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Submarine and research-submersible design

Study submerged buoyancy, trim, external pressure, resistance and life-support energy budgets using civilian research-submersible examples.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Submerged buoyancy and ballast

Definitions & inputs. V displaced external volume,m mass,ρ water density,Δm ballast change.

  1. Net vertical force is buoyancy minus weight.

    Fz=(ρV−m)gF_z=(\rho V-m)g

  2. Neutral buoyancy requires matching total mass to displaced-water mass.

    mneutral=ρVm_{neutral}=\rho V

  3. Added ballast decreases upward force if external displacement is unchanged.

    ΔFz=−gΔm\Delta F_z=-g\Delta m

Interpretation. Compressibility, salinity and temperature alter buoyancy; emergency ascent requires a validated independent system.

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2. Trim and submerged static stability

Definitions & inputs. zB,zG centers of buoyancy and gravity,θ pitch,weights mi at positions xi.

  1. Mass moments determine longitudinal center of gravity.

    xG=∑imixi/∑imix_G=\sum_i m_ix_i/\sum_i m_i

  2. A buoyancy center above the gravity center gives a restoring couple.

    Mrestore≃mg(zB−zG)sin⁡θM_{restore}\simeq mg(z_B-z_G)\sin\theta

  3. Transferring ballast shifts trim moment.

    ΔM=Δm g Δx\Delta M=\Delta m\,g\,\Delta x

Interpretation. Do not use surface-ship metacentric formulas unchanged for a fully submerged body.

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3. Pressure hull: stress versus collapse

Definitions & inputs. h depth,p external pressure,R radius,t wall thickness,E modulus,ν Poisson ratio.

  1. Integrate hydrostatic equilibrium at constant density.

    p−psurface=ρghp-p_{surface}=\rho gh

  2. Spherical membrane compression follows force balance on a hemisphere.

    σmem=−pR/(2t)\sigma_{mem}=-pR/(2t)

  3. Classical elastic buckling is an ideal-shell benchmark, not an allowable pressure.

    pcr,ideal=2E3(1−ν2)(t/R)2p_{cr,ideal}=\frac{2E}{\sqrt{3(1-\nu^2)}}(t/R)^2

Interpretation. Real shells have imperfections, openings, joints, residual stresses and fatigue. Cylinders have different buckling formulas and boundary sensitivity; use validated standards and tests.

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4. Drag, power and endurance

Definitions & inputs. CD drag coefficient,A frontal area,V speed,η propulsive efficiency,Eusable stored energy,Pload total power.

  1. Pressure and viscous effects are combined in a characterized drag coefficient.

    D=12ρV2ACDD=\tfrac12\rho V^2AC_D

  2. Mechanical power rises rapidly with speed if coefficients remain fixed.

    Pshaft=DV/ηP_{shaft}=DV/\eta

  3. Include propulsion, control, sensors, thermal management and life-support loads in an endurance budget.

    t=Eusable/Ploadt=E_{usable}/P_{load}

Interpretation. Battery reserve, oxygen, carbon-dioxide removal and thermal conditions can impose different endurance limits.

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Graphical worked example

Constant-density seawater pressure. This is not a hull collapse or operating-depth limit. X axis: Depth below surface (m). Y axis: Hydrostatic gauge pressure (MPa).
Constant-density seawater pressure. This is not a hull collapse or operating-depth limit. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. Neutral mass

Definitions & inputs. V10m³,ρ1025kg/m³.

  1. Choose the governing model and isolate the requested quantity.

    m=ρVm=\rho V

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1025(10)1025(10)

  3. Evaluate the expression; the result uses the units shown.

    Result=10250 kg\mathrm{Result}=10250\ \mathrm{kg}

Interpretation. Fully submerged displacement.

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Example 02. Positive buoyancy

Definitions & inputs. SameV,m10000kg,g9.81.

  1. Choose the governing model and isolate the requested quantity.

    F=(ρV−m)gF=(\rho V-m)g

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (10250−10000)(9.81)(10250-10000)(9.81)

  3. Evaluate the expression; the result uses the units shown.

    Result=2452.5 N\mathrm{Result}=2452.5\ \mathrm N

Interpretation. Upward net force.

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Example 03. Ballast for neutrality

Definitions & inputs. Neutral10250kg,current10000kg.

  1. Choose the governing model and isolate the requested quantity.

    Δm=mn−m\Delta m=m_n-m

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    10250−1000010250-10000

  3. Evaluate the expression; the result uses the units shown.

    Result=250 kg\mathrm{Result}=250\ \mathrm{kg}

Interpretation. Fixed external volume.

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Example 04. Density sensitivity

Definitions & inputs. V10m³,Δρ5kg/m³.

  1. Choose the governing model and isolate the requested quantity.

    Δm=VΔρ\Delta m=V\Delta\rho

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    10(5)10(5)

  3. Evaluate the expression; the result uses the units shown.

    Result=50 kg\mathrm{Result}=50\ \mathrm{kg}

Interpretation. Neutral mass changes with water density.

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Example 05. Pressure at10m

Definitions & inputs. ρ1025,g9.81,h10m.

  1. Choose the governing model and isolate the requested quantity.

    pg=ρghp_g=\rho gh

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1025(9.81)(10)1025(9.81)(10)

  3. Evaluate the expression; the result uses the units shown.

    Result=100552.5 Pa\mathrm{Result}=100552.5\ \mathrm{Pa}

Interpretation. Gauge pressure, not absolute.

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Example 06. Pressure at100m

Definitions & inputs. Same water.

  1. Choose the governing model and isolate the requested quantity.

    pg=ρghp_g=\rho gh

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1025(9.81)(100)1025(9.81)(100)

  3. Evaluate the expression; the result uses the units shown.

    Result=1005525 Pa\mathrm{Result}=1005525\ \mathrm{Pa}

Interpretation. Constant density approximation.

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Example 07. Absolute pressure

Definitions & inputs. Gauge1MPa,surface101325Pa.

  1. Choose the governing model and isolate the requested quantity.

    pabs=pg+psp_{abs}=p_g+p_s

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    106+10132510^6+101325

  3. Evaluate the expression; the result uses the units shown.

    Result=1101325 Pa\mathrm{Result}=1101325\ \mathrm{Pa}

Interpretation. Distinguish inside–outside differential from absolute pressure.

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Example 08. Sphere membrane stress

Definitions & inputs. Δp1MPa,R1m,t.02m.

  1. Choose the governing model and isolate the requested quantity.

    ∣σ∣=pR/(2t)|\sigma|=pR/(2t)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    106/(2(.02))10^6/(2(.02))

  3. Evaluate the expression; the result uses the units shown.

    Result=2.5×107 Pa\mathrm{Result}=2.5\times10^{7}\ \mathrm{Pa}

Interpretation. Not a buckling allowable.

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Example 09. Thickness scaling

Definitions & inputs. Doublet at fixed p,R.

  1. Choose the governing model and isolate the requested quantity.

    σ′/σ=t/t′\sigma'/\sigma=t/t'

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1/21/2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 \mathrm{Result}=0.5\ {}

Interpretation. Membrane stress scaling only.

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Example 10. Ideal buckling scaling

Definitions & inputs. Doublet/R at fixed material.

  1. Choose the governing model and isolate the requested quantity.

    pcr′/pcr=(2)2p'_{cr}/p_{cr}=(2)^2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    44

  3. Evaluate the expression; the result uses the units shown.

    Result=4 \mathrm{Result}=4\ {}

Interpretation. Real imperfection knockdowns not included.

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Example 11. Longitudinal CG

Definitions & inputs. Mass100kgat0m and300kgat2m.

  1. Choose the governing model and isolate the requested quantity.

    xG=∑mx/∑mx_G=\sum mx/\sum m

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    (100(0)+300(2))/400(100(0)+300(2))/400

  3. Evaluate the expression; the result uses the units shown.

    Result=1.5 m\mathrm{Result}=1.5\ \mathrm m

Interpretation. Common reference datum.

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Example 12. Trim transfer moment

Definitions & inputs. Δm20kg,Δx2m.

  1. Choose the governing model and isolate the requested quantity.

    ΔM=ΔmgΔx\Delta M=\Delta mg\Delta x

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    20(9.81)(2)20(9.81)(2)

  3. Evaluate the expression; the result uses the units shown.

    Result=392.4 N m\mathrm{Result}=392.4\ \mathrm{N\,m}

Interpretation. Sign depends on transfer direction.

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Example 13. Submerged restoring moment

Definitions & inputs. m10000kg,zB−zG.1m,pitch5°.

  1. Choose the governing model and isolate the requested quantity.

    M=mg(zB−zG)sin⁡θM=mg(z_B-z_G)\sin\theta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    10000(9.81)(.1)sin⁡5∘10000(9.81)(.1)\sin5^\circ

  3. Evaluate the expression; the result uses the units shown.

    Result=854.9978 N m\mathrm{Result}=854.9978\ \mathrm{N\,m}

Interpretation. Positive separation gives restoring tendency.

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Example 14. Drag

Definitions & inputs. ρ1025,V2m/s,A2m²,CD.2.

  1. Choose the governing model and isolate the requested quantity.

    D=ρV2ACD/2D=\rho V^2AC_D/2

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    .5(1025)(4)(2)(.2).5(1025)(4)(2)(.2)

  3. Evaluate the expression; the result uses the units shown.

    Result=820 N\mathrm{Result}=820\ \mathrm N

Interpretation. Coefficient is illustrative.

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Example 15. Tow power

Definitions & inputs. D820N,V2m/s.

  1. Choose the governing model and isolate the requested quantity.

    P=DVP=DV

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    820(2)820(2)

  3. Evaluate the expression; the result uses the units shown.

    Result=1640 W\mathrm{Result}=1640\ \mathrm W

Interpretation. No propulsive loss yet.

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Example 16. Shaft power

Definitions & inputs. Tow1640W,η.6.

  1. Choose the governing model and isolate the requested quantity.

    Ps=P/ηP_s=P/\eta

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    1640/.61640/.6

  3. Evaluate the expression; the result uses the units shown.

    Result=2733.333 W\mathrm{Result}=2733.333\ \mathrm W

Interpretation. Aggregate efficiency assumption.

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Example 17. Speed-power ratio

Definitions & inputs. Double speed,constant CD andη.

  1. Choose the governing model and isolate the requested quantity.

    P′/P=(V′/V)3P'/P=(V'/V)^3

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    232^3

  3. Evaluate the expression; the result uses the units shown.

    Result=8 \mathrm{Result}=8\ {}

Interpretation. Approximate cubic scaling.

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Example 18. Electrical endurance

Definitions & inputs. Usable30kWh,load5kW.

  1. Choose the governing model and isolate the requested quantity.

    t=E/Pt=E/P

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    30/530/5

  3. Evaluate the expression; the result uses the units shown.

    Result=6 h\mathrm{Result}=6\ \mathrm h

Interpretation. No reserve added implicitly.

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Example 19. Reserve energy

Definitions & inputs. Nominal40kWh,reserve25percent.

  1. Choose the governing model and isolate the requested quantity.

    Eu=E(1−f)E_u=E(1-f)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    40(.75)40(.75)

  3. Evaluate the expression; the result uses the units shown.

    Result=30 kWh\mathrm{Result}=30\ \mathrm{kWh}

Interpretation. Capacity degradation may reduce further.

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Example 20. Heat rejection

Definitions & inputs. Input5kW,useful mechanical3kW.

  1. Choose the governing model and isolate the requested quantity.

    Q=Pin−PoutQ=P_{in}-P_{out}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    5−35-3

  3. Evaluate the expression; the result uses the units shown.

    Result=2 kW\mathrm{Result}=2\ \mathrm{kW}

Interpretation. Energy balance, not temperature prediction.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.