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3.15: Untitled Page 41

  • Page ID
    18174
  • Chapter 3

    control volumes and we can apply it to the arbitrary moving control volume shown in Figure 3‐13. The mass within the moving control volume is given by

     mass contained in 

     

    dV

    (3‐43)

    any control volume

    ( )

    a

    V t

    and the time rate of change of the mass in the control volume is expressed as

     time rate of change 

    d

    of the mass contained  

    dV

    (3‐44)

    dt

    in any control volume

    ( )

    a

    V t

    In order to determine the net mass flow leaving the moving control volume, we simply repeat the development given by Eqs. 3‐18 through 3‐29 noting that the velocity of the fluid determined by an observer on the surface of the control volume illustrated in Figure 3‐13 is the relative velocity v r . This concept is illustrated in Figure 3‐14 where we have shown the volume of fluid leaving the Figure 3‐14. Volume of fluid crossing a moving surface dA during a time  t surface element dA during a time  t. On the basis of that representation, we express the macroscopic mass balance for an arbitrary moving control volume as

    Single component systems

    67

    d

    dV

      dA  0

    r

    v n

    (3‐45)

    dt V ( t)

    A ( t)

    a

    a

    The relative velocity is given explicitly by

    v

    v w

    (3‐46)

    r

    thus the macroscopic mass balance for an arbitrary moving control volume takes the form

    d

    dV

    ( 

    ) dA

    v w n

    0

    (3‐47)

    dt V ( t)

    A ( t)

    a

    a

    Here it is helpful to think of w as the velocity of an observer moving with the surface of the control volume illustrated in Figure 3‐13 and it is important to recognize that this result contains our previous results for a fluid body and for a fixed control volume. This fact is illustrated in Figure 3‐15 where we see that the arbitrary velocity can be set equal to zero, w  0 , in order to obtain Eq. 3‐29, and we see that the arbitrary velocity can be set equal to the fluid velocity, w v , in order to obtain Eq. 3‐7.

    d

    r dV +

    r( - )× dA = 0

    ò

    ò v w n

    dt

    a

    V (t)

    Aa(t)

    w = 0

    w = v

    d

    r

    d

    dV +

    r × dA = 0

    ò

    ò v n

    r dV = 0

    ò

    dt

    dt

    V

    A

    ( )

    m

    V t

    Figure 3‐15. Axiomatic forms for the conservation of mass Clearly Eq. 3‐47 is the most general form of the principle of conservation of mass since it can be used to obtain directly the result for a fixed control volume and for a body.

    EXAMPLE 3.4. Water level in a storage tank

    A cylindrical API open storage tank, 2m in diameter and 3m in height,

    index-77_1.png

    index-77_2.png

    68