Stress Analysis of the Steam-Side Oxide of Boiler Tubes: Contributions from Thermal Strain, Interface Roughness, Creep, and Oxide Growth

Abstract

Stresses in the steam-side oxide of boiler tubes are evaluated based on analytical derivations and numerical simulations. With only thermal strain considered, analytical solutions of stress distribution are obtained for a cylindrical geometry representative of boiler tubes and a flat-plate geometry—a typical simplification assumed in the literature to represent boiler tubes with extremely thin oxide scale. In these analytical derivations, the substrate metal has finite dimensions or is assumed to be very thick. The solutions under the approximations of flat-plate geometry and very thick substrate are employed to examine the accuracy of various approximations adopted in the literature to analyze the stress distribution in boiler tubes. For more complicated situations where the contributions from thermal strain, interface roughness, creep of the oxide and metal, and oxide growth are considered, numerical simulations are performed for a cylindrical geometry to evaluate the stress distribution in boiler tubes. The simulation results reveal that: (1) the local radial stress at curved oxide–metal interfaces is enhanced by interface roughness with implications about interfacial crack growth; (2) contrary to the general belief that creep relieves the oxide stresses, creep may actually increase the stresses in the oxide due to different creep rates of the oxide and substrate metal; and (3) the geometrically induced oxide growth strain substantially increases the magnitudes of the hoop and axial stresses in the oxide. Based on the assumption that the failure of the oxide scales is caused by crack growth which is dominated by the stress intensity factor, damage maps are plotted directly using the hoop, axial, and radial stresses as the critical variables. Our work provides a quantitative understanding of the interactions between different thermomechanochemical processes and oxide scale failure in boiler tubes.

Graphic Abstract

Appendices

Appendix 1: Analytical Solutions When Only Considering Thermal Strain

Following the analytical approach in the previous study [13], we adopt the three-concentric-circle model to describe the cross section of boiler tubes. As shown in Fig. 12, the oxide scale is grown on the inner surface of the metal. The inner and outer radii of the oxide scale are a and b, and the outer radius of the metal is c.

Fig. 12

Schematic for the three-concentric-circle model. The two annuli represent oxide scale and metal on the cross section of boiler tubes

First, the plane-stress assumption is used for the out-of-plane direction. After cooling down, the oxide and metal develop different thermal strains, given by ε_ox,th and ε_met,th. A radial stress σ_b is required at the interface r = b to maintain the attached interface. Assuming that the oxide and metal surfaces are stress-free, i.e., $\sigma_{a} = \sigma_{c} = 0$ at r = a and r = c, the radial and hoop stresses are given by [13]

$$\sigma_{{{\text{ox}},r}} = \frac{{(r^{2} b^{2} - a^{2} b^{2} )\sigma_{b} }}{{r^{2} (b^{2} - a^{2} )}},$$

(A1)

$$\sigma_{{{\text{ox}},\theta }} = \frac{{(r^{2} b^{2} + a^{2} b^{2} )\sigma_{b} }}{{r^{2} (b^{2} - a^{2} )}},$$

(A2)

$$\sigma_{{{\text{met}},r}} = \frac{{b^{2} (c^{2} - r^{2} )\sigma_{b} }}{{r^{2} (c^{2} - b^{2} )}},$$

(A3)

$$\sigma_{{{\text{met}},\theta }} = - \frac{{b^{2} (c^{2} + r^{2} )\sigma_{b} }}{{r^{2} (c^{2} - b^{2} )}},$$

(A4)

where r is the radial coordinate, the subscripts ox and met denote oxide and metal and the subscripts r and θ denote radial and hoop components, respectively.

In this appendix, we only consider the contribution from thermal strain. The total hoop strain includes two parts, the elastic strain and thermal strain, i.e.,

$$\varepsilon_{{{\text{ox}},\theta }} = \frac{{\sigma_{{{\text{ox}},\theta }} - \nu_{\text{ox}} \sigma_{{{\text{ox}},r}} }}{{E_{\text{ox}} }} + \varepsilon_{{{\text{ox}},{\text{th}}}} ,$$

(A5)

$$\varepsilon_{{{\text{met}},\theta }} = \frac{{\sigma_{{{\text{met}},\theta }} - \nu_{\text{met}} \sigma_{{{\text{met}},r}} }}{{E_{\text{met}} }} + \varepsilon_{{{\text{met}},{\text{th}}}} ,$$

(A6)

where $E_{\text{ox}}$, $E_{\text{met}}$, $\nu_{\text{ox}}$, and $\nu_{\text{met}}$ are Young’s modulus and Poisson’s ratio of the oxide and metal, respectively.

To maintain the attachment of the oxide and metal at the interface, continuity of the radial displacement is required at the interface. Since the hoop strain is proportional to the radial displacement in the cylindrical coordinate system [14], the continuity condition results in

$$\varepsilon_{{{\text{ox}},\theta }} = \varepsilon_{{{\text{met}},\theta }} ,\quad \left( {at \:r = b} \right),$$

(A7)

By solving Eqs. (A1) to (A7), we obtain the interfacial radial stress σ_b as

$$\sigma_{b} = - \frac{{E_{\text{ox}} E_{\text{met}} (a^{2} - b^{2} )(b^{2} - c^{2} )(\varepsilon_{{{\text{ox}},{\text{th}}}} - \varepsilon_{{{\text{met}},{\text{th}}}} )}}{{E_{\text{ox}} (b^{2} + c^{2} )(b^{2} - a^{2} ) + (b^{2} - c^{2} )(b^{2} P_{1} - a^{2} P_{2} )}},$$

(A8)

where $P_{1} = - E_{\text{met}} + E_{\text{met}} \upsilon_{\text{ox}} - E_{\text{ox}} \upsilon_{\text{met}}$ and $P_{2} = E_{\text{met}} + E_{\text{met}} \upsilon_{\text{ox}} - E_{\text{ox}} \upsilon_{\text{met}}$. If we assume that the oxide and metal share the same Young’s modulus and Poisson’s ratio, i.e., $E_{\text{ox}} = E_{\text{met}} = E$, $\upsilon_{\text{ox}} = \upsilon_{\text{met}} = \upsilon$, Eq. (A8) is simplified to

$$\sigma_{b} = \frac{{E(a^{2} - b^{2} )(b^{2} - c^{2} )(\varepsilon_{{{\text{ox}},{\text{th}}}} - \varepsilon_{{{\text{met}},{\text{th}}}} )}}{{2b^{2} (a^{2} - c^{2} )}},$$

(A9)

Equations (A8) and (A9) are the solutions for the plane-stress situation, which corresponds to thin circular disks. The long circular cylinders correspond to the plane-strain assumption (cylinders with fixed ends) and generalized plane-strain (GPS) assumption (cylinders with free deformed ends).

For the plane-strain case, the total strain in the z-direction $\varepsilon_{z} = 0$. In Eqs. (A8) and (A9), we need to do the following substitutions: $E^* \to \frac{E}{{1 - \upsilon^{2} }},\:$$\upsilon ^* \to \frac{\upsilon }{1 - \upsilon }$, $\varepsilon ^*_{{{\text{ox}},{\text{th}}}} \to (1 + \upsilon )\varepsilon_{{{\text{ox}},{\text{th}}}} ,$ and $\varepsilon ^*_{{{\text{met}},{\text{th}}}} \to (1 + \upsilon )\varepsilon_{{{\text{met}},{\text{th}}}}$ (the symbols with * superscripts denote the effective properties under the plane strain assumption) [14]. The plane-strain version of Eq. (A9) becomes

$$\sigma_{b} = \frac{{E(a^{2} - b^{2} )(b^{2} - c^{2} )(\varepsilon_{{{\text{ox}},{\text{th}}}} - \varepsilon_{{{\text{met}},{\text{th}}}} )}}{{2b^{2} (a^{2} - c^{2} )(1 - \upsilon )}},$$

(A10)

For the GPS assumption, a uniform and nonzero $\varepsilon_{z}$ exists. When the overall axial force is zero, we have [7]

$$2\pi \int_{a}^{c} {\sigma_{z} r{\text{d}}r = 0} ,$$

(A11)

Based on the Saint–Venant’s principle, the differences of the stress distributions between the GPS solution and three-dimensional free long cylinders only exist locally at the tube ends [14]. Equations (A1) to (A4) are still valid for the GPS case. The stress–strain relations for the GPS are modified to

$$\varepsilon_{z} - \varepsilon_{{{\text{ox}},{\text{th}}}} = E_{\text{ox}} [\sigma_{{{\text{ox}},z}} - \upsilon_{\text{ox}} (\sigma_{r} + \sigma_{\theta } )],$$

(A12)

$$\varepsilon_{z} - \varepsilon_{{{\text{met}},{\text{th}}}} = E_{\text{met}} [\sigma_{{{\text{met}},z}} - \upsilon_{\text{met}} (\sigma_{r} + \sigma_{\theta } )],$$

(A13)

$$\varepsilon_{{{\text{ox}},\theta }} = (1 + \nu_{\text{ox}} )\left[ {\frac{{(1 - \nu_{\text{ox}} )\sigma_{{{\text{ox}},\theta }} - \nu_{\text{ox}} \sigma_{{{\text{ox}},r}} }}{{E_{\text{ox}} }} + \varepsilon_{{{\text{ox}},{\text{th}}}} } \right] - \nu_{\text{ox}} \varepsilon_{z} ,$$

(A14)

$$\varepsilon_{{{\text{met}},\theta }} = (1 + \nu_{\text{met}} )\left[ {\frac{{(1 - \nu_{\text{met}} )\sigma_{{{\text{met}},\theta }} - \nu_{\text{met}} \sigma_{{{\text{met}},r}} }}{{E_{\text{met}} }} + \varepsilon_{{{\text{met}},{\text{th}}}} } \right] - \nu_{\text{met}} \varepsilon_{z} ,$$

(A15)

Combining Eqs. (A1)–(A4), (A7), (A11), and (A12)–(A15), we can solve the expressions for $\sigma_{b}$ and $\varepsilon_{z}$. The general solution is unwieldy, and here we only show the solution when $E_{\text{ox}} = E_{\text{met}} = E$ and $\upsilon_{\text{ox}} = \upsilon_{\text{met}} = \upsilon$, which is

$$\sigma_{b} = \frac{{E(a^{2} - b^{2} )(b^{2} - c^{2} )(\varepsilon_{{{\text{ox}},{\text{th}}}} - \varepsilon_{{{\text{met}},{\text{th}}}} )}}{{2b^{2} (a^{2} - c^{2} )(1 - \upsilon )}},$$

(A16)

$$\varepsilon_{z} = \frac{{(b^{2} - a^{2} )\varepsilon_{{{\text{ox}},{\text{th}}}} + (c^{2} - b^{2} )\varepsilon_{{{\text{met}},{\text{th}}}} }}{{c^{2} - a^{2} }},$$

(A17)

Equation (A10) is the same as Eq. (A16), which indicates that under the assumption of homogeneous elastic constants, the radial and hoop stresses are independent of the axial strain under the GPS assumption (the plane-strain can be treated as a special case of GPS). Note that this conclusion will not hold when $E_{\text{ox}} \ne E_{\text{met}}$ or $\upsilon_{\text{ox}} \ne \upsilon_{\text{met}}$. Equation (A17) indicates that along the axial direction, the system is relaxed based on the average thermal strain of the oxide and metal. The axial stress is uniform within the oxide and is given by

$$\sigma_{{{\text{ox}},z}} = \frac{{E(c^{2} - b^{2} )(\varepsilon_{{{\text{met}},{\text{th}}}} - \varepsilon_{{{\text{ox}},{\text{th}}}} )}}{{(c^{2} - a^{2} )(1 - \upsilon )}}$$

(A18)

Appendix 2: Creep Rate of the Oxide Considering the Effect of Oxygen Partial Pressure

It is well known that the creep rate of an oxide is controlled by its oxygen vacancy concentration and consequently by the oxygen partial pressure in the environment [36]. In the earlier report, the creep rate of magnetite is measured in the CO/CO₂ mixture with the controlled ratios between CO and CO₂ partial pressures [32]. Here, we derive the expression about how the creep rate is dependent on the oxygen partial pressure based on the data provided in Ref. [32].

Based on the thermodynamic data, the oxygen partial pressure is related to the ratio between CO and CO₂ by the following relation [37]

$$RT\ln P_{{{\text{O}}_{2} }} = \Delta {}^{0}H - T\Delta {}^{0}S - 2RT\ln \frac{{P_{\text{CO}} }}{{P_{{{\text{CO}}_{2} }} }} = - 5 6 2 , 9 2 7+ \left( { 1 7 2. 0 2 { - }2R\ln \frac{{P_{\text{CO}} }}{{P_{{{\text{CO}}_{2} }} }}} \right)T,$$

(A19)

where R is the gas constant, $P_{{{\text{O}}_{2} }}$, $P_{\text{CO}}$, and $P_{{{\text{CO}}_{2} }}$ are the partial pressure of O₂, CO, and CO₂, $\Delta {}^{0}H$ and $\Delta {}^{0}S$ are the standard enthalpy and entropy of formation. Eq. (A19) can be rearranged to

$$P_{{{\text{O}}_{2} }} = 9.7736 \times 10^{8} \exp ( - 67741/T)\left( {\frac{{P_{{{\text{CO}}_{2} }} }}{{P_{\text{CO}} }}} \right)^{2} ,$$

(A20)

According to Ref. [32], with 1.5% CO and 98.5% CO₂, the creep rate of magnetite follows an Arrhenius law as

$$\dot{\varepsilon } = 3.36 \times 10^{3} \sigma^{3} \exp \left( {\frac{{ - 264\;{\text{kJ/mol}}}}{RT}} \right)\;{\text{s}}^{ - 1} ,$$

(A21)

where $\sigma$ denotes the von Mises stress in this appendix. Also, Ref. [32] shows that the creep rate is a function of the CO₂-to-CO partial pressure ratio as

$$\dot{\varepsilon } = k\left( {\frac{{P_{{{\text{CO}}_{2} }} }}{{P_{\text{CO}} }}} \right)^{ - 0.56} ,$$

(A22)

Combining Eqs. (A20)–(A22), we have

$$\dot{\varepsilon } = A_{cr} \sigma^{3} \exp \left( {\frac{{ - 421.62\;{\text{kJ/mol}}}}{RT}} \right)(P_{{{\text{O}}_{2} }} )^{ - 0.28} ,$$

(A23)

where A_cr is the creep rate coefficient to be determined. Note that with the fixed CO₂-to-CO pressure ratio, the oxygen partial pressure $P_{{{\text{O}}_{2} }}$ is a function of temperature, and thus the values of Q_cr in Eqs. (A21) and (A23) are different. When the CO-to-CO₂ pressure ratio is 1.5%/98.5%, from Eq. (A20) $P_{{{\text{O}}_{2} }} = 1.6041 \times 10^{ - 17} {\text{ Atm}} = 1.6202 \times 10^{ - 18} \;{\text{MPa}}$ at 1000 K. By letting Eq. (A23) equal Eq. (A21) under the same condition, we can obtain the value of A_cr, as $A_{\text{cr}} = 1.1516 \times 10^{7} \;{\text{s}}^{ - 1}$.

We also need to estimate the oxygen partial pressure in the boiler steam. In the air environment, $P_{{{\text{O}}_{2} }} = 0.02\;{\text{MPa}}$, which can be the upper limit of the oxygen partial pressure. For the lower limit, we assume that the liquid water at room temperature is in equilibrium with oxygen in the air, and 8 mg oxygen (2.5 × 10⁻⁴ mol) is dissolved in 1 kg (55.556 mol) of water [38], resulting in the oxygen mole fraction of 4.5 × 10⁻⁶. After the temperature increases and the water vaporizes as steam, we assume that the same amount of oxygen is maintained within the steam. At full load, the steam pressure is 17 MPa, and we have $P_{{{\text{O}}_{2} }} = 7.6500 \times 10^{ - 5} \;{\text{MPa}}$, which is the lower limit of the oxygen partial pressure in the steam. In the oxide scales grown on boiler tubes, the oxygen partial pressure decreases with increasing depth. In our simulations, we use the lower limit of the oxygen partial pressure in the steam to represent the average oxygen partial pressure in the oxide (the creep rate data for the Fe–Cr spinel are not available, and we assume that the creep rate of the spinel is the same as that of magnetite [8]). The creep rate is given by

$$\dot{\varepsilon } = 8. 5 8 7 5\times 10^{7} \sigma^{3} \exp \left( {\frac{{ - 421.62\;{\text{kJ/mol}}}}{RT}} \right){\text{ s}}^{ - 1} ,$$

(A24)

Note that since the data-measuring environment for Eq. (A21) has a much lower oxygen partial pressure ($P_{{{\text{O}}_{2} }} = 1.6202 \times 10^{ - 18} \;{\text{MPa}}$) than the estimated value in the oxide ($P_{{{\text{O}}_{2} }} = 7.6500 \times 10^{ - 5} \;{\text{MPa}}$), Eq. (A21) overestimates the creep rate of the oxide if it is directly used in the numerical simulations.