2017-01-24

Lecture Note of Naval Architectural Calculation

Ship Stability

Ch. 6 Free Surface Effect

Spring 2016

Myung-Il Roh

Department of Naval Architecture and Ocean Engineering Seoul National University

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Contents

 Ch. 1 Introduction to Ship Stability  Ch. 2 Review of Fluid Mechanics  Ch. 3 Transverse Stability Due to Cargo Movement  Ch. 4 Initial Transverse Stability  Ch. 5 Initial Longitudinal Stability  Ch. 6 Free Surface Effect  Ch. 7 Inclining Test  Ch. 8 Curves of Stability and Stability Criteria  Ch. 9 Numerical Integration Method in Naval Architecture  Ch. 10 Hydrostatic Values and Curves  Ch. 11 Static Equilibrium State after Flooding Due to Damage  Ch. 12 Deterministic Damage Stability  Ch. 13 Probabilistic Damage Stability

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Ch. 6 Free Surface Effect

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Free Surface Effect

G: Center of total mass (gravity) B: Center of buoyancy z z B1: New position of center buoyancy after the ship has been inclined * M(Metacenter): The intersection of a If the cargo in hold is fixed, e.g. a vertical line through the center of buoyancy at initial position(B) with a solid cargo, this keeps its original vertical line through the new M position of the center of position when a ship is inclined. buoyancy(B1) after the ship has been inclined transversally through a small angle 

y

FG  If there is a liquid cargo in hold, what will happen when y the ship is inclined? G Z

B B 1

FB K

* Hold or Cargo Hold: Space for loading the cargo in the ship 4 Naval Architectural Calculation, Spring 2016, Myung-Il Roh

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Definition of Free Surface Effect

G: Center of total mass (gravity) G1: New position of center of total • When the ship is inclined, the liquid in mass (gravity) z z B: Center of buoyancy the tank is also inclined. And the center B1: New position of center of buoyancy after the ship has been inclined of gravity of the liquid shifts toward the g: Center of the emerged volume M g1: Center of the submerged volume inclined side. b: Center of liquid in tank

b1: New position of center of liquid in tank • This causes the ship’s center of gravity  to move toward the inclined side, reducing the righting arm. y

FG 

y Z G 1 Definition of Free Surface Effect G1 • The motion of the liquid in a tank that is B B 1 partially full causes the ship’s center of g1 gravity to move. g b • This reduces the righting arm and b 1 FB reduces the ship’s stability. K

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Assumption to Evaluate the Effect of Free Surface

G: Center of total mass (gravity) G1: New position of center of total  The effects of free surface depend on mass (gravity) z z B: Center of buoyancy the dimensions of the surface of the liquid. B1: New position of center of buoyancy after the ship has been inclined g: Center of the emerged volume M What assumption is appropriate g1: Center of the submerged volume b: Center of liquid in tank to evaluate the effect of free b1: New position of center of liquid in tank surface in a ship’s tank? G0: Virtual risen center of gravity 

y Assumption: Small inclination  The metacentric height (GM) provides a y fairly accurate evaluation of the righting G moment up to 7~10 degrees. G1 B B 1 And, surface of the liquid does not reach g1 the top or bottom of the tank during this g inclination. b b 1 K

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Transverse Righting Moment When there is no Effect of Free Surface

G: Center of total mass (gravity) B: Center of buoyancy z B1: New position of center of buoyancyz after the ship has been inclined b: Center of liquid in tank  Transverse Righting Moment M Righting arm: GZ

 • Transverse Righting Moment   FGZ y righting B  GZ  GM sin y Z G GM KB BM KG

B B 1

b K

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 GZ F The Influence of Free Surface restoring B GZ GM sin on the Initial Stability at Small Angle (1/5) GM KB BM KG

G: Center of total mass (gravity) G1: New position of center of total  Evaluation of effect of free surface on mass (gravity) z z B: Center of buoyancy metacentric height B1: New position of center of buoyancy after the ship has been inclined g: Center of the emerged volume M g1: Center of the submerged volume b: Center of liquid in tank If a ship is inclined, the plane of liquid in b1: New position of center of liquid in tank tank is changed to be parallel to water  plane.

y bb11// gg // GG 1  And the relation between change in total y center of gravity and center of liquid in Z tank is G G w 1 GG bb 11F B G B 1 g1 g b b 1 K

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 GZ F The Influence of Free Surface restoring B GZ GM sin on the Initial Stability at Small Angle (2/5) GM KB BM KG z z bb11// gg // GG 1 M  Evaluation of effect of free surface on metacentric height  From the geometric shape, the righting y arm G1Z1 is expressed as follows:  GZ11GZG G1 y Z M G  G1 GZ(cossin) yGG  z  B B 1 This term has negative effect to the restoring moment arm. g1 g  b : It means that the shift of center of total b 1 gravity GG1 causes reduction of righting K arm. And it causes stability to be worse. G: Center of total mass (gravity) G0 G0: Virtual risen center of gravity : Free Surface Effect (FSE) G1: New position of center of total mass (gravity)  G1’: The intersection of the line GZ with G0G1 Reduction of righting arm B: Center of buoyancy (It causes stability to be worse.) B1: New position of center of buoyancy after the ship has been inclined  Z g: Center of the emerged volume G g : Center of the submerged volume 1  z  sin 1 Z1 G b: Center of liquid in tank G1 ' b1: New position of center of liquid in tank  y  cos m: Metacenter of cargo hold G G G1 G1 G   zG   yG GG0 : free surface correction 9 Naval Architectural Calculation, Spring 2016, Myung-Il Roh

: free surface correction GG0  GZ F The Influence of Free Surface restoring B GZ GM sin on the Initial Stability at Small Angle (3/5) GM KB BM KG z z bb11// gg // GG 1 M  Evaluation of effect of free surface on metacentric height  : Free Surface Effect (FSE) y  Reduction of righting arm  (It causes stability to be worse.) y Z M Assume that    1 , then from the G G1 geometric shape, the righting arm G1Z1 is B expressed as follows: B 1 g 1 GZ GZG G g  11 1 b b 1 GMsin GG0 sin K ()sinGM GG0  G: Center of total mass (gravity) G G : Virtual risen center of gravity 0 0  GM0 sin G1: New position of center of total mass (gravity) G1’: The intersection of the line GZ with G0G1 B: Center of buoyancy It means that the shift of center of total B1: New position of center of buoyancy after gravity GG is considered as the elevation the ship has been inclined  Z 1 g: Center of the emerged volume G G G1 of center of total gravity from to 0. g1: Center of the submerged volume Z b: Center of liquid in tank 1 GG : free surface correction b1: New position of center of liquid in tank 0 m: Metacenter of cargo hold G G1 GM KB BM KG

KG0 10 Naval Architectural Calculation, Spring 2016, Myung-Il Roh

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 GZ F The Influence of Free Surface restoring B GZ GM sin on the Initial Stability at Small Angle (4/5) GM KB BM KG z z bb11// gg // GG 1 KG0 M  Evaluation of effect of free surface on

 M righting arm

y G0  

 y GG11// bb , GG0 // bm , GG10// bm 1 Z G G0 G1 m GG G B Triangle 10 is similar to triangle bb 1 m . B 1 g  1  g Z GG w w b G1 1 b 1 Z  GG11 bb 1 F G bb1 FG G K G 1 G: Center of total mass (gravity) m G0: Virtual risen center of gravity G1: New position of center of total mass (gravity) G1’: The intersection of the line GZ with G0G1 B: Center of buoyancy B1: New position of center of buoyancy after the  ship has been inclined b g: Center of the emerged volume b 1 g1: Center of the submerged volume b: Center of liquid in tank

b1: New position of center of liquid in tank m: Metacenter of cargo hold w: Weight of liquid in tank

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w  restoringGZ F B The Influence of Free Surface GG bb 11F GZ GM sin on the Initial Stability at Small Angle (5/5) G GM KB BM KG z z bb11// gg // GG 1 KG0 M  Evaluation of effect of free surface on

 M righting arm y Because ϕ is small, (small inclination) G 0    GG11 GG , GG10  GG sin ,

 y GG10 GG sin , Z G G0 G1 In the same manner, bb1  bmsin m B B 1 ww g GG10 GGsin bb 1 bm sin  1  FF g Z GG b G1 b 1 Z w 1 GG bm K G1 0 G FG G: Center of total mass (gravity) m w iT F gv iT F iT G0: Virtual risen center of gravity    G1: New position of center of total mass (gravity) Fv  g v   G1’: The intersection of the line GZ with G0G1 G SW SW B: Center of buoyancy B1: New position of center of buoyancy after the ship  has been inclined b g: Center of the emerged volume b 1  i : Free Surface Moment g1: Center of the submerged volume F T b: Center of liquid in tank

b1: New position of center of liquid in tank The elevation of center of total gravity due to m: Metacenter of cargo hold the shift of center of gravity of liquid in tank is w: Weight of liquid in tank related to the density of liquid in tank and iT: Moment of inertia of liquid plane area in tank about x' axis moment of inertia of liquid plane area in tank. F: Density of liquid in tank, SW: Density of sea water  : Displacement volume (It’s not related to weight of liquid in tank.) v: Volume of liquid in tank 12 Naval Architectural Calculation, Spring 2016, Myung-Il Roh

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G: Center of total mass (gravity) F: Density of liquid in tank

G1: New position of center of total mass (gravity)z z SW: Density of sea water The Effects of Free Surface w: Weight of liquid in tank M : Displacement volume iT: Moment of inertia of liquid plane area in tank v: Volume of liquid in tank about x' axis  • Free Surface Effect y G0 GM KB BM KG0 , KG0  KG  GG0  w i T  y GG0  Z Wv G G1  gvi  i m F T F T B   B 1 SW g v SW  g1 g b : free surface correction b 1

F iT : free Surface Moment K • The effects of free surface do not depend on the amount of liquid in the tanks.

• The weight and vertical position of the liquid which have an effect on transverse stability is not associated with free surface effects.

• The effects of free surface depend on the ratio of density of the liquid in the tank to the density of the liquid in which the ship is floating.

• The breadth of liquid, which almost wholly accounts for free surface effects, changes when inclined, depending on the height of the liquid in the tank, the degree of inclination, and the breadth depth ratio of the tank.

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w iiTFT GG0  Question) Effect of Tank Size on Free Surface Wv SW  : free surface correction • Effect of the breadth of liquid in a tank which almost wholly accounts for free surface effects.

Question) Sea water is filled partially in tank with water plane of rectangular shape. If a longitudinal bulkhead is installed in center of tank, how much GM will be changed? Installation of longitudinal bulkhead

z y x z y x

Hint) Answer) lb 3 lb333 l(/2) b lb 1 iydA2 ii, 2 i T  12 TT1212 12 48 4 T 1

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w iiTFT The Influence of Free Surface GG0  Wv SW  at Large Angles of Heel (1/5) : free surface correction

• Effect of filling ratio of a ship and on free surface effects 10% full  In making the loading calculations, free surface corrections to GM must be made on the basis of reasonable assumptions regarding the condition of all tanks.

Example) 50% full Departure condition & Arrival condition - Fuel oil system: Settling tank: assumed to be half full. Fuel oil tank: assumed to be 98% full or empty in merchant ship - Water ballast tank: 90% full Empty or nominally completely full (to have no effect)

* Settling tank: Tank for settling impurities in fuel oil before using it. 15 Naval Architectural Calculation, Spring 2016, Myung-Il Roh

w iiTFT The Influence of Free Surface GG0  Wv SW  at Large Angles of Heel (2/5) : free surface correction

7° Inclination

• Small angle of inclination

 It has been pointed out 10% full previously that the metacentric height (GM) provides a fairly accurate evaluation of the righting moment up to 7~10 degrees.

50% full  In the same manner, when the metacentric height is reduced by the effect of free liquid, this assumption of small angle provides that the surface of the liquid does not reach the top or 90% full bottom of the tank during this inclination.

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w iiTFT The Influence of Free Surface GG0  Wv SW  at Large Angles of Heel (3/5) : free surface correction

7° Inclination 25° Inclination 50° Inclination

10% full

50% full

90% full

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w iiTFT The Influence of Free Surface GG0  Wv SW  at Large Angles of Heel (4/5) : free surface correction

7° Inclination 25° Inclination 50° Inclination

10% full

If the tank is nearly full or nearly 50% full empty, the breadth will decrease rapidly to a degree that will make free surface effects almost negligible.

90% full

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w iiTFT The Influence of Free Surface GG0  Wv SW  at Large Angles of Heel (5/5) : free surface correction

7° Inclination 25° Inclination 50° Inclination

10% full

50% full

If the tank is half full, a maximum increase of breadth is 90% full produced, thus increasing free surface effects up to a large angle of inclination.

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