M. G. Choudhury and S. S. DARGAN, KBR, Gurugram, India; and V. NELLURI, Black Cat Consulting & Engineering Services W.L.L., Doha, Qatar
Refractory lining is a critical component in industrial piping systems and is utilized extensively across various sectors, including the petrochemicals, chemicals, metallurgical and power generation industries. Its primary function is to lower skin temperature and enable the use of cost-effective materials like carbon steel (CS). By providing thermal insulation and protection against corrosion and erosion, refractory lining plays a pivotal role in maintaining the integrity and longevity of piping systems subjected to high temperatures and harsh operating conditions.
In applications ranging from fluid catalytic cracking units (FCCUs) in refineries to furnaces in metallurgical plants and boilers in power generation facilities, refractory lining serves as a barrier between hot gases or fluids and the structural components of the piping system. This insulation helps to mitigate thermal stress, prevent corrosion and ensure optimal operating temperatures.
Efforts are often made to maintain skin temperatures around 150°C, as this range balances thermal insulation, energy efficiency and operational requirements. Deviations from this optimal temperature range can lead to inefficiencies in heat transfer, increased energy consumption or potential risks such as thermal stress on the piping system.
Therefore, accurately estimating skin temperature and understanding the impact of refractory lining on thermal performance are crucial aspects of piping system design, operation and maintenance. By recognizing the multifaceted role of refractory lining in enhancing operational efficiency, mitigating risks and extending the lifespan of industrial piping systems, industries can optimize their processes and ensure the reliability and safety of their infrastructure.
Calculating skin temperature using a manual calculation theory. One common method for estimating the skin temperature of refractory lined pipes is based on manual calculations using a simplified one-dimensional heat transfer model. The model assumes that the heat transfer is steady-state, radial and uniform along the pipe length. The model also neglects the effects of contact resistance between the layers. The model can be represented by a series of thermal resistances, as shown in FIG. 1.
The skin temperature is the temperature at the outer surface of the pipe that is exposed to the ambient air. The pipe has two layers: the inner refractory lining and the outer metal shell. The pipe is subjected to a fluid flow inside, which has a temperature of Tf and an external heat transfer coefficient of ho.
To find the skin temperature, the energy balance equation must be applied at each layer, assuming steady-state and one-dimensional heat transfer. The energy balance equation states that the heat transfer rate through each layer is equal. The heat transfer rate can be written as the product of the overall heat transfer coefficient U, the outer surface area Ao, and the temperature difference between the fluid and the ambient air (Tf – Ta ). We can also express the heat transfer rate as the sum of the thermal resistances of each layer multiplied by the same temperature difference. The thermal resistances are inversely proportional to the heat transfer coefficients and the surface areas of each layer, and directly proportional to the thicknesses and thermal conductivities of each layer. For a cylindrical pipe, natural logarithms should be used in the conduction resistances.
The energy balance equation can be written as Eq. 1:1
Q = [(Tf – Tw) / (1 / (hi Ai ))] = [(Tw – Tr ) / ((ln(r0 / ri )) / (2π kr L))] = [(Tr – Ts ) / ((ln(r0s / ris )) / (2π ksL ))] = [(TS – Ta ) /(1 / ho Ao)] = [(Tf – Ta ) / (1 / UAo )] (1)
where Tf is the fluid temperature inside the pipe, Ta is the ambient temperature surrounding the pipe, Tw is the temperature at the inner wall, Tr is the temperature at the refractory-metal interface, Ts is the skin temperature, hi is the internal heat transfer coefficient, Ai is the inner surface area, ri is the inner radius, ro is the outer radius, kr is the thermal conductivity of the refractory, L is the length of the pipe, ris is the inner radius of the shell, ros is the outer radius of the shell, ks is the thermal conductivity of the shell, Ao is the outer surface area, ho is the outside heat transfer coefficient and U is the overall heat transfer coefficient.
This equation can be simplified by defining some terms (Eq. 2):1
[(Tf – Tw) / Ri ] = [(Tw – Tr ) / Rr ] = [(Tr – Ts ) / Rs ] = [(TS – Ta ) / R0 ] = [(Tf – Ta ) / RT ] (2)
where Ri is the internal convection resistance, Rr is the refractory conduction resistance, Rs is the shell conduction resistance, R0 is the external resistance and RT is the total thermal resistance.
To solve for Ts , all the other parameters in this equation must be known. Some of them are given by design specifications or measurements, such as Tf, Ta, ho, ri, ro, ris, ros, kr, ks and L. The only unknown parameter is hi, which depends on the fluid properties and flow conditions inside the pipe.
For internal forced convection in a pipe (turbulent flow), a correlation can be used to estimate hi (Eq. 3):1
Nu = 0.023 × Re0.8 × Pr0.3 (3)
where Nu is the Nusselt number (hi × D/kf), Re is the Reynolds number (ρ × u × D/μ), Pr is the Prandtl number (μ × cp /kf ), ρ is the fluid density, u is the fluid velocity, D is the hydraulic diameter (2 × ri ), μ is the fluid viscosity, cp is the fluid specific heat, and kf is the fluid thermal conductivity.
Re and Pr can be calculated from fluid properties at a reference temperature. Then, Nu and hi can be solved.
To calculate the heat transfer coefficient ho for external natural convection and radiation to the surrounding air, both modes of heat transfer must be considered and added together as (Eq. 4):1
ho = hc + hr (4)
where hc is the convection heat transfer coefficient and hr is the radiation heat transfer coefficient. The convection heat transfer coefficient hc can be estimated using another empirical correlation based on the Grashof number Gr and the Prandtl number Pr, which are dimensionless parameters that describe the buoyancy-driven flow and the fluid properties. For a horizontal cylinder in air, one of the commonly used correlations is Eq. 5:1
hc = ka/Dc × 0.53 × (Gr × Pra )0.25 (5)
where Gr = g × β × (Ts – Ta ) × Dc3/ν2 is the Grashof number, g is the gravitational acceleration, β is the thermal expansion coefficient of air, Ts is the surface temperature of the cylinder, Ta is the ambient temperature of air, ν is the kinematic viscosity of air, ka is the thermal conductivity of air, μa is the air dynamic viscosity, cpa is the gas specific heat and the Prandtl number is Pra = μa × cpa /ka , respectively.
The radiation heat transfer coefficient hr can be calculated using the Stefan-Boltzmann law as (Eq. 6):2
hr = ε × σ × (Ts4 – Ta4) / (Ts – Ta ) (6)
where ε is the emissivity of the cylinder surface, σ is the Stefan-Boltzmann constant, and Ts and Ta are the surface and ambient temperatures of air, respectively.
Here, ε represents the effective emissivity influenced by the surface paint. Different types of paint (e.g., black silicone paint, aluminum paint) can vary in their emissivity values. For example, black silicone paint may have an emissivity of ~0.9, while aluminum paint may not exceed 0.65.
Once hi and ho are determined, they can be plugged into the energy balance equation and Ts can be solved using any numerical method.
It is pertinent to note that these considerations exclude personal protection insulation. For areas accessible to personnel, the implementation of open mesh metal guards has been factored in for protection. Consequently, external insulation effects are not accounted for in these calculations. This distinction ensures that the thermal analysis is tailored to the specific conditions and safety measures implemented within the industrial environment.
Manual calculation. Given the geometric and thermal properties, fluid and ambient temperatures, and fluid properties, an iterative process is outlined. The steps involve defining geometry and properties, applying the energy balance equation, calculating internal convection resistance, and determining external convection and radiation coefficients. The solution is obtained iteratively by rearranging the energy balance equation until convergence.
Given: Inner radius of steel pipe: ri = 0.915 m
Outer radius of steel pipe: ros = 0.931 m
Inner radius of refractory layer: ris = 0.8 m
Outer radius of refractory layer: ro = 0.915 m
Length of pipe: L = 1 m
Thermal conductivity of steel: ks = 44.928 W/mK
Thermal conductivity of refractory: kr = 1.23432 W/mk
Emissivity of steel: ε = 0.8
Fluid temperature: Tf = 500°C
Ambient temperature: Ta = 25°C
Density of fluid flowing inside the pipe: ρf = 1.027 kg/m3
Dynamic viscosity of fluid, μ = 0.0000038 kg/m/sec
Specific heat of the fluid, cp = 1,266.8 J/kgK
Thermal conductivity of fluid, k = 0.026 W/mK
Velocity of fluid, V = 15 m/sec
Characteristic length, Dc = 2 × ros = 1.862 m
Boltzmann constant, σ = 5.67 e-8 J/m2 K4
Kinematic viscosity air, νa =15.89 e-6 m2/sec
Density of air, ρa = 1.168 kg/m3
Thermal conductivity of fluid, ka = 0.0262 W/mK
Dynamic viscosity of air, μa = 0.00001855 kg/m/sec
Specific heat of the air, cpa = 1,001 J/kgK
Step 1: Calculate hi using Nusselt number correlation:
Re = ρvD/μ = (1.027) (15) (1.6) / (3.8e-6) = 6,436,352 Pr = μcp/k = (3.8e-6) (1,266.8) / (0.026) = 0.18731 Nu× = 0.023 × Re0.8 × Pr0.3 = 0.023 × (6,436,352)0.8 × (0.18731)0.3 = 3,894
hi = Nu × k/D = (3894)(0.026) / (1.6) = 63 W/m2K
Step 2: Calculate ho using radiation and free convection:
hr = εσ(Ts4 – Ta4 ) / (Ts – Ta) hc = C × (Gr × Pra )m × ka/Dc
Iteration1: Assume Ts = 100°C as a first guess, then:
Ts = 373 K Ta = 298 K
T = (Ts + Ta ) / 2 = 671 / 2 = 335.5°C
β = 1/T = 1/335.5 = 2.98e–3
ν = 15.89e-6 m2/sec
hr = (0.8)(5.67e-8)(3734 – 2984 ) / (373 – 298) = 5.76 W/m2K Gr = gβ(Ts – Ta)Dc3 / ν2 = (9.81)(2.98e-3)(373 – 298)(1.862)3 / (15.89e-6)2 = 5.61e10
Pra = μa x cpa/ka = 0.00001855 × 1001 / 0.0262 = 0.71
Assume the pipe is horizontal, then C = 0.53 and m = 0.25 Nu = C × (Gr × Pra)m = (0.53)(1.22e9 × 0.71)0.25 = 91
hc = Nu × ka /Dc = (91)(0.0262) / (1.862) = 1.28 W/m2K
ho = hr + hc = 5.76 + 1.28 = 7.04 W/m2K
Step 3: Solve for Ts by rearranging the energy balance equation and using a trial-and-error method, shown in Eqs. 7–17:
[(TS – Ta ) / R0 ] = [(Tf – Ta ) / RT ] (7)
TS = [((Tf – Ta ) × R0 ) / RT ] + (Ta ) (8)
Ri = [1 / (hi × Ai )], R0 = [1 / (h0 × A0 )], Rr = [(ln(r0 /ri )) / 2π kr L ], Rs = [(ln(r0s /ris )) / 2π ks L ] (9)
Ai = 2π × 0.8 × 1m = 5.02 m2 (10)
A0 = 2π × 0.931 × 1m = 5.85 m2 (11)
Ri = [1 / (hi × Ai )] = [1 / (63 × 5.02)] = 0.00316 k/w (12)
R0 = [1 / (h0 × A0 )] = [1 / (7.04 × 5.85)] = 0.0242 k/w (13)
Rr = [(ln(r0 /ri )) / 2π kr L ] = [(ln(0.915 / 0.8)) / (2π × 1.23432 × 1)] = 0.0234 k/w (14)
Rs = [(ln(r0s /ris )) / 2π ks L ] = [(ln(0.931 / 0.915)) / (2π × 44.928 × 1)] = 0.0000585 k/w (15)
RT = Ri + Rr + Rs + R0 = 0.0508K/W (16)
TS = [((Tf – Ta ) × R0 ) / RT ] + (Ta ) = [((500 – 25) × 0.0242) / 0.0508] + 25 = 251°C (17)
Continue this iterative process until Ts stabilizes and converges to a specific value.
Continuing the iterative process until the surface temperature (Ts ) stabilizes and converges to a specific value, TABLE 1 presents a comprehensive overview of each iteration's surface temperature and the corresponding outside heat transfer coefficient (ho). The iterations are performed with varying surface temperatures, ranging from 149°C–176.6°C, and the respective outside heat transfer coefficients fluctuate accordingly. The stabilization and convergence of the Ts will be observed over subsequent iterations, providing valuable insights into the thermal behavior of the system.
CALCULATION OF SKIN TEMPERATURE USING FEA
Finite element analysis (FEA) modeling procedure. FEA is a powerful numerical method used to solve complex heat transfer problems. In the case of refractory pipes, FEA involves dividing the structure into finite elements, each with defined material properties and boundary conditions. Commercial software packagesa were used for the simulation of heat transfer phenomena within the pipes. The results obtained from FEA provide a detailed temperature distribution along the surface of the pipes.
The finite element model within this article was conducted in accordance with the following design specifications:
Model assembly and analysis steps followed in the softwarea. In the engineering analysis, the software of choicea was used for its robust capabilities in modeling complex geometries. With its comprehensive toolset, the intricate structures were accurately represented and analyzed. Differentiating the refractory lining from the pipe itself, thermal conductivity properties were assigned to each component. Mesh generation, conducted with softwarea tools, ensured detailed representation while maintaining computational efficiency. Defining contact and interfaces between components allowed precise modeling of thermal and mechanical interactions. Boundary conditions were meticulously set to mirror theoretical calculations. Employing a static steady-state approach, the analysis aimed for equilibrium insights into thermal behavior.
Model results. The temperature distribution for the refractory-lined pipe is shown in FIG. 2. The highest temperature is the inside surface of the refractory and the temperature decreases in a radial direction to a minimum of approximately 181.29°C at the outer surface of the pipe.
The inner surface temperature of the refractory lining is slightly lower than the flue gas temperature due to the internal convection resistance. The heat transfer coefficient (hi ) depends on the fluid properties and flow conditions inside the pipe. A lower hi value means a higher thermal resistance and a lower heat transfer rate. Therefore, the inner surface temperature of the refractory lining is not equal to the flue gas temperature, but slightly lower. In this case, the hi value is calculated as 63 W/m2K, which results in a temperature difference of about 20°C between the flue gas and the inner surface of the refractory lining.
Results comparison and discussion. Skin temperature can be calculated using two methods: the manual approach and FEA. The manual method yields a skin temperature of 176.6°C, while FEA predicts a slightly higher value of 181.29°C, representing a mere 3% difference. This variance highlights the remarkable accuracy of FEA, attributed to its computational sophistication and incorporation of temperature-dependent properties in the analysis. By harnessing FEA's capabilities, a realm of precision can be unlocked that underscores its superiority in temperature calculations.
Takeaway. This study has introduced a manual calculation method designed to estimate temperature distribution in refractory-lined pipes. The methodology employs an iterative approach, treating the outside heat transfer coefficient as a composite of convection and radiation. The application of this method to a standard refractory pipe, characterized by specific dimensions, material properties and operational parameters is detailed. Comparative analysis with results obtained through FEA utilizing proprietary softwarea revealed that the manual calculation method offers a balance of accuracy and efficiency. The outcomes demonstrate that this method can effectively determine skin temperatures with a narrow margin, approximately a 3% variation from the skin temperature arrived using the FEA method. HP
NOTES
LITERATURE CITED
Mrinmoy Ghosh Choudhury is an accomplished engineering professional with extensive experience in process plant engineering, specializing in piping and plant engineering from initial concept to final commissioning. His expertise encompasses various crucial aspects, including concept layout, stress and support, and materials. Choudhury has a proven track record in resolving complex issues during plant startups, addressing problems such as vibration, rotary equipment alignment and piping/equipment system failures. He has also significantly contributed to leading engineering journals with numerous published articles. Throughout his career, Choudhury has contributed his expertise to prominent companies like Reliance Engineering, Toyo Engineering India, Chemtex Engineering India, Engineers India Ltd., and DCPL. He is currently working in KBR Gurgaon & Air Product India as an advisor.
Sukhjinder Singh Dargan is a seasoned engineering professional with a rich 18-yr background in the piping stress analysis domain, focusing on process plants, refineries and fertilizer plants. Since 2018, he has been an integral part of KBR in Gurgaon, India. Prior to his tenure at KBR, Dargan worked at L&T Chiyoda, India, and Fluor Daniel, India, where he honed his expertise in this field.
Venkatrao Nelluri serves as a Senior Piping Stress Design Engineer at Black Cat Consulting & Engineering Services W.L.L. With a robust 16 yr of industry experience, he specializes in designing and engineering piping systems and pipe supports across a spectrum of industries, including refining, fertilizer, metals and mining, power, petrochemical sectors and desalination plants. Before joining Black Cat Consulting & Engineering Services, his previous affiliations include roles at KBR, Fluor Corp. and M.N. Dastur. Nelluri has also been published in several professional publications in his field.