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

**A TECHNIQUE FOR MASS BALANCE OF PROCESS DATA USING LINEAR
COMBINATION OF LSQ AND LRSQ**

**A. I. A.
Salama and T. Dabros**

CanmetENERGY-Devon

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

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**Abstract**

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).

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** 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 p_{1}
and p_{2} and the corresponding estimates are x_{1} and x_{2}
as shown in Fig 3. The constraint, x_{1}+x_{2}=1 is imposed
and labeled the “Solution line (SL)” (see Fig
3). In the LSQ case the objective function is defined as f=(x_{1}-p_{1})^{2}+(x_{2}-p_{2})^{2}.
The LSQ case solutions are concentric circles around the raw data point (p_{1},
p_{2}). The optimal solution is located when a concentric circle
becomes tangent to the SL and the solution is labeled X_{E}*. 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 p_{1} or p_{2 }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 (p_{1}, p_{2})
to the SL and is labeled X_{R}* (see Fig 3). The X_{R}*
solution indicates that: a) the corrections between X_{R}* and P are
proportional to the ratio of p_{1} and p_{2 }squared, and b) if
one of SCs, p_{1} or p_{2 },_{ }becomes zero the
corresponding optimal SCs, x_{1}* or x_{2}*, 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}.

**Extended Abstract:**File Uploaded

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