dy t
dt1 y t1 4y t y t
2 3
0 04 10
( )
= - .( )
+( ) ( )
dy t
dt2 y t1 4y t y t y t
2 3 7
2
0 04 10 3 10 2
( )
= .( )
-( ) ( )
- ´( )
dy t
dt3 7y t2
3 10 2
( )
= ´( )
The Robertson reaction model is a typical example of the stiffness problem. Solve the Robertson ODE system from t = 0 to 3 and plot y2(t) versus time t. The initial conditions are y1(0) = 1 and y2(0) = y3(0) = 0.
2.20 Consider a catalytic fluidized bed in which an irreversible gas phase reaction A →B occurs.
The mass and energy balances along with the kinetic rate constant for this system are given as6,24:
dP
d P P dT
d T T
p p
t=0 1 320. + -321 , t =1752 269- +267 , dP
d P P K dT
d T T KP
p
p p
p p
t =1 88 10. ´ 3
{
-(
1+) }
, t =1 3.(
-)
+1 04 10. ´ 4 ,T0
T1 T2 T3
w0
M
Steam, Ts Steam, Ts Steam, Ts
M M
T1
w1
T2
w2
T3
w3
FIGURE P2.18 Continuous stirred heating tanks.
K exp
Tp
= ´ æ -
èç ö
ø÷ 6 10-4 20 7. 1500
where
T is the absolute temperature (°R)
P is the partial pressure of the reactant in the fluid (atm) Tp is the temperature of the reactant at the catalyst surface (°R) Pp is the partial pressure of the reactant at the catalyst surface (atm) K is the dimensionless reaction rate constant
τ is the dimensionless time
Solve the equation system in the range of 0 ≤τ≤ 1500 and plot changes of dependent variables as functions of time. Use the initial values of P(0) = 0.1, T(0) = 600, Pp = 0, and Tp = 761.
2.21 Estimate the amount of heat (cal) required to heat 1 g mole of propane from 200°C to 700°C at 1 atm. The heat capacity of propane is given by
Cp=2 41 0 057195. + . T-4 3 10. ´ -6T2
(
T:K d)
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3 Physical Properties
Physical property data of chemical compounds are parameters frequently required for simulating chemical processes and in designing chemical plants. However, such data are not readily available outside of certain proprietary simulation design packages or large commercial databases. Physical property data can be obtained by conducting experiments to measure the properties of individual substances or of mixtures. However, because of the multitude of chemical compounds that are of interest to the chemical engineer and the vast number of thermodynamic states in which they appear in combination, it is impossible to obtain all physical property data by experiments alone. Therefore, it is necessary to establish generalizations for estimating physical properties of mixtures using only limited available experimental information.
The estimation of thermodynamic properties of vapors and liquids plays a very important role in chemical engineering computations. Numerous predictions and correlations of thermodynamic physical property data have been presented—and students, scientists, and practicing engineers alike have utilized these correlations and interpolations to obtain data of sufficient accuracy for their individual purposes.
In this chapter, physical property data and correlations for typical liquids and gases are intro- duced and reviewed. The correlations described in this chapter are limited to those that the author has deemed to have the greatest value in common practical application. The main objectives of this chapter are to provide readers with various estimation procedures and correlations for a limited set of properties of typical chemical compounds and to present MATLAB® programs that analyze prop- erties for a range of variables and correlation constants.