How much cargo did the draft survey find?

Formula, assumptions, and limits

A draft survey finds the cargo as the difference between two net displacements, each worked from six draft readings by the sequence on Form “C” of the UN/ECE Draught Survey Code, ECE/ENERGY/19 of 3 February 1992. The code’s Part 1 recommends the procedure “for calculating the weight of dry cargoes at both loading and discharge ports”, so its title’s coal is not a limit on its use.

Formula

• $d_F$, $d_A$, $d_M$: the means of the port and starboard readings at the forward, aft and midship marks, m.
• $t$: trim between the marks, $d_A - d_F$, positive by the stern, m; $L_{BM}$ the length between the forward and aft marks, m.
• $l_F$: the forward marks’ distance from the forward perpendicular, positive aft of it; $l_A$ the aft marks’ distance from the after perpendicular, positive forward of it; $l_M$ the midship marks’ distance from amidships, positive aft, all m.
• $T$: trim between the perpendiculars, $D_{AP} - D_{FP}$, positive by the stern, m.
• $\Delta_{MMM}$, $TPC$, $LCF$: displacement in t, tonnes per centimeter in t/cm and the longitudinal center of flotation from amidships in m, positive aft, all read from the tables at $MMM$.
• $MTC_{+}$, $MTC_{-}$: moment to change trim one centimeter at $MMM$ plus and minus 0.50 m, t·m/cm.
• $\rho_{dock}$: dock water density read in air, t/m³; $\rho_{table}$ the density the tables are worked for, usually 1.025 t/m³.
• $\Delta_2$: the displacement corrected for trim, t.
• $W_{ded}$: ballast, fresh water, fuel oil, diesel oil, lube oil and other weights aboard, t.

For a discharge the cargo is the net displacement before less the net displacement after.

Derivation

Draft marks are seldom at the perpendiculars, so each mean reading is moved to its perpendicular along the waterline: the trim between the marks over the length between them, times the mark’s distance from the perpendicular. Form “C” lines 133, 140 and 148 give the correction as $\pm (T/D_1) \times D_2$ and leave the sign to “the observation of the slope of the waterplane”; the signed form above gives the same answer for every case. Line 150 then takes the midships draft less the mean of the forward and aft drafts: a positive result is sag, a negative one hog.

The hydrostatic tables give displacement for a ship floating level, so the code reads them at the M/M/M draft of lines 151 and 152, which weights amidships six times for a hull bent into a parabola. A trimmed ship pivots about its center of flotation, not about amidships, so the mean draft is off the true level draft by trim × LCF ÷ LBP; that, times the TPC converted to tonnes per meter, is the first trim correction of line 162. Its sign follows the code’s table: trim by the stern with the LCF forward of amidships is minus. The second correction of line 163, named for Nemoto, allows for the center of flotation moving as the ship trims; the code says it “is always +”.

The density correction of line 167 is written as a difference, (table density − dock density) ÷ table density × displacement, with the sign set by which is larger. The ratio form above gives the same figure in one step.

Assumptions

• The hull bends as a parabola, which the M/M/M assumes (code p. 36). An approved deflection table or a draft survey manual’s own method takes precedence where the ship has one; the code also gives station factors and a waterplane-coefficient factor (pp. 24 and 25) as other methods.
• The table figures are read by linear interpolation at the M/M/M draft (code p. 25), from the ship’s approved hydrostatic data.
• The two MTC figures are read 0.50 m above and below the M/M/M. The code defines them about the mean observed draft (p. 27), which differs from the M/M/M by the corrections for the marks and for deflection.
• The second trim correction is always added (p. 27), so the MTC 0.50 m above must be at least the MTC 0.50 m below; the calculator refuses the two the other way round, which is how they are most often mistyped.
• The dock water density is the average of samples in air, with no temperature correction (code p. 58).
• The ship is upright to within ½° and her trim lies inside her trim correction tables (code pp. 15 and 19).
• Every deductible is measured at the time of the drafts; the calculator takes the figures as entered.

Worked example

The example is constructed for this page; ECE/ENERGY/19 contains no worked example. A ship of 225.00 m between perpendiculars has her forward marks 1.50 m aft of the forward perpendicular, her aft marks 5.00 m forward of the after perpendicular and her midship marks 0.50 m aft of amidships, with tables at 1.025 t/m³ and a lightship of 11,250 t.

The cargo loaded is 76,163.66 − 11,617.66 = 64,546.00 t, and the ship’s constant is 11,617.66 − 11,250.00 = 367.66 t, 3.6% above the 355 t from the ship’s earlier surveys and inside the code’s ±10%. At the code’s 0.5% the cargo lies within ±322.73 t.

Edge cases and limits

At even keel both trim corrections are zero, and the M/M/M is the corrected mean draft. Marks outside a perpendicular take a negative distance. Where the tables are referred to the molded base line, the keel plate thickness comes off the M/M/M before the tables are read (line 153); tables to the underside of the keel need nothing. Outside the hydrometer range of 0.990 to 1.040 kg/l that the code specifies on page 56, the calculator refuses the density. The calculator does not interpolate the tables or store any ship’s data.

Regulatory basis

ECE/ENERGY/19 was adopted by the ECE Working Party on Coal in October 1991 and endorsed by the ECE Committee on Energy in November 1991. Its own background note calls it “a set of recommendations which may be applied in full or in part”. A charter party or sale contract makes the code binding between the parties, and the code’s pages 39 and 40 give a model clause.

Common errors

• An LCF taken from the after perpendicular, as some tables give it, used as if it were from amidships: the first correction then comes out more than ten times too large.
• An LCF from tables that count it positive forward of amidships entered here, where it is positive aft, without changing its sign. The first trim correction of each survey then turns the wrong way, +499.99 t before loading and −29.88 t after it in the worked example: the cargo comes out at 63,493.38 t instead of 64,546.00 t, 1,052.63 t short, and the constant at 1,360.80 t against 355 t.
• The trim between the marks used for the trim corrections, or the trim between the perpendiculars used to move the marks: lines 133 to 141 take the first, lines 162 and 163 the second.
• A density corrected to 15 °C, or a single sample taken at the surface near a river mouth, where fresh water lies over salt (code p. 36).
• The second trim correction dropped at large trim. It grows with the square of the trim: 47.06 t at 2.657 m in the example, 0.37 t at 0.288 m.
• Ballast tanks entered as “full” or “empty” without sounding, which the code rules out on page 30.

Signs of the trim corrections

The first trim correction takes its sign from the trim and the position of the center of flotation, and the second is always added. The table is the code’s line 162 table, with the trim measured between the perpendiculars.

Figures the UN/ECE code sets

The two-decimal rule is for recording. Rounding the M/M/M to two decimals before reading the tables moves the result: in the example, 12.48 m instead of 12.483 m is worth about 17 t at 67.40 t/cm. The calculator carries every figure at full precision and rounds only what it shows.

Where each figure comes from

The ship’s particulars, the length between perpendiculars and the positions of the draft marks, come from the approved hydrostatic tables and the draft mark plan. The lightship weight is the one in the approved trim and stability booklet.

The drafts are read from a boat or the quay at all six marks, as near the water as possible. The code has the midships drafts determined by measuring the freeboard from the deck line to the water on each side, whether or not the ship has midship marks; the depth that freeboard is taken from is the summer draft plus the summer freeboard on the ship’s International Load Line Certificate.² The density is read on a certified glass hydrometer, first amidships near the surface and near the keel, then at the bow and the stern: one reading at each end when the two amidships readings agree, two levels at each end when they do not. The density is the average of all the readings (code p. 27).

Deductibles come from sounding every ballast tank, with the ballast’s own density from a sample, and from the chief engineer’s figures for fuel, diesel, lube oil and fresh water, checked against soundings. At the survey after cargo work the code has only the ballast measured again, and the stores taken as the first figures less what was used in port, unless they have been moved or bunkered.

Each centimeter of draft is worth one TPC: on the example’s loaded ship, 67.40 t/cm, or 67.07 t after the density ratio. The hydrometer’s kg/l and the tables’ t/m³ are the same figure, since a tonne is 1,000 kg and a liter a thousandth of a cubic meter.³ A density error of 0.001 t/m³ is worth 76.94 t on the same survey, which is why the code wants four decimals.