for simulating elastoplastic adhesives, may be inadequate due to its inability to account for the plastic deformation of the adhesive. In the present work, the pre-cracks were virtually generated in DCB FE models so that their effect on the stresses within the joint could be examined, making it a novel contribution to the field. The components were assigned with appropriate thermal expansion coefficients, and a simulation of the cool-down process was conducted to determine the thermal residual stresses. Furthermore, the Drucker-Prager plasticity criteria were used to capture the elastoplastic behaviour of adhesives in the FE simulations. Concurrently, the T-stresses were assessed through numerical investigations. For validation, experiments were conducted on DCB specimens made of two cross-ply composite laminates bonded with a ∼ 10 mm thick layer of an epoxy-based adhesive. A good agreement between computational and experimental results was observed, confirming the effectiveness and reliability of the proposed approach.

Thermodynamic analysis and optimization of a condensation-driven dilution refrigerator

Dilution refrigerators are widely used in the fields of condensed matter physics and quantum technology. The condensation-driven dilution refrigerator uses a condenser operating at temperatures below the Still to liquefy the 3He vapor and achieve the circulation of 3He, which has the advantages of compact structure, lightweight, low cost, and low vibration. The published research primarily focuses on the system architecture and performance, and the research and analysis on the thermodynamic cycle are still incomplete. In this paper, the condensation-driven dilution refrigeration cycle has been clarified as a thermally driven refrigeration cycle, and the thermodynamic performances including the figure of merit and thermodynamic perfectibility are investigated. Optimizations have been made to the key components of the previous prototype, and the lowest no-load temperature of 68 mK and a cooling power of 4 μW at 100 mK were achieved. Compared with the previous prototype, the thermodynamic perfectibility of the system increased from 7.63% to 17.83%. The impact of the Still heating strategies on the system was analyzed, and a start-up strategy was proposed to speed up the cooldown process.

Co-simulation of cool down process from 300 to 80 K for ITER Tokamak

Dynamic simulation of Tokamak superconducting magnet system has been conducted to investigate the cool-down process from 300 to 80 K. The simulation focuses on the cool-down speed variations with respect to the global temperature gradients in the different coils; Toroidal Field (TF) coil, TF STructure (TF-ST), Central Solenoid (CS) and Poloidal Field (PF)/Correction Coil (CC) systems. As imposing the maximum temperature gradients dT max ≤ 50K, the speed should be adjusted, ensuring the limited mechanical stresses due to thermal contractions. So far, the process simulation of Tokamak cryogenic system has been concentrated on the Deuterium Tritium (DT) operation phase; therefore, it is necessary to revise the model to extend its capability for the cool-down; for example, the thermo-hydraulic properties of Cable-in-Conduit Conductor (CICC) for each coil, mechanical properties of materials for the magnet system. The cool-down process is implemented at the helium refrigerator by utilizing LN2 heat exchanger up to 80 K. Its speed is set at -0.8 K/hr as a baseline, which can be controlled by the global temperature gradients in the magnet system. The process will be on hold as dT max ≥ 50K, and resumed once dT max < 50K. The paper discusses the cool-down processes of the Tokamak and identifies the impact on the speed, dT i/dt, of each component.

Simulation of the cool-down process for the ESS cryogenic moderator system

The ESS cryogenic Moderator System (CMS) has been designed to supply a subcooled liquid hydrogen (17.5 K and 1.0 MPa) to two hydrogen moderators where the nuclear heating is estimated to be 6.7 kW (to be increased to four in the future). The CMS is cooled by a helium refrigerator system with a cooling power of 30.3 kW at 15 K. In this study, a one-dimensional simulation code has been developed to understand temperature behaviors during the cool-down process. The operational methods and parameters have been optimized. The CMS cool-down process is divided into three phases (I: gaseous hydrogen above 36 K, II: condensation from 36 to 31 K, and III: liquid hydrogen below 31 K). It turns out from the simulation result that the CMS would be able to be cooled down to the nominal condition of 17 K within 27 hours.

Study of operational modes of cool-down and warm-up processes for the ESS large-scale 20 K helium refrigeration system

In the beginning, the European Spallation Source (ESS) will install two hydrogen moderators where the nuclear heating is estimated to be 6.7 kW at 5 MW proton beam power. The ESS cryogenic moderator system (CMS) has the function to circulate a subcooled liquid hydrogen through the moderatos. It is cooled through a heat exchanger by a large-scale 20 K helium refrigeration system with a cooling capacity of 30.3 kW at 15K, which is called Target Moderator Cryoplant (TMCP). The commissioning of the TMCP cold box and the two compressor skids was completed in 2019. Cryogenic transfer lines with a length of 300 m (CTL) and a valve box located next to the CMS cold box were installed during the summer of 2022. In this paper, the TMCP cool-down and warm-up processes have been studied and the operational parameters have been optimized based on the result of the CMS cool-down simulation.

Numerical modelling of the cool-down of the helium transfer-lines for FRIB LINAC and experimental systems

The cryogenic distribution system at FRIB is extensive, encompassing three Linac segments, fourteen experimental system superconducting magnets, cross-connect between FRIB and the reconfigured NSCL legacy system, and the A1900 magnets, as well as many other user experimental loads. The cool-down or warm-up of these transfer-lines is an inherently transient process and must be conducted at a gradual enough rate to keep local temperature gradients and corresponding thermal stress on the piping system within acceptable limits. To this end, estimation of this transient process is very helpful in operational planning. A two temperature (fluid and pipe wall) model was developed to capture the transient characteristics of the cool-down or warm-up process with real fluid properties, and considering the effects of heat in-leak, momentum, flow resistance, and piping components. It was employed to estimate the cool-down of different FRIB transfer line sections, and the results were compared against data obtained during actual cool-down of the transfer line.

Low-noise thermal shielding around the cryogenic payloads in the Einstein Telescope

The Einstein Telescope (ET) is a planned third-generation gravitational-wave detector that includes a low-frequency (LF) and a high-frequency (HF) laser interferometer. Cryogenic operation of ET-LF is imperative for exploiting the full scientific potential of ET, with mirrors operated at temperatures of 10 K to 20 K in order to reduce the thermal noise. Thermal shielding around the optics is essential to support the cooldown process and to decrease the heat load. Additionally in steady-state operation, mechanical vibrations must be kept to an absolute minimum in order to limit noise contributions from scattered light.We present a cooling concept for a thermal shield surrounding the cryogenic optics of ET-LF, which considers rapid cooldown and low vibration in steady-state operation. During cooldown, cooling tubes enable the flow of supercritical helium, driving the shield temperature decrease by forced convection. For steady-state operation, the shield cooling mechanism is converted to static heat conduction in He-II within the same tubes. A first mechanical model is presented that fulfills the thermal and vibrational requirements. Thermal characteristics of the shield are demonstrated by means of analytical and numerical modeling results. Modal and dynamic analyses are performed to obtain natural frequencies and transfer functions.

Dynamic acoustic optimization of pulse tube refrigerators for rapid cooldown

Pulse tube refrigerators are a critical enabling technology for many disciplines that require low temperatures. These refrigerators dominate the total power consumption of most modern cryostats, including those that reach millikelvin temperatures using additional cooling stages. In state-of-the-art commercial pulse tube refrigerators, the acoustic coupling between the driving compressor and the refrigerator is fixed and optimized for operation at base temperature. We show that this optimization is incorrect during the cooldown process, which results in wasted power consumption by the compressor and slow cooldown speed. After developing analytic expressions that demonstrate the need for acoustic tuning as a function of temperature, we dynamically optimize the acoustics of a commercial pulse tube refrigerator and show that the cooldown speed can be increased to 1.7 to 3.5 times the original value. Acoustic power measurements show that loss mechanism(s)—and not the capacity of the compressor—limit the maximum cooling available at high temperatures, suggesting that even faster cooldown speeds can be achieved in the future. This work has implications for the accessibility of cryogenic temperatures and the cadence of research in many disciplines such as quantum computing.

Automation of superconducting cavity cooldown process using two-layer surrogate model and model predictive control method

Superconducting cavity is the key equipment of the superconducting accelerator, which provides higher acceleration voltage and higher frequency power per unit length, and saves equipment space. Superconducting cavities need to be gradually cooled from ambient temperature (300 K) to the superconducting temperature (4.2 K or below) during the test and operation. The temperature difference on the cavity must be strictly limited during the cooldown process to prevent excessive thermal stress on the surface of the superconducting cavity. Since this cooldown process for the superconducting cavity is a typical large hysteresis, non-linear process that is difficult to control automatically using decoupled proportion integral derivative (PID) methods directly, a less efficient manual control scheme is normally adopted. In this paper, 3D numerical simulation, 1D pipe and 0D tank model with artificial neural network (ANN) were combined to generate a two-layer surrogate model that can balance computational accuracy and speed, to improve the automation and cooling efficiency of the superconducting cavity cooldown process. In order to achieve automatic control of the cooling procedure for the superconducting cavity, a model predictive control (MPC) approach was also built on the basis of this two-layer surrogate model. According to the results of the experiment test, the improved method could realize a quick and smooth cooldown process of the superconducting cavity, during which the temperature difference on the cavity could satisfy the requirements. Additionally, the improved automatic cooldown method was more adaptable and saved 29 % more time than the original manual control method. The foundation for a more intelligent automated control of future large cryogenic systems or other system with the large hysteresis, non-linear properties, was laid.

The Simons Observatory: Cryogenic half wave plate rotation mechanism for the small aperture telescopes.

We present the requirements, design, and evaluation of the cryogenic continuously rotating half-wave plate (CHWP) for the Simons Observatory (SO). SO is a cosmic microwave background polarization experiment at Parque Astronómico de Atacama in northern Chile that covers a wide range of angular scales using both small (⌀0.42m) and large (⌀6m) aperture telescopes. In particular, the small aperture telescopes (SATs) focus on large angular scales for primordial B-mode polarization. To this end, the SATs employ a CHWP to modulate the polarization of the incident light at 8Hz, suppressing atmospheric 1/f noise and mitigating systematic uncertainties that would otherwise arise due to the differential response of detectors sensitive to orthogonal polarizations. The CHWP consists of a 505mm diameter achromatic sapphire HWP and a cryogenic rotation mechanism, both of which are cooled down to ∼50K to reduce detector thermal loading. Under normal operation, the HWP is suspended by a superconducting magnetic bearing and rotates with a constant 2Hz frequency, controlled by an electromagnetic synchronous motor. We find that the number of superconductors and the number of magnets that make up the superconducting magnetic bearing are important design parameters, especially for the rotation mechanism's vibration performance. The rotation angle is detected through an angular encoder with a noise level of 0.07 μrad s. During a cooldown process, the rotor is held in place by a grip-and-release mechanism that serves as both an alignment device and a thermal path. In this paper, we provide an overview of the SO SAT CHWP: its requirements, hardware design, and laboratory performance.

An adaptive metastructure concept using bistable composite laminates

Concepts involving adaptive and morphing structures offer us the possibility to realise shape transformation with each shape imparting an individual functionality. There is an increased demand in the design of structural components that are suitable for diverse operational conditions rather than limited to a unique one. However, most concepts of shape adaptive structures require a “holding force” or a continuous supply of energy to maintain a targeted 3D shape. As a result, structures that can lead to desired shape transformation with self-locking capabilities are more desirable.In this work, the concept of multistability is employed to demonstrate a novel class of metastructures that can tackle this challenge. The metastructure is constructed with a periodic arrangement of bistable unit cells made of highly anisotropic composite laminates. Each unit cell comprises multi-sectioned rectangular composite plates exhibiting bistable behaviour due to the thermal residual stresses engendered during the cool-down process from curing to room temperature. To analyse the proposed metastructure, a finite element model has been developed at three hierarchical levels: a plate level, a unit cell level, and a lattice level. At the plate level, a corresponding semi-analytical model using the Rayleigh–Ritz method has also been formulated to validate the finite element models. By carefully tuning the size and spacing of the unit cell, a desired response of the metastructure can be achieved. Additionally, by changing the ply layup and the fibre orientation of each layer of constituent composite plates, conflicting requirements, including load-carrying, shape-adaptive and lightweight, at the same time can be addressed simultaneously. From an extensive parametric study, few designs have been selected for its application in a load-carrying morphing structure.

Adenine’s band strength at cryogenic temperatures

Infrared observations of the interstellar medium have unveiled the existence of numerous solid-phase molecules. The recent detection of uracil within the (162173) Ryugu asteroid has opened new prospects for the discovery of additional compounds, including nitrogenous bases, in the near future. To facilitate the comprehension, identification, and observation of these substances within astrophysical environments, precise measurements of their infrared band strengths are crucial. In this study, we present the outcomes of laboratory experiments conducted to investigate the behavior of adenine at various temperatures. The measured band strength from 3600 to 1970 cm−1 was determined to be 3.21⋅10−16 cm molecule−1. Additionally, as the temperature decreased, its behavior exhibited a well-fitted second-degree polynomial function. Notably, the experiments indicated no permanent phase changes or hysteresis during the cool-down and warm-up processes, further elucidating the thermal properties of adenine.

Experimental study on a closed-cycle Joule-Thomson cryocooler working at liquid hydrogen temperature

Because of high reliability, simplicity and no moving parts in the cold end, the Joule-Thomson (JT) cryocooler working at liquid hydrogen temperature has the potential to be used in the field of hydrogen liquefaction, liquid hydrogen storage and electronic devices cooling. However, studies on JT cryocoolers with cooling capacities above 10 W at liquid hydrogen temperature are rare. Therefore, in order to study cooling characteristics of a closed-cycle JT cryocooler working at liquid hydrogen temperature with a cooling capacity above 10 W, a laboratory prototype of a simplified JT cryocooler driven by an oil-lubricated scroll-type compressor has been built and tested. Pure hydrogen gas is used as the working fluid. The cool-down process of this closed-cycle JT cryocooler is illustrated and analyzed. The cooling temperature of 20.7 K is approached at the end of the cool-down process. Using the heat balance method, a stable cooling capacity of 10.9 W at 20.8 K is measured. The stability of temperature, pressure and mass flow rate is analyzed. Over a period of 0.5 h, the maximum differences in cooling temperature and pressure are lower than 0.1 K and 0.001 MPa, respectively. Mass flow rate fluctuation which is 0.44 mg/s has little influence on the cooling temperature.

Experimental and dynamic simulation study of the Joule-Thomson cryocooler working at liquid hydrogen temperature

With advantages of long-life time, low level vibration and continuous refrigeration, precooling Joule-Thomson cryocooler (PJTC) working at liquid hydrogen temperature has the potential to be applied to liquid hydrogen storage system in space. For the large-cooling-power PJTC working at liquid hydrogen temperature, its dynamic response characteristic to heat load has a great importance on liquid hydrogen storage system, especially in aerospace applications. In order to study dynamic characteristics of PJTC working at liquid hydrogen temperature, an experimental setup for an open-cycle PJTC is built and its dynamic model is established by MATLAB software. Cool-down process and temperature variation characteristic with a heat load of 10.19 W are simulated and verified with experimental results. Meanwhile, the effect of precooling power on the performance of JT cryocooler is analyzed. It is found that if the precooling power is lower than 50% of the original precooling power, the recovery time of JT cryocooler will increase significantly.

Estimation of Cool-Down Speed Affected by Mass Flow Rate and Inlet Pressure in JT-60SA CS Module

The JT-60SA central solenoid (CS) module is cooled to 4.5 K using liquid helium. This cool-down operation of the coil is planned to take about one month. During cool-down, the maximum temperature difference in the coil (dT <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C-M</sub> ) must be kept below 40 K to avoid excessive thermal stress. Thermometers are attached only at the inlet and outlet on the outer perimeter of the coil. However, because the maximum temperature inside the coil occurs near its inner edge, dT <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C-M</sub> cannot be measured directly. Hence, estimating dT <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C-M</sub> is important to allow the cool-down process to be completed within one month without excessive thermal stress on the CS. In the present study, based on the results of the JT-60SA cool-down test performed in 2020, a temperature simulation model of the CS module, including the strands, jacket, and He in the bundle and hole regions was created using MATLAB, and dT <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C-M</sub> was estimated during the cool-down test. This model was then used to evaluate the effect of the mass flow rate and inlet pressure on the cool-down speed.

Comparative thermodynamic analysis of China Spallation Neutron Source second phase (CSNSII) SRF system cooling Scheme

The China Spallation Neutron Source second phase (CSNS-II) accelerator upgrades consist of three major components: linac energy upgrade from 80 MeV to 300 MeV, injection system upgrade, and new magnetic alloy dual harmonic cavities. There are ten double spoke cavities cryomodules and eight elliptical cavities cryomodules in total. A comparative thermodynamic analysis of the superconducting radio frequency (SRF) systems can be of great importance to select a reliable, cost-effective and easy-maintenance cooling schemes. This is particularly critical for SRF systems that require long-term operation. A comparative thermodynamic analysis of both types of cooling schemes for SRF systems, referred to as Scheme 1 (a cryogenic distribution system with distributed heat exchangers) and Scheme 2 (a cryogenic distribution system with central heat exchangers), was presented in this paperThe length and size of process transfer lines, as well as the heat loads subjected to their components, of cryogenic distribution systems were introduced. In addition, the unsteady state cooldown process for the CSNS-II SRF cryomodules was investigated. The analysis results that included temperature and pressure profiles along the linac distance, as well as distributions of JT expansion effectiveness, mass flow rate, and cryomodule exegetic efficiency. Furthermore, the exergy analysis and refrigeration power are investigated using a neural network optimized by a novel sparrow search algorithm genetic algorithm (SSA-GA). Results show that, Scheme 1 is more advantageous than Scheme 2 because it can save maintenance time. From 300 K to 4.5 K, the total cooldown time of SRF cryomodules is approximately 28 h. The maximum ΔP is around 1.1 bara, which can be considered in the acceptable range. The prediction of ηex is stable at the 7th epoch after being optimized by SSA-GA, and the MSE is 1 × 10-5. And the error between the predicted and true values of the BP neural network after SSA optimization is very small (less than 0.1%), approaching zero, indicating that the BP optimized by SSA achieves good prediction. This paper's work can serve as a solid foundation for the construction and operation of the CSNS-II.

Experimental study on a two-stage large cooling capacity stirling cryocooler working below 30 K

Developing superconducting technology and hydrogen energy system requires large cooling capacity devices working below 30 K. Two-stage Stirling cryocooler has the advantages of large cooling capacity, high efficiency, fast cool-down process, and compact configuration, etc. A two-stage Stirling cryocooler driven by crank-rod mechanism was developed and studied experimentally. The operating characteristics, including the cooling down processes, the pressure characteristics, the heat transfer characteristics of the ambient heat exchanger, and the cooling performances, are analyzed. Under 2.2 MPa charging pressure, a no-load refrigeration temperature of 15.84 K was obtained. With no heat load on the first stage, the cryocooler was capable of providing a cooling power of 138.1 W@30 K with a relative Carnot efficiency of 12.87 %, which is one of the best results of Stirling-type cryocoolers at present. Additionally, a cooling capacity of 47.4 W at 20 K with the relative Carnot efficiency of 6.44 % was achieved. During the cooling down process, an acceleration of cooling down rate of the second stage was observed, leading to a distinct acceleration phase of the declining rate of compression space’s mean pressure. It was found that the influence of the heat load on the first stage on the second stage’s cooling performance was not remarkable, and the cryocooler can provide around 131 W@30 K or 40 W@20 K cooling power on the second stage while the first stage can provide 100 W of cooling power at 75.01 K or 78.42 K, respectively. These experimental results have laid a solid foundation for a deeper understanding of the cryocooler’s operating mechanism and practical application.

Experimental study on a simplified precooled JT cryocooler for liquid hydrogen zero boil-off storage

For a hydrogen zero boil-off storage system in space, active cooling technique is one of the key technologies that requires effective cryocoolers. The precooled Joule-Thomson (JT) cryocooler working at liquid hydrogen temperature has no moving parts at low temperature with continuous refrigeration, making it reliable for space applications. In this study, a new simplified JT cryocooler with a large cooling capacity is proposed to approach liquid hydrogen refrigeration. In order to clarify cooling capacity characteristics of the simplified JT cryocooler, a laboratory prototype precooled by a Gifford-Mcmahon (GM) cryocooler is designed, fabricated and tested. The cool-down process of the simplified JT cryocooler is presented. A stable cooling capacity of 10.06 W at 20.7 K is provided by the cryocooler. The availability of the simplified JT cryocooler without a bypass assembly is verified.

Machine-Learning and Physics-Based Models Compared in Downhole Pressure Prediction in Deepwater Reservoirs

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper OTC 31758, “Downhole Pressure Prediction for Deepwater Gas Reservoirs Using Physics-Based and Machine-Learning Models,” by Jincong He, SPE, Matthew Avent, and Mathieu Muller, Chevron, et al. The paper has not been peer reviewed. Copyright 2022 Offshore Technology Conference. Reproduced by permission. Pressure measurements from permanent downhole gauges (PDHGs) during extended shut-ins often are used for model calibration and reserves estimation in deepwater gas reservoirs. A key challenge in practical operation has been the failure of PDHGs within the first few years of operation. To overcome this challenge, modeling solutions have been developed to enable accurate bottomhole well pressures to be calculated. In the complete paper, novel physics-based models and machine-learning (ML) models are presented and compared for estimating PDHG measurement from wellhead measurements. Introduction One way to predict PDHG shut-in pressure is with a full-physics wellbore simulator. However, those simulators usually require input parameters such as heat conductivity through the reservoir, itself highly uncertain in the field. In addition, wellbore simulation can be expensive computationally to run. The authors investigate two alternative modeling approaches to predict future PDHG shut-in data based on historical PDHG measurement from the same, or different, wells. The first approach is a physics-based data-driven (PBDD) method in which the temperature profile along the wellbore is characterized by a piecewise linear model, the cooldown process during shut-in is characterized by a decline-curve model, and the dependency of the cooldown on previous gas production is characterized by a linear model. During training, the parameters in these models are calibrated by historical PDHG shut-in data before gauge failure. Once trained, the model can be used to predict downhole measurement during future extended shut-in events. The second approach is to use ML methods. Using historical data, this approach trains a black-box model or function that directly projects the wellhead measurements and time since shut-in to downhole measurements. Both approaches are shown to be accurate when calibrated on historical data and used to predict future shut-ins of the same well. However, when they are used to predict shut-in pressure and temperatures of a new well based on other existing wells, the behavior of the ML model can be nonphysical and erratic, while the behavior of the PBDD model is more constrained and interpretable.

Numerical Simulations of Lightning Strikes to Carbon Fiber Reinforced Polymers Using a Multi-Variable Approach

Carbon fiber-reinforced polymer (CFRP) composite materials are being employed widely in the manufacturing of next-generation aircraft because of their superior mechanical properties compared to those of metallic-based materials. Owing to growing interest in CFRP materials from large commercial and military aircraft manufacturers, such as Boeing and Airbus, research that investigates lightning strikes and associated damage is of paramount importance, as the fiber-polymer composite is highly resistive to electrical current. The high resistivity of traditional CFRP materials means that they are susceptible to extreme damage caused by the Joule effect, also known as resistive heating. The aim of the study presented in this paper is to determine and quantify the effectiveness of a developed multi-variable pyrolysis approach to model the transient material degradation during a lightning strike event. The study employs the finite element method (FEM) in the form of a coupled thermal–electrical analysis in conjunction with a user subroutine available in Abaqus. A parametric study was conducted to compare three different peak current values (20[Formula: see text]kA, 30[Formula: see text]kA and 40[Formula: see text]kA) based on a model that utilized pyrolysis-dependent material constituents as well as a novel model for specific heat capacity. The study also included a heat transfer step for a cool-down process. The results obtained from the model developed have strong agreement with the surface damage area and depth of damage compared with those from experimental analysis in literature.