A new approach on estimating the fluid temperature in a multiphase flow system using particle filter method
AA new approach on estimating the fluid temperature in a multiphaseflow system using particle filter method
Zhuoran Dang ∗ School of Nuclear EngineeringPurdue UniversityWest Lafayette, IN 47907 A BSTRACT
Fluid temperature is important for the analysis of the heat transfers in thermal hydraulics. An accuratemeasurement or estimation of the fluid temperature in multiphase flows is challenging. This is due tothat the thermocouple signal that mixes with temperature signals for each phase and non-negligiblenoises. This study provides a new approach to estimate the local fluid temperature in multiphaseflows using experimental time-series temperature signal. The thermocouple signal is considered tobe a sequence with Markov property and the particle filter method is utilized in the new method toextract the fluid temperature. A complete description of the new method is presented in this article. K eywords Particle filter · fluid temperature · multiphase flow · Monte Carlo
Fluid temperature is fundamental to the analysis of heat transfers in heating and cooling systems. It is a parameterthat can be experimentally measured or calculated and mainly used in the calculation of the heat transfer coefficients.Thus, accurate fluid temperature is very important for the estimation of heat transfer rates or coefficients. Heat transferwith phase change has been widely used in the current industrial applications such as boiling water reactors. This isbecause the heat transfer rate is larger in a flow system with phase change than one without phase change. However, theestimation of the heat transfer coefficient with phase change, namely, the interfacial heat transfer coefficient is harderthan that of the signal phase heat transfer coefficient. This is mainly because the measurement of the fluid temperaturein the multiphase flow system is challenging. The temperatures of each phase are different and the temperature signalsare fluctuating. In this sense, the estimation of the fluid temperature in a multiphase flow system requires more studies.Despite it is a topic with several decade histories, the optimal approach of fluid temperature estimation is still notclear. Although some methods have been proposed, there is still no method that is recognized in consensus to bereliable. One simple and direct way to estimate the fluid temperature is to consider it to be equal to the lowest valuesin the temperature signals.[1, 2] This is easy to process yet it lacks either theoretical or experimental validations.Besides, the lowest values are not the fluid temperature if there are non-negligible inherent noise in the data acquisitionsystem. A more reliable and also more widely used method is to statistically extract the fluid temperature from thethermocouple signal by plotting the probability density function or histogram.[3, 4, 5] In a steam-water flow system, forexample, there are two characteristic peaks in the histogram and they correspond to the steam and water temperatures.This method is more reasonable and more robust than the above method. However, it highly relies on the quality ofthe thermocouple signal, that is, the thermocouple should be fast-response and signal-to-noise-ratio should be small.Recently, an algorithm that directly processes the thermocouple signal is developed [6] that it discriminates the phasesbased on self-tuned thresholds and calculates the temperature of each phase based on the energy balance equation. Thisalgorithm is more elaborated than the above two methods yet it still requires high-quality thermocouple signals.The recently drastic development of computer science provides new solutions for every other subject related to it. Instatistics and control theory, Kalman filtering, also known as linear quadratic estimation has been widely applied.[7] ∗ [email protected] a r X i v : . [ phy s i c s . d a t a - a n ] J a n his algorithm uses time-series signals containing statistical noise and other inaccuracies and it estimates the unknownvariable using a joint probability distribution over the variables for each time frame. It tends to be more accurate thanthose based on a single measurement alone. While the Kalman filter usually uses in a linear system with Gaussiandistributed noises, Particle filter can extend the application range to the non-linear systems with non-Gaussian distributednoises.[8] The particle filter is a common application of the Sequential Monte Carlo method, which is a set of MonteCarlo algorithms used to solve filtering problems arising in signal processing and Bayesian statistical inference. Theobjective is to compute the posterior distributions of the states of some Markov process, given some noisy and partialobservations. In this sense, it is possible to use the Particle filter for the estimation of fluid temperature using thethermocouple measurements in the multiphase flows.In this study, a new method for fluid temperature estimation based on particle filtering is designed. The theoretical basisof this method and a detailed description of this new method is presented. This new method is tested with time-seriesdata generated using a microthermocouple measurements in subcooled boiling two-phase flow. The particle filter method is developed for the problem of a stochastic process with Markov property. This processcontains both observable and hidden variables, and the observable variables are related to the hidden variables in someways that can be described in known forms or models. In the hidden Markov model (HMM), the observable variablesand hidden variables are used to describe the observation process and state process, respectively. The goal of the particlefilter is to estimate the posterior state given the observable state.[8]Define X n is the state of a Markov process and the observations Z n is the observation. The state X t is changedbased on the transition probability function, X t | X t − = x t ∼ p ( x t | x t − ) (1)The observation Z t is assumed to be only related with the state X t , Z t | X t = z t ∼ p ( z t | x t ) (2)The system can be modeled in the following state space equations: X t = g ( X t − ) + W t − Z t = h ( X t ) + V t (3)where g and f are both known functions. If g and f are linear and W and V are Guassian, then the system can bemodeled with the Kalman filters to obtained the unbiased, optimal estimation. If not, we can design a particle filter ifwe could assume that we can create sample set for the transition of X t to X t +1 and compute the weights/probabilitiesof the set.Suppose that we are processing a time-series signal with a length of T. The number of the independent particle is Nand the particles X N T are independent of each other with density distributions of p ( X T | Z T ) . Z represents theobservations. Note that each particle is a hypothesis about the current state.For nonlinear filtering, recall that the conditional probability can be expressed with Bayes Rule: [9] p ( X t | Z t ) = p ( Z t | X t ) p ( X t ) p ( Z t ) (4)where p ( Z t ) = (cid:82) p ( Z t | x t ) p ( x t ) dx t p ( Z t | x t ) = (cid:81) tt =0 p ( z t | x t ) p ( x t ) = p ( x ) (cid:81) tt =1 p ( x t | x t − ) (5)Based on the above equations, the nonlinear particle filtering equation is given as follows, p ( X t +1 | Z t ) = (cid:82) p ( x t +1 | x t ) p ( x t | Z t ) dx t p ( X t +1 | Z t +1 ) = p ( Z t +1 | x t +1 ) p ( X t +1 | Z t ) || p ( X t +1 | Z t +1 ) || = p ( x t +1 | Z t +1 ) (cid:82) p ( Z t | x (cid:48) t ) p ( x (cid:48) t | y t − ) dx (cid:48) t (6)The three equations in equation (6) corresponds to the evaluation, propagation, and normalization stage, which will bediscussed in the later section. Besides, one of the important stage in the particle filtering that should be mentioned isresampling. Resampling allows us to obtain samples distributed approximately according to p ( x n ) .[10] It removesparticles with low weights and keeps the particles with a high weights. This is useful in a time-series modeling orsequential framework because the particles with low weights could be likely to turn into a high-weighted particle at thenext time step and affects the estimation.[10] In the next section, the details of how to implement fluid temperatureestimation using particle filtering are presented. 2 Description of the new method
The measurement of temperature using thermocouple is considered to be a stochastic process with Markov property. Itmeans that the fluid temperature in the next state is assumed to be only related to the parameters in the current state.The fluid temperature estimation using the thermocouple signal can be regarded as a process of tracking with dynamics:given a model of expected fluid temperature in the current state, predict the value of fluid temperature in the next state.Therefore, the key idea of the particle filter is to generate a number of hypotheses, namely, particles, about the fluidtemperature at time step t-1, x t − , and keep the most likely ones and propagate them further to x t with the measurementat time step t. Repeat these process again using x t to x t +1 .An important consideration is how to choose the model that describes the fluid temperature, which corresponds tothe observation model. The mixed temperature in multiphase flow is simply modeled using a steady-state relation asfollows, T m = α T + α T + . . . + α n T n α + α + . . . + α n = 1 (7)where α and T are the local void fraction and temperature for each phase, respectively. In a two-phase flow systemsuch as subcooled boiling flow, this equation reduces to a two-partition formulation and the steam temperature canbe assumed to be equal to the saturation temperature. The mixed temperature is obtained from the thermocouplemeasurement and the local void fraction can be measured by various instrumentations such as conductivity probe. Fromthe equation, the model calculates a constant value in a short measured period, which will definitely produce errors.However, the fluid temperature fluctuation is very complex to model even in a steady-state condition. This modelapproximates the fluid temperature and when it combines with the new method proposed in this study, the estimationaccuracy can be improved.An example of this process is shown in the conceptual figure 1. The left part of the figure contains the averagethermocouple measurement values marked in black dot and line in 5-time steps. The right part of the figure is a pipelinediagram that briefly explains the main procedures included for a single time step process. In the time step t-1, initially,the measured density distribution is represented by particle positions and weights, p ( X t − | Z t − ) . The particles withrelatively large weights are chosen and evaluated with the fluid temperature model, while the particles with smallweights are discarded. This step is called as evaluation or prediction. In this process, the evaluation is to calculate themean square error (MSE) between the particle value and the fluid temperature model. Then the evaluated particlesare resampled based on their weights that particles with largely weights are represented by more particles, which willbe evaluated in the next time step. In the propagation stage, some dynamics and random noise are added in eachresampled particle. In this study, the dynamic model is simplified to a Gaussian distribution function with the centerof the distribution equals to zero. In the final stage of measurement, the resampled particles are evaluated with themeasurement density distribution at time step t. The algorithm is summarized below. Algorithm 1 initialize x i ∼ p ( x ) , w = N . loop for each time step t initialize S t = ∅ , η = 0 . evaluate w it = p (cid:0) x it | x i − t , u t (cid:1) for i = 1 to N . resample and propagate S t = S t ∪ [ x it ] Ni =1 from [ x it , w it ] Ni =1 . compute weights with measurement w it = p (cid:0) z t | x it (cid:1) , η = η + w it . normalize weights w it = w it /η . end The method is tested with the time-series signals measured using a microthermocouple in a steady-state, subcooledboiling, two-phase flow condition. The local void fraction is measured using a conductivity probe sensor near the samelocation. The fluid temperature is estimated using this time-averaged void fraction and mixed fluid temperature, aspresented above.The performance of the particle filters is largely related with the observation model, dynamic model, and the numberof the particles.[] The observation model and the dynamic model varies from different modeling subject and setup,3igure 1: Conceptual diagram of fluid temperature estimate process with particle filter.therefore, analyzing their effect on the model performance may not be worthwhile. The effect of the number of particlesis analyzed in this study, which is shown in the results figures 2. It can be easily seen that with a larger number ofparticles, the oscillation of the estimation curve is suppressed. However, the effect on the accuracy of the estimation isnot clearly observed when comparing the results, yet it is reported in many studies. This may due to that the changes inthe thermocouple signal are not significant.
This study provides a new method to estimate the local fluid temperature in multiphase flows using experimentaltime-series temperature signal. This new method is designed based on the particle filter algorithm and it assumes thatthe thermocouple signal is with Markov property, that is, the future state of the temperature conditioned on both thepast and present states depends only on the present state. [11] In this method, a batch of particles/hypothesis about thefluid temperature is generated and evaluated. The particles with a high probability or weight are kept and propagate tothe next time step. This method is tested with the time-series temperature signal generated with a microthermocouple ina subcooled boiling two-phase flows and it is observed that the oscillation of the estimation curve is strongly relatedwith the number of particles.
The author is currently a Ph.D. student in thermal hydraulics and reactor safety laboratory (TRSL) at Purdue Universityand under the supervision of Dr. Mamoru Ishii, Walter Zinn Distinguished Professor of Nuclear Engineering. Theauthor would like to deeply thank his support and guidance in the theory of thermo-fluid dynamics and two-phase flow.
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