THE WEDGE FORMULA

Dear Petrol Inspector,

In my two decades or so of petroleum inspection, one of the daring challenges I have seen inspectors and Loss Adjusters face - most avoid, few face with trepidation is when (On Board Quantity – OBQ or Remaining On Board – ROB) on cargo tanks are so small that it is not captured by the MMC/UTI or the sounding tape or rod.

Most inspectors turn to the most popular, but now cliché "Unpumpable". They avoid the actual professional route to follow - and that is using the Wedge Formula.

What is Wedge Formula? 

Wedge Formula is a calculation to determine the small quantity of liquid or non-liquid cargo (On Board Quantity – OBQ or Remaining On Board – ROB) on cargo tanks. However, note that it should fulfil the following conditions:

• The vessel has huge trim, but there is no trim correction value available on the tank table,
• The liquid or cargo on tank is not touch one or more of the tank bulkheads,
• Observed sounding tanks at several points to ensure the cargo is not touch one of the tank bulkheads

See below diagram of a typical wedge condition:

In order to standardise the OBQ/ROB calculations on board the Crude Oil or white products carrying tanker vessels, the following geometric form of the Wedge Formula shall be used and this form of the formula assumes that the cargo tank is ‘box shaped’ with no internal ‘deadwood’ or pipeline systems, heating coils etc. which would impact the accuracy of the volume calculated from the sounding.

Furthermore this wedge formula calculation makes the enormous assumption that any ‘liquid’ found in a cargo tank is in the form of a regular wedge shape with its base at the aft bulkhead of the cargo tank.

It is obvious that such a series of assumptions normally can invalidate the absolute accuracy of the calculation immediately given, amongst other issues, the shape of the wing tanks (the turn of the bilge) and in particular those wing tanks at the fore and aft parts of the vessel.

The calculation method for the Geometric edition of the Wedge Formula: 

Assumption:

Given the small angle involved with the trim of the vessel, then the "Sine" of an angle can be considered as the same as the "Tangent"(Tan) of an angle and consequently: 

Step 1:
Correct the position of the sounding position with respect to the aft bulkhead of the cargo tank due to the trim of the vessel, distance = A
A = Tank Reference Height (Observed Height) x Tan X; 
    where X = the Trim angle of the vessel and;
           Tan X = (Aft draft – Forward draft) / Length Between Perpendiculars (L.B.P.) of the vessel.


Step 2: 
Determine the distance of the apex of the wedge from the aft bulkhead for obtaining information whether:

(1) Should a Wedge Formula be used at all (kindly note that a wedge formula is not applicable if:
     a. The liquid surface covers the total cargo tank bottom or the calculated apex of the wedge is at or          beyond the forward bulkhead of the cargo tank or:

     b. it is sludge ROB volumes only;

And

(2) Whether the wedge is a regular wedge (which can be checked by comparison with alternative soundings being taken).
       S = Observed Sounding;
       F (Distance of the apex of the wedge from the sounding position) = S x Tan X;
       E (Distance of the apex of the wedge to the aft bulkhead) = (F – A) + B;
          where B is the distance on deck from the point of sounding to the aft bulkhead.


Step 3:
Determine the depth of the wedge at the aft bulkhead of the cargo tank, depth = D;
D = E x Tan X


Step 4:
Knowing D (sounding depth at the aft bulkhead) and E (the distance from the aft bulkhead to the apex of the wedge), then the area of the longitudinal cross section of the wedge may be calculated,

Thus as the area of a triangle = (Base x Height) / 2 then;

(D x E) / 2 = cross sectional area of wedge.

Step 5:
Having obtained the cross sectional area of the wedge, the volume of the wedge is calculated by multiplication by the breadth of the cargo tank (please note that the breadth of the cargo tank should be measured at the bottom of the tank at the aft bulkhead position and not at deck level or elsewhere within the cargo tank).
Volume of theWedge = Cross sectional Area x Breadth of Tank

Throughout this calculation it is very important that all distances are in metres. Do not use centimetres for the observed sounding. 

Alternatives
Regardless above stated requirement, an I.S.O. standard method is also available in the event that any Cargo/Petroleum Inspector does not accept the geometric edition of the wedge formula. This method depends upon the accuracy of the vessel’s tank ullage calibration tables for the larger ullages / smaller soundings in the cargo tank. If the tank calibration tables are accurate for this region of the cargo tanks, then this method will give added accuracy to the general method of calculating tank residues after discharge.

This method is as follows:

Step 1: 
Calculate DA (the Corrected liquid sounding at the aft bulkhead position);
DA = D + {f(Y – (H x f))}
  where: D is the observed liquid sounding;
              f is the Trim factor ( TS / LS ); TS is the vessel’s trim, LS is the vessel’s Length Between                      Perpendiculars.;
             Y is the distance of the sounding point to the aft bulkhead;
             H is the reference height of the cargo tank;


Step 2: 
Calculate Ct (the Tank constant);
Ct = LS / ( 2 x TS x Lt ) (where Lt is the Length of the Cargo Tank).


Step 3: 
Calculate the ‘k‘ coefficient;
k = DA x Ct 
if k > 0.5 wedge is not required to be carried out;
if k = 0.5 wedge must be carried out.


Step 4:
if k > 0.5 then calculate the volume of the liquid contained in the cargo tank from the calibration tables using the Observed sounding, D, applying the trim corrections.


Step 5: 
if k = 0.5 then calculate DX (the wedge sounding).
DX = DA / 2


Step 6: 
Enter the cargo tank calibration tables with DX, without applying trim corrections to equivalent volume VO.


Step 7: 
Calculate the liquid wedge volume V1;
V1 = VO x 2 x k

In addition to above methods it should be noted that if the procedures as specified in the vessel’s COW manual are being followed for the determination of the ‘Dryness’ of a cargo tank, namely, the sounding of the residues in four (4) differing locations within the cargo tank, then the foregoing methods of calculations can be avoided.
Assuming the shape of the individual cargo tanks is fairly regular / constant in a fore and aft direction and, notwithstanding the fact that the vessel will be significantly trimmed by the stern, then the four measurements, as suggested in the COW Manual guidelines, as obtained by sounding can be used to calculate an average sounding so as to obtain a single sounding. The single average sounding can be used directly in order to obtain an equivalent volume from the vessel’s tank ullage calibration tables. Such a method will provide a clearer indication as to the type and nature of the residues on the cargo tank floor as well as provide much clearer indications as to the profile of the residues within the cargo tanks.

I understand that due to the technical mathematical computation involved in all the steps, it could be cumbersome - but there lies the difference between the truly professional inspector and the lazy charlatans.

Which are you?

Till next time -

....Keep inspecting properly & professionally!