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Stability & CargoPage 6 of 12

GM with a slack tank: the free surface correction

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GM with a slack tank: the free surface raises G virtually by the free surface correctionMidship section looking forward, heeled a little to starboard for illustration. One centre tank is slack: breadth 22 m, length 34 m, cargo of 0.845 t/m³. As she heels the cargo's surface stays level, a wedge runs to the low side and G rises virtually to Gv: FSC = (34 × 22³ ÷ 12) × 0.845 ÷ 148500 = 0.172 m. Solid GM = KM 19.86 − KG 17.42 = 2.44 m; fluid GM = 2.44 − 0.172 = 2.27 m.b = 22 ml = 34 m fore and aftρ 0.845 t/m³gg₁MKM 19.86 mGvGKfluid GM2.27 mFSC 0.172 mKG17.42 msolid GM2.44 mWLW₁L₁PSfluid GM = 2.44 − 0.172 = 2.27 mGv = G corrected for free surface (fluid G)Looking forward · heel drawn for illustration · not to scale

The rule

  • As she heels, the liquid in a slack tank runs to the low side; the effect is the same as G rising — that virtual rise is the free surface correction (FSC).
  • Moment of inertia of a rectangular surface: i = l × b³ ÷ 12 — the breadth is cubed, so the breadth decides.
  • Free surface moment = i × ρ, with ρ the density of the liquid IN the tank; FSC = free surface moment ÷ displacement.
  • Fluid GM = solid GM (KM − KG) − FSC. Press the tank up or empty it to get the GM back; lowering the level a little does not.

Formula

  • i = l × b³ ÷ 12
  • FSC = free surface moment (i × ρ of the liquid in the tank) ÷ displacement
  • Fluid GM = KM − KG − FSC

Worked example

Displacement148500 t
KM at this draft19.86 m
KG in the loaded condition17.42 m
Slack tank, length34 m
Slack tank, breadth22 m
Density of the cargo in the slack tank0.845 t/m³
  1. i = 34 × 22³ ÷ 12 = 30169 m⁴
  2. Free surface moment = 30169 × 0.845 = 25493 t·m
  3. FSC = 25493 ÷ 148500 = 0.172 m
  4. Solid GM = 19.86 − 17.42 = 2.44 m
  5. Fluid GM = 2.44 − 0.172 = 2.27 m

Answer2.27 m

Common mistake

Putting sea water's 1.025 into the correction instead of the 0.845 of the cargo in the tank.

It is the cargo that moves when she heels, so its density sets the moment: with 1.025 the correction grows to 0.208 m and the GM shrinks to 2.23 m — a GM the ship does not have.