261049 A Technique for Mass Balance of Process Data Using Linear Combination of LSQ and Lrsq

Tuesday, October 30, 2012: 5:20 PM
404 (Convention Center )
Ahmed I.A. Salama, CanmetENERGY-Devon, Natural Resources Canada, Devon, AB, Canada


A. I. A. Salama and T. Dabros


Natural Resources Canada

Suite A202, #1 Oil Patch Drive, Devon, Alberta, Canada T9G 1A8

Phone (780) 987-8635, Fax (780) 987-8676

E-mail: Ahmed.Salama@NRCan-RNCan.gc.ca



A technique is introduced for mass balance of raw data of a flume process (used to simulate depositions of solids on the beach of oil sands tailing ponds). The proposed objective function of the optimization problem is a linear combination of the absolute and the relative error squares.  In the conventional least-error-squares (LSQ) case, the stream component (SC) errors are scaled by the standard deviations of the raw SC data errors, however, in the least-relative-error-squares (LRSQ) case; the errors are referenced to the raw SC values.  In the LRSQ case zero stream-components in the raw data remain zeros in the optimal estimates which is not a feature in the LSQ case.  Such advantage is important in applications where some SCs are single component (i.e., water, steam, hydrocarbon solvent).


Keywords: Least-error-squares (LSQ) technique, Least-relative-error-squares (LRSQ) technique, Mass balance technique, Flume test.


      Let us consider a two-dimensional case. The raw SCs are p1 and p2 and the corresponding estimates are x1 and x2 as shown in Fig 3.  The constraint, x1+x2=1 is imposed and labeled the “Solution line (SL)” (see Fig 3).  In the LSQ case the objective function is defined as f=(x1-p1)2+(x2-p2)2.  The LSQ case solutions are concentric circles around the raw data point (p1, p2).  The optimal solution is located when a concentric circle becomes tangent to the SL and the solution is labeled XE*.  Close observation of the LSQ solution indicates that: a) for all raw points on the lines parallel above and below the SL generates the same adjustments and b) either one of the SCs p1 or p2 becomes greater than one the optimal solutions have negative values.  Both results are disadvantages.  The LRSQ solution is the directed line from the raw data point (p1, p2) to the SL and is labeled XR* (see Fig 3).  The XR* solution indicates that: a) the corrections between XR* and P are proportional to the ratio of p1 and p2 squared, and b) if one of SCs, p1 or p2 , becomes zero the corresponding optimal SCs, x1* or x2*, are also zero.  In this case both results are advantageous.  It should be emphasized that Fig 3 is only given as demonstration but no further conclusions can be drawn.

A raw data set collected around the CanmetENERGY-Devon flume unit is used to demonstrate the results.  The proposed technique is implemented in spread sheet format using Microsoft Excel TM.

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