Nuclear Reactor Types and Layouts
Table 1-3 Typical characteristics of the fuel for six reference power reactor types
The primary function of a reactor is to transfer energy from the nuclear fission that takes place in the within the fuel to the moderator which is normally in the form of water. In light water reactors (LWRs), the moderator also acts as the coolant. Typically, LWRs have a small fuel to water volume ratio, otherwise known as metal to water ratio, and a rather large fuel rod centerline to centerline spacing. This spacing is referred to as the rod pitch (P) and spans the distance between fuel rods. A considerable difference between LWRs, heavy water reactors (HWRs), and advanced gas reactors (AGRs) is that HWRs and AGRs are designed for on-line refueling. Typically with LWRs, a refueling outage takes place every 24 months where 1/3 of the fuel is taken out and replaced with new fuel. In HWRs and AGRs, fuel assemblies are stacked within circular pressure tubes and due to their circular geometry, an irregular array is designed. In Liquid metal cooled fast breeder reactors (LMFBRs), a moderator is not needed. These reactors achieve high power densities due to their compact hexagonal fuel rod packing. The most popular reactors in the United States are the LWRs which consist of the Boiling Water Reactors (BWRs) and the Pressurized Water Reactors (PWRs) (Nuclear Systems I). Some examples of basic layouts of fuel assemblies are shown below:
Figure 1: Westinghouse Electric Company PWR Fuel Assembly
Figure 2: General Electric Company BWR Fuel Assembly
When designing a nuclear fuel assembly, the most important design parameter is to ensure the cladding integrity is always kept. Due to transients, steady state conditions, high temperatures, vibrations, and radiation, it is impractical to create design limit criteria that take all of these into account. As a result, design limit criteria are created for clad temperatures and heat fluxes. Typically, the outer clad temperatures are limited to a range slightly above the coolant saturation temperature. It is important to remember that the saturation temperature is governed by the pressure of the core and losses associated with the fluid traveling through the core. The outer clad temperature limits corresponds to the instability of water and two phase boiling. This will be elaborated in a later section. There are also certain design requirements that are associated with accidents such as a loss of coolant accident (LOCA). In the event of a loss of coolant, temperatures suddenly rise because the linear heat generation stays the same while the fluid evaporates and leaves the system. This creates excessive temperatures in the core and threatens the integrity of the cladding. As a result, it is important to keep temperatures below the Zircaloy clad’s melting temperature of 2200⁰F.
A major factor in limiting the outer clad temperature to slightly above the saturated temperature is the phenomenon of critical heat flux (CHF). This results from a sudden reduction in heat transfer capability of the two phase coolant. For PWRs, this design limit is known as departure from nucleate boiling condition and the critical power condition for BWRs (Nuclear Systems I). The bulk temperature of the coolant and the outer clad temperature can be related to the heat energy generated by the fuel with equation .
Tco – Tb = q”/h = q”’*Rfo2/(h*Dco) 
Tco = Outer Clad Temperature Tb = Coolant Bulk Temperature
q” = Heat Flux h = Heat Transfer Coefficient
q”’ = Heat generated Rfo = Radius of Fuel Pellet
Dco = Diameter of Outer Clad
For the design of fuel rods, typically q”’ is an independent parameter and the bulk fluid temperature of the coolant is predetermined. This causes a reduction in the heat transfer capability because of the relatively limited coolant temperature which results in the outer clad temperature to rise. For PWRs, nucleate boiling typically is the means of cooling as a result of the low void fractions during operating conditions. The void fraction is the fraction of steam to water by unit volume. As the outer clad temperature continues to arise from a result of decreased heat transfer ability, the coolant becomes vapor-blanketed and results in spiking outer clad temperatures. Under this condition, liquid water cannot penetrate the vapor covering the clad which inhibits the ability of the coolant to transfer heat energy away from the cladding. This condition is known as departure from nucleate boiling (DNB). In similar fashion, a high void fraction occurs in BWR operating conditions, which is normally cooled by a liquid film and the risk of dryout can occur. Film dryout occurs when the liquid film cooling overheats from high outer clad temperatures and physically dries out yielding only vapor (steam). This condition depends heavily on the conditions in the channel upstream of the location of dryout where as DNB is a localized problem (Nuclear Systems I).
Figure 3: Departure from Nuclate Boiling
After understanding these conditions, the design of a reactor becomes critically dependent on generating a power curve that does not cause DNB or dryout. As a result, the critical heat flux ratio (CHFR) and the minimum departure from nuclear boiling ratio (MDNBR) were developed. It is common practice to design a reactor that has a MDNBR less than 1.3 for PWRs and a CHFR of 1 for BWRs.
Core Thermal Performance
The performance of a reactor can be characterized by the power density (Q’’’) and the specific power. The power density is the measure of the energy generated relative to the core volume. This indicator is used as a factor on maximizing capital cost given that the larger the reactor vessel, the more capital is needed. The power density is also an important factor for applications past commercial power generation such as space and naval environments. This is a function of the fuel pin arrangement. For example, a triangular array increases the power density by 15.5% over a square array but each type of arrays offer unique advantages other than power densities (Nuclear Systems I).
(Q”’)square array = (4*(1/4*π*Rfo2)q”’dz)/(P2dz) = q’/P2 
(Q”’)triangular array = (3*(1/6*π*Rfo2)q”’dz)/(P/2*(31/2/2*P)dz) = q’/(31/2/2*P2) 
The specific power is the measure of the energy generated per unit mass of fuel material and is typically expressed in watts per gram of heavy atoms. This variable indicates the fuel cycle cost and core inventory requirements such as waste storage.
Specific power = Q(rate)/mass of heavy atoms = q’/(π*(Rfo + δg)2ρsmeared*f  where..
Ρsmeared = (π*Rfo2ρpellet)/(π*(Rfo + δg)2) 
f = mass fraction of heavy atoms in the fuel = grams of fuel heavy atoms/grams of fuel
Thermal Analysis of Fuel Pellets
As the fuel undergoes its life cycle, there are many variables that effect the temperature distribution, linear heat rate, and structure of the pellet. The temperature of the fuel pellet depends on the heat generation rate, material properties, and the coolant temperature. In LWRs, the primary fuel is Uranium Dioxide (UO2) with a cladding material of Zircaloy or Stainless Sttel 316. There are many factors that affect the thermal conductivity of UO2. Experimental research has shown that as the temperature increases, the thermal conductivity (k) dramatically increases until about 1700⁰C where it slightly increases again. Another factor that affects the thermal conductivity of the fuel is its porosity or density. By nature of the manufacturing process, sintering pressed powder of UO2, the density can be controlled up to about 90% of theoretical density. As the number of voids, higher porosity, increases, the thermal conductivity decreases. Therefore, it would be desireable to have 90% density although during operation, gasses are released and can cause swelling and deform the pellet. This makes some level of porosity desireable to limit swelling which can cause contact between the pellet and the cladding material. These are known as jumps and can cause decreased resistance and differences in expected operation. There is usually a small gap between the pellet and the cladding that is filled with helium. In addition to swelling, pellet cracking and fragment relocation can occur and alter the gap conductance. Another factor that can affect the performance is the oxygen to metal atomic ratio of the uranium and plutonium oxides that can vary from theoretical. Generally, burnup of the fuel causes this ratio to decrease during operation and can reduce the thermal conductivity of the fuel. When the fuel is irradiated, the fuel undergoes the changes discussed previously in addition to burnup. Burnup occurs over the life cycle of the fuel. As fission products are introduced into the fuel, there is a slight decrease in conductivity. As the fuel reaches a temperature higher than about 1400⁰C, the pellet undergoes a sintering process and reduces the porosity. This leads to restructuring where the fuel can create two to three zones of conductivity. The first zone is typically a void that is created around the centerline and surrounded by a columnar grain structure. The columnar grain structure occurs when the temperature reaches temperatures above 1800⁰C and experiences densification of about 98 to 99%. The next zone is the Equiaxed grain structure and occurs between 1600⁰C to 1800⁰C and experiences densification of 95 to 97%. The last zone is the “As-fabricated” zone and remains unchanged. The heat conduction can be modeled with the energy transport equation below (Nuclear Systems I).
ρ*Cp(r,T)*(∂T(r,T)/∂t)=Δ ∙k(r,T)*ΔT(r,t) + q”’(r,t) where Δ is the gradient 
The pellet and clad assembly can also be modeled as resistances where the temperature difference between the centerline and bulk temperature of the coolant equals the linear heat rate q’ times the sum of resistances. The equation below shows this relation and the sum of the resistances starting with the fuel resistance + the gap resistance + the cladding resistance + the coolant resistance.
Tcl – Tm = q’*[1/(4π*kf)+1/(2πRghg)+1/(2πkc)*ln(Rco/Rci)+1/(2πRco*h)] 
Tcl = Fuel Centerline temperature Tm = Coolant bulk temperature kf = fuel conductivity
Rg = Average gap radius hg = heat transfer coefficient of gap kc = clad conductivity
Rco = Outer radius of clad Rci = Inner radius of clad
h = heat transfer coefficient of the coolant
Single Phase Heat Transfer and Pressure Losses
To understand the phenomenon that is associated with two phase theory, it is essential to understand what happens under single phase conditions. The above heat transfer equations rely on the heat transfer coefficient of the coolant. This coefficient is not only a function of coolant properties but also the mass flow rate. As the mass flow rate is increased, the heat transfer coefficient is increased as a result of increasing the coolant’s ability to transfer units of energy away from the cladding surface. A turbulent flow is very desirable because it is easier to transfer energy with random eddies as opposed to laminar flow where the heat transferred between layers depends on the pulling or shearing from layer to layer. The heat transfer coefficient (h) is determined by the Nusselt number in the relation Nu = h*D/k. There are many experimental and empirical correlations to describe fluid flow over the fuel assemblies. One popular relation was developed for the radiator business and is called the Dittus-Boelter equation (Aumiller, David).
Nuoo = 0.023*Re0.8PrN where N = 0.4 when the fluid is heated and 0.3 when the fluid is cooled 
The Reynolds Number is Re = ρVD/µ and the Prandtl number Pr = µCp/k
For rod bundles, a correction factor Ψ is used to accommodate for geometry differences with the use of experimental graphs for triangular, square, and circular arrays. As the fluid travels up the length of the fuel assembly, there is pressure losses associated with frictional, gravity, form, and acceleration factors. The largest factor is the frictional pressure drop. This consists of a relation of a frictional factor, geometry, and velocity of the fluid. For turbulent flow, the frictional term can be solved for using the McAdams relation for smooth pipes of f = 0.184*Re-0.2. It is important to remember to use a hydraulic diameter and area associated with fuel rods. For example, for a square array, the area equals the pitch squared minus the cross sectional area of the rod and the hydraulic diameter is four times the area divided by the perimeter of a rod. When grid spacers are introduced, the pressure loss is greatly increased. Grid spacers are used for support of the fuel rods from vibration but also external loads of the coolant. Another advantage of spacers is that they increase the heat transfer coefficient by creating additional mixing and direction of the coolant. Modern day spacers provide proper mixing to allow for a swirling affect along the length of a section of the rod. As the mixing increases, the pressure loss increases. A delicate balance is sought by vendors such as Westinghouse to research and provide the NRC with data for licensing. As the pressure drop through the core is increased, more pump head is required which increases the amount of energy required to operate the plant. Form losses associated with the spacers as well as other pressure losses are shown below (Esposito, Vinny).
Two Phase Heat Transfer and Pressure Losses
To understand how the coolant changes as it travels along the fuel assembly and transports heat energy, it is important to understand the different flow regimes.
Figure 4: Flow Regimes
As the coolant travels along the fuel rod, heat energy is added because of the difference between the bulk temperature of the coolant and the wall temperature. When the wall temperature is very high, boiling can occur on the surface of the rod even though the bulk temperature of the fluid is below saturation temperature. This condition is called Sub-Cooled Boiling. Although it is very intermittent and unpredictable, it is a very efficient way of transferring energy and can cause the linear heat rate to double or triple. The liquid soon penetrates the vapor barrier and again makes contact with the wall. As the coolant rise through the core, the pressure decreases as a result of increasing frictional and spacer losses. This causes the saturation temperatures to decrease and increases the instability of the system. Modeling this phenomenon is very difficult and companies invest millions in researching and designing optimum systems. To analyze the frictional losses, there are relations developed by Martinelli and Lockhart to solve for the friction term of the changing two phase coolant. To have a basic understanding of the single phase relations discussed above, a rough estimate for system requirements can be developed. Further analysis is needed to develop accurate modeling with real data from experiments. To control these conditions and provide a safe guard in accidents, an extensive control system is developed are discussed in the following section (Aumiller, David).
A control rod is a rod, plate, or tube used to control the power of a nuclear reactor. By absorbing neutrons, a control rod prevents the neutrons from causing further fissions. Control rods are used to control the amount of energy emission in a nuclear reactor. When a control rod is in normal operating position, it is not in any contact with the fuel element. The amount of area submitted to the fuel element is what is used to control the amount of fissions occurring. By fully inserting the control rod, a reactor is shut down, since all nuclear fission is prevented (NRC: Glossary – Control Rod).
Figure 5: Control Rod positions in a Nuclear Reactor.
The material of a control rod must be capable of absorbing the stray neutrons that cause nuclear fission. The rods can contain elements such as silver, indium, cadmium, boron, or hafnium. The nuclear reactor uses an automated system to control the location of the control rods in the reactor core, and thus alter the nuclear fission rate.
Aumiller, David. "Heat Transfer and Fluid Flow in Nuclear Reactors." University of Pittsburgh, Pittsburgh.
Esposito, Vinny. "Heat Transfer and Fluid Flow in Nuclear Reactors." University of Pittsburgh, Pittsburgh.
“NRC: Glossary – Control Rod” nrc.com. 26 June 2007. U.S. NRC 05 Dec. 2007
Todreas, Neil E., and Mujid S. Kazimi. Nuclear Systems I. New York: Taylor & Francis, 1990. 1-34.