- Email: [email protected]
Modelling of the PBMR Using a Pipe Network Analysis Approach
Copyright ~ IFAC Technology Transfer in Developing Countries, Pretoria, South Africa, 2000
MODELLING OF THE PBMR USING A PIPE NETWORK ANALYSIS APPROACH
Prof. G.P.Greyvenstein and Mr. F.N. Emslie
School ofMechanical and Materials Engineering, Potchefstroom University for Christian Higher Education
Abstract: This paper deals with some of the processes involved in modelling the thermal-hydraulic behaviour of the Pebble Bed Modular Reactor (PBMR). The PBMR is an inherently safe gas-cooled nuclear power plant that utilized the Recuperative Brayton cycle to convert the heat energy into work. To model the main pow~r generation cycle a pipe network analysis approach is used. The met?0d ~ed IS presented in this paper and critical sub-models of the PBMR are also mvesngated. Copyright © 2000 IFAC Keywords: Nuclear power stations, numerical methods, subsystems, power systems.
Light Water reactor (L WR) yielding efficiencies of over 40%. The PBMR uses helium as a working fluid and not water, which greatly simplifies design. Since a gas is used the regenerative Brayton cycle is used instead of the Rankine cycle as in LWR's.
1. INTRODUCTION
1.1 The PBMRproject The use of fossil fuels as a sustainable source of energy is depleting at an alarming rate. The use of nuclear energy as an alternative has thus received a lot of attention in the recent past. Even though nuclear is very dangerous in the case of accidents it is a convenient and environmentally-friendly fonn of power.
The power released by one fuel sphere can provide the electricity demands of an average sized South African household for one year. These balls measure 6cm in diameter and consist of a inner fuel core and an outer protective graphite layer. The fuel core comprises a graphite matrix that is impregnated with thousands of small fuel cells. Each cell again consists of an inner core of enriched uranium fuel and is coated by three protective layers.
In 1995 the 1ST engineering group in conjunction with South Africa's primary electricity provider ESKOM decided to look into using high temperature gas-cooled nuclear reactor (HTGR) technology for future power plants. Subsequently feasibility studies were conducted and so the Pebble Bed Modular Reactor (PBMR) project was born. The PBMR being a special kind ofHTGR.
The pebble bed reactor's main advantage is that it is inherently safe, i.e. there is no possibility of a core meltdoWD. This safety characteristic is due to the design of the fuel spheres that have a negative temperature gradient. If there is a rupture in the pressure boundary that surrounds the system and all the helium leaks to atmosphere the core will not over heat but instead will start to lose reactivity. Therefore one could say that the more coolant is aloud to flow through the core the more reactive it will get.
But what makes the PBMR so different from the conventional nuclear power plant? A major difference is evident by the name "Pebble Bed" that refers to the fuel spheres (pebbles) used instead of fuel rods. The PBMR is furthermore much more thermally efficient than the conventionally used
67
Furthermore, because the PBMR uses fuel spheres (pebbles), instead of the traditional fuel rods, it is much easier and safer to dispose of the used fuel spheres. Conventional fuel rods need to be buried within specially built concrete bunkers deep under ground. This make disposal costly and there is still the danger of radiation leaking out. The PBMR uses a special fuel handling system to do refueling and fuel maintenance. The system used high-pressure helium to move the fuel spheres from the core to a fuel-handling unit. This eliminates the need to shutdown the entire plant if there is a problem with the fuel or for refueling (about every two years). All the fuel used during the life of the plant (40 years) is stored within the reactor building. After the 40 years the plant is left sealed for a certain time before it is decommissioned to allow the fuel radiation levels to decrease. By the time decommissioning takes place the used fuel is fairly easy and inexpensive to dispose of.
temperature to 900°C before it is sent through the turbines. There are three turbines: a high and a low pressure turbine that supply power to the two compressors respectively and a power turbine that provides power to an electric generator. Cooling then takes place as the fluid goes through the recuperator and a pre-cooler upon which it exits at a temperature of 27°C. The Temperature-Entropy diagram for the cycle is shown in Figure 1. The next section describes the numerical simulation of the plant in more detail.
1.2 The recuperative Brayton Cycle
The implicit and explicit methods are two ways of modelling thermal hydraulic systems and they both have distinct advantages and disadvantages. The Method of Characteristics (MOC) is a well known explicit method. It is a fast, accurate solver that can handle phenomena such as water hammer very well. A disadvantage of the MOC is that for high wave propagation velocities a very small time step must be used. This prolongs computation times if long simulations have to be conducted. Implicit methods on the other hand are more robust and are not restricted by the time step/distance relationship. Although slower per time step than explicit methods one can cut down drastically on execution time if longer time steps are used. In many cases the longer time steps do not compromise accuracy.
1.3 Modelling as a design tool
Although the steady-state analysis of the ideal Brayton cycle is quite straight forward the accurate simulation of a real system taking pressure drops, actual performance characteristics of components such as heat exchangers, compressors, turbines and leak flows into account is a challenging task.
The pebble-bed modular reactor (PBMR) system uses a recuperative Brayton cycle with helium as the working fluids to convert the heat transferred to the helium in the nuclear reactor into electrical power. Figure 1 gives a schematic diagram showing the PBMR reactor unit (RU) and power conversion unit (PCU). Starting at I, helium at a pressure of 25 bar is compressed by a low pressure compressor to 40 bar after which it is cooled to 27°C by an intercooler. A high pressure compressor then takes the pressure up to 70 bar while the temperature also rises to 104°C. The high pressure gas then goes through a recuperator where it takes up heat from the hotter low pressure gas. The pebble-bed reactor then adds heat to the fluid (approximately 265 MW) raising the
HP
Compressor
3
5
Intercooler
Temperature
7
LP Compressor
8 5 9
4
Pre-cooler
10
Recuperator Fig. 1. A schematic layout of the power conversion unit of the PBMR Commercially there are a number of thermalhydraulic simulation codes that utilize both explicit and implicit methods. The PCIM method is used in a new code named Flownet that is being developed as
Entropy
the design tool for the PBMR. The code has already been used in the petrochemical, gas-turbine and mining industries and has proven to be very accurate and fast. The Flownet code uses a pipe network 68
3. THE IMPLICIT PRESSURE CORRECTION METHOD (IPCM)
modelling approach that is discussed in the following section.
3.1 Introductory remarks
2. PIPE NETWORK MODELING
When designing fluid networks a steady state analyses must be carried out. Therefore an accurate modeling algorithm is of utmost importance. These algorithms fall into three categories depending on with they consider node, element or loop equations. Node equations are used in the IPCM method as they use less computer space and are much more flexible than the other two methods. The method is based on the simultaneous Newton Raphson technique.
A pipe network is a arbitrary structured network of thermo-flow components such as pipes, heat exchangers and turbo-machines. Examples of networks include city gas networks, gas-turbines, ventilation systems and power systems. A network consists of a number of orientated elements connected to nodes as shown in Figure 2. Elements are connected to one another at nodes and each element is associated with an upstream and downstream node. Numbers are used to identify each node and element. For the purpose of the computational scheme the network is divided into a number of meshes. A mesh is a node with all its branch elements and neighbouring nodes as shown in Figure 2.
In the IPCM the three governing equations are the conservation of mass (continuity equation), momentum and energy equations. The solution algorithm used in the method starts by estimating the pressures for all the network nodes. The mass flows are then calculated using the pressure-drop/ volumetric flow rate relationship. With the new mass flows continuity is tested at each node with the pressures being corrected to satisfy continuity. The mass flows are then updated using the new pressures and the sequence is repeated until convergence is reached. The energy equation is then solved to obtain a temperature distribution for the network. The process is then repeated until convergence of the temperature field is also achieved. When doing a transient analysis one would now proceed to the next time step.
In Figure 2 node i is connected through elements eij to neighbouring nodes nij • withj = 1,2, ... ,J. J is the number of branches associated with node i. The indices i. nij and eij are global indices while j is a local index. If the assumed positive flow direction for element eij is towards node i. eij is assigned a positive value in an element connectivity matrix. Otherwise it is assigned a negative value. In the solver the element connectivity matrix contains all the indices associated with the particular element and similarly a node connectivity matrix contains the associated indices for the nodes in the network.
Node
Element
Node
~ Branch 1
Fig. 2. Graphical representation and notation used to model a thermal-hydraulic network The specific mathematical derivation of the method falls beyond the scope of this paper but is discussed in greater detail in the references.
To model a thermal-hydraulic network one needs a reliable solver that will give accurate results at an acceptable speed of solution. The solver used to model the PBMR network solves the the mass and energy equations at each node in the network. Element pressure drop/ flow rate models are used to solve the pressure drop across each element. Specific elements in the cycle are simulated using specialized sub-models. Each sub-model is derived from the best available numerical techniques for the specific component.
The characteristics of the IPCM method used in Flownet include the following. It is very fast due to the use of very efficient sparse matrix solvers, conservation of both mass and energy are satisfied and the code is second-order accurate in time due to Crank-Nicholson time integration.. Furthermore it is unconditionally stable as it is not limited by time/distance step relationships (as in explicit methods) and the code can accurately deal with both
69
fast and slow transients. The following section shifts the focus to the modelling of the PBMR.
Applying Newton's Second Law to the boundary velocities in the LP and HP gas streams we get:
- ~ _ PL,., - PI., PI., --;]t -!iX
4. SUBMODELS
f - V2
2D PI., I., L
I
A shown previously in the paper the PBMR consists of various specialized components. These are modeled in the simulation code as sub-models. This section discusses the theory behind the creation of these sub-models. There is a large variety of submodels components such as pipes, diffusers, valves, fans, etc. A exhaustive discussion of all of the submodels is beyond the scope of this paper but the submodels for heat exchangers, turbomachines and nuclear reactors are briefly discussed in this section.
(3)
(4)
where
p
=
f
density at control volume boundary,
friction factor and boundary values. =
the overbar denotes
Applying the First Law of Thenoodynarnics to the gas and wall control volumes we get:
4. J Recuperator The recuperator is used to recover heat from the outlet of the power turbine and transfer it to the high pressure gas entering the reactor. This greatly increases the efficiency of the whole plant. Figure 3 represents a control volume around a heat exchange surface.
(6)
Cw, d~;
The following equations represent a second order scheme that takes both heat transfer and flow transients on the gas side into account. Consider a control volume as shown in Figure 3. Applying mass balance to the gas control volumes we may write: A Ax dp L, = rh - rh L ' dt L, L, . ,
(1)
A Ax dp H , = rh - rh H ' dt H, H,. ,
(2)
where = constant volume specific heat of gas, c p = constant pressure specific heat of gas, CWi = product of lump mass and specific heat of wall per segment and HH and HL = product of heat transfer area and heat transfer coefficient per segment . Equations (1) to (7) can all be written in the following fono :
dx} _ ( Tt - f X" X where
I
~x1 +1
I
PL,
PI<
I;,
v"
J:,
Li
PL' . I
';' ~.'
---.
~
mH t
+
1
~,
VH,
PH,
ii•.
.......
j = 1, 2....n
(8)
is the variable under consideration.
(9) The approach followed with Flownet is to use a second order Crank-Nicholson approximation which gIves:
HI
PH I • 1
+
HP
Side X
t+1 }
.,.
)
Side Wall
m
"" X'
LP
Tr ,
~,
Xj
2
With a first order implicit integration one would write
I
:~:~: I
m
(7)
- Tw,)- H H , (TA', - TH , )
Cv
where A is the cross sectional area.
I
= H L, (TL,
-x
~t
.... .,.
I j
= 0.5
V' + ft+' )
(10)
Applying equation (10) to equations (1) to (7) gives a set of algebraic equations which is solved by a matrix technique.
Fig. 3: Control volume of heat exchange surface
70
4.2 Compact gaslliquid heat exchangers The PBMR uses compact gas/liquid heat exchangers for the pre-cooler and intercooler. These heat exchangers are serpentine shaped, cross-flow heat exchangers and are modelled as shown in Figure 4. In the modelling process the heat exchanger is discretized to look similar to the example in Figure 4. Then, as shown in the figure, every segment can be considered as a small heat exchanger on its own and the flow and heat transfer can be solved for each section in a similar approach as used for the recuperator.
Compressor
mCL1 P 01
999
Fig. 5. Compressor performance map
o-C~:J-&O-o?-O ti :
.
:--\
4.4 Reactor
:
~
r~
O-·~0t·
-{)i:rJ
y
The pebble bed reactor is modelled by discretizing it into layers. Each layer consists of a certain amount of fuel spheres which are lumped into one ' sub'element or increment of the total PBR element. In each layer every fuel sphere is again broken up as a function of the radius from its center. Figure 6 shows the basic modelling layout for the reactor. The average reactivity, temperature and heat transfers is calculated for a layer and these values are used to generate a power profile in the core as a function of core height. The neutronic model used to calculate all the variables is of complicated structure and falls beyond the scope of this paper.
Gas
n". ~
:'-l
r:
0
t':
{\
o-··()i:·[j--():r o-··()r-O j j (]
Heat
exchanger
mtW+ Liquid flow
Fig. 4. Discretized heat exchanger model
4.3 Turbo-machines Both the turbines and compressors of the PBMR are modelled in Flownet using performance maps and certain operating point parameters. Performance maps, such as that for a compressor as shown in Figure 5, consist of pressure ratio and efficiency verses non-dimensional mass flow for different speeds. These maps are further enhanced by adding a third dimension namely the guide vane angle. Flownet uses a third order interpolation scheme to find the operating point on the map during a simulations.
Internal network
Fuel layer
The capabilities of the model include the following. Any number of compressor and/or turbine disks (stages) can be on a single shaft and there can be multiple shafts. Generator and mechanical losses are taken into account. There is speed control on the shaft and speed transients can be modelled. The model uses characteristic curves from which it interpolates for the right shaft speed for a given mass flow and pressure ratio. Where other techniques have difficulty the solver can solve on positive slopes of a constant N"To curve.
Fuel pebble
Fig. 6. Model of Pebble Bed reactor
71
5. RESULTS
The following steady state results were obtained after solving a pipe model of the PBMR with Flownet.
The code has undergone extensive validations as part of the requirements set by nuclear safety authorities. An example of the accuracy of the code can be seen in Figure 7 that illustrates the pressure waves in a pipe. The 5km long pipe has a valve at the end that is 'slammed' shut allowing pressure waves to develop. The fluid in the pipe is helium moving at 120 kg/so The compressibility of the fluid is well illustrated by the "line packing" at the peaks of the pressure waves. The Flownet results compare very favourably with benchmark analytical results.
• • •
• • •
Reactor power = 265 MW Grid power delivered = 110 MW Efficiency = 45,8 % Mass flow = 138 kg/s Max. Temperature = 900°C Max. Pressure = 70 bar 6. CONCLUSION
New sources of electricity are of utmost importance to the future of technological growth. Nuclear energy is one of the most unexploited fields in this industry. The introduction of the PBMR project has enormous potential for South Africa's economic growth as well as for the international power generation industry. The importance and capabilities of good thermal-hydraulic models to the development of the PBMR has been shown in this .,#.-..,; n •• n..... These models not only function as simulators to observe the working of the plant but also greatly to the design, optimization and r;;~~d control of the plant.
Flownet's application in the PBMR project is that of a design, simulation and optimization tool. It is used to solve both steady state and transient problems, both of which are discussed in the following sections. 10000 9OIX)
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7. REFERENCES
Greyvenstein, G. P. and Laurie, D. P. (1994) A segregated CFD approach to pipe network analysis. International. Journal for Numerical Methods Engineering., 37, 3685 - 3705.
1 - • ..-1 Fig. 7. Pressure waves in a pipe
72
MODELLING OF THE PBMR USING A PIPE NETWORK ANALYSIS APPROACH
Prof. G.P.Greyvenstein and Mr. F.N. Emslie
School ofMechanical and Materials Engineering, Potchefstroom University for Christian Higher Education
Abstract: This paper deals with some of the processes involved in modelling the thermal-hydraulic behaviour of the Pebble Bed Modular Reactor (PBMR). The PBMR is an inherently safe gas-cooled nuclear power plant that utilized the Recuperative Brayton cycle to convert the heat energy into work. To model the main pow~r generation cycle a pipe network analysis approach is used. The met?0d ~ed IS presented in this paper and critical sub-models of the PBMR are also mvesngated. Copyright © 2000 IFAC Keywords: Nuclear power stations, numerical methods, subsystems, power systems.
Light Water reactor (L WR) yielding efficiencies of over 40%. The PBMR uses helium as a working fluid and not water, which greatly simplifies design. Since a gas is used the regenerative Brayton cycle is used instead of the Rankine cycle as in LWR's.
1. INTRODUCTION
1.1 The PBMRproject The use of fossil fuels as a sustainable source of energy is depleting at an alarming rate. The use of nuclear energy as an alternative has thus received a lot of attention in the recent past. Even though nuclear is very dangerous in the case of accidents it is a convenient and environmentally-friendly fonn of power.
The power released by one fuel sphere can provide the electricity demands of an average sized South African household for one year. These balls measure 6cm in diameter and consist of a inner fuel core and an outer protective graphite layer. The fuel core comprises a graphite matrix that is impregnated with thousands of small fuel cells. Each cell again consists of an inner core of enriched uranium fuel and is coated by three protective layers.
In 1995 the 1ST engineering group in conjunction with South Africa's primary electricity provider ESKOM decided to look into using high temperature gas-cooled nuclear reactor (HTGR) technology for future power plants. Subsequently feasibility studies were conducted and so the Pebble Bed Modular Reactor (PBMR) project was born. The PBMR being a special kind ofHTGR.
The pebble bed reactor's main advantage is that it is inherently safe, i.e. there is no possibility of a core meltdoWD. This safety characteristic is due to the design of the fuel spheres that have a negative temperature gradient. If there is a rupture in the pressure boundary that surrounds the system and all the helium leaks to atmosphere the core will not over heat but instead will start to lose reactivity. Therefore one could say that the more coolant is aloud to flow through the core the more reactive it will get.
But what makes the PBMR so different from the conventional nuclear power plant? A major difference is evident by the name "Pebble Bed" that refers to the fuel spheres (pebbles) used instead of fuel rods. The PBMR is furthermore much more thermally efficient than the conventionally used
67
Furthermore, because the PBMR uses fuel spheres (pebbles), instead of the traditional fuel rods, it is much easier and safer to dispose of the used fuel spheres. Conventional fuel rods need to be buried within specially built concrete bunkers deep under ground. This make disposal costly and there is still the danger of radiation leaking out. The PBMR uses a special fuel handling system to do refueling and fuel maintenance. The system used high-pressure helium to move the fuel spheres from the core to a fuel-handling unit. This eliminates the need to shutdown the entire plant if there is a problem with the fuel or for refueling (about every two years). All the fuel used during the life of the plant (40 years) is stored within the reactor building. After the 40 years the plant is left sealed for a certain time before it is decommissioned to allow the fuel radiation levels to decrease. By the time decommissioning takes place the used fuel is fairly easy and inexpensive to dispose of.
temperature to 900°C before it is sent through the turbines. There are three turbines: a high and a low pressure turbine that supply power to the two compressors respectively and a power turbine that provides power to an electric generator. Cooling then takes place as the fluid goes through the recuperator and a pre-cooler upon which it exits at a temperature of 27°C. The Temperature-Entropy diagram for the cycle is shown in Figure 1. The next section describes the numerical simulation of the plant in more detail.
1.2 The recuperative Brayton Cycle
The implicit and explicit methods are two ways of modelling thermal hydraulic systems and they both have distinct advantages and disadvantages. The Method of Characteristics (MOC) is a well known explicit method. It is a fast, accurate solver that can handle phenomena such as water hammer very well. A disadvantage of the MOC is that for high wave propagation velocities a very small time step must be used. This prolongs computation times if long simulations have to be conducted. Implicit methods on the other hand are more robust and are not restricted by the time step/distance relationship. Although slower per time step than explicit methods one can cut down drastically on execution time if longer time steps are used. In many cases the longer time steps do not compromise accuracy.
1.3 Modelling as a design tool
Although the steady-state analysis of the ideal Brayton cycle is quite straight forward the accurate simulation of a real system taking pressure drops, actual performance characteristics of components such as heat exchangers, compressors, turbines and leak flows into account is a challenging task.
The pebble-bed modular reactor (PBMR) system uses a recuperative Brayton cycle with helium as the working fluids to convert the heat transferred to the helium in the nuclear reactor into electrical power. Figure 1 gives a schematic diagram showing the PBMR reactor unit (RU) and power conversion unit (PCU). Starting at I, helium at a pressure of 25 bar is compressed by a low pressure compressor to 40 bar after which it is cooled to 27°C by an intercooler. A high pressure compressor then takes the pressure up to 70 bar while the temperature also rises to 104°C. The high pressure gas then goes through a recuperator where it takes up heat from the hotter low pressure gas. The pebble-bed reactor then adds heat to the fluid (approximately 265 MW) raising the
HP
Compressor
3
5
Intercooler
Temperature
7
LP Compressor
8 5 9
4
Pre-cooler
10
Recuperator Fig. 1. A schematic layout of the power conversion unit of the PBMR Commercially there are a number of thermalhydraulic simulation codes that utilize both explicit and implicit methods. The PCIM method is used in a new code named Flownet that is being developed as
Entropy
the design tool for the PBMR. The code has already been used in the petrochemical, gas-turbine and mining industries and has proven to be very accurate and fast. The Flownet code uses a pipe network 68
3. THE IMPLICIT PRESSURE CORRECTION METHOD (IPCM)
modelling approach that is discussed in the following section.
3.1 Introductory remarks
2. PIPE NETWORK MODELING
When designing fluid networks a steady state analyses must be carried out. Therefore an accurate modeling algorithm is of utmost importance. These algorithms fall into three categories depending on with they consider node, element or loop equations. Node equations are used in the IPCM method as they use less computer space and are much more flexible than the other two methods. The method is based on the simultaneous Newton Raphson technique.
A pipe network is a arbitrary structured network of thermo-flow components such as pipes, heat exchangers and turbo-machines. Examples of networks include city gas networks, gas-turbines, ventilation systems and power systems. A network consists of a number of orientated elements connected to nodes as shown in Figure 2. Elements are connected to one another at nodes and each element is associated with an upstream and downstream node. Numbers are used to identify each node and element. For the purpose of the computational scheme the network is divided into a number of meshes. A mesh is a node with all its branch elements and neighbouring nodes as shown in Figure 2.
In the IPCM the three governing equations are the conservation of mass (continuity equation), momentum and energy equations. The solution algorithm used in the method starts by estimating the pressures for all the network nodes. The mass flows are then calculated using the pressure-drop/ volumetric flow rate relationship. With the new mass flows continuity is tested at each node with the pressures being corrected to satisfy continuity. The mass flows are then updated using the new pressures and the sequence is repeated until convergence is reached. The energy equation is then solved to obtain a temperature distribution for the network. The process is then repeated until convergence of the temperature field is also achieved. When doing a transient analysis one would now proceed to the next time step.
In Figure 2 node i is connected through elements eij to neighbouring nodes nij • withj = 1,2, ... ,J. J is the number of branches associated with node i. The indices i. nij and eij are global indices while j is a local index. If the assumed positive flow direction for element eij is towards node i. eij is assigned a positive value in an element connectivity matrix. Otherwise it is assigned a negative value. In the solver the element connectivity matrix contains all the indices associated with the particular element and similarly a node connectivity matrix contains the associated indices for the nodes in the network.
Node
Element
Node
~ Branch 1
Fig. 2. Graphical representation and notation used to model a thermal-hydraulic network The specific mathematical derivation of the method falls beyond the scope of this paper but is discussed in greater detail in the references.
To model a thermal-hydraulic network one needs a reliable solver that will give accurate results at an acceptable speed of solution. The solver used to model the PBMR network solves the the mass and energy equations at each node in the network. Element pressure drop/ flow rate models are used to solve the pressure drop across each element. Specific elements in the cycle are simulated using specialized sub-models. Each sub-model is derived from the best available numerical techniques for the specific component.
The characteristics of the IPCM method used in Flownet include the following. It is very fast due to the use of very efficient sparse matrix solvers, conservation of both mass and energy are satisfied and the code is second-order accurate in time due to Crank-Nicholson time integration.. Furthermore it is unconditionally stable as it is not limited by time/distance step relationships (as in explicit methods) and the code can accurately deal with both
69
fast and slow transients. The following section shifts the focus to the modelling of the PBMR.
Applying Newton's Second Law to the boundary velocities in the LP and HP gas streams we get:
- ~ _ PL,., - PI., PI., --;]t -!iX
4. SUBMODELS
f - V2
2D PI., I., L
I
A shown previously in the paper the PBMR consists of various specialized components. These are modeled in the simulation code as sub-models. This section discusses the theory behind the creation of these sub-models. There is a large variety of submodels components such as pipes, diffusers, valves, fans, etc. A exhaustive discussion of all of the submodels is beyond the scope of this paper but the submodels for heat exchangers, turbomachines and nuclear reactors are briefly discussed in this section.
(3)
(4)
where
p
=
f
density at control volume boundary,
friction factor and boundary values. =
the overbar denotes
Applying the First Law of Thenoodynarnics to the gas and wall control volumes we get:
4. J Recuperator The recuperator is used to recover heat from the outlet of the power turbine and transfer it to the high pressure gas entering the reactor. This greatly increases the efficiency of the whole plant. Figure 3 represents a control volume around a heat exchange surface.
(6)
Cw, d~;
The following equations represent a second order scheme that takes both heat transfer and flow transients on the gas side into account. Consider a control volume as shown in Figure 3. Applying mass balance to the gas control volumes we may write: A Ax dp L, = rh - rh L ' dt L, L, . ,
(1)
A Ax dp H , = rh - rh H ' dt H, H,. ,
(2)
where = constant volume specific heat of gas, c p = constant pressure specific heat of gas, CWi = product of lump mass and specific heat of wall per segment and HH and HL = product of heat transfer area and heat transfer coefficient per segment . Equations (1) to (7) can all be written in the following fono :
dx} _ ( Tt - f X" X where
I
~x1 +1
I
PL,
PI<
I;,
v"
J:,
Li
PL' . I
';' ~.'
---.
~
mH t
+
1
~,
VH,
PH,
ii•.
.......
j = 1, 2....n
(8)
is the variable under consideration.
(9) The approach followed with Flownet is to use a second order Crank-Nicholson approximation which gIves:
HI
PH I • 1
+
HP
Side X
t+1 }
.,.
)
Side Wall
m
"" X'
LP
Tr ,
~,
Xj
2
With a first order implicit integration one would write
I
:
m
(7)
- Tw,)- H H , (TA', - TH , )
Cv
where A is the cross sectional area.
I
= H L, (TL,
-x
~t
.... .,.
I j
= 0.5
V' + ft+' )
(10)
Applying equation (10) to equations (1) to (7) gives a set of algebraic equations which is solved by a matrix technique.
Fig. 3: Control volume of heat exchange surface
70
4.2 Compact gaslliquid heat exchangers The PBMR uses compact gas/liquid heat exchangers for the pre-cooler and intercooler. These heat exchangers are serpentine shaped, cross-flow heat exchangers and are modelled as shown in Figure 4. In the modelling process the heat exchanger is discretized to look similar to the example in Figure 4. Then, as shown in the figure, every segment can be considered as a small heat exchanger on its own and the flow and heat transfer can be solved for each section in a similar approach as used for the recuperator.
Compressor
mCL1 P 01
999
Fig. 5. Compressor performance map
o-C~:J-&O-o?-O ti :
.
:--\
4.4 Reactor
:
~
r~
O-·~0t·
-{)i:rJ
y
The pebble bed reactor is modelled by discretizing it into layers. Each layer consists of a certain amount of fuel spheres which are lumped into one ' sub'element or increment of the total PBR element. In each layer every fuel sphere is again broken up as a function of the radius from its center. Figure 6 shows the basic modelling layout for the reactor. The average reactivity, temperature and heat transfers is calculated for a layer and these values are used to generate a power profile in the core as a function of core height. The neutronic model used to calculate all the variables is of complicated structure and falls beyond the scope of this paper.
Gas
n". ~
:'-l
r:
0
t':
{\
o-··()i:·[j--():r o-··()r-O j j (]
Heat
exchanger
mtW+ Liquid flow
Fig. 4. Discretized heat exchanger model
4.3 Turbo-machines Both the turbines and compressors of the PBMR are modelled in Flownet using performance maps and certain operating point parameters. Performance maps, such as that for a compressor as shown in Figure 5, consist of pressure ratio and efficiency verses non-dimensional mass flow for different speeds. These maps are further enhanced by adding a third dimension namely the guide vane angle. Flownet uses a third order interpolation scheme to find the operating point on the map during a simulations.
Internal network
Fuel layer
The capabilities of the model include the following. Any number of compressor and/or turbine disks (stages) can be on a single shaft and there can be multiple shafts. Generator and mechanical losses are taken into account. There is speed control on the shaft and speed transients can be modelled. The model uses characteristic curves from which it interpolates for the right shaft speed for a given mass flow and pressure ratio. Where other techniques have difficulty the solver can solve on positive slopes of a constant N"To curve.
Fuel pebble
Fig. 6. Model of Pebble Bed reactor
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5. RESULTS
The following steady state results were obtained after solving a pipe model of the PBMR with Flownet.
The code has undergone extensive validations as part of the requirements set by nuclear safety authorities. An example of the accuracy of the code can be seen in Figure 7 that illustrates the pressure waves in a pipe. The 5km long pipe has a valve at the end that is 'slammed' shut allowing pressure waves to develop. The fluid in the pipe is helium moving at 120 kg/so The compressibility of the fluid is well illustrated by the "line packing" at the peaks of the pressure waves. The Flownet results compare very favourably with benchmark analytical results.
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Reactor power = 265 MW Grid power delivered = 110 MW Efficiency = 45,8 % Mass flow = 138 kg/s Max. Temperature = 900°C Max. Pressure = 70 bar 6. CONCLUSION
New sources of electricity are of utmost importance to the future of technological growth. Nuclear energy is one of the most unexploited fields in this industry. The introduction of the PBMR project has enormous potential for South Africa's economic growth as well as for the international power generation industry. The importance and capabilities of good thermal-hydraulic models to the development of the PBMR has been shown in this .,#.-..,; n •• n..... These models not only function as simulators to observe the working of the plant but also greatly to the design, optimization and r;;~~d control of the plant.
Flownet's application in the PBMR project is that of a design, simulation and optimization tool. It is used to solve both steady state and transient problems, both of which are discussed in the following sections. 10000 9OIX)
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7. REFERENCES
Greyvenstein, G. P. and Laurie, D. P. (1994) A segregated CFD approach to pipe network analysis. International. Journal for Numerical Methods Engineering., 37, 3685 - 3705.
1 - • ..-1 Fig. 7. Pressure waves in a pipe
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