Biomedical Engineering Reference
In-Depth Information
Conservation of mass, charge, energy, and
momentum can be very useful in developing
mathematical models for analysis of engineering
artifacts. In addition to conserved properties, there
are other extensive properties for which we know
limits on the generation/consumption terms. The
classic example of this is the
Second Law of
Thermodynamics
and its associated property,
entropy.
When written as a conservation equation, it is easy to
see that entropy can only be produced within
a system. Furthermore, in the limit of an internally
reversible process, the entropy production rate
reduces to zero.
3.
Determine the time period to be analyzed: When
a system undergoes a change in state, we say that the
system has undergone a
process.
It is frequently the
goal of engineering analysis to predict the behavior of
a system, i.e., the path of states that result, when it
undergoes a specified process. Processes can be
classified in three ways based on the time interval
involved: steady-state, finite-time, and transient
processes.
4.
Formulate a mathematical expression of the
conservation law: Experience has taught us that the
amount of an extensive property within a systemmay
change with time. This change can only occur by two
mechanisms:
Transport of the extensive property across the
system boundary
Generation (production) or consumption
(destruction) of the extensive property inside
the system
Thus, the change of an extensive property within
a system can be related to the amount of the
extensive property transported across the boundary
and the amount of the extensive property generated
(and/or consumed) within the system. This ''ac-
counting principle'' for an extensive property is
known to engineers as a
balance equation.
Although
this principle can be applied to a system for any
extensive property, it will be especially useful for
properties that are conserved. There are two forms
of a balance equation: the accumulation form and the
rate form.
When accounting for the input and output, the total
amount that enters the system (during the time period) is
computed and the total amount that exits in the same time
is subtracted. The accounting statement is:
*
+
¼
*
+
Net amount
accumulated
inside the system
Net amount
transported
into the system
þ
*
+
Net amount
generated
inside the system
or, for a conserved property
P
:
P
final
inside
P
initial
inside
¼ðP
i
P
o
ÞþðP
G
P
C
Þ
where the left-hand side is the net amount accumulated
inside; the first difference term on the right-hand side is the
netamounttransportedintothesystem(input-output);
and the second difference term on the right is the net
amount of the property generated (generated - consumed).
The advantage of using an accumulation form of the
conservation or accounting laws is that the mathematical
expression is in the form of either algebraic or integral
equations. The disadvantage of the accumulation form of
the law is that it is not always possible to determine the
amount of the property of interest entering or exiting
from the system.
The rate form
The rate form of a balance equation is similar to the ac-
cumulation form, except that the time period is in-
finitesimally small, so in the limit the net amounts
become rates of change. The mathematical relationship
between the accumulation form and the rate form can
easily be developed by dividing the accumulation form
through by the time interval
Dt
and taking the limit as
Dt
/ 0. That is:
*
Rate of change
inside the system
at time
t
+
*
Transport rate
into the system
at time
t
+
¼
*
Generation rate
into the system
at time
t
+
þ
2.1a.3.2 Formulating the mathematical
expression of conservation
or, for a conserved property
P
:
dP
dt
¼ð
_
P
i
_
P
o
Þþð
_
P
G
_
P
C
Þ
The accumulation form
In the accumulation form, the time period used in
the analysis is finite and is therefore used to formulate
equations thatmodel steady-state or finite-time processes.
where the first term on the right-hand side is the dif-
ference in transport rates across the system boundary,
and the second term is the difference in the rates at
