- Email: [email protected]
Reactor thermal-hydraulics
CHAPTER
Reactor thermalhydraulics
10
10.1 Introduction Temperature and pressure of reactor fluids and solids are important variables in steady state and transient operation. Along with associated coefficients of reactivity, they determine the magnitude of reactivity feedbacks. Conservation of mass, energy, and momentum are the basis for thermal-hydraulics models. However, since pressure transients reach a new steady state so much faster in a transient than mass or energy, the differential equation for momentum is usually unnecessary. Much of the information in this chapter comes from Ref. [1]. This chapter includes short descriptions of major heat transport systems in nuclear power reactors. Chapters 12, 13, and 14 provide more detailed information about power reactor characteristics and control systems.
10.2 Heat conduction in fuel elements Most of the reactors addressed in this book use cylindrical UO2 fuel rods clad in zircaloy and have a gas-filled gap between the UO2 and the cladding. Fuel heat transfer in reactors with non-cylindrical fuel elements (such as high temperature gas cooled reactors and molten salt reactors).is not addressed here. A complete heat transfer model would require a partial differential equation in two spatial dimensions (axial and radial … azimuthal is not required) and time. Solutions of such a model equation are known, but they are not suitable for coupling with models for cooling fluid. Instead, lumped parameter models (also, sometimes referred to as nodal models) must be used. Lumped parameter models involve breaking the system into regions with uniform internal properties and coupling to adjacent regions. It is possible to model fuel rods with concentric radial lumps, but the simplest and most common lumped parameter model for cylindrical fuel elements uses a single radial node. Consider a single radial node model for a fuel element with heat transfer to fluid coolant. The fuel has mass, Mf, and specific heat capacity, Cf. The model equation is as follows: Mf Cf
dTf ¼ UA Tf θavg + Pf dt
Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00010-X # 2019 Elsevier Inc. All rights reserved.
(10.1)
111
112
CHAPTER 10 Reactor thermal-hydraulics
where Mf ¼ mass of fuel Cf ¼ specific heat capacity of fuel Tf ¼ fuel temperature Pf ¼ power released in the fuel node U ¼ overall fuel-to-coolant heat transfer coefficient A ¼ fuel cylinder surface area (fuel-to-coolant heat transfer area) θavg ¼ average coolant temperature in the adjacent coolant node Eq. (10.1) may be rewritten as dTf Pf UA ¼ Tf θavg + dt Mf Cf Mf Cf
(10.2)
The quantity, (Mf Cf /UA) has the units of time. It is the time constant for fuel-tocoolant heat transfer. Typical values for LWRs and CANDU reactors are 4 to 5 s.
10.3 Heat transfer to liquid coolant The core heat transfer model also requires heat balance equations for the coolant. A general model requires mass and energy balances. If the coolant density and node volume are constant, a mass balance is not needed (see Section 10.4 for a discussion of heat transfer in a model with a moving boundary). As with the fuel model, a nodal model for the coolant is needed. Consider the system shown in. Fig. 10.1 The figure shows that there are five variables as defined below: Pc ¼ power generated within the node (as by interaction of radiation with coolant atoms) Tf ¼ temperature of adjacent fuel node θin ¼ inlet coolant temperature θout ¼ outlet coolant temperature θavg ¼ average coolant temperature in the node θout
Fuel Node
Tf
θavg
θin
FIG. 10.1 Heat transfer to a liquid coolant lump (node).
Pc
Coolant Node
10.3 Heat transfer to liquid coolant
The nodal internal power generation, Pc, the fuel temperature, Tf, and the inlet coolant temperature, θin, are defined by other subsystem equations. That leaves two variables, but the coolant equation provides only one. An assumption is required to eliminate θout. θavg must be retained because it appears in the equation for heat transfer from fuel to coolant. The average temperature is given by the following: θavg ¼ ðθin + θout Þ=2
(10.3)
θout ¼ 2 θavg θin
(10.4)
or There is a problem with this formulation. Note in Eq. (10.4) that a sudden increase in inlet temperature would cause a sudden decrease in outlet temperature. This is an unphysical feature, causing consideration of an alternate formulation. Another possibility is the “well-stirred-tank” formulation. That is, the outlet temperature from the node is set equal to the average node temperature. This solves the problem in the previous formulation, but equating average and outlet temperatures does not represent actual behavior very well. Ray Mann of Oak Ridge National Laboratory addressed the problem [2]. Mann’s formulation uses two coolant nodes adjacent to a single fuel node; Fig. 10.2 shows this arrangement. Well-stirred-tank models represent each pair of coolant nodes. The outlet temperature of the first coolant node (assumed equal to the average temperature of that node) serves as the coolant temperature that provides the driving force for heat transfer from the fuel. Each coolant node receives half of the heat transfer from the fuel node. Therefore, the model equations for Mann’s formulation are as follows: Mc C c
dθ1 UA ¼ W Cc ðθin θ1 Þ + Tf θ1 + Pc1 dt 2
(10.5)
Mc Cc
dθ2 UA ¼ W C c ðθ 1 θ 2 Þ + Tf θ1 + Pc2 dt 2
(10.6)
∼ Θ2
∼ Θ2 ∼ Θ1
Tf ∼ Θ1
Qin
FIG. 10.2 Schematic of Mann’s model with one fuel node and two coolant nodes.
113
114
CHAPTER 10 Reactor thermal-hydraulics
where Mc ¼ mass of coolant in a node Cc ¼ specific heat capacity of coolant θ1 ¼ temperature in the first coolant node θ2 ¼ temperature in the second coolant node W ¼ coolant mass flow rate U ¼ overall heat transfer coefficient from fuel to coolant A ¼ total fuel surface area for fuel to coolant heat transfer Pc1 ¼ heat generation rate in the first coolant node Pc2 ¼ heat generation rate in the second coolant node. Note that the fuel-to-coolant heat transfer is given by the difference between the fuel temperature and the temperature of the first coolant node (assumed equal to the outlet temperature of that coolant node). The fuel-to-coolant heat transfer is divided equally to both coolant nodes in this formulation. The simplest formulation involves representing all of the fuel in a single node and all of the coolant as a pair of Mann’s nodes. A more detailed formulation uses a series of axial fuel nodes, each coupled with a pair of Mann’s nodes. See Fig. 10.3 for the case of two fuel nodes. ∼ ∼ Θ4 = Θout
∼ Θ4 Tf2
∼ Θ3
∼ Θ3 ∼ Θ2
∼ Θ2 Tf1
∼ Θ1
∼ Θ1 ∼ Θin
FIG. 10.3 Series of Mann’s models for more than one fuel node (two fuel nodes) representation.
10.5 Plenum and piping models
10.4 Boiling coolant Modeling channels with boiling fluid is more complex than modeling channels with single-phase fluid. Boiling heat transfer models are needed for BWRs and steam generators. Consider a heated channel into which liquid fluid is introduced into a channel with boiling fluid. The fluid temperature in the sub-cooled region at the entrance increases as it travels through the channel. At some point, the hotter fluid adjacent to the heated surface reaches saturation temperature even though the fluid farther from the heated surface remains sub-cooled. Boiling begins near the surface. The heat transfer coefficient for this region is greater than in the sub-cooled region. As the fluid continues through the channel, the bulk fluid temperature reaches saturation and vigorous boiling throughout the fluid occurs. High heat transfer coefficients occur in this region. A mixture of liquid and vapor exists in this region. This liquid-vapor mixture exits the core in BWRs and in U-tube steam generators. If all of the liquid boils before exiting the heated region (as in once-through steam generators), continuing heat transfer results in superheating of the steam. In a transient, the boundaries between the different regions move. Models have been developed in which these boundaries are state variables. Moving boundary models for BWRs and steam generators are quite complex, too complex for inclusion in this introductory book. But the concept is important. Appendix F illustrates the approach with a moving boundary model for the subcooled node in a heated channel.
10.5 Plenum and piping models Plenums are regions in which fluid enters and mixes with the existing fluid inventory. Plenums exist at core inlets and outlets and at steam generator inlets and outlets. Well-stirred-tank models are usually used for plenums. The model is as follows: Mp Cp
dθp ¼ W Cp θpin θp dt
(10.7)
where Mp ¼ mass of fluid in the plenum Cp ¼ specific heat capacity of the fluid W ¼ fluid flow rate θp ¼ fluid temperature in the plenum (equal to plenum outlet temperature because of well-stirred-tank approximation) θpin ¼ temperature of fluid entering the plenum Piping carries fluid from one reactor subsystem to another. Piping models are used for hot leg and cold leg piping in PWRs and CANDU reactors, and feedwater for BWRs and steam generators.
115
116
CHAPTER 10 Reactor thermal-hydraulics
Pure time delay models are often used. That is the formulation is simply as follows: θout ðtÞ ¼ θin ðt τÞ
(10.8)
where τ ¼ residence time in the piping. Well-stirred-tank models are also used. The rationale for a well-stirred-tank model is that some axial mixing does occur in piping. The more general dynamic formulation of the temperature of the liquid in the piping has the form dθout 1 ¼ ðθin θout Þ dt τ
(10.9)
The resident time is approximated as the ratio between the mass of fluid (Mp) in the piping and the fluid flow rate (Wp). τ ¼ Mp =Wp
(10.10)
10.6 Pressurizer A PWR pressurizer is a vessel with liquid water in the bottom section and saturated steam in the top section. A pressurizer is used to regulate the primary coolant pressure ( 150 bars) in PWRs and CANDU reactors. The pressurizer is connected to one of the hot leg pipings with a long surge line. Fig. 10.4 shows a typical pressurizer. Because of the contact between steam and liquid water, the water is also at the saturation temperature at steady state. Spray of cooler water enters from the top and electrical heaters at the bottom heat the liquid water. The steady state can be disturbed by water inflow or outflow, changes in inlet water temperature, changes in spray flow or changes in heater power. A PWR pressurizer control system can alter the pressure by modulating heater power and/or spray flow. A schematic representation of a pressurizer model structure appears in Fig. 10.5. As shown in Chapter 14, PWR and CANDU reactor pressurizers are slightly different.
10.7 Heat exchanger model Mann’s formulation also may be used in liquid-liquid heat exchangers. Fig. 10.6 shows the model. The equations are as follows: Mp1 Cp
UA dθp1 ¼ Wp Cp θpin θp1 + Tt θp1 dt 2
(10.11)
Mp2 Cp
UA dθp2 ¼ Wp Cp θp1 θp2 + Tt θp1 dt 2
(10.12)
10.7 Heat exchanger model
FIG. 10.4 Schematic of a pressurizer used in a PWR. Courtesy of Westinghouse Electric Company (The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Corporation, Pittsburgh, 1984).
117
118
CHAPTER 10 Reactor thermal-hydraulics
FIG. 10.5 Schematic of a pressurizer model.
Wp Θp,in
Mp1 Θp1
Θp1
Ms2 Θs2
Mt Tt
Mp2 Θp2
Θs1
Ms1 Θs1
Ws Θs,in
FIG. 10.6 Schematic of a liquid-liquid heat exchanger model.
10.8 Steam generator model
dTt ¼ U A Tt θp1 + U A ðTt θs1 Þ dt
(10.13)
Ms1 Cs
dθs1 UA ðTt θs1 Þ ¼ Ws Cs ðθsin θs1 Þ + dt 2
(10.14)
Ms2 Cs
dθs2 UA ðTt θs1 Þ ¼ Ws Cs ðθs1 θs2 Þ + 2 dt
(10.15)
Mt Ct
where Mp1 ¼ mass of primary fluid node-1 Mp2 ¼ mass of primary fluid node-2 Mt ¼ mass of metal node (heat exchanger tubing) Ms1 ¼ mass of secondary fluid node-1 Ms2 ¼ mass of primary fluid node-2 Wp ¼ flow rate of primary fluid Ws ¼ flow rate of secondary fluid Cp ¼ specific heat capacity of the primary fluid Cs ¼ specific heat capacity of the secondary fluid Ct ¼ specific heat capacity of the tube metal U ¼ overall heat transfer coefficient from primary fluid to metal, or from metal to secondary fluid A ¼ heat transfer area from primary fluid to metal node, or from metal node to secondary fluid θpin ¼ temperature of primary fluid inflow θsin ¼ temperature of secondary fluid inflow θp1 ¼ temperature of primary fluid node-1 θp2 ¼ temperature of primary fluid node-2 θs1 ¼ temperature of secondary fluid node-1 θs2 ¼ temperature of secondary fluid node-2 Tt ¼ temperature of metal (tube) node
10.8 Steam generator model Steam generator modeling can be simple or complex. Fig. 10.7 shows a typical U-tube steam generator. Different requirements for secondary system modeling determine the detail required for steam generators. For example, if the simulation focuses on reactor behavior, then a simple steam generator model that adequately simulates heat removal is adequate. A simple model, called the “teakettle model” represents steam generator dynamics with only three Eqs. A schematic representation of a teakettle model appears in Fig. 10.8. This approach has been found to represent heat transfer to the secondary fluid quite well in overall system simulations [1].
119
120
CHAPTER 10 Reactor thermal-hydraulics
STEAM OUTLET TO TURBINE GENERATOR DEMISTERS SECONDARY MOISTURE SEPARATOR
SECONDARY MANWAY
UPPER SHELL
FEEDWATER RING
ORFICE RINGS SWIRL VANE PRIMARY MOISTURE SEPARATOR
FEEDWATER INLET
ANTIVIBRATION BARS
TUBE BUNDLE
LOWER SHELL
WRAPPER
TUBE SUPPORT PLATES
SECONDARY HANDHOLE
BLOWDOWN LINE TUBE SHEET
TUBE LANE BLOCK PRIMARY MANWAY
PRIMARY COOLANT OUTLET
PRIMARY COOLANT INLET
FIG. 10.7 Schematic of a typical U-tube steam generator (UTSG). Courtesy of Westinghouse Electric Company (The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Corporation, Pittsburgh, 1984).
10.8 Steam generator model
FIG. 10.8 Schematic representation of a teakettle model of a steam generator.
10.8.1 U-Tube steam generator (UTSG) More detailed models are needed if steam generator performance itself is under consideration. For example, feedwater control system simulation is often an objective. Finite difference models with parameter updates in each node as a transient proceeds are used, but are quite complex. The moving boundary approach described in Section 10.4 also be used for steam generator modeling and is somewhat simpler to implement [3–6]. The nodal structure for a moving boundary model for a U-tube steam generator appears in Fig. 10.9. The model uses fourteen coupled differential equations and additional algebraic equations. The complete model is described in Ref. [6].
10.8.2 Once-through steam generator (OTSG) Once-through steam generators for PWRs are shell and tube heat exchangers with liquid water inside the tubes and water and steam on the shell side. Fig. 10.10 is a schematic of a once-through steam generator (OTSG) [7]. A schematic for a moving boundary model appears in Fig. 10.11 [8]. Note that two boundaries move (the boundary between the subcooled water section and the boiling section, and the boundary between the boiling section and the superheating section). A detailed model of an OTSG was developed in Ref. [8].
121
122
CHAPTER 10 Reactor thermal-hydraulics
FIG. 10.9 Lumped parameter (nodal) structure of a moving boundary U-tube steam generator model [6].
10.9 Balance-of-Plant (BOP) system models The components of a balance-of-plant (BOP) system are similar in nuclear power plants that generate steam using reactor thermal power. These include PWRs, BWRs, CANDU reactors, sodium fast reactors, gas-cooled reactors, molten salt reactors, and small modular reactors. Fig. 10.12 shows the major components of a BOP system and associated state variables [8]. The figure shows the following BOP components: • • • • • • •
Steam chest High pressure and low pressure turbines Moisture separators and steam reheaters Condenser Feedwater heaters Condenser pressure controller Feedwater temperature controller.
10.9 Balance-of-Plant (BOP) system models
FIG. 10.10 Schematic of a once-through steam generator (OTSG). Courtesy of The Babcock & Wilcox Company (Steam, its generation and use, The Babcock & Wilcox Company, forty second ed., 2015).
123
124
CHAPTER 10 Reactor thermal-hydraulics
FIG. 10.11 A moving boundary model of OTSG [8].
The dynamics of these components are represented by using mass and energy conservation Eqs. A complete description of the governing equations is given in Refs. [6, 8] with associated plant parameters. Valve and pump models are not included in this representation. For a complete representation of the BOP dynamics it is necessary to include dynamic models of these components (and actuators).
10.10 Reactor system models Chapter 12 addresses PWR dynamics. An appendix to Chapter 12 illustrates the formulation of a complete PWR reactor system model using models described in Chapter 3 for neutronics and this chapter for heat transfer. Another appendix (Appendix K) addresses molten salt reactor dynamics. Appendix K provides an illustration of the use of modeling techniques for a reactor with very different properties than current commercial reactors.
Wv Hs
From OTSG
Win Hs
Ws Hs
Nozzle chest
Wnc Hnc Pnc
Whpt Hhpt Phpt
Reheater Moisture separator
Wms Wrh Hrh Prh
Steam valve
Wlpt Hlpt Plpt
HP turbine IP & LP turbine
Wbhp
Generator TCWin
TCWout
WlIq WbIp
Condenser
To OTSG
Wfw HP feedwater heater Whp→lp
FIG. 10.12 Schematic of the BOP system showing important components and associated state variables [8].
Wco Hco LP feedwater heater
10.10 Reactor system models
cooling water
Wvrh
125
126
CHAPTER 10 Reactor thermal-hydraulics
Exercises 10.1. Reformulate Mann’s model for an assumption that the average coolant temperature is the average of the temperature in each node rather than the temperature in the first node. How would this affect simulation results? 10.2. Compare the computational differences for modeling a boiling channel with the moving boundary approach and a model with fixed boundaries and updating of coefficients during a transient. 10.3. Consider a moving boundary model for a once through steam generator with superheat. How many boundaries are needed for a dynamic model? How would the boundaries move (up or down) following an increase in primary side fluid temperature? 10.4. Consider a PWR with dissolved boron in the coolant. Describe differences in the reactor power response to an increase in inlet coolant temperature for a condition of low boron concentration and a condition of high boron concentration. 10.5. A reactor with a negative fuel temperature coefficient of reactivity and a positive coolant temperature coefficient of reactivity can be stable even if the magnitude of the positive coefficient is larger than the magnitude of the negative coefficient. Explain how this is possible and why it might be counterintuitive. 10.6. A change in boiling rate caused by a disturbance in a Boiling Water Reactor causes changes in coolant density as it moves along the channel. BWRs have negative coolant density coefficients. This effect can be destabilizing. a. Explain how this happens. b. Would increasing the coolant flow rate make the system more stable or less stable? Explain.
References [1] T.W. Kerlin, Dynamic analysis and control of pressurized water reactors, in: C.T. Leondes (Ed.), Control and Dynamic Systems, vol. 14, Academic Press, 1978. [2] S.J. Ball, Approximate models for distributed parameter heat transfer systems, ISA Trans. 3 (1) (1964) 38–47. [3] M.R.A. Ali, Lumped-Parameter, State Variable Dynamic Model for U-Tube Recirculation Type Steam Generators, PhD dissertation, The University of Tennessee, 1976. [4] A. Ray, H.F. Bowman, A nonlinear dynamic model for a once-through subcritical steam generator, J. Dyn. Syst. Meas. Control 98 (3) (1976) 332–339. Series G. [5] T.W. Kerlin, E.M. Katz, J. Freels, J.G. Thakkar, Dynamic modeling of nuclear steam generators, in: Proceedings of the Second International Conference sponsored by the British Nuclear Energy Society, Bournemouth, England, 1979. October.
Further reading
[6] M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, University of Tennessee Research Report, 1991. DOE/NE/88ER12824-02, June. [7] Steam, its generation and use, The Babcock & Wilcox Company, forty second ed., 2015. [8] V. Singh, Study of Dynamic Behavior of Molten Salt Reactors, MS Thesis, The University of Tennessee, Knoxville, 2019.
Further reading [9] A.T. Chen, A Digital Simulation for Nuclear Once-Through Steam Generators, PhD dissertation, The University of Tennessee, 1976. [10] The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Corporation, Pittsburgh, 1984.
127
Reactor thermalhydraulics
10
10.1 Introduction Temperature and pressure of reactor fluids and solids are important variables in steady state and transient operation. Along with associated coefficients of reactivity, they determine the magnitude of reactivity feedbacks. Conservation of mass, energy, and momentum are the basis for thermal-hydraulics models. However, since pressure transients reach a new steady state so much faster in a transient than mass or energy, the differential equation for momentum is usually unnecessary. Much of the information in this chapter comes from Ref. [1]. This chapter includes short descriptions of major heat transport systems in nuclear power reactors. Chapters 12, 13, and 14 provide more detailed information about power reactor characteristics and control systems.
10.2 Heat conduction in fuel elements Most of the reactors addressed in this book use cylindrical UO2 fuel rods clad in zircaloy and have a gas-filled gap between the UO2 and the cladding. Fuel heat transfer in reactors with non-cylindrical fuel elements (such as high temperature gas cooled reactors and molten salt reactors).is not addressed here. A complete heat transfer model would require a partial differential equation in two spatial dimensions (axial and radial … azimuthal is not required) and time. Solutions of such a model equation are known, but they are not suitable for coupling with models for cooling fluid. Instead, lumped parameter models (also, sometimes referred to as nodal models) must be used. Lumped parameter models involve breaking the system into regions with uniform internal properties and coupling to adjacent regions. It is possible to model fuel rods with concentric radial lumps, but the simplest and most common lumped parameter model for cylindrical fuel elements uses a single radial node. Consider a single radial node model for a fuel element with heat transfer to fluid coolant. The fuel has mass, Mf, and specific heat capacity, Cf. The model equation is as follows: Mf Cf
dTf ¼ UA Tf θavg + Pf dt
Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00010-X # 2019 Elsevier Inc. All rights reserved.
(10.1)
111
112
CHAPTER 10 Reactor thermal-hydraulics
where Mf ¼ mass of fuel Cf ¼ specific heat capacity of fuel Tf ¼ fuel temperature Pf ¼ power released in the fuel node U ¼ overall fuel-to-coolant heat transfer coefficient A ¼ fuel cylinder surface area (fuel-to-coolant heat transfer area) θavg ¼ average coolant temperature in the adjacent coolant node Eq. (10.1) may be rewritten as dTf Pf UA ¼ Tf θavg + dt Mf Cf Mf Cf
(10.2)
The quantity, (Mf Cf /UA) has the units of time. It is the time constant for fuel-tocoolant heat transfer. Typical values for LWRs and CANDU reactors are 4 to 5 s.
10.3 Heat transfer to liquid coolant The core heat transfer model also requires heat balance equations for the coolant. A general model requires mass and energy balances. If the coolant density and node volume are constant, a mass balance is not needed (see Section 10.4 for a discussion of heat transfer in a model with a moving boundary). As with the fuel model, a nodal model for the coolant is needed. Consider the system shown in. Fig. 10.1 The figure shows that there are five variables as defined below: Pc ¼ power generated within the node (as by interaction of radiation with coolant atoms) Tf ¼ temperature of adjacent fuel node θin ¼ inlet coolant temperature θout ¼ outlet coolant temperature θavg ¼ average coolant temperature in the node θout
Fuel Node
Tf
θavg
θin
FIG. 10.1 Heat transfer to a liquid coolant lump (node).
Pc
Coolant Node
10.3 Heat transfer to liquid coolant
The nodal internal power generation, Pc, the fuel temperature, Tf, and the inlet coolant temperature, θin, are defined by other subsystem equations. That leaves two variables, but the coolant equation provides only one. An assumption is required to eliminate θout. θavg must be retained because it appears in the equation for heat transfer from fuel to coolant. The average temperature is given by the following: θavg ¼ ðθin + θout Þ=2
(10.3)
θout ¼ 2 θavg θin
(10.4)
or There is a problem with this formulation. Note in Eq. (10.4) that a sudden increase in inlet temperature would cause a sudden decrease in outlet temperature. This is an unphysical feature, causing consideration of an alternate formulation. Another possibility is the “well-stirred-tank” formulation. That is, the outlet temperature from the node is set equal to the average node temperature. This solves the problem in the previous formulation, but equating average and outlet temperatures does not represent actual behavior very well. Ray Mann of Oak Ridge National Laboratory addressed the problem [2]. Mann’s formulation uses two coolant nodes adjacent to a single fuel node; Fig. 10.2 shows this arrangement. Well-stirred-tank models represent each pair of coolant nodes. The outlet temperature of the first coolant node (assumed equal to the average temperature of that node) serves as the coolant temperature that provides the driving force for heat transfer from the fuel. Each coolant node receives half of the heat transfer from the fuel node. Therefore, the model equations for Mann’s formulation are as follows: Mc C c
dθ1 UA ¼ W Cc ðθin θ1 Þ + Tf θ1 + Pc1 dt 2
(10.5)
Mc Cc
dθ2 UA ¼ W C c ðθ 1 θ 2 Þ + Tf θ1 + Pc2 dt 2
(10.6)
∼ Θ2
∼ Θ2 ∼ Θ1
Tf ∼ Θ1
Qin
FIG. 10.2 Schematic of Mann’s model with one fuel node and two coolant nodes.
113
114
CHAPTER 10 Reactor thermal-hydraulics
where Mc ¼ mass of coolant in a node Cc ¼ specific heat capacity of coolant θ1 ¼ temperature in the first coolant node θ2 ¼ temperature in the second coolant node W ¼ coolant mass flow rate U ¼ overall heat transfer coefficient from fuel to coolant A ¼ total fuel surface area for fuel to coolant heat transfer Pc1 ¼ heat generation rate in the first coolant node Pc2 ¼ heat generation rate in the second coolant node. Note that the fuel-to-coolant heat transfer is given by the difference between the fuel temperature and the temperature of the first coolant node (assumed equal to the outlet temperature of that coolant node). The fuel-to-coolant heat transfer is divided equally to both coolant nodes in this formulation. The simplest formulation involves representing all of the fuel in a single node and all of the coolant as a pair of Mann’s nodes. A more detailed formulation uses a series of axial fuel nodes, each coupled with a pair of Mann’s nodes. See Fig. 10.3 for the case of two fuel nodes. ∼ ∼ Θ4 = Θout
∼ Θ4 Tf2
∼ Θ3
∼ Θ3 ∼ Θ2
∼ Θ2 Tf1
∼ Θ1
∼ Θ1 ∼ Θin
FIG. 10.3 Series of Mann’s models for more than one fuel node (two fuel nodes) representation.
10.5 Plenum and piping models
10.4 Boiling coolant Modeling channels with boiling fluid is more complex than modeling channels with single-phase fluid. Boiling heat transfer models are needed for BWRs and steam generators. Consider a heated channel into which liquid fluid is introduced into a channel with boiling fluid. The fluid temperature in the sub-cooled region at the entrance increases as it travels through the channel. At some point, the hotter fluid adjacent to the heated surface reaches saturation temperature even though the fluid farther from the heated surface remains sub-cooled. Boiling begins near the surface. The heat transfer coefficient for this region is greater than in the sub-cooled region. As the fluid continues through the channel, the bulk fluid temperature reaches saturation and vigorous boiling throughout the fluid occurs. High heat transfer coefficients occur in this region. A mixture of liquid and vapor exists in this region. This liquid-vapor mixture exits the core in BWRs and in U-tube steam generators. If all of the liquid boils before exiting the heated region (as in once-through steam generators), continuing heat transfer results in superheating of the steam. In a transient, the boundaries between the different regions move. Models have been developed in which these boundaries are state variables. Moving boundary models for BWRs and steam generators are quite complex, too complex for inclusion in this introductory book. But the concept is important. Appendix F illustrates the approach with a moving boundary model for the subcooled node in a heated channel.
10.5 Plenum and piping models Plenums are regions in which fluid enters and mixes with the existing fluid inventory. Plenums exist at core inlets and outlets and at steam generator inlets and outlets. Well-stirred-tank models are usually used for plenums. The model is as follows: Mp Cp
dθp ¼ W Cp θpin θp dt
(10.7)
where Mp ¼ mass of fluid in the plenum Cp ¼ specific heat capacity of the fluid W ¼ fluid flow rate θp ¼ fluid temperature in the plenum (equal to plenum outlet temperature because of well-stirred-tank approximation) θpin ¼ temperature of fluid entering the plenum Piping carries fluid from one reactor subsystem to another. Piping models are used for hot leg and cold leg piping in PWRs and CANDU reactors, and feedwater for BWRs and steam generators.
115
116
CHAPTER 10 Reactor thermal-hydraulics
Pure time delay models are often used. That is the formulation is simply as follows: θout ðtÞ ¼ θin ðt τÞ
(10.8)
where τ ¼ residence time in the piping. Well-stirred-tank models are also used. The rationale for a well-stirred-tank model is that some axial mixing does occur in piping. The more general dynamic formulation of the temperature of the liquid in the piping has the form dθout 1 ¼ ðθin θout Þ dt τ
(10.9)
The resident time is approximated as the ratio between the mass of fluid (Mp) in the piping and the fluid flow rate (Wp). τ ¼ Mp =Wp
(10.10)
10.6 Pressurizer A PWR pressurizer is a vessel with liquid water in the bottom section and saturated steam in the top section. A pressurizer is used to regulate the primary coolant pressure ( 150 bars) in PWRs and CANDU reactors. The pressurizer is connected to one of the hot leg pipings with a long surge line. Fig. 10.4 shows a typical pressurizer. Because of the contact between steam and liquid water, the water is also at the saturation temperature at steady state. Spray of cooler water enters from the top and electrical heaters at the bottom heat the liquid water. The steady state can be disturbed by water inflow or outflow, changes in inlet water temperature, changes in spray flow or changes in heater power. A PWR pressurizer control system can alter the pressure by modulating heater power and/or spray flow. A schematic representation of a pressurizer model structure appears in Fig. 10.5. As shown in Chapter 14, PWR and CANDU reactor pressurizers are slightly different.
10.7 Heat exchanger model Mann’s formulation also may be used in liquid-liquid heat exchangers. Fig. 10.6 shows the model. The equations are as follows: Mp1 Cp
UA dθp1 ¼ Wp Cp θpin θp1 + Tt θp1 dt 2
(10.11)
Mp2 Cp
UA dθp2 ¼ Wp Cp θp1 θp2 + Tt θp1 dt 2
(10.12)
10.7 Heat exchanger model
FIG. 10.4 Schematic of a pressurizer used in a PWR. Courtesy of Westinghouse Electric Company (The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Corporation, Pittsburgh, 1984).
117
118
CHAPTER 10 Reactor thermal-hydraulics
FIG. 10.5 Schematic of a pressurizer model.
Wp Θp,in
Mp1 Θp1
Θp1
Ms2 Θs2
Mt Tt
Mp2 Θp2
Θs1
Ms1 Θs1
Ws Θs,in
FIG. 10.6 Schematic of a liquid-liquid heat exchanger model.
10.8 Steam generator model
dTt ¼ U A Tt θp1 + U A ðTt θs1 Þ dt
(10.13)
Ms1 Cs
dθs1 UA ðTt θs1 Þ ¼ Ws Cs ðθsin θs1 Þ + dt 2
(10.14)
Ms2 Cs
dθs2 UA ðTt θs1 Þ ¼ Ws Cs ðθs1 θs2 Þ + 2 dt
(10.15)
Mt Ct
where Mp1 ¼ mass of primary fluid node-1 Mp2 ¼ mass of primary fluid node-2 Mt ¼ mass of metal node (heat exchanger tubing) Ms1 ¼ mass of secondary fluid node-1 Ms2 ¼ mass of primary fluid node-2 Wp ¼ flow rate of primary fluid Ws ¼ flow rate of secondary fluid Cp ¼ specific heat capacity of the primary fluid Cs ¼ specific heat capacity of the secondary fluid Ct ¼ specific heat capacity of the tube metal U ¼ overall heat transfer coefficient from primary fluid to metal, or from metal to secondary fluid A ¼ heat transfer area from primary fluid to metal node, or from metal node to secondary fluid θpin ¼ temperature of primary fluid inflow θsin ¼ temperature of secondary fluid inflow θp1 ¼ temperature of primary fluid node-1 θp2 ¼ temperature of primary fluid node-2 θs1 ¼ temperature of secondary fluid node-1 θs2 ¼ temperature of secondary fluid node-2 Tt ¼ temperature of metal (tube) node
10.8 Steam generator model Steam generator modeling can be simple or complex. Fig. 10.7 shows a typical U-tube steam generator. Different requirements for secondary system modeling determine the detail required for steam generators. For example, if the simulation focuses on reactor behavior, then a simple steam generator model that adequately simulates heat removal is adequate. A simple model, called the “teakettle model” represents steam generator dynamics with only three Eqs. A schematic representation of a teakettle model appears in Fig. 10.8. This approach has been found to represent heat transfer to the secondary fluid quite well in overall system simulations [1].
119
120
CHAPTER 10 Reactor thermal-hydraulics
STEAM OUTLET TO TURBINE GENERATOR DEMISTERS SECONDARY MOISTURE SEPARATOR
SECONDARY MANWAY
UPPER SHELL
FEEDWATER RING
ORFICE RINGS SWIRL VANE PRIMARY MOISTURE SEPARATOR
FEEDWATER INLET
ANTIVIBRATION BARS
TUBE BUNDLE
LOWER SHELL
WRAPPER
TUBE SUPPORT PLATES
SECONDARY HANDHOLE
BLOWDOWN LINE TUBE SHEET
TUBE LANE BLOCK PRIMARY MANWAY
PRIMARY COOLANT OUTLET
PRIMARY COOLANT INLET
FIG. 10.7 Schematic of a typical U-tube steam generator (UTSG). Courtesy of Westinghouse Electric Company (The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Corporation, Pittsburgh, 1984).
10.8 Steam generator model
FIG. 10.8 Schematic representation of a teakettle model of a steam generator.
10.8.1 U-Tube steam generator (UTSG) More detailed models are needed if steam generator performance itself is under consideration. For example, feedwater control system simulation is often an objective. Finite difference models with parameter updates in each node as a transient proceeds are used, but are quite complex. The moving boundary approach described in Section 10.4 also be used for steam generator modeling and is somewhat simpler to implement [3–6]. The nodal structure for a moving boundary model for a U-tube steam generator appears in Fig. 10.9. The model uses fourteen coupled differential equations and additional algebraic equations. The complete model is described in Ref. [6].
10.8.2 Once-through steam generator (OTSG) Once-through steam generators for PWRs are shell and tube heat exchangers with liquid water inside the tubes and water and steam on the shell side. Fig. 10.10 is a schematic of a once-through steam generator (OTSG) [7]. A schematic for a moving boundary model appears in Fig. 10.11 [8]. Note that two boundaries move (the boundary between the subcooled water section and the boiling section, and the boundary between the boiling section and the superheating section). A detailed model of an OTSG was developed in Ref. [8].
121
122
CHAPTER 10 Reactor thermal-hydraulics
FIG. 10.9 Lumped parameter (nodal) structure of a moving boundary U-tube steam generator model [6].
10.9 Balance-of-Plant (BOP) system models The components of a balance-of-plant (BOP) system are similar in nuclear power plants that generate steam using reactor thermal power. These include PWRs, BWRs, CANDU reactors, sodium fast reactors, gas-cooled reactors, molten salt reactors, and small modular reactors. Fig. 10.12 shows the major components of a BOP system and associated state variables [8]. The figure shows the following BOP components: • • • • • • •
Steam chest High pressure and low pressure turbines Moisture separators and steam reheaters Condenser Feedwater heaters Condenser pressure controller Feedwater temperature controller.
10.9 Balance-of-Plant (BOP) system models
FIG. 10.10 Schematic of a once-through steam generator (OTSG). Courtesy of The Babcock & Wilcox Company (Steam, its generation and use, The Babcock & Wilcox Company, forty second ed., 2015).
123
124
CHAPTER 10 Reactor thermal-hydraulics
FIG. 10.11 A moving boundary model of OTSG [8].
The dynamics of these components are represented by using mass and energy conservation Eqs. A complete description of the governing equations is given in Refs. [6, 8] with associated plant parameters. Valve and pump models are not included in this representation. For a complete representation of the BOP dynamics it is necessary to include dynamic models of these components (and actuators).
10.10 Reactor system models Chapter 12 addresses PWR dynamics. An appendix to Chapter 12 illustrates the formulation of a complete PWR reactor system model using models described in Chapter 3 for neutronics and this chapter for heat transfer. Another appendix (Appendix K) addresses molten salt reactor dynamics. Appendix K provides an illustration of the use of modeling techniques for a reactor with very different properties than current commercial reactors.
Wv Hs
From OTSG
Win Hs
Ws Hs
Nozzle chest
Wnc Hnc Pnc
Whpt Hhpt Phpt
Reheater Moisture separator
Wms Wrh Hrh Prh
Steam valve
Wlpt Hlpt Plpt
HP turbine IP & LP turbine
Wbhp
Generator TCWin
TCWout
WlIq WbIp
Condenser
To OTSG
Wfw HP feedwater heater Whp→lp
FIG. 10.12 Schematic of the BOP system showing important components and associated state variables [8].
Wco Hco LP feedwater heater
10.10 Reactor system models
cooling water
Wvrh
125
126
CHAPTER 10 Reactor thermal-hydraulics
Exercises 10.1. Reformulate Mann’s model for an assumption that the average coolant temperature is the average of the temperature in each node rather than the temperature in the first node. How would this affect simulation results? 10.2. Compare the computational differences for modeling a boiling channel with the moving boundary approach and a model with fixed boundaries and updating of coefficients during a transient. 10.3. Consider a moving boundary model for a once through steam generator with superheat. How many boundaries are needed for a dynamic model? How would the boundaries move (up or down) following an increase in primary side fluid temperature? 10.4. Consider a PWR with dissolved boron in the coolant. Describe differences in the reactor power response to an increase in inlet coolant temperature for a condition of low boron concentration and a condition of high boron concentration. 10.5. A reactor with a negative fuel temperature coefficient of reactivity and a positive coolant temperature coefficient of reactivity can be stable even if the magnitude of the positive coefficient is larger than the magnitude of the negative coefficient. Explain how this is possible and why it might be counterintuitive. 10.6. A change in boiling rate caused by a disturbance in a Boiling Water Reactor causes changes in coolant density as it moves along the channel. BWRs have negative coolant density coefficients. This effect can be destabilizing. a. Explain how this happens. b. Would increasing the coolant flow rate make the system more stable or less stable? Explain.
References [1] T.W. Kerlin, Dynamic analysis and control of pressurized water reactors, in: C.T. Leondes (Ed.), Control and Dynamic Systems, vol. 14, Academic Press, 1978. [2] S.J. Ball, Approximate models for distributed parameter heat transfer systems, ISA Trans. 3 (1) (1964) 38–47. [3] M.R.A. Ali, Lumped-Parameter, State Variable Dynamic Model for U-Tube Recirculation Type Steam Generators, PhD dissertation, The University of Tennessee, 1976. [4] A. Ray, H.F. Bowman, A nonlinear dynamic model for a once-through subcritical steam generator, J. Dyn. Syst. Meas. Control 98 (3) (1976) 332–339. Series G. [5] T.W. Kerlin, E.M. Katz, J. Freels, J.G. Thakkar, Dynamic modeling of nuclear steam generators, in: Proceedings of the Second International Conference sponsored by the British Nuclear Energy Society, Bournemouth, England, 1979. October.
Further reading
[6] M. Naghedolfeizi, B.R. Upadhyaya, Dynamic Modeling of a Pressurized Water Reactor Plant for Diagnostics and Control, University of Tennessee Research Report, 1991. DOE/NE/88ER12824-02, June. [7] Steam, its generation and use, The Babcock & Wilcox Company, forty second ed., 2015. [8] V. Singh, Study of Dynamic Behavior of Molten Salt Reactors, MS Thesis, The University of Tennessee, Knoxville, 2019.
Further reading [9] A.T. Chen, A Digital Simulation for Nuclear Once-Through Steam Generators, PhD dissertation, The University of Tennessee, 1976. [10] The Westinghouse Pressurized Water Reactor Nuclear Power Plant, Westinghouse Electric Corporation, Pittsburgh, 1984.
127