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    sm5 110

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    Sadie Hnatow
    The document summarizes the derivation of the finite-difference equations for the surface node of a one-dimensional wall subjected to uniform volumetric heating and convective surface conditions using explicit and implicit methods. For the explicit method, the resulting equation relates the future surface temperature to the present temperatures, heat generation, and Fourier and Biot numbers. The stability criterion requires the coefficient of the present surface temperature be positive. For the implicit method, the resulting equation relates the future surface temperature to the present and future temperatures, heat generation, and Fourier and Biot numbers.

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    sm5 110

    Uploaded by

    Sadie Hnatow
    15 views1 page
    The document summarizes the derivation of the finite-difference equations for the surface node of a one-dimensional wall subjected to uniform volumetric heating and convective surface conditions using explicit and implicit methods. For the explicit method, the resulting equation relates the future surface temperature to the present temperatures, heat generation, and Fourier and Biot numbers. The stability criterion requires the coefficient of the present surface temperature be positive. For the implicit method, the resulting equation relates the future surface temperature to the present and future temperatures, heat generation, and Fourier and Biot numbers.

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    sm5-110

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    The document summarizes the derivation of the finite-difference equations for the surface node of a one-dimensional wall subjected to uniform volumetric heating and convective surface conditions using explicit and implicit methods. For the explicit method, the resulting equation relates the future surface temperature to the present temperatures, heat generation, and Fourier and Biot numbers. The stability criterion requires the coefficient of the present surface temperature be positive. For the implicit method, the resulting equation relates the future surface temperature to the present and future temperatures, heat generation, and Fourier and Biot numbers.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    15 views1 page

    sm5 110

    Uploaded by

    Sadie Hnatow
    The document summarizes the derivation of the finite-difference equations for the surface node of a one-dimensional wall subjected to uniform volumetric heating and convective surface conditions using explicit and implicit methods. For the explicit method, the resulting equation relates the future surface temperature to the present temperatures, heat generation, and Fourier and Biot numbers. The stability criterion requires the coefficient of the present surface temperature be positive. For the implicit method, the resulting equation relates the future surface temperature to the present and future temperatures, heat generation, and Fourier and Biot numbers.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 1

    PROBLEM5.

    110
    KNOWN: One-dimensional wall suddenly subjected to uniform volumetric heating and
    convective surface conditions.
    FIND: Finite-difference equation for node at the surface, x = -L.
    SCHEMATIC:
    ASSUMPTIONS: (1) One-dimensional transient conduction, (2) Constant properties, (3)
    Uniform q.
    ANALYSIS: There are two types of finite-difference equations for the explicit and implicit
    methods of solution. Using the energy balance approach, both types will be derived.
    Explicit Method. Perform an energy balance on the surface node shown above,
    p+1 p
    o o
    in out g st conv cond
    T T
    E E E E q q qV cV
    t


    + = + + =
    A

    (1)
    ( )
    ( )
    ( )
    p p
    p+1 p
    o p
    o o 1
    o
    T T
    T T x x
    h 1 1 T T k 1 1 q 1 1 c 1 1 .
    x 2 2 t


    A A ( (
    + + =
    ( (
    A A

    (2)
    For the explicit method, the temperatures on the LHS are evaluated at the previous time (p).
    The RHS provides a forward-difference approximation to the time derivative. Divide Eq. (2)
    by cAx/2At and solve for
    p+1
    o
    T .
    ( ) ( )
    p+1 p p p p
    o o o o
    1
    2
    h t k t t
    T 2 T T 2 T T q T .
    c x c
    c x

    A A A
    = + + +
    A
    A

    Introducing the Fourier and Biot numbers,
    ( )
    2
    Fo k/ c t/ x Bi h x/k A A A
    ( )
    2
    p+1 p p
    o o
    1
    q x
    T 2 Fo T Bi T 1 2 Fo 2 Fo Bi T .
    2k

    (
    A
    = + + + (
    (

    (3)
    The stability criterion requires that the coefficient of
    p
    o
    T be positive. That is,
    ( ) ( ) 1 2 Fo 2 Fo Bi 0 or Fo 1/2 1 Bi . > s + (4) <
    Implicit Method. Begin as above with an energy balance. In Eq. (2), however, the
    temperatures on the LHS are evaluated at the new (p+1) time. The RHS provides a backward-
    difference approximation to the time derivative.
    ( )
    p+1 p+1
    p+1 p
    o p+1
    o o 1
    o
    T T
    T T x x
    h T T k q c
    x 2 2 t


    A A ( (
    + + =
    ( (
    A A

    (5)
    ( ) ( )
    2
    p+1 p+1 p
    o o
    1
    q x
    1 2 Fo Bi 1 T 2 Fo T T 2Bi Fo T Fo .
    k

    A
    + + = + +

    (6) <
    COMMENTS: Compare these results (Eqs. 3, 4 and 6) with the appropriate expression in
    Table 5.3.

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