Introduction to Nuclear System - Chapter 2-2
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What does the general solution for temperature distribution in the conduction equation represent?

  • An exponential relationship between temperature and distance
  • A linear relationship between temperature and distance
  • A quadratic relationship between temperature and distance (correct)
  • A constant temperature value throughout the material

    • What boundary condition leads to C1 being equal to 0?

  • The temperature gradient is at maximum at the boundary
  • The heat flow through the boundary is constant
  • The temperature is constant at the boundary
  • The rate of change of temperature is zero at the boundary (correct)

    • How is the temperature at the surface of the fuel (Ts) calculated?

  • Ts = Tm - q''/ka
  • Ts = Tm - (q''' * a)/(2kf) (correct)
  • Ts = Tm - (kf * a)/q''
  • Ts = Tm - q''A/(2kf)

    • What assumption is made when applying Fourier's law to find the heat flow out of the surface of the fuel?

    <p>Heat flow through the surface is proportional to the temperature gradient</p> Signup and view all the answers

    In the rearrangement of the equation for heat flow (q), what does the variable 'a' represent?

    <p>The thickness of the material</p> Signup and view all the answers

    What is the primary cause of conduction in heat removal from a reactor?

    <p>Temperature difference</p> Signup and view all the answers

    Which equation represents Fourier's law for heat conduction?

    <p>q'' = -k∇T</p> Signup and view all the answers

    In a steady state heat conduction scenario, what does Laplace's equation signify?

    <p>No heat source present</p> Signup and view all the answers

    What does the symbol 'k' represent in the conduction equation?

    <p>Thermal conductivity</p> Signup and view all the answers

    What does the term 'q''''' signify in the context of heat conduction?

    <p>Heat generation density</p> Signup and view all the answers

    What is the relationship between the net rate of heat flow out of volume V and the rate of heat production within volume V?

    <p>Net flow minus production equals zero</p> Signup and view all the answers

    In the conduction of heat through a plate, what direction does the heat flow primarily occur?

    <p>X-direction</p> Signup and view all the answers

    What does the term ∇²T represent in the steady-state heat conduction equation?

    <p>Laplacian of temperature</p> Signup and view all the answers

    What is the general solution for temperature distribution in the case of no heat source in the cladding?

    <p>T = C1 x + C2</p> Signup and view all the answers

    What boundary condition relates the temperature at x = a to Ts?

    <p>C1 a + C2 = Ts</p> Signup and view all the answers

    In the equation for heat balance, what does the term q' represent?

    <p>Heat generation density</p> Signup and view all the answers

    What is the expression for the heat transfer q in the cladding?

    <p>q = k_c A (Ts - Tc) / b</p> Signup and view all the answers

    What does the equation T_m - T_c represent in terms of thermal resistance?

    <p>Temperature difference across all layers</p> Signup and view all the answers

    In the cylindrical fuel rod, what is the primary direction of heat flow?

    <p>R-direction only</p> Signup and view all the answers

    Which term indicates the contribution of multiple materials to thermal resistance?

    <p>Linearly adding thermal resistances</p> Signup and view all the answers

    What does the term d²T/dx² equal when there is no heat source?

    <p>0</p> Signup and view all the answers

    What is the boundary condition for temperature at the origin of the cylinder, T(x = 0)?

    <p>T(0) = Tm</p> Signup and view all the answers

    Which equation represents the temperature distribution in the cladding?

    <p>T = C1 ln r + C2</p> Signup and view all the answers

    What is the form of the general solution for temperature T in a cylinder with heat sources?

    <p>T = -q''r^2 + C1 ln r + C2</p> Signup and view all the answers

    What is the relationship between the heat transfer rate q and thermal resistance Rf?

    <p>q = (Tm - Ts) / Rf</p> Signup and view all the answers

    What must be true for C1 in the general solution concerning non-singular temperature?

    <p>C1 must equal 0</p> Signup and view all the answers

    How is the temperature at the surface of the fuel Ts expressed?

    <p>Ts = -q''/4k f + Tm</p> Signup and view all the answers

    What condition is needed for C1 when deriving temperature T at radius r = a?

    <p>C1 must satisfy C1 ln a + C2 = Ts</p> Signup and view all the answers

    What does the term 'q' represent in the context of heat transfer in a cylinder?

    <p>The heat flow rate</p> Signup and view all the answers

    In a cylinder, which equation correctly represents the relationship between temperature difference and thermal resistance?

    <p>Tm - Tc = q(Rf + Rc)</p> Signup and view all the answers

    Which parameter influences the thermal resistance in a cylindrical heat transfer system?

    <p>All of the above</p> Signup and view all the answers

    What does the term 'Rf' represent in the equations related to heat transfer?

    <p>Thermal resistance of the fuel</p> Signup and view all the answers

    Which factor does NOT lead to non-uniform temperature distribution along the z-direction in a fuel rod?

    <p>Negligible heat generation variations</p> Signup and view all the answers

    Which statement is true regarding the exponential heat source in shielding and structure?

    <p>Radiative energy is deposited exponentially.</p> Signup and view all the answers

    What can be concluded about the temperature profile within a slab where heat generation is exponentially distributed?

    <p>Temperature could be higher than both T1 and T2.</p> Signup and view all the answers

    Which equation represents Fourier's law of thermal conduction?

    <p>q'' = -k∇T</p> Signup and view all the answers

    In a cylindrical heat conduction problem, what effect does increasing 'a' and 'b' have on the thermal resistance?

    <p>Increases thermal resistance</p> Signup and view all the answers

    What role does 'k' play in describing heat conduction?

    <p>It is a constant for specific materials.</p> Signup and view all the answers

    Which condition is implied by 'steady state' in heat conduction scenarios?

    <p>Temperature distributions are constant over time.</p> Signup and view all the answers

    In a fuel rod, what does the term 'Tc' typically denote?

    <p>The cladding temperature</p> Signup and view all the answers

    Which expression defines the general solution for a temperature profile in a material under heat conduction?

    <p>T = C1 * x + C2</p> Signup and view all the answers

    Which statement regarding heat conduction in cylindrical coordinates is correct?

    <p>Only radial heat flow is considered significant.</p> Signup and view all the answers

    Study Notes

    Introduction to Nuclear System and Operation - Chapter 2-2

    • This chapter focuses on heat removal from nuclear reactors using conduction.
    • Conduction is heat transfer caused by temperature differences, without macroscopic matter movement.
    • Convection is heat transfer due to temperature differences involving moving fluids (liquid or gas).
    • Radiation is heat transfer via thermal radiation.
    • Fourier's law describes heat conduction: q" = -k∇T, where q" is heat flux, k is thermal conductivity, and ∇T is the temperature gradient.
    • This law is similar to Fick's law, comparing the mechanisms of heat and mass transfer.

    Energy Conservation

    • In a unit volume, the net rate of heat flow out equals the heat production rate (steady state).
    • The equation derived is ∇ ⋅ q" + q" = 0.
    • This is the steady state heat conduction equation or Poisson's equation.
    • If there is no internal heat generation, it reduces to Laplace's equation (∇²T = 0).

    Plate-type Fuel Element Solutions

    • Fuel elements are plates with a central fueled strip.
    • Cladding surrounds the fuel strip.
    • Heat generation density is uniform (q").
    • Heat flow is only in the x-direction.
    • Poisson's equation is used to model heat transfer: ∇²T + q" / k = 0.
    • The general solution for temperature is T = (q"/2k)x² + C₁x + C₂.
    • Boundary conditions determine constants: T(x=0) = Tm, dT/dx|x=0 = 0.

    Temperature Distribution and Heat Flow

    • The temperature distribution is described by T = (q"/2k)x² + Tm .
    • Heat flow out of the fuel surface is calculated as q = q"A.
    • The temperature at the surface of fuel can be expressed as Ts = (q"/2k)a² + Tm.
    • Thermal resistance can be used to simplify calculations.

    Temperature Distribution in Cladding

    • Cladding temperature distribution is determined without internal heat generation (∇²T = 0).
    • Cladding general solution is T = C₁ x + C₂.
    • Boundary conditions are used to determine constants: T(x=a) = Ts, T(x=a+b) = Tc.
    • Heat balance ensures the heat flow into the cladding equals the heat generated in the fuel.

    Temperature Difference (Fuel and Cladding)

    • The heat flow in fuel meat (Tm - Ts) is calculated by q=a/2kA.
    • Heat flow in cladding (Ts - Tc) is given by q = k A (Ts - Tc)/ b.
    • The total temperature difference (Tm-Tc) can be simplified by adding the thermal resistance of the fuel and cladding.

    Cylindrical Fuel Rod Solutions

    • A cylindrical fuel rod has a central fueled rod of radius 'a' surrounded by cladding of radius 'b'.
    • Uniform heat generation density (q").
    • Heat flow is steady in the radial direction only.
    • Poisson's equation is used to model the heat flow: ∇²T + q" / k = 0.
    • The general solution for temperature is T = (q"/4k)r² + C₁lnr + C₂.
    • Boundary conditions determine constants: T(r=a) = Ts, T(r=b) = Tc.

    Temperature Distribution (Cylindrical)

    • The general solution for a cylindrical fuel rod is given, and the expressions for temperature distribution and heat flow are determined, incorporating natural logs and thermal resistances.

    Space Dependent Heat Source

    • The discussion involves systems with varying heat generation across the component, emphasizing radial non-uniformity within a given fuel rod.
    • The formulation focuses on a specific z-coordinate within a finite core and calculates the heat flow in a more generalized form.

    Exponential Heat Source

    • The heat source attenuates exponentially, like in a shield or structural material.
    • The heat flow out of the surface of the fuel is calculated for exponential heat sources in slab structures using energy balance and boundary conditions.
    • The temperature distribution can exceed the surface temperatures.

    Homework Assignment

    • The assignment is due one week before the next class.

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    Related Documents

    NUCE 402: Introduction to Nuclear System and Operation Lecture 4 PDF

    Description

    This chapter delves into the principles of heat removal in nuclear reactors through conduction. It covers key concepts such as Fourier's law of heat conduction and the equations governing steady-state heat flow. Understanding these fundamentals is crucial for comprehending nuclear reactor operations and energy conservation principles.

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