168
L. Carlsen and K. Abit
in order to have the same orientation for all indicators, i.e. the higher the better
(Bruggemann and Carlsen 2006; Bruggemann and Patil 2011)
2. The eventual yield of AC (Yield)
3. The surface area of the produced AC (Surf) as an indicator for the sorption
characteristics of the AC
4. The method of activation (MoA) that can be either physical, i.e., without applying
any chemicals (apart from possibly water/steam) or chemical, where specific
chemicals are applied as part of the activation process. As itis believed that
avoiding using chemicals is preferable. Thus, the indicator values for chemical
and physical activation are set to 1 and 2, respectively.
Table 1 presents the characteristics of some studies of various materials used in
the production of AC, as a result of which a comparative characteristic was compiled
for various methods of obtaining AC.
2.2 Partial Ordering
Partial ordering appears in its basis pretty simple as the only mathematical relation
among the elements is “≤” (Bruggemann and Carlsen 2006; Bruggemann and
Münzer 1993; Bruggemann and Patil 2011). The basis for a comparison of elements
is a group of indicators (vide infra) that characterizes the elements. This series of
indicators, r j , is selected in order to be characteristic for the processes studied, i.e.,
the preparation of activated carbon. Thus, characterizing one of the methods for AC
production (x) by the a set of indicators r j (x), j = 1, . . . , m, where m is the number of
indicators, can be compared to another method (y), characterized by the indicators
r j (y), when
r j (y) ≤ r j (x) for all j = 1, . . . , m
( 1 )
Equation 1 is a very hard and strict requirement for establishing a comparison. It
demands that all indicators of method x should be better (or at least equal) than those
of method y. Further, let X be the group of methods included in the analyses, x will
be ordered higher (better) than y, i.e., x > y, if at least one of the indicator values for x
is higher than the corresponding indicator value for y and no indicator for x is lower
than the corresponding indicator value for y. On the other hand, if r j (x) > r j (y) for
some indicator j and r i (x) < r i (y) for some other indicator i, the methods x and y will
be denoted incomparable (notation: x | y) expressing the mathematical contradiction
due to conflicting indicator values. A set of mutual incomparable elements is called
an antichain. When all indicator values for x are equal to the corresponding indicator
values for y, i.e., r j (x) = r j (y) for all j, the two methods will have identical rank and
will be considered as equivalent, i.e., x ~ y. The analysis of Eq. 1 results in a graph,
L. Carlsen and K. Abit
in order to have the same orientation for all indicators, i.e. the higher the better
(Bruggemann and Carlsen 2006; Bruggemann and Patil 2011)
2. The eventual yield of AC (Yield)
3. The surface area of the produced AC (Surf) as an indicator for the sorption
characteristics of the AC
4. The method of activation (MoA) that can be either physical, i.e., without applying
any chemicals (apart from possibly water/steam) or chemical, where specific
chemicals are applied as part of the activation process. As itis believed that
avoiding using chemicals is preferable. Thus, the indicator values for chemical
and physical activation are set to 1 and 2, respectively.
Table 1 presents the characteristics of some studies of various materials used in
the production of AC, as a result of which a comparative characteristic was compiled
for various methods of obtaining AC.
2.2 Partial Ordering
Partial ordering appears in its basis pretty simple as the only mathematical relation
among the elements is “≤” (Bruggemann and Carlsen 2006; Bruggemann and
Münzer 1993; Bruggemann and Patil 2011). The basis for a comparison of elements
is a group of indicators (vide infra) that characterizes the elements. This series of
indicators, r j , is selected in order to be characteristic for the processes studied, i.e.,
the preparation of activated carbon. Thus, characterizing one of the methods for AC
production (x) by the a set of indicators r j (x), j = 1, . . . , m, where m is the number of
indicators, can be compared to another method (y), characterized by the indicators
r j (y), when
r j (y) ≤ r j (x) for all j = 1, . . . , m
( 1 )
Equation 1 is a very hard and strict requirement for establishing a comparison. It
demands that all indicators of method x should be better (or at least equal) than those
of method y. Further, let X be the group of methods included in the analyses, x will
be ordered higher (better) than y, i.e., x > y, if at least one of the indicator values for x
is higher than the corresponding indicator value for y and no indicator for x is lower
than the corresponding indicator value for y. On the other hand, if r j (x) > r j (y) for
some indicator j and r i (x) < r i (y) for some other indicator i, the methods x and y will
be denoted incomparable (notation: x | y) expressing the mathematical contradiction
due to conflicting indicator values. A set of mutual incomparable elements is called
an antichain. When all indicator values for x are equal to the corresponding indicator
values for y, i.e., r j (x) = r j (y) for all j, the two methods will have identical rank and
will be considered as equivalent, i.e., x ~ y. The analysis of Eq. 1 results in a graph,