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3.14: Untitled Page 40

  • Page ID
    18173
  • Chapter 3

    dm

      22 . 27 kg/day

    (9)

    dt

    This equation is easily integrated, and the initial condition imposed to obtain,

    m( t)  m( t  0)  22 2

    . 7 kg / day t

    (10)

    For an initial mass of 750 kg we find that the tank will be empty ( m  0 ) when t  33 7

    . days . For these circumstances, we require one delivery per onth

    m

    to insure that the propane tank will never be empty.

    3.3 Moving Control Volumes

    In Figure 3‐12 we have illustrated the transient process of a liquid draining from a cylindrical tank, and we want to be able to predict the depth of the fluid in the tank as a function of time. When gravitational and inertial effects dominate the flow process, the volumetric flow rate from the tank can be expressed as

    Q C A o 2

    D

    gh

    (3‐42)

    Here Q represents the volumetric flow rate, A o is the cross‐sectional area of the orifice through which the water is flowing, and C is the discharge coefficient that D

    must be determined experimentally or by using the concepts presented in a subsequent course on fluid mechanics. Equation 3‐42 is sometimes referred to as Torricelli’s efflux principle (Rouse and Ince, 1957, page 61) in honor of the Italian scientist who discovered this result in the seventeenth century. We would like to use Torricelli’s law, along with the macroscopic mass balance to determine the height of the liquid as a function of time. The proper control volume to be used in this analysis is illustrated in Figure 3‐12 where we see that the top portion of the control surface is moving with the fluid, and the remainder of the control surface is fixed relative to the tank.

    In order to develop a general method of attacking problems of this type, we need to explore the form of Eq. 3‐8 for an arbitrary moving control volume. In Sec. 3.1 we illustrated how Eq. 3‐8 could be applied to the special case of a moving, deforming fluid body, and our analysis was quite simple since no fluid crossed the boundary of the control volume as indicated by Eq. 3‐9. To develop a general mathematical representation for Eq. 3‐8, we consider the arbitrary moving control volume shown in Figure 3‐13. The speed of displacement of the surface of this control volume is w n which need not be a constant, i.e., our control volume may be moving, deforming, accelerating or decelerating. An observer moving

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    Single component systems

    65

    Figure 3‐12. Draining tank

    with the control volume determines the rate at which fluid crosses the boundary of the moving control volume,

    ( ) , and thus observes the relative velocity.

    a

    V t

    Figure 3‐13. Arbitrary moving control volume

    Previously we used a word statement of the principle of conservation of mass to develop a precise representation of the macroscopic mass balance for a fixed control volume. However, the statement given by Eq. 3‐8 is not limited to fixed

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    66