lipsettualbertaca Abstract The governing equations of a system can be formulat ed using a set of state variables and constitutive relationships for the elements of the system yielding a set of differential equations that models how energy is distribu ID: 25468 Download Pdf

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Embed / Share - Incorporating Damage Accumulation in Deterministic System Models to Model Reliability of TimeVarying Systems Michael G Lipsett University of Alberta Edmonton Alberta TG G Canada Telephone Telefax E

Incorporating Damage Accumulation in Deterministic System Models to Model Reliability of Time-Varying Systems Michael G Lipsett University of Alberta Edmonton, Alberta, T6G 2G8, Canada Telephone: 780-492-9494 Telefax: 780-492-2200 E-mail: michael.lipsett@ualberta.ca Abstract The governing equations of a system can be formulated using a set of state variables and constitutive relationships for the elements of the system, yielding a set of differential equations that models how energy is distributed, stored, transformed, and dissipated within the system. While there is extensive literature on specific damage mechanisms under established conditions, a general framework has been lacking for linking the damage accumulation to the process itself. Dissipative elements represent losses in a system, such as electrical resistance, pipe flow resistance, and mechanical damping. In mixed dynamic models, actuator elements transform energy from one type to another. In this work, process modeling is combined with damage modeling, in lumped-parameter mixed models that include dissipative elements that represent the energy lost in damage accumulation. The key challenges in modeling the effects of system dynamics on damage accumulation within the system are to develop appropriate constitutive relationships for the energy lost in damage to system components, and to develop methods to validate these relationships while systems are in service. These relationships may be related to the time derivatives of variables of state rather than the variable itself, or to a nonlinear function. Empirical damage models are typically only valid in a fairly narrow range of conditions. Often these models are not expressed in terms of the variable pairs for energy (or power) expressions (force & velocity, current & voltage difference, flow rate & pressure drop, etc.). The system model has to reflect the relationship between overall measurement of process variables and local conditions where damage occurs. A general formulation is presented, with examples of damage mechanisms of interest in reliability of oilsands processes head loss in a slurry pipeline section compared to energy transfer in abrasion and impact at the pipe wall, and loading the bucket of an excavator compared to the speed and load variation at a mechanical joint that is prone to damage. 1. Lumped-parameter system modeling The governing equations of a system can be formulated based on state variables and constitutive relationships, which describe the overall physics of the system comprised of a set of connected elements within the system. In dynamical systems, the lumped-parameter physics of efforts and flows through and across each element contribute to a set of differential equations. These equations model how energy is distributed within the elements of the system, how elements are storing and releasing energy, how energy is transformed to other forms, and how energy is dissipated within the system [1,2]. Dissipative elements represent losses in a system, such as electrical resistance, pipe flow resistance, and mechanical damping. In mixed dynamic models, actuator elements transform energy from one type to another. A familiar transducer is a motor in an electromechanical system. If there is a dissipative load in the mechanical subsystem, such as a brake, then the electrical energy eventually is lost as heat. Ideally the dissipative element remains intact, but in reality the element will eventually become damaged. System models generally neglect the degradation in the system, which typically accumulates over a long time interval compared to the time frame of interest for modeling and simulating the process dynamics. 1.1 Challenges in modeling damage accumulation In many cases, parametric models have been developed that describe the rate at which damage accumulates. Examples include how different wear mechanisms have been characterized for a range of materials at different conditions, corrosion rates, and the effect of radiation on material strength. These damage models are difficult to integrate into more general system models, for several reasons. A typical damage model is an empirical relationship that neglects the physics of how energy inputs to the system result in damage. The effect of conditions outside the measured range cannot be predicted. Empirical relationships are usually based on equilibrium testing conditions, and transient dynamic effects on damage are therefore ignored. Some damage relationships do include time-varying inputs: in fatigue with alternating loads, Miner’s Rule is used in rainflow calculations to relate damage accumulation with respect to alternating stresses over a period of time [3]. While there is extensive literature on specific damage mechanisms under established conditions, a general framework has been lacking for linking the damage accumulation to the process itself. The variables used for characterization are important for local effects such wear from solid particles impinging on a solid material, but these variables often do not directly map to process variables such as mean pipeline fluid velocity. This is the biggest challenge in creating damage models that can be linked to process dynamics. A damage mechanism can be incorporated into a lumped-parameter system process model, with a state variable that directly relates to the failure mode of damage process of interest, such as material loss rate in wear, crack length, etc. This variable will have a complementary variable that allows an energy dissipation rate to be determined for the damage mechanism itself rather than attendant energy losses such as heat, which is only indirectly related to the damage. In this way, a process mode can be combined with damage modeling, in lumped-parameter mixed models that include dissipative elements that represent the energy lost in damage accumulation. The key challenges in modeling the effects of system dynamics on damage accumulation within the system are to develop appropriate constitutive relationships for the energy lost in damage to system components, and to develop methods to validate these relationships while systems are in service. These relationships may be related to the time derivatives of variables of state rather than the variable itself, or to a nonlinear function. Empirical damage models are typically only valid in a fairly narrow range of conditions; and often these models are not expressed in terms of the variable pairs for energy (or power) expressions (force & velocity, current & voltage difference, flow rate & pressure drop, etc.). 1.2 Constitutive relationships Variables acting on elements of a system always appear in pairs. Through variables sum at nodes in a system; across variables are measured across an element. For many types of physical systems, through variables are flows, and across variables are efforts; but in mechanical systems, the through variables are forces (which may be moments) and the across variables are displacements. Energy in an energy-storing element is TdA 0 (1) For example, in a linear mechanical spring, the constitutive relationship between force F and deflection is F=kx, where is the across variable and is the through variable. The energy stored is E= kx/2. A spring is a one-port device that connects through a pair of terminals to the rest of the system. Rate-dependent elements will have corresponding power storage or dissipation functions. For a linear mechanical damper, the constitutive relationship isxb = and the energy dissipation rate is , where E is power. When modeling a part of the system, the modeling convention is for power to go in the same direction as the through variable. When the through and across variables act in opposite directions, then power is being consumed by that element, as it acts as a load on the system. A generic dissipative element is shown in Figure 1, with a through variable related to an across variable by . The direction of determines that the left side of the element is the input side. In general, the functional relationship between and may be nonlinear. Figure 1. Generic damage element dissipating power from a system A two-port device connects two parts of a system, and allows power to go from one part of a system to another. A generic two-port element is shown in Figure 2, with power flowing from the left side of the element to the right side. Figure 2. Generic two-port element When power out equals power in, there is ideal energy transfer and no losses. When power out is less than power in, then there is an instantaneous storage of energy within the element. Conversely, when power out is greater than power in, then there is an instantaneous release of energy from the element into the system. Power out plus power dissipated equals power into the element, thus representing the rate of energy lost from the system through that element. A transformer element allows energy to flow from one post to the other and from one through/across ratio to another, in the same medium (same variable types), and with little loss of energy. Examples include gears, levers, and electrical transformers. Modeling a damage mechanism for an element within a system will require a physical model for the damage, likely expressed as a function of time and some system process parameters. When the energy dissipation rate associated with the damage mechanism is expressed in terms of variables other than those of the nominal plant, then additional expressions need to be added to the governing equations of the system to reflect the relationship between overall measurement of process plant variables and variables for local conditions where damage occurs. This variable transformation can be achieved by modeling the combined plant and damage process as a mixed system, using transducers rather than transformers. 2. Mixed Systems Mixed systems include more than one energy type in the system. For example, a mixed system with a mechanical linkage and load driven by an electric motor constitutes a mixed system. The mechanical subsystem has flows of mechanical energy (force & velocity); and the electrical subsystem has flows of electrical energy (current and voltage) [2]. 2.1Transducers Subsystems are connected through a transducer, which is a two-port device that converts power from one type to another. In a lumped-parameter model of an electro-mechanical system, a motor is a transducer (also called an actuator) that transforms electrical power to mechanical power. An electric motor is shown in Figure 3. Electrical power drives the transducer, and so the magnitudes of voltage in and current through the device are both in the same direction. The mechanical subsystem acts as the load, and so the angular velocity and the torque act in opposite directions. Figure 3. Electromechanical transducer For an ideal motor (or generator), electrical power equals mechanical power, and so ie i = W t . (2) An ideal transducer converts 100% of the power with no losses. (A sensor is a type of transducer, which has very low power consumption.) The relationship between the across variables is W = a t e , (3) where is the coupling coefficient that converts the electrical across input variable to the mechanical across output variable. Real transducers have losses, which are modeled as additional elements within the subsystem in which the type of energy dissipation occurs: friction as mechanical power loss or as heat, electrical resistance, hydraulic resistance, etc. A constant loss within the transducer itself can be modeled by a modified coupling coefficient is * . The combined electromechanical system is illustrated in Figure 4. Figure 4. Mixed electromechanical system The energy transfer through the transducer entails both types of through and across variables; but the governing differential equations for the overall system dynamics are written in terms of state variables and expressions for the entire system. For example, the state variables for the electromechanical mixed system can be chosen as current and angular velocity . The concept of subsystems connected by a transducer can be extended to a nominal plant that has flows of energy for some process, combined with one or more other parasitic processes that dissipate energy through ongoing damage accumulation. Entropy in the system will increase due to increased dissipation of energy, through both incurring damage (increasing the disorder in the system) and reducing the efficiency of the plant process (reducing the amount power that gets converted into useful work as defined by the process). A process model generally entails fluid flows and pressures, while a damage mechanism will be due to a different type of physical process, such as heat damage or stress-induced loss of solid material. While a great deal of power may have been transmitted through the system, very little of the energy rate translates into damage to the components. 3. Constitutive Relationships for Damage Mechanisms 3.1 Energy loss and damage accumulation assessment A constitutive relationship defines how energy is stored or dissipated within an element of a system. Damage increases the disorder within the system, and so it is represented by a dissipative element. For a linear element, the total energy dissipated in time interval intis intdtAT, (4)where is the through variable (such as current and flow), and is the across variable (such as voltage difference and pressure difference). This is consistent with dissipative elements such as electrical resistors, with current (rate of charge) passing through the element, a voltage difference v across the element, and a rate of energy dissipation of vi. In a system component that undergoes damage, some fraction of the energy transmitted by the component is converted to a form that creates damage, through permanent deformation of a material, ablation, heat damage, change in chemical composition, etc. In this case, the integral can be interpreted as the cumulative energy dissipated due to damage. Integration over time of the damage rate D results in an expression for the cumulative damage between time and int, which is a measurable indication of the condition of the component: intintdt (5) The difference in damage over time does not necessarily have a linear relationship to a process state variable. The damage rate may be a nonlinear function of a process state variable, in transforming to variables pertinent to the physics of the damage mechanism. In any case, damage accumulation will be monotonically non-decreasing over time. 3.2 Damage models involving no change in through variables in a single system When damage rate is directly related to a measurable process variable that acts through the system elements (including an element representing the damage process) without being affected by the energy dissipation during damage, then the damage element can be included as an additional element within the system. The energy dissipation rate is driven by a change in the complementary across variable (which is measured across the damage element). In this case, the plant and damage systems do not comprise a mixed system: the damage element has the same type of through and across variables as those of the plant process. Figure 5 illustrates a system with flow as the through variable and pressures acting across a plant element (1) and acting across a damage element (2). Figure 5. Damage element with no change in through variable The additional element in the system will change the governing equations for the nominal system with no damage accumulation. An unreliable system will have a change in over time that eventually meets a condition indicating functional failure of the system. A degradation in the pressure difference across a pump or valve with no change in the flow would be an example of such a failed component. 3.3 Damage models involving no change in across variables in a single system A single system may have a damage mechanism that involves the plant variable types, but with some of the through variable distributed into the damage element as with no change in the overall across variable in the system. The damage element (2) is added in parallel with the plant element (1) that is associated with , shown in Figure 6. Figure 6. Damage element with no change in across variable DDAs for the first case, the additional element in the system will change the governing equations for the nominal system with no damage accumulation; but now, an unreliable system will have a change in over time that eventually meets a condition indicating functional failure of the system. A leak would be an example of such a failure. If both variables change as a result of the damage mechanism, then multiple elements can be employed to model the damage, such as a transformer element plus a dissipative load element. 3.4 Damage models involving change of variable type When the damage model is of a different physical process than the nominal plant process, then the overall system model must include a variable transformation between the variables of the plant and the damage process, using a combination of a transducer and a damage element. Figure 7 shows a nominal plant process element (1) with through variable , which then enters the two-port transducer element (2) with coupling coefficient and transforms to through variable y. Note the direction of the through variable changes, indicating that output side of the transducer is the load: the dissipative element (3). The across variable in the nominal plant subsystem is and the across variable in the damage subsystem is . Figure 7. Damage element with change in variables The transducer can also be modeled to allow a loss in the plant through variable as well as a dissipative damage load. In that case, the through variable out of the transducer is q* q with a nonideal coupling coefficient *.A simple example of a dissipation damage element is a pipe that is corroding with an active replenishment zone, in other words, corrosion that does not self-arrest due to formation of a protective oxide or other change that leads to a reduction in the corrosion rate, all other factors being equal. From an overall system viewpoint, the state variables of interest are the flow rate and the pressure change across the piping element. Corrosion does depend on the flow rate of the fluid in the pipe, assuming that bulk flow rate is same as or at least proportional to flow rate at the wall. Generally this damage mechanism depends on factors other than simply process parameters, for instance, dissolved oxygen, electrochemical potential, and flow variations that affect the transport phenomena in the fluid. As flow progresses through a pipe network, the level of dissolved oxygen in the fluid will change (lower downstream as corrosion occurs, unless refreshed). where is the chemical potential and is the mole flow rate, which relates to the mass flow rate at the wall but is dependent on the flow regime and other transport phenomena that are not process state variables. From this simple example, it is clear that a damage mechanism often involves different physical phenomena than that of the process, and may be dynamic rather than in equilibrium. As well, the damage process may not be proportional to process variables, in which case the damage process must be described using a different set of variables. 4. Using propagation simulation to estimate time to failure Once a mechanistic model of a damage mechanism has been developed, then the expected inputs to the system determine the damage accumulation rate, which integrated over time results in the damage amount in an element of the system. Once damage amount exceeds a predetermined threshold threshold that compromises the plant process performance in some way, then a specific failure has occurred and the system is now unreliable: D t � D threshold . (6) The time at which this failure occurs can be found by simulating the behaviour of the system (including the damage accumulation) as an initial-value problem solved as a propagation simulation. 4.1 Stochastic models for random time-varying inputs In most cases the system inputs will be not be completely known, but rather some inputs will be random variables that represent variability in loading conditions, flow rates, impact forces, and other variables that affect a damage rate. There may be a higher-time-derivative relationship for damage to occur; or there may be a threshold value below which no damage occurs (for example, fatigue in steel - but not aluminum). The damage accumulation may appear to be stochastic in nature, because the nature of the interaction may not be properly captured in the process model (flow-induced vibration). For such systems, a Monte Carlo simulation can be used to map the probability distribution functions of the stochastic inputs through the plant process and the damage process to yield an output distribution of the damage variable. Care must be taken to ensure that the bandwidth of the random inputs is below the Nyquist frequency of the simulated system for the time step being used for the propagation simulation. Stochastic models are often used when there is no mechanistic model for the actual damage. Nonparametric models that related state variables of the plant process to the damage output are commonly employed. These data-driven models often assume linear input-output relationships, and in some cases deal poorly with time delays and time integrated effects. This simulation approach will not give a good prediction of the time to failure if the input probability distribution function is based on a set of conditions that are not actually present in the current system. This is similar to choosing the wrong order of model or an incorrect constitutive relationship. There are also potential errors associated with using quasi-static damage models if dynamic effects are significant. In that case, a single dissipative element is not an appropriate model for the damage mechanism. Model validation will rely on training data sets of input-output data for the mixed system, which means that the damage condition indicator must be measurable. 5. Example: wear in a slurry pipeline system 5.1 Lumped-parameter process model The set of processes in a section of pipeline transporting a dense oilsand slurry is shown in Figure 8. The system model comprises a network of elements through variables of mass flow rate, and across variables of pressure difference. Four nodes are defined and designated as A, B, C, and D. Node A lies at the beginning of the pipeline section where the average pressure in the upper layer of bulk fluid flow is . Flow divides between the two layers, with a pressure difference at the high pressure end of the pipe [4,5]. The average pressure in the low-concentration, top-layer portion of the downstream end of the pipe is , with a pressure differential at the downstream end between the bed and the top layer of . Intermediate pressure in the dense bed where bitumen released from the oilsand flows to the top layer is ; and the pressure in the top layer where bitumen enters the top layer is . For convenience, the flow separates coming into the pipe section and recombines leaving the section. Figure 8. System model of slurry pipeline section Across-type system variables are , , , and ; and through-type variables are the flow in the upper layer , the flow in the lower layer , and the transition flow between layers . The overall flow in the pipe is (7)The linear constitutive relationship for steady-state resistance to flow is D = (8)where is the coefficient of resistance, which is related to wall friction, density, viscosity, and pipe length. The rate of energy dissipation in a process element is 2/ (9) Each of the process elements in the network dissipates energy, with the exception of the pressure sources, which are assumed to have the same pressure difference regardless of the flow through the element (i.e., the top layer and the bottom layer). From the energy equation for the system, BDBDCDCDCBCBACACABAB (10)the governing equations of the equilibrium system are )( (11) and (12) (13)where ,1DABAPPPPx- - (14) ,2DADBPPPPx- - (15) .3DACBPPPPx- - (16) 5.2 Damage model For a moving bed the friction force is ),cos)(sin)( 2 gD (17) where is the coefficient of sliding friction (usually assumed to be 0.5), is the density of solids, f is the density of the fluid, is the concentration of the bed, is the concentration of the bulk, is the pipe diameter, and is the angle that defines the (modeled) interface between the bed and the bulk fluid layer. In reality there is not a sharp transition between the dense bed and the less dense top layer [6,7]. It is hypothesised that the damage rate is directly proportional to the energy dissipation due to friction on the wall. The candidate damage model is (18) where D is the damage rate as a function of wall friction and bulk velocity of the dense bed . This is a simplification because the actual flow rate at the wall is likely to be different from the average bed flow rate. This simplification means that the governing equations do not have to be reformulated, because the damage element is merely a multiplier of one of the energy dissipating process elements. A similar relationship was reported from experimental slurry studies [7]. Cumulative damage is measured as the loss of pipe wall thickness over an interval of time from t=0 to : dt (19) Damage due to impacts of particles against the wall during mixing of the bed is neglected, under the assumption that the relative velocity of small particles in a hindered medium is not likely to result in large impacts against the wall, at least in straight pipe segments. The dynamics of flow variations are also neglected. In industrial applications, the rate of change of flows is quite slow, and so a quasi-static model is thought to be a reasonable approximation. 5.3 Friction measurement To determine whether this damage model is correct, a set of pipeline wear tests has been undertaken. A laboratory-scale slurry pipeflow loop has been constructed with a test section for monitoring process variables such as flow rate and density, as well as pipe wall thickness as a condition indicator. The loop length is 20 m and the pipe internal diameter is ~ 5 cm (2 inches). Characterising the constitutive relationship of the damage process requires an estimate of both the flow rate and the wall friction. A floating element has been designed and constructed to measure the friction on an area along the bottom of the pipe wall, using a pair of cantilevers with strain gauges to measure the shear force on the free plate. The plate was cut by EDM and is mounted flush with the rest of the pipe to reduce discontinuities at the wall. Gland water is injected through the narrow gap between the plate and the rest of the pipe wall to prevent plugging with particulate matter. The wall flow rate will be estimated using a boroscope and a camera-based method for tracking particle velocity. 6. Conclusions and future work This paper describes a general framework for including damage elements in lumped-parameter process systems. Further work is needed to generate specific elements for different types of damage processes, with the relevant variables and constitutive relationships. A slurry loop incorporating the floating element sensor is undergoing commissioning trials, after which the element will have to be calibrated. The boroscope flow measurement device is being assembled and will be calibrated with separate particle tracking by particle image velocimetry. A more complete description of the apparatus and test plan will appear elsewhere. Full testing is expected to begin in late Summer 2009. Acknowledgments Funding support from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. This project is also supported by Syncrude Canada Ltd. Suheil El-Sayed developed the floating element; and Victor Jaimes designed the test loop. References 1.BT Kulakowski, JF Gardner, J Lowen Shearer. Dynamic Modeling and Control of Engineering Systems (3rd Ed.). Cambridge University Press New York 2007. 2.SH Crandall. Engineering Analysis: A Survey of Numerical Procedures. McGraw-Hill New York 1956. 3.Anonymous. ASTM E 1049-85. Standard practices for cycle counting in fatigue analysis. ASTM International, re-approved 2005. 4.J Schaan, RJ Sumner, RG Gillies, CA Shook, ‘The Effect of Particle Shape on Pipeline Friction for Newtonian Slurries of Fine Particles,’ Can J Chem Eng, Vol 78, pp 717-725, August 2000. 5.BEA Jacobs. "Design of Slurry Transport Systems." Elsevier New York 1991. 6.CA Shook, M McKibben, M Small, ‘Experimental Investigation of Some Hydrodynamic Factors Affecting Slurry Pipeline Wall Erosion,’ Can J Chem Eng Vol 68, pp 17-23, February 1990. 7.R Gillies, CA Shook, ‘Modeling High Concentration Slurry Flows,’ Can J Chem Eng Vol 78 (4), pp 709-716, August 2000. 8.M.C. Roco, G.R. Addie. Erosion wear in slurry pumps and pipes. Powder Technology, 50(1), 35-46, 1987.