Hemodialysis Home Modeling
Hemodialysis Therapy
Properties of
Rate Equations Example of Rate
Equation Derivation
Derivation of rate equations: conservation principles
Conservation
principles, in conjunction with the thermodynamic concept that potential differences
drive fluxes, can be used to develop rate equations that describe the behavior
of physical systems. Recall, from
physics, some of the conservation principles:
• conservation of
mass
• conservation of
momentum (linear and angular)
• conservation of
energy
• conservation of
charge
A more mathematical way to
state a conservation principle is
You can fill in the blank with the conserved quantity of
your choice. If we consider a finite
amount of time, say Dt, then we can write the conservation principle as
for any conserved quantity. A more specific mathematical notation for the accumulation is
That is, the accumulation in time period Dt is equal to the amount present at the end of the time
period, i.e., at time (t+Dt), less the amount present at the start of the time
period, i.e., at time t. Likewise, we can write for the gain and loss
terms
and
Thus, a conservation balance can be written as
This, in turn, can be rearranged to the form
Note, that if we take the limit as (Dt ® 0) of both sides of the above relation, the left hand
side looks suspiciously like the derivative of the conserved quantity, i.e.,
while the right hand side, which does not depend on Dt, remains unchanged.
Thus, the conservation balance produces the rate equation
Mathematical
application of a conservation principle, i.e., finding appropriate mathematical
expressions for bookkeeping to keep track of conserved quantities, leads
directly to formulation of rate equations through the definition of the
derivative. However, simple bookkeeping
is not sufficient. Understanding and
elucidating the basic physical and chemical phenomena of a situation are
required before one can develop a reasonable mathematical model to describe the
situation. As a result, mathematical models of physical systems are
only as good as the physics and chemistry which go into developing the
description. When developing
mathematical models one needs to keep in mind the actual situation versus model
approximations and simplifications.
Understanding and applying the thermodynamic concepts about potential
differences driving fluxes will aid in developing physically meaningful rate
equations.
Hemodialysis Home Modeling
Hemodialysis Therapy
Properties of
Rate Equations Example of Rate
Equation Derivation
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2001, John F Patzer II
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