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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 635140, 12 pages

http://dx.doi.org/10.1155/2013/635140

## Influence of Heat Flux and Friction Coefficient on Thermal Stresses in Risers of Drum Boilers under Dynamic Conditions of Steam Demand

^{1}Mechanical Engineering Department, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia^{2}Consulting Services Department, Saudi Aramco, Saudi Arabia

Received 29 December 2012; Revised 23 September 2013; Accepted 24 September 2013

Academic Editor: Seung-Bok Choi

Copyright © 2013 M. A. Habib et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Boiler swing rate, which is the rate at which the boiler load is changed, has significant influence on the parameters of the boiler operating conditions such as drum water pressure and level, steam quality in the riser tubes, wall temperatures of riser tubes, and the associated thermal stresses. In this paper, the thermal stresses developed in boiler tubes due to elevated rates of heat transfer and friction are presented versus thermal stresses developed in tubes operated under normal conditions. The differential equations comprising the nonlinear model and governing the flow inside the boiler tubes were formulated to study different operational scenarios in terms of resulting dynamic response of critical variables. The experimental results and field data were obtained to validate the present nonlinear dynamic model. The calculations of the heat flux and the allowable steam quality were used to determine the maximum boiler swing rates at different conditions of riser tube of friction factor and heat flux. Diagrams for the influence of friction factor of the boiler tubes and the heat flux, that the tube is subjected to, on the maximum swing rate were examined.

#### 1. Introduction

Thermal shocks and metal fatigue due to thermal stresses have always been known as main failure reasons for components of steam boilers. Design of boilers against failure due to thermal stresses went through a number of improvements going from trial and error type development to numerical and experimental analyses. Wolf and Neill [1] presented some protective measures against thermally induced stress cycling. Among these measures, the most important is the difference between boiler supply-water temperature and the system return-water temperature. As a rule of thumb, it is suggested that this difference shall not exceed 40°F. Among the important experimental works is that of Kudrayavtsev et al. [2] who developed a method for an accelerated sample testing of boiler materials. In their tests, the experimental conditions were made similar to the operating conditions. The results from fatigue testing of two different high strength boiler steels exposed to different heats were used to compare the materials with regard to their sensitivity to asymmetric loading. The authors recommended that boiler manufacturers should perform similar tests on the materials they intend to use for building boilers and components. Krüger et al. [3] developed an optimal control for fast boiler start-ups. Noting that the major limiting factor relevant to power plant start-ups is the thermal stress for thick-walled components, they have incorporated stress evaluation modules in their nonlinear model thermal. They presented a start-up control simulation in which drum and superheater maximum thermal stresses are set as predefined constraints and showed by simulation that their model will result in drastic reduction of start-up time. In a later work, the same authors [4] developed a simple equation for estimating thermal stresses from temperature during boiler start-ups. They stated that the major limiting factor relevant to fast power plant start-ups is the maximum admissible thermal stress for thick-walled components such as headers of superheaters and reheaters and boiler drum. They suggested that the control task during the boiler start-up should consider the current values of thermal stress and coordinate the boiler inputs in order to avoid any violation of the individual stress limits of each involved thick-walled component. Pronobis et al. [5] used finite element method to calculate the equivalent stress distribution in the cross-section of the tubes of the superheater in a boiler having steam capacity of 1150 ton/h considering nonuniform heat flux distribution resulting from radiation and convection. The stress evaluation modules in the tube material were due to both the inner pressure and local temperature gradients. The authors used the stresses to estimate tube lives showing that small increases in steam temperatures will reduce the tube life tremendously. They also showed that the assumption of uniform heating of the tube leads to unrealistically long expected lifetime.

The critical heat flux is a phenomenon that occurs in tube assemblies of thermal-hydraulic systems leading to covering the tubes by vapor blankets. This results in significantly reducing the local heat transfer coefficient which overheats the tubes and consequently causes higher levels of thermal stresses. Several authors have addressed the issue of critical heat transfer [6–10]. The empirical correlations for critical heat flux prediction are usually jointly employed with other methods such as table look-up and artificial neural network methods for predicting the critical heat flux [11]. Groeneveld et al. [12] extended the original look-up table method, which was developed by Doroshchuk et al. [13]. Due to its accuracy and simplicity in use, the extended work was widely implemented in several thermal-hydraulic analysis applications. The artificial neural network technique has also been usefully applied to critical heat flux data bases and analysis of the critical heat flux as a thermal-hydraulic phenomenon [14–16]. Several analytical models were developed by considering specific physical mechanisms leading to expressions describing certain critical heat flux conditions [17–19].

The problem of the evaluation of stresses due to the effects of both cyclic temperature and pressure was considered by Kandil [20]. He presented a complete analysis of stresses within the wall of a cylindrical pressure vessel subjected to cyclic internal pressure and temperature. The time-dependent stress distribution was obtained using a numerical model on the basis of the forward finite difference technique. The influence of mean pressure and mean temperature, pressure and temperature amplitudes, and diameter ratio on the effective stress was studied. The relation between the mean stress and stress amplitude was obtained for different working conditions. An approximate expression for the relation between the working parameters was introduced in a simple and direct form. The results of the approximate solution are found to fit well with the numerical findings. It was concluded that the internal surface of a cylinder subjected to cyclic pressure and cyclic temperature is exposed always to maximum effective stress. The outside surface was found to be exposed to the minimum effective stress. The problem of thermally induced stresses in boilers has been apparent for many years. These stresses can cause failures which can be abrupt, termed as “thermal shock,” or over a period of time, termed as fatigue failures. The latter are caused by repeated thermal expansions and contractions within the boiler or its components such as the riser pipe. The impact type failure “shock” is usually preceded by fatigue of the metal, but it has been observed to happen in short-term overheating. Sharp radius corners and abrupt changes in thickness of metals can amplify thermally induced stresses.

Sharp radius corners and abrupt changes in thickness of metals can amplify thermally induced stresses. In this paper, the thermal stresses are calculated for the case of abnormal riser tubes having the following features: one entrance (mud drum to riser), one exit (riser to steam drum), one expansion, one contraction, one 90-degree bend, and one 100-degree bend. Effects of the heat flux and friction are considered to derive a generic formula for the steam quality of the abnormal riser tube. The friction factor and heat flux value combinations are selected in a safe region to avoid critical heat flux situations. The swing heat rate condition is also analyzed for an abnormal riser tube case.

The present work aims at studying the influence of heat flux and friction coefficient on thermal stresses in risers of drum boilers under dynamic conditions of steam demand. This is achieved through the development of two computational models. The first is for the prediction of the heat flux along the riser tubes of the main water circulation circuits. The second is a nonlinear dynamic model for the investigation of influence of changes in operating conditions on the response of natural circulation. The dynamic response of the system’s state variables due to rapid changes in steam demand is investigated. The state variables include the pressure in the drum and steam quality at the exit of the riser tubes. The limits of boiler load swing rate are developed and used as guide to prevent boiler problems such as overheating of the riser tubes. The derivation of a general formula for the steam quality in an abnormal tube as a function of steam quality in a normal tube is presented. The influence of the heat flux and friction factor of the riser tubes on the allowable boiler swing rates is presented. The system under consideration includes the drum, the riser, and downcomer of the main circuits of a natural circulation boiler as its major components.

The differential equations comprising the nonlinear model and governing the flow inside the boiler tubes were formulated to study different operational scenarios in terms of resulting dynamic response of critical variables. The model is based on applying the basic laws of mass, energy, and momentum balance along with the thermodynamic properties of the water-steam mixture.

#### 2. Modeling

##### 2.1. Boiler Dynamics Model

The boiler under consideration here is of the water-tube natural circulation type. The operational and physical data of the drum boiler used in the study are presented in Table 1. The main components of the boiler (Figure 1) are the steam drum, the downcomer, and riser tubes which represent the complete water circulation loop. Most of thermal energy is added to the fluid while flowing in the riser tubes and thus boiling takes place. The distribution of water flow among the riser tubes is governed by the pressure differences along the tubes. In the abnormal operation where some of these tubes are partially blocked, these tubes exhibit abnormal conditions with low flow rates. To differentiate these tubes from other normal tubes, they will be referred to as “target (abnormal) riser tubes.” The following are the governing equations. The governing equations (1) consist of conservation of mass and energy of the total system, the phase change in the drum including the steam and water volumes inside the drum, and the rate of steam condensation, in addition to the equations for the flow circulation in the riser-downcomer loop which describe the transport of the mass, energy, and momentum [22]. Thus, a set of nonlinear differential equations representing the time dependence of the state variables of the pressure, , total volume of water, , and steam quality at the exit of the riser tube, , can be presented in a matrix form as follows: The model derived parameters are given by where is the average value of the void fraction along the riser tube and is the volume of the riser tubes. is the total mass of the system and is the metal temperature. , , , and are the density and specific enthalpy of the water and steam and is the specific enthalpy of evaporation .

##### 2.2. Steam Quality Model for an Abnormal Riser Tube

The model presented by (1) considers the steam-water flow to be equally distributed among the riser tubes. The model also assumes that the heat flux (heat transfer per unit area of the riser tubes) is the same for all the riser tubes. In practice, the heat flux distribution to the riser tubes is dependent on the location of the tube within the furnace. This influences the rate of radiation heat transfer from the flame and the hot gases. Consequently, the heat flux level and distribution along the riser tube differs from one tube to another. On the other hand, the riser tubes differ in their length and geometry. Some tubes may have more bends, contractions, and expansions than others. As a result, the riser tubes exhibit different friction losses. This results in nonuniform mass flow rates inside the different tubes since all the riser tubes are subjected to same pressure at their ends of the steam and mud drums. These two parameters of elevated heat flux and friction factor in the target (abnormal) riser tube affect the steam quality at exit of this target (abnormal) riser tube.

In order to investigate the influence of the heat transfer and friction losses on the thermal stresses of the riser tubes, the present work considers a case in which one of the riser tubes has a different geometrical condition and is also subjected to higher heat flux. This tube, called *target (abnormal) riser tube*, is longer and has more bends than the normal riser tube. As a result of this, it has a higher loss coefficient than a normal tube. The mass flow through the target (abnormal) riser tube is governed by the principle that the head loss through the target (abnormal) riser tube is the same as that of the normal tube, since they are parallel. For this target (abnormal) riser tube, the heat flux and friction factor can be higher than other tubes. The abnormal tube will, therefore, exhibit steam quality conditions that are different from the normal tube. In this study, the target (abnormal) riser tube is considered as shown in Figure 1 with more bends, reduction in diameter close to inlet and exit and contraction at mud drum and expansion at steam drum.

The steam quality for a target (abnormal) riser pipe, , is calculated from (3) of which derivation is presented in the Appendix as follows:
where one has the following.: ratio of the heat flux/unit area on the target tube as compared to the heat flux/unit area for all the tubes (). The value of is found, based on the given boiler capacity and the heat transfer area of the boiler considered in the present study, to be 60 kW/m^{2}. This value is based on a CFD numerical investigation of the flow and energy inside the boiler furnace in which the heat flux along all riser tubes was calculated. The heat flux distributions along the riser tubes at different locations from the burner plane indicated that the heat transfer could range from less than 60 up to around 300 kW/m^{2}. On this basis, it was decided to employ in this study the values of 1 to 5 for . : ratio of the length of the target (abnormal) riser tube to the nominal length of the normal riser tubes (). : ratio of the friction coefficient of the target (abnormal) riser tube to the nominal friction coefficient of the target (abnormal) riser tubes ().

The calculation of the friction coefficient of the flow inside the riser-downcomer loop is based on the losses due to the tube length and the other minor losses due to pipe contractions and expansions as well as tube bends. The loss coefficients for the minor losses are obtained, considering the riser to have the following features: one entrance (mud drum to riser), one exit (riser to steam drum), one expansion, one contraction, one 90-degree bend, and one 100-degree bend. Based on the actual boiler layout drawings, each downcomer tube is assumed to have minor losses due to one entrance (steam drum to downcomer), one exit (downcomer to mud drum), one expansion, one contraction, and two 120-degree bends. Based on these parameters, it was found that the value of is taken as 25, which presents the friction coefficient in the riser tubes. As the number of bends, pipe contractions and expansions, and the tube length are increased, the value of is, consequently, increased. The geometry of the tubes of the present boiler indicates a maximum of twice as much as the losses of the normal tube. Thus, the effect of was considered in the range of to be equal to 1 to 2.

*Thermal Model.* The following equations are used to calculate the inside and outside wall temperatures of the target (abnormal) riser tube:
where is the heat flux at the outside surface of the tubes, is the pipe outer diameter, and and are the wall temperatures at the inner and outer walls of the pipe. is the water saturation temperature. and are resistances of the inner water film and the pipe wall. They are calculated from the following relations [21]:
with as the thermal conductivity of the pipe material and is the heat transfer coefficient of the water film inside the tube.

The heat transfer coefficient including convection evaporation and nucleate boiling terms is expressed as [21] where is the single phase heat transfer coefficient and is calculated from the Dittus-Boelter equation [23] as follows: where is the Reynolds number of the flow inside the tube Consider and is the Prandtl number for water. is a fluid-dependent parameter (= 1 for water) and where is the density of liquid phase and is the density of the vapor phase. The constants , , , and and are given in Table 2 [21].

##### 2.3. Combined Stresses Problem

The finite element method has been employed for solving the heat conduction equation to determine the temperature distribution in the riser pipe wall. Resulting temperature profiles are integrated numerically for finding the effective thermal stresses in the wall. Grid size independency tests have been conducted for finding the optimum number of elements and for obtaining results with maximum accuracy. The riser pipe wall was divided into SOLID98 ANSYS elements. The SOLID98 ANSYS element is a 10-node tetrahedral coupled-field solid version of the 8-node SOLID5 ANSYS element [24]. The element has a quadratic displacement behavior and is well suited to model either regular or irregular mesh. Each of the ten nodes has six degrees of freedom at each node. The effective stresses in the cylinder wall are calculated from the solved obtained temperature and pressure distributions according to Von-Mises theory [25] as follows: where where , , and are the tangential, longitudinal, and radial stresses, respectively, and where , , and are, respectively, the thermal tangential, radial, and longitudinal stresses and are given as [26] where is the modulus of elasticity, is Poisson’s ratio, and is the thermal expansion coefficient of the riser tube material. The tangential, longitudinal, and radial stresses caused by pressure stresses , , and are given as The transient thermal stresses part of the total stresses of the riser tube are calculated based on the identified inner and outer wall temperatures of the tube using (12). The temperature histories are determined at different time points from the values of heat transfer coefficients at the inner surface of the tube using (4) and (5). Both the tube pressure and the inner and outer wall temperatures were employed in the FEM simulation.

#### 3. Solution Procedure

The present model then solves the differential equations of pressure, quality, and total water volume. Equations (1) were solved simultaneously using an explicit method with a time step of 1 s for a total time of 1500 s. The coefficients in these equations were obtained from (2). In order to integrate the differential equations of the system, a MATLAB subroutine was developed. The computed pressure and temperature values are then used to calculate numerically the effective von-Mises stresses (see (10)) at each time step.

#### 4. Results and Analysis

This section presents analysis for effects of heat flux and friction factors on stress distributions for determining the swing rate considering a riser tube of severe conditions of high heat flux and high friction factor. Thus, the influences of the parameters of heat flux and friction factor on temperature and stress distributions are presented. The model allows the computation of boiler pressure, quality of steam at exit of the riser, and the temperature of riser tubes under various operating conditions including loading conditions representing different swing rates. The boiler under consideration here is of the water-tube natural circulation type. The main components of the boiler are the steam drum and the downcomer-riser tubes which represent the complete water circulation loop.

##### 4.1. Validation (Open Loop)

In order to validate the present nonlinear dynamic model, the responses of the different boiler parameters to step variations in firing rate (known as the rate of heat addition to the boiler and is equal to the chemical energy in the fuel) and steam flow rate were compared with available experimental data. Validation of the model was performed through comparison of the present results against the data of [22, 27]. The measurements provide firing rate and steam flow variations with time. The responses of drum pressure and drum water level to these variations [22] were recorded. The results of the present calculations are shown in Figure 2 for response to step variations in steam flow rate. The results show that the model captures the variations in drum pressure. A reasonably good agreement has been achieved.

##### 4.2. Validation (Closed Loop)

The data from the literature included response variations of drum pressure and water level to time variations in steam demand for the open loop case and were used for validation of the nonlinear dynamic model for this case. Experimental results and field data were obtained for two boilers of Shedgum plant of Saudi Aramco. The data include variations of firing rate, drum pressure, drum water level, and feedwater flow rate in addition to other boiler operating parameters in response to time variations in steam demand. These data formed the basis for the validation of the developed nonlinear dynamic model for closed loops (with control).

The calculations of the measured field data were conducted. The calculations include the drum pressure and the firing rate and the feedwater flow rate. Reasonable agreement between the field data and calculated results was obtained. The influence of the swing rate on the boiler operating conditions such as drum pressure, firing rate, riser wall temperature, steam quality, and riser wall stresses was conducted.

Calculations of the field results were performed to validate the equations governing the control loops. Thus, the steam flow rate of the field results was supplied as input data, and the response of drum pressure and fuel flow rate were calculated. The obtained results for drum pressure and firing rate are shown in Figures 3 and 4. Figure 3 presents time variation of the measured and calculated drum pressure in response to time variations in steam demand. As the pressure changes, the control system responds by changing the firing rate as shown in Figure 4. The figure presents the comparison of the measures and calculated time variations in fuel flow rate. The results indicate that the procedure, in general, predicts the boiler transients with a good accuracy.

##### 4.3. Effects of Heat Flux

Figure 5 shows the riser pressure variation with time for the different heat flux factors, Table 3, and shows that the heat flux has insignificant effect on the drum pressure. It should be noted that the heat flux increase is applied to the target (abnormal) riser tube only, while the pressure is controlled by the changes along all the riser and downcomer tubes. Accordingly, the pressure is not affected by increasing the heat flux along the riser tube. Figure 6 illustrates the time variations of the inner and outer wall temperatures with time and factor for 20% swing rate and . The calculated inner and outer wall initial temperature conditions are shown in Table 4, while the initial pressure was taken as 4480 kPa. As expected, the temperature level increased with increasing heat flux factor, while the inner tube pressure is unaffected by the heat flux factor; thus only thermal stresses will be affected by . The resulting effective stresses were calculated for different heat flux factors of 1, 3, 4, and 5. It should be noted here that the base case is which corresponds to heat flux of 60 kW/m^{2}. The stress distributions are illustrated in Figure 7. It is worth noting that the maximum stress and the stress distribution are mainly affected by the increase in temperature resulting from the increase in heat flux according to (14)–(16). The maximum stress for on the inner tube surface reaches a value of 123 MPa.

##### 4.4. Effects of Friction Factor

The case of and 20% swing rate is selected as a base case for which the effects of friction factor are to be studied. Figure 8 shows that similar to the heat flux factor the friction factor does not have any significant influence on the drum pressure variation. The inner and outer temperature distributions are however slightly affected by the friction factor as illustrated in Figure 9. The calculated inner and outer wall initial temperature conditions are shown in Table 5, while the initial pressure was taken as 4480 kPa. The maximum temperature of the inner wall increases by about 1.5°C, going from 270.5 to 272°C. This small increase in temperature has an insignificant effect on the effective stress value and variation with time as shown in Figure 10. It can, thus, be concluded that for the case of and 20% swing rate there is no much significant influence of the friction factor on the effective stresses.

##### 4.5. Swing Rate Effects at High Flux and Friction Factor

The curves of Figure 11 represent the variation of riser tube inner pressure with time and swing rate for the case of and . As shown in Figure 11, the highest swing rate (40%) is what causes the largest perturbation in the pressure distribution (from 4300 kPa to 4550 kPa). The pressure stabilizes after about 800 seconds for the present case. Figure 12 presents the influence of swing rate on the time variation of effective stresses at the base case of (i.e., heat flux of 60 kW/m^{2}) and indicates a maximum value of 53.5 MPa corresponding to swing rate of 40%.

The curves of Figure 11 represent the variation of riser tube inner pressure with time and swing rate for the case of and . It can be seen that similar to what was observed for the baseline heat flux and base friction of a normal tube (Figure 12) the highest swing rate (40%) is what causes the largest perturbation in the pressure distribution (from 4300 kPa to 4550 kPa). The pressure stabilizes after about 800 seconds for the present case.

As shown in Figure 13, the inner and outer wall temperatures and temperature variations with time are influenced by the higher flux and friction factors. The temperature of the inner wall in the transition zone is observed to vary from a minimum of 267.8 to a maximum of 272°C in about 240 seconds, and the outer wall temperature swung from 293 to 308°C in the same period of time. This increase in temperature has caused a noticeable rise in the effective stresses reaching a maximum of 105 MPa. For the present case, the thermal components of the tangential and radial stresses (see (14) and (15)) are dominant over the stresses caused by pressure.

The effect of the present stress levels on the yield and fatigue failure of the riser is examined in the following. As was shown in Figure 14, the critical case of 40% swing rate, and , resulted in a maximum stress of 105 MPa and an estimated maximum stress range of 25 MPa, occurring at the inner wall. The equivalent stress amplitude is calculated from (29) and used to estimate the expected life of the riser tube. To include the mean stress effect for asymmetric cycles, Goodman’s equation is used here as follows:

The mean stress is taken here as the steady state stress of 97 MPa; the ultimate strength MPa [27] of A 178 at room temperature is used here because carbon steel strength is not affected by temperatures below 645 K (700°F) [28]. The equivalent stress amplitude is estimated from (29) to be 15.6 MPa. This stress level is below the endurance limit for steels (about 70 MPa (10 Ksi)) with less than 550 MPa (80 Ksi) and at temperatures less than 645 K (700°F) [28]. This will result in a safety factor, guarding against fatigue failure, as high as 4.5.

As shown above, the stress variation in the riser tube due to 40% swing rate, and , is not expected to result in tube failure by fatigue cycles provided that the boiler and tube were initially at steady state and that the start-up and shutdown cycle was taken care of in the design of the boiler system. Tube material rupture or yielding is also not expected to occur at these stresses. The safety factor guarding against yielding can be estimated at 2.1 by taking the design yield stress to be equal to 0.4 [28]_{.}

Figure 15 shows the maximum swing rates as limited by the coefficient of friction (-factor) of the riser tubes and heat flux factor (-factor) subjected to the riser tube. The region below the curves refer to the safe operation and that above the curve refers to the unsafe operation. For a certain tube of a fixed heat flux, as the -factor increases due to more bends in tube or blockage, in general, the maximum swing rate decreases. Also for a certain tube with a given friction factor, as the heat flux increases the maximum swing rate decreases. It should be noted that the swing rate is relative to the operating pressure and not to maximum continuous rating (MCR). Figure 15 shows that, for a tube of a given heat flux, the maximum swing rate decreases as the friction factor increases. Thus, as an example, a tube having a heat flux factor of three and a friction factor of two (an effective length of two times a regular riser tube) can withstand a maximum swing rate of 10%.

#### 5. Conclusions and Remarks

The dynamic model for the system of drum, riser, and downcomer was developed. The established nonlinear dynamic model includes the governing equations for mass and energy of the global system, the riser balance equations, and the drum governing equations. The governing equations for the flow and combustion processes in the boiler were developed, and the calculations are presented in this report. Experimental results and field data were used to validate the present nonlinear dynamic model for closed loops. Reasonable agreement between the field data and calculated results was obtained. The influence of the swing rate on the boiler operating conditions such as pressure, drum water level, firing rate, feedwater flow rate, riser wall temperature, steam quality, and stresses are presented. The combined temperature and pressure induced stress fluctuations in the riser tube due to swing rate is found to be too low to result in riser tube damage provided that the boiler and tube were initially at steady state and that the start-up and shutdown cycle was taken care of in the design of the boiler system. The calculations of the critical heat flux and the allowable steam quality were used to determine the maximum boiler swing rates at different conditions of riser tube of friction factor and heat flux. Diagrams for the influence of friction factor of the boiler tubes and the heat flux, that the tube is subjected to, on the maximum swing rate were presented. Thus, for any tube of a given effective length and known heat flux as calculated by the heat flux model, the maximum boiler swing rate that ensures safe operation can be determined.

#### Appendix

#### Calculation of Steam Quality in the Target (Abnormal) Riser Tube

Consider the riser tubes of Figure 1 which include a target (abnormal) riser tube and the rest of the tubes being normal tubes. The mass flow rates in the normal tube and target (abnormal) riser tubes are and , where is the sum of the mass flow rate in the normal tubes and is the mass flow rate in the abnormal tubes. The conservation of mass in the global system is given as follows: As indicated by Astrom and Bell [22], the downcomer mass flow rate is given as (4) or Solving (A.1) and (A.3), thus, The heat flux per unit area is given as where is the surface area of the pipe.

Thus, is the surface area of pipe and is the surface area of the rest of the pipes. and are the distances along the riser tubes. Consider rewriting (A.7) as From (A.4) and (A.8), one has where is the number of tubes. A general formula can be obtained from (A.9) as follows:

#### Nomenclature

: | Riser pipe inner area |

: | Downcomer pipe inner area |

: | Specific heat of metal |

: | Riser pipe inner diameter |

: | Riser pipe outer diameter |

: | Modulus of elasticity |

: | Mass flux |

: | Heat transfer coefficient specific enthalpy |

: | Gravitational acceleration |

: | Specific enthalpy of feedwater |

: | Specific enthalpy of saturated liquid water |

: | Specific enthalpy of saturated water vapor |

: | Specific enthalpy of steam leaving the boiler |

: | Specific condensation enthalpy |

: | Dimensionless friction coefficient in the downcomer-riser loop |

: | Thermal conductivity |

: | Drum pressure |

: | Heat flow rate to the risers |

: | Heat flux |

: | Total mass of the system |

: | Mass flow rate of feedwater supplied to the drum |

: | Mass flow rate of steam exiting the boiler |

: | Steam flow rate through the liquid surface in the drum |

: | Mass flow rate through riser |

: | Nusselt number |

: | Drum pressure |

: | Prandtl number |

: | Reynolds number |

: | Time |

: | Temperature |

: | Metal temperature |

: | Riser tube steam temperature |

: | Riser tube outer wall temperature |

: | Riser tube inner temperature |

: | Metal temperature |

: | Steam saturation temperature |

: | Riser pipe radius |

: | Riser pipe inner radius |

: | Riser pipe outer radius |

: | Drum volume |

: | Volume of riser tubes |

: | Total volume of steam in the system |

: | The total volume of the drum, downcomer, and risers; |

: | Total volume of water in the system. |

: | Steam quality. |

*Greek Symbols*

: | Thermal expansion coefficient |

: | Average volume fraction of steam in the riser |

: | Poisson’s ratio |

: | Dynamic viscosity |

: | Saturated water density |

: | Saturated steam density |

: | Liquid phase density |

: | Longitudinal stress |

: | Tangential stress |

: | Radial stress. |

#### Conflict of Interests

The authors of this paper confirm having no financial interests or conflict of interests in this paper.

#### Acknowledgments

The authors wish to acknowledge the support received from King Fahd University of Petroleum & Minerals and Saudi Aramco during this study. This work was partially supported by KACST through the Project no. 10-ENE1371-04.

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