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Problem 3.12

# Problem 3.12 - Problem 3.12[Difficulty 4 Given Model...

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Problem 3.12 [Difficulty: 4] Given: Model behavior of seawater by assuming constant bulk modulus Find: (a) Expression for density as a function of depth h. (b) Show that result may be written as ρ = ρ o + bh (c) Evaluate the constant b (d) Use results of (b) to obtain equation for p(h) (e) Determine depth at which error in predicted pressure is 0.01% Solution: From Table A.2, App. A: SG o 1.025 = E v 2.42 GPa 3.51 10 5 × psi = = Governing Equations: dp dh ρ g = (Hydrostatic Pressure - h is positive downwards) (Definition of Bulk Modulus) E v dp d ρ ρ = Then dp ρ g dh = E v d ρ ρ = or d ρ ρ 2 g E v dh = Now if we integrate: ρ o ρ ρ 1 ρ 2 d 0 h h g E v d = After integrating: ρ ρ o ρ ρ o g h E v = Therefore: ρ E v ρ o E v g h ρ o = and ρ ρ o 1 1 ρ o g h E v = (Binomial expansion may be found in a host of sources, e.g. CRC Handbook of Mathematics ) Now for ρ o g h E v <<1, the binomial expansion may be used to approximate the density: ρ ρ o 1

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