Free Surface Moment of a Slack Tank
What a part-full tank costs a vessel in stability.
Example
You enter
- Tank length, fore and aft (ft) 10
- Tank breadth, across the vessel (ft) 8
- Liquid density (lb/cu ft; seawater 64, fresh 62.4, diesel 53) 64
- Compartments across (1 = undivided) 1
- Identical slack tanks 1
- Vessel displacement (lb, 0 to skip) 60000
You get
- Surface inertia ft4 426.667
- Free surface moment 27,307 ft-lb
- Loss of metacentric height 0.455 ft (5.5 in)
- Moment if split (ft-lb) 6826.67
No published worked example is on file for this calculation. It is checked against a hand-worked derivation of its method and an independent reference test.
Details, formula, and sources
ScopeThe free surface moment is the liquid density times tank length times breadth cubed over twelve, and dividing it by the displacement gives the loss of metacentric height. A 10 by 8 ft seawater tank has a moment of 27,307 ft-lb, which takes 0.455 ft of GM from a 60,000 lb vessel; one centerline division cuts that to a quarter. The depth of liquid does not matter. Rectangular tanks at small angles of heel. The stability booklet governs.
Formula or tableFree surface moment = liquid density x i, with i = l b^3 / 12 the second moment of area of a rectangular liquid surface about its own fore-and-aft centreline (l fore and aft, b across). n equal fore-and-aft compartments give l b^3 / (12 n^2). Identical tanks add. Loss of metacentric height = free surface moment / displacement. Units: lb per cubic foot x ft^4 = ft-lb; divided by lb gives ft.
Source and editionThe free surface correction as naval architecture states it: Principles of Naval Architecture (SNAME), intact stability, and standard ship stability texts such as Derrett, Ship Stability for Masters and Mates, cited by name.
Free accessThe second moment of area of a rectangle and one division; no table is reproduced. The n-squared rule follows from the cube: n compartments of breadth b/n give n (b/n)^3 = b^3 / n^2.
What governsEstimate. AHJ and licensed professional govern.
API field namestank_length_ft, tank_breadth_ft, liquid_density_lb_ft3, compartments_across, tanks, displacement_lb, surface_inertia_ft4, free_surface_moment_ftlb, free_surface_correction_ft, moment_if_split_ftlb
- Rectangular surface i = l b^3 / 12; a tank of another plan is not this modelthe builder's tank tables
- Liquid density the liquid in the tank: seawater about 64 lb/cu ft, fresh water 62.4, diesel about 53entered
- Slack tank surface reaches both sides and clears the top and bottom through the heelscope of this tile
- Divisions are watertight a swash plate with openings does not divide the surfacescope of this tile