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20 October, 12:58

Experimental data indicate that in a region downstream of a given louvered supply vent the velocity of the emitted air is defined by τ = 0.18τo/x where τ and x are expressed in m/s and meters, respectively, and τo is the initial discharge velocity of the air. For τo 3.6 m/s. Determine:

a. The acceleration of the air at x = 2m.

b. The time required for the air to flow from x=1 to x = 3m.

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  1. 20 October, 13:02
    0
    a) a = - 0.0524 m/s²

    b) t = 6.17 s

    Explanation:

    Given

    τ = 0.18*τ₀ / x

    τ₀ = 3.6 m/s

    Determine:

    a) The acceleration of the air at x = 2m.

    Knowing

    a = dτ / dt

    Multiplying both the numerator and denominator by dx

    a = (dτ / dx) (dx / dt)

    Substituting τ for dx / dt

    a = τ * (dτ / dx)

    ⇒ a = τ * (-0.18*τ₀ / x²) = (0.18*τ₀ / x) (-0.18*τ₀ / x²)

    ⇒ a = - 0.18²*τ₀² / x³ = - 0.18² * (3.6) ² / x³

    ⇒ a = - 0.4199 / x³

    If x = 2m

    ⇒ a = - 0.4199 / (2) ³

    ⇒ a = - 0.0524 m/s²

    b) The time required for the air to flow from x=1 to x = 3m.

    If

    τ = dx / dt = 0.18*τ₀ / x

    ⇒ dt = (x / 0.18*τ₀) dx

    ⇒ t = (1 / 0.18*τ₀) ∫x dx

    ⇒ t = (1 / 0.18*τ₀) * (1 / 2) * x²

    then

    t = (1 / (0.18*3.6)) * (1 / 2) * ((3) ² - (1) ²)

    ⇒ t = 6.17 s
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