Nuclear Thermal Hydraulics for Exam 2

PWR Reactor Core Properties

Control Volume contains the primary coolant

Assume steady flow and incompressible fluid

Mass equation for all loop components in a simplified PWR

0 = m^o_in - m^o_out

so,

m^o_in = m^o_{out (Steady Flow)}

Energy equation for a simplified PWR Core

Q^o_th = m^o_p(h_out - h_in)

Equation of state for an incompressible fluid

ρ = constant

Equation of state for an Ideal Gas

PV = mRT

ρ = P/RT

Equation for the specific internal energy of an incompressible fluid

u = u₀ + c(T - T₀)

Equation for the specific internal energy of an Ideal gas

u = u₀ + c_v(T - T₀)

Equation for the specific enthalpy of an Ideal gas

h = h₀ + c_p(T - T₀)

Equation for the specific enthalpy of an Incompressible fluid

h h= ₀ + c(T - T₀) + (P - P₀) / ρ

Equation for specific entropy of an Incompressible fluid

s = s₀ + (c)ln(T/T₀)

Equation for specific entropy of an Ideal gas

s = s₀ + (c)ln(T/T₀) - (R)ln(P/P₀)

Relation between specific ideal gas constant and the universal gas constant

R = R*/A
R - specific gas constant [J/kg*K]
R* - universal gas constant [J/mol*K]
A = molar mass [kg/mol]

Specific heat capacity relationship

c_v = c_p + R
c_v - heat capacity at constant volume

c_p - heat capacity at constant pressure

Best equation to use when solving for mass flow rate

m^o_p = Q^o_th / cΔT_core
Comes from: Q^o_th = m^o_pcΔT_core

Properties of a simplified PWR Steam Generator

1. Control volume for primary and secondary coolant

2. Assume Steady Flow

3. Incompressible fluid not valid assumption for secondary coolant since it is a saturated or superheated vapor
   
   

What is a pure substance

A pure substance is a material with a consistent and uniform chemical composition. It can exist in different phases (solid, liquid, gas) but its chemical identity remains unchanged.

Critical Point Definition

The exact end of the liquid-gas boundary line where the liquid and gas phases become completely identical and form a single fluid

Triple Point Definition

The condition where all three states are present at the same time

Steam Quality Definition Equation

x = M_vapor / ( M_vapor + M_liquid)
= (h - h_f) / (h_g - h_f)
This would be the same for internal energy and entropy

Energy Equation for the Steam Generator in a simplified PWR

Q^o_th = m^o_s(h_g(P₃) - h₂) = m^o_p(h_out - h_in)
m^o_s - mass flow rate of the secondary coolant in the heat generator
m^o_s = Q^o_th / (h_g(P₃) - h₂(P₂,T₂)

Properties of a Turbine in a simplified PWR

Control Volume that contains secondary coolant
Steady Flow
Converts thermal energy carried by the steam into mechanical work (shaft rotation)
Inlet: Saturated Vapor
Outlet: liquid-vapor mix

Entropy Equation for a Turbine in a simplified PWR

0 = m^o_s(s₃ - s₄) + S^o_gen
If S^o_gen = 0, there are no irreversibilities and s_4s = s₃ (isentropic)
If S^o_gen > 0, s_4s > s₃ (realistic due to friction, heat loss, and mixing)

Isentropic Efficiency of a Turbine

Not going to give the full equation, but it is typically between 70-90% and is the number given by the manufacturer

Is high steam quality wanted in the turbine? Why so?

Yes, because it maximizes the amount of useful work that can be extracted and lower steam quality can lead to erosion of the turbine and other components.

How to increase steam quality: superheat, reheat, moisture separation

Properties of a Pump in a simplified PWR

Control volume with secondary coolant
Assume steady flow
(Vapor is extremely bad for pumps, can lead to cavitation)

Inlet: Saturated liquid
Outlet: Subcooled liquid

Entropy equation for a Pump in a simplified PWR

0 = m^o_s(s₁ - s₂) + S^o_gen

Overall Plant Efficiency Equation

η_cycle = Net Electric Power / Thermal Power
= (h₃ - h₄ + h₁ - h₂) / (h₃ - h₂)
Typical efficiency for PWRs and BWRs: η_cycle = 30-35%

What is cavitation and why is it important to avoid?

Cavitation is the rapid formation and collapse of air bubbles in a fluid. This occurs in the pumps when there is air still in the fluid. This damages the pump and causes replacements to happen more frequently.