Sensor fault compensation via software sensors: Application in a heat pump's helical evaporator

Highlights

• An evaporator model with phase change has been developed.

• The evaporator's model reduces the computational cost without precision lose.

• The fault compensation system allows tolerance against sensor faults.

• The fault compensation improves the evaporator performance.

• The FTC allows operation even in fault occurrence.

Abstract

This work presents a sensor fault compensation system, applied to a heat pump's helical evaporator. The mathematical model of the evaporator is given by algebraic and differential equations. These equations were selected according to the phases and regime of the fluid to be evaporated. The sensor fault compensation is based on fault detection and isolation system and a MPC (model predictive control) strategy. The fault detection isolation system is based on a bank of two high-gain observers which have two main purposes. The first one is to generate adequate residuals when a sensor fault occurs. The second purpose is to act as a software sensor, meaning, the measure estimated by the observer replaces the measure given by physical sensor when a sensor fault occurs. The high-gain observers were selected because they are easy to implement and tune. Furthermore, they provide an adequate estimation of the process outputs (when a sensor fault occurs) to a model predictive control (MPC) strategy. The MPC has been implemented to regulate the steam outlet temperature. Several experiments were carried out to show that the MPC regulates the process output even if a fault occurs. The experiments in the evaporator of the absorption heat pump have shown reliability of the method presented in this work to detect a sensor failure, isolate the sensor failure and regulate the steam temperature when a total or partial failure occurs.

Introduction

Control theory and its application in industrial processes arises from the need to provide regulated operating conditions to ensure repeatability of the results in processes and to provide security for the operators of industrial equipment. The heat exchangers are an important part of industrial processes (food, textiles, chemicals, etc.) (Kakac and Liu, 2002). Some heat exchangers are designed for cooling or heating substances; while other applications of this equipment are evaporation or crystallization. In this kind of processes (evaporation and crystallization) to keep temperature conditions under control is not an easy task, because of the close interaction with the pressure.

The heat exchange systems have been the subject of study in order to propose strategies that will maintain the systems under controlled conditions and/or under supervision (Astorga-Zaragoza et al., 2007, Jonsson et al., 2007, Karsten and Ballé, 2000, Hangos et al., 2004, Weyer et al., 2000). Zavala-Río and Hernández-González (2007) presented a bounded positive adaptive control for counter-flow heat exchangers. The advantage of this control law is that it does not depend on a model of the system and its implementation is easy. The authors in (Zavala-Río and Hernández-González, 2007, Maidi et al., 2008) proposed techniques of control in heat exchangers without change of phase in fluids. Qu et al. (2006) consider necessary an exact model of the heat exchanger in order to realize an accurate control of its temperatures. In Maidi et al. (2008) it is presented an optimal linear PI fuzzy controller. The design of the controller is based on the use of a finite-dimensional approximation model of high order. Another application of the fuzzy controllers in thermodynamics process is presented in Hua and Fei (2011), where the authors present a fuzzy control with feedforward compensator to save energy and improve COP (coefficient of performance) of the system. Maidi et al. (2009) presented a model by partial differential equations to design a boundary geometric control law with an application in a counter-current heat exchanger.

Works by Cardona et al., 2007, Aguado, 2006, Giraldo et al., 2006, Pérez et al., 2004 present researches about control applications in evaporators. In Aguado, 2006, Giraldo et al., 2006, it is presented a model predictive control using ARX models. In these works it is mentioned that employing this kind of models is a good choice to control multivariable systems with delays. Some authors (Pérez et al., 2004) suggest that the evaporators are complex systems that are required to be linearized in the equilibrium point to control the process. Another work in the control of heat exchangers is presented in Rasmussen and Larsen (2011), in this work a low order nonlinear model of the evaporator is developed and used in a backstepping design of a nonlinear adaptive controller. The proposed method allows work in a wide range of operating points.

In large systems, as it is the case of heat pumps, each component is designed to provide a specific service to its operation. The overall system works satisfactorily if, and only if, all components provide the adequate service for which they were designed. A faulty component such as the evaporator usually changes the overall behavior of the system. Generally, a failure is an event that changes the behavior of a system such that the system does not suit its purpose. The failures are the leading cause of changes in the behavior of the system or parameters, and they will eventually lead to a degraded system performance or even loss of control of the system (Blanke et al., 2006). Accordingly, several researches had been published, addressing fault detection and isolation (FDI) schemes. For instance Du and Mhaskar (2012) propose a fault detection isolation (FDI) scheme focused on sensor faults. This approach is based on analytical redundancy by using of a bank of high-gain observers. The authors have shown the effectiveness of FDI system in a chemical process.

The absorption heat pumps present several interactions and nonlinearities in their behavior, so it is difficult to design a controller for this system (Ohgata et al., 1997). The nature of this process makes it difficult to control it manually. In the work presented by Escobar et al. (2008), it is shown a strategy to estimate the COP (coefficient of performance) of the absorption heat pump, however the authors did not present a control strategy to ensure the steady state of the heat pump, but they indicate that it is possible to implement a control strategy from the estimation of the COP. Experimental results (Olarte-Cortés, 2010, Escobar et al., 2009, Rodriguez, 2008, Bonilla, 2007) show that it is difficult to maintain steady state conditions in the process due to the lack of a suitable automatic control system. In addition, due to equipment failure caused by sensors or actuators, it cannot work for long periods.

Therefore, the main goal of this research is the implementation of a sensor fault compensation via high-gain observers, which is applied in the outlet sensors of the absorption heat pump's evaporator (Fig. 1). The high-gain observers were designed from an energy model commutated by the states, the commutations allow the estimation of the temperatures in the three operating regions of the evaporation process, subcooled liquid, biphase and superheated steam. The high-gain observer was selected because it is easy to implement and tune. Furthermore, this kind of observer provides an adequate estimation of the process outputs. The estimation time can be decreased by increasing the value of the tuning parameter of the observer. The main idea to use high-gain observer to develop a FDI scheme is to reconstruct the full system states. Also use the properties of the observer as precision and speed to the convergence, in order to isolate faster the faulty sensor, allowing a trade-off between precision and speed. This feature ensures a fast estimation in consequence, a fast detection of the failure. A fault detection system (FDI) based on physical redundancy implies an increment in the cost (in the majority of cases). The proposed method in this research, offers an alternative to implement a FDI system based on a virtual sensor (software solution) that will keep the continuous operation of the process. In addition, a model predictive control (MPC) was implemented, in order to regulate the heat supplied to the absorber. The MPC allows the continuous regulated operation of the heat pump even in faults presence of output sensors.

A double-pipe heat exchanger is formed by two concentric circular pipes with a fluid flowing in the internal pipe and another fluid flowing in the external section or annular space between the pipes. The dynamics of these systems can be modeled by coupling a finite number of first order differential equations (Khalil, 2002).

The heat exchanger dynamics is obtained by a balance of energy for each side of the heat exchanger (Fazlur-Rahman and Devanathan, 1994) as shown in Eq. (1). This model has been used in different works (Escobar et al., 2010, Zavala-Río and Hernández-González, 2007, Weyer et al., 2000, Varga et al., 1995) to represent the heat exchange in systems which do not have change of phase in any fluid.

In this work the model presented in Eq. (1) was used to represent the dynamic of an evaporator. The heat exchanger model describes the temperature profile in the evaporation process. To realize this task it was used a thermodynamic model which is composed by heat transfer equations. The heat transfer equations are in function of the flow rate ($W_v$), temperature, and pressure (T, P) in the evaporator. These equations change according to the flow rate ($W_v$), depending if the flow regimen is laminar or turbulent, or by the relation-ship between the temperature and pressure of the fluid (T, P). In this case, the model employs the equation according to the operation zone of the fluid. The different operation zones of the fluid are: subcooled liquid, biphase and superheated steam

$$\begin{array}{c}{\dot{T}}_{\mathrm{co}}(t)=+\left(\frac{U(T,W_{\mathrm{vc}})A_i}{C_{p_c}{\rho }_c{V}_c}\right)(T_{\mathrm{ho}}(t)-T_{\mathrm{co}}(t))+\frac{W_{\mathrm{vc}}}{V_c}(T_{\mathrm{ci}}(t)-T_{\mathrm{co}}(t))\hfill \\ {\dot{T}}_{\mathrm{ho}}(t)=-\left(\frac{U(T,W_{\mathrm{vh}})A_o}{C_{p_h}{\rho }_h{V}_h}\right)(T_{\mathrm{ho}}(t)-T_{\mathrm{co}}(t))+\frac{W_{\mathrm{vh}}}{V_h}(T_{\mathrm{hi}}(t)-T_{\mathrm{ho}}(t))\hfill \end{array}$$

The model of the evaporator Eq. (1) can be approximated by a set of three interconnected cell scheme (Weyer et al., 2000) as shown in Fig. 2 where each pair of cells represents a section of the heat exchanger and is described by a pair of ordinary differential equations. This configuration cell represents the heat exchanger with input and output flows (Varga et al., 1995).

Studies concerning heat exchangers (Weyer et al., 2000, Varga et al., 1995, Steiner, 1989) suggest that the overall heat transfer coefficient (U) can be used in the heat exchanger model Eq. (1) as a constant. But, in a real process this assumption is incorrect, when temperature variations are presented in the process. If this is the case, the valid range of the model to estimate the states variables is limited, and the estimation error is bigger, since a constant overall heat transfer coefficient U does not correspond to the flow rate and temperature conditions, where the system operates. In such a way, the overall heat transfer coefficient (U) has to be calculated when the system operates in several operating regions, considering it as dependent of the temperature and the flow U(T, W) in order to have precise estimates of the temperatures over a wide range of operation.

The mathematical model, given in Eq. (1), takes into account the following assumptions:

• The volumes in the pipe and in the annular section are constant.

• The convective coefficient is a variable dependent upon the temperature and the flow rate h(T, W).

• There is no heat transfer between the outer pipe and the environment.

• The water physical properties are evaluated as function of the temperature and pressure by empiric correlations.

• There is no energy accumulation in the pipe walls.

• The system inputs are measurable.

The overall heat transfer coefficient (U(T, W)) is calculated by the convective heat transfer coefficients of the external side h_o(T, W) and the internal side h_i(T, W). The relationship between these coefficients is given in Eq. (2):

$$U=\frac{1}{\frac{1}{h_o}+\frac{r_{o}ln(r_o/r_i)}{{\lambda }_w}+\frac{1}{h_i}}$$

where r_o, r_i are the external and internal radius of the pipes; h_o, h_i, and, ${\lambda }_w$ is the thermal conductivity of the wall.

In Escobar (2012) it is shown that applying the considerations in the model related to the temperature and flow to estimate the convective heat transfer coefficient h(T, W), an adequate approximation of the evaporator temperatures is ensured.

In order to predict the temperatures precisely, in this work it is proposed to compute the convective coefficients (h_o, h_i), which are calculated by Eq. (3) for each side.

Then the convective coefficient (h_φ) can be calculated for each side of the heat exchanger by

$$h_{\phi }=\frac{{\mathrm{Nu}}_{\phi }}{D/{\lambda }_{\phi }}$$

Henceforth, the subscript φ will be used to refer to either external or an internal side. For a single phase, the convective coefficient is determined through the Nusselt number (Nu), which is calculated for each side of the heat exchanger.

The model employs the equations of laminar flow if Re < 2300 or the equations of turbulent flow if the Re > 4000. Additionally, some terms of the Nusselt number related to geometry were recalculated, as it is described in the next paragraphs.

For laminar flow, the Churchill correlation was used as basis to calculate the Nusselt number:

$$\mathrm{Nu}={\left[{\left(\frac{48}{11}+\frac{51/11}{1+(1342/{\mathrm{PrHe}}^2)}\right)}^3+1.816{\left(\frac{\mathrm{He}}{1+(1.15/\mathrm{Pr})}\right)}^{3/2}\right]}^{1/3}$$

In order, to increase the model approximation to the measure data, it was necessary to adjust the exponents of the first and second term of the correlation (see Eq. (5)).

$$\mathrm{Nu}={\left[{\left(\frac{48}{11}+\frac{51/11}{1+(1342/{\mathrm{PrHe}}^2)}\right)}^{0.1}+1.816{\left(\frac{\mathrm{He}}{1+(1.515/\mathrm{Pr})}\right)}^{3/2}\right]}^{0.742}$$

For turbulent flow the Sebas-McLaughlin (Colorado-Garrido et al., 2009) correlation was taken as basis to calculate the Nusselt number

$$\mathrm{Nu}=0.023{\mathrm{Re}}^{0.8}{\mathrm{Pr}}^{0.4}{\left[\mathrm{Re}{\left(\frac{d}{D}\right)}^2\right]}^{0.05}$$

However, to increase the precision of the model, the correlation exponent of this equation (see Eq. (7)) was readjusted:

$$\mathrm{Nu}=0.020{\mathrm{Re}}^{0.9}{\mathrm{Pr}}^{0.8}{\left[\mathrm{Re}{\left(\frac{d}{D}\right)}^2\right]}^{0.0001}$$

where Re is the Reynolds number given by Eq. (8) and Pr is the Prandtl number given by Eq. (9)

$$\mathrm{Re}=\frac{\rho D{V}_s}{\mu }$$

$$\mathrm{Pr}=\frac{\mathrm{Cp}\mu }{\lambda }$$

where ρ is the density, μ is the dynamical viscosity, λ the thermal conductivity, V_s the velocity of the fluid, and, D is the diameter of the pipe.

To calculate the convective coefficient in biphase zone (h_bf) the Kozeki's correlation given in Eq. (10) (Colorado-Garrido et al., 2009) was used, where h_l is calculated by Eq. (3)

$$\frac{h_{\mathrm{bf}}}{h_l}={\left(\frac{1}{X_{\mathrm{tt}}}\right)}^{0.75}$$

$$X_{\mathrm{tt}}={\left(\frac{1-x_g}{\mathrm{xg}}\right)}^{0.9}{\left(\frac{{\rho }_g}{{\rho }_l}\right)}^{0.5}{\left(\frac{{\mu }_g}{{\mu }_l}\right)}^{0.1}$$

The Kozeki's correlation involves the parameter of Martinelli (X_tt) in Eq. (11) and it is employed in laminar flow and in turbulent flow.

The selection of the convective heat transfer coefficient is done according to the phase of the fluid, temperature and pressure (T, P). There are three zones along the inner pipe, which are subcooled liquid, biphase and superheated steam. In the subcooled liquid zone and in the superheated steam zone the convective heat transfer coefficient is calculated by Eq. (3). In the biphase zone the convective heat transfer coefficient takes the form of Eq. (10).

To have accuracy nonlinear model, it is necessary to estimate the variables of the process with its dependencies. In a heat exchanger, it is necessary estimate the physical properties (density, specific heat, viscosity) as a function of the pressure and temperature, because, the heat exchanger depends on the pressure and temperature. In a closed system the saturation temperature depends on the system temperature. Also, the heat exchanger model considers the flow regime; laminar o turbulent, calculated by Eqs. (5), (7). Therefore, if all these criteria are taking account to estimate the Overall Heat Transfer Coefficient it can say that such coefficient depends on the Temperature and Flow rate.

The issue concerning the sensor fault compensation with application in an absorption heat pump's helical evaporator is presented in this work. The equipment operates under conditions of negative manometric pressure. The evaporator has two inlet temperatures and two outlet temperatures which are measured. In Fig. 3 it is presented the proposed scheme for sensor fault compensation (SFC). The SFC system is based on a fault diagnosis and isolation system (FDI) which depends on an analytical redundancy of the sensors. A model predictive control (MPC) control law was implemented. In the FDI system, the analytical redundancy is generated by a bank of two high-gain observers. Each observer was designed to estimate both outlet temperatures of the evaporator (T_co, T_ho), from only one outlet measured temperature (y = T_co or y = T_ho). The inputs of the observer 1 are (T_ci, T_hi, T_co), and the inputs of the observer 2 are (T_ci, T_hi, T_ho). In the FDI system it is performed a residual evaluation. In the case that a fault occurs, the FDI system sends a signal to the reconfiguration system and the alarm is displayed in the human interface. If the reconfiguration system detects a faulty signal, then the system switches the measured temperature signal to the estimated temperature by the observer, allowing that the system be in continuous operation until the fault is corrected.

In Fig. 4 it is presented the scheme of the absorption heat pump's helical evaporator in which the sensor fault compensation is applied. The evaporator is a double pipe counter-current heat exchanger. Through the evaporator, there are two fluids: one fluid is going to be evaporated; while the other is the heating fluid. The first one flows through the inner side of the evaporator, called pipe. The inlet pipe temperature is T_ci and its outlet temperature (steam temperature) is T_co. The second fluid flows through the annular section and it is the heating fluid. The inlet temperature of the annular section is T_hi and the outlet temperature is T_ho.

The nomenclature used to define the temperatures of inlets and outlets of the evaporator is described in Fig. 4.

To implement the sensor fault compensation (fault tolerant system) it is necessary:

• To design a model that describes the profile of temperatures in the evaporator.

• To design a fault detection and isolation system.

• To design a sensor fault compensation system.