Hemodialysis Home Modeling
Hemodialysis Therapy
Derivation of
Rate Equations Example of Rate
Equation Derivation
Properties of rate equations
Rate
equations reveal much about how the dependent variable depends on the
independent variable, usually time, without actually solving for the functional
relationship between the two variables.
The usual form of the derivative in a rate equation is
which is read as the
rate of change of the dependent variable with respect to the independent
variable.
When the
independent variable is time, we can determine whether there is a steady-state
region where the dependent variable does not change with time. For some types of first-order rate
equations, we can use the time constant to determine how quickly the system
will approach such a steady-state region.
For some types of second-order rate equations, we can determine whether
the approach to steady-state will be direct or will oscillate above and below
the steady-state solution. Using the
technique of nondimensionalization, we can determine the dimensionless groups
which govern the system described by the rate equation.
Steady-State
Solutions of Rate Equations
First-order
rate equations describe the rate of change of the dependent variable with
respect to time. The question arises
whether there are regions or values for the dependent variable where it does
not change with time. If such regions
exist, what is the physical meaning associated with them?
Regions
where the value of the dependent variable does not change with respect to time
are referred to as steady-state solutions of the rate
equation. As the name steady-state implies,
the dependent variable value is "steady" and does not change. Occasionally, you will see a steady-state
solution referred to as an equilibrium
solution. When systems are at
thermodynamic equilibrium, there is no net driving force (potential) and values
do not change. However, steady-states
can exist even where externally imposed potential differences create variations
in thermodynamic variables such as temperature, pressure, or electric
potential. Think about a manufacturing
process that uses a conveyor belt moving through an oven. Such systems are in steady-state but not in
equilibrium. Likewise, steady-state
solutions are often referred to as the ultimate
or limiting solution of the rate
equation. This terminology results from
the fact that frequently the steady-state solution occurs when time becomes
infinitely great (which is indeed an ultimate or limiting value!). However, depending upon the rate equation,
steady-state solutions can occur at values of time other than infinity. To avoid confusion, it is preferable to use
the terminology of steady-state solution.
The
steady-state solution for a rate equation can be found by employing the
definition of the steady-state: a region where the value of the dependent
variable does not change with time, i.e., a region where the dependent variable
is constant. If the dependent variable is constant, then
the derivative of the dependent variable must be zero! Thus, the steady state solution is found
from
for any first-order rate equation.
First-Order
Approach to the Steady-State
Frequently, first-order rate equations
can be rearranged by simple algebraic manipulations to the form
where t is referred to as
the time constant and yss is the steady state
solution. (What are the units
associated with t?). This rate equation describes a system
initially at steady-state y0 that is suddenly subjected to a
stimulus which causes the system to move to the new steady-state yss. The time response of such a system is
depicted in the figure to the right.
Although shown for a new steady-state at a higher value than the initial
steady-state, the new steady-state could be at a lower value. The total change in the system, d,
is given by the difference between the two steady-states as
The time
constant determines how long the system takes to approach the new
steady-state. After one time constant,
the system is about 63% of the way to the new steady-state. After two time constants, the system is
about 86% of the way to the new steady-state.
After three time constants, the system is about 95% of the way to the
new steady-state. And, after four time
constants, the system is almost at the new steady-state. Thus, we can use the time constant to
estimate approximately how much time will elapse before the new steady-state is
achieved.
The
relationship between the time constant and approach to steady-state for rate
equations which can be put into the general form above comes from the
analytical solution to the equation.
The analytical solution is the exponential relation
(How might you show that is the analytical solution to the differential equation?) The change in y with respect to time is
embodied in the exponential term
The table at the right shows the value of this term for
values of time spaced one time constant apart. The last column in the table was
obtained by rearranging the analytical solution to the form
Hemodialysis Home Modeling Hemodialysis
Therapy
Derivation of
Rate Equations Example of Rate
Equation Derivation
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2001, John F Patzer II
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