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    Unsteady State Conduction

    Uploaded by

    Crislyn Akilit Bayawa
    The document discusses unsteady-state heat conduction where the rate and duration of heating and cooling are design parameters. It presents the general conduction equation and equations for rectangular geometry. It discusses transient conduction where temperature varies with location and time based on internal and surface resistance. The Biot number characterizes the system and environment by comparing convection heat transfer at the surface to conduction within. For Biot numbers less than 1, the system has negligible internal resistance and temperature is uniform. Thermal time constant and Fourier number are also introduced to characterize transient behavior.

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    Unsteady State Conduction

    Uploaded by

    Crislyn Akilit Bayawa
    168 views18 pages
    The document discusses unsteady-state heat conduction where the rate and duration of heating and cooling are design parameters. It presents the general conduction equation and equations for rectangular geometry. It discusses transient conduction where temperature varies with location and time based on internal and surface resistance. The Biot number characterizes the system and environment by comparing convection heat transfer at the surface to conduction within. For Biot numbers less than 1, the system has negligible internal resistance and temperature is uniform. Thermal time constant and Fourier number are also introduced to characterize transient behavior.

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    Heat transfer

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    The document discusses unsteady-state heat conduction where the rate and duration of heating and cooling are design parameters. It presents the general conduction equation and equations for rectangular geometry. It discusses transient conduction where temperature varies with location and time based on internal and surface resistance. The Biot number characterizes the system and environment by comparing convection heat transfer at the surface to conduction within. For Biot numbers less than 1, the system has negligible internal resistance and temperature is uniform. Thermal time constant and Fourier number are also introduced to characterize transient behavior.

    Copyright:

    © All Rights Reserved
    Download as PPT, PDF, TXT or read online from Scribd
    Download as ppt, pdf, or txt
    168 views18 pages

    Unsteady State Conduction

    Uploaded by

    Crislyn Akilit Bayawa
    The document discusses unsteady-state heat conduction where the rate and duration of heating and cooling are design parameters. It presents the general conduction equation and equations for rectangular geometry. It discusses transient conduction where temperature varies with location and time based on internal and surface resistance. The Biot number characterizes the system and environment by comparing convection heat transfer at the surface to conduction within. For Biot numbers less than 1, the system has negligible internal resistance and temperature is uniform. Thermal time constant and Fourier number are also introduced to characterize transient behavior.

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    UNSTEADY-STATE HEAT CONDUCTION

    Applications where rate/duration of heating/cooling


    is a Design Parameter

    General Conduction Equation


    The general form of these equations in multidimensions is:

    T
    g ( x, t )
    2
    T
    t
    C p
    For Rectangular Geometry:

    T T T g ( x, y , z : t )
    T

    2 2
    2
    t
    y
    z
    C p
    x
    2

    Solar Drying

    Oil at T

    Transient-conduction
    Temperature will vary with location
    within a system and with time.
    Temperature and rate heat transfer
    variation of a system are dependent on
    its internal resistance and surface
    resistance.
    If we have a ball with initial temperature
    of T0 and it is left in fluid at T.
    Heat is transferred by convection at the
    surface.
    As the surface temperature decreases,
    heat is transferred from the center of the
    ball to the surface, then to the fluid.

    Oil at T

    Thermal Analysis of Transient System


    Rate of Change in energy of system = Rate of Heat transfer
    Rate of Heat Transfer = Rate of Convection /radiation at surface =
    Rate of Conduction transfer in the system

    dU
    Rate of change in enegy :
    dt
    dU

    dt

    d cTdV
    V

    dt

    d c 4r 2 T (r , t )dr
    0

    dt

    At any instant:
    R

    d c 4r 2 T (r , t )dr
    0

    dt

    For Convection heat transfer across surface.

    dT r , t
    kAcond
    dr

    hAconv T ( R, t ) Te
    r R

    For Radiation heat transfer across surface.


    R

    d c 4r 2 T (r , t )dr
    0

    dt
    R

    d c 4r 2 T (r , t )dr
    0

    dt

    dT r , t
    kAcond
    dr
    dT r , t
    kAcond
    dr

    Arad T 4 ( R, t ) T 4 e
    rR

    hrad Arad T ( R, t ) Te
    r R

    Characterization of System & Environment


    If the system conductivity is high or the system volume is small,
    the temperature response is very fast.
    The response has to do with what is called the internal resistance
    of the material.
    If the convection/radiation coefficient is very high, then the
    surface temperature of the system almost becomes identical to
    the fluid temperature quickly.
    Alternatively, for a low convection/radiation coefficient a large
    temperature difference exists between the surface and the fluid.
    The value of the coefficient controls what is known as the
    surface resistance to heat transfer.
    The instantaneous temperature variation within the system is
    dependent on the internal and surface resistances.
    The larger internal resistance or the smaller surface resistance,
    the larger temperature variation within the system, and vice
    versa.

    Total thermal resistance of the system: Rtot

    T0

    Rcond

    T0

    Rconv

    A Biot number is defined as:

    Convection/radiation heat transfer rate at the surface


    Bi
    Conduction heat transfer rate within the body

    hAconv T ( R, t ) T
    Bi
    dT r , t
    kAcond
    dr r R
    Bi

    hAconv
    kAcond
    Lcharacteristic

    hAconv T0 T
    Bi
    T0 T
    kAcond

    hLcharacteristic

    k solid

    Bi

    hAconv
    kAcond
    Lcharacteristic

    Lcharacteristic
    kAcond
    Rcond

    1
    Rconv
    hAconv

    Thermal Characterization of System & Environment using


    Biot number

    System with negligible internal resistance


    For this case Bi 1.0.and the temperature profile within the body is
    quite uniform.
    The rate of change in internal energy of the
    body is equal to the rate of heat taken away
    from the surface by convection:

    d cT (t )dV
    V

    dt

    hAconv T (t ) T

    d cVT
    hAconv T (t ) T
    dt

    dT
    cV
    hAconv T (t ) T
    dt
    Rearranging:

    hAconv
    dT

    dt
    T (t ) T
    cV
    Integrating to any time interval:
    T

    dT
    hAconv
    T T (t ) T cV 0 dt
    0

    T T
    hA
    ln


    cV
    T0 T

    T T
    e
    T0 T
    Define, Thermal Time Constant

    T T
    e
    T0 T

    hA

    cV

    cV
    c
    hA

    T T
    T
    T0 T
    *

    t
    thA
    t
    c cV
    *

    Thermal Time Constant

    1
    c
    CV
    hA

    c Rth Cth
    The total energy transferred in time t

    hLcharacteristic
    hV
    Bi

    k solid
    k solid Asurface
    The Fourier number (Fo) or Fourier modulus, named after Joseph
    Fourier, is a dimensionless number that characterizes transient behavior
    of a system.
    Conceptually, it is the ratio of the heat conduction rate to the rate of
    thermal energy storage. It is defined as:

    kAT

    Lcharacteristic

    Fo
    CVT

    kt

    CL

    2
    characteristic

    t
    L2characteristic

    Hot Rolling of Steel Sheets

    hLcharacteristic
    Bi
    1
    k solid

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