(371f) 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

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).

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

 
Demonstration

      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 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 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 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 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.