1
CHAPTER
FIVE
REACTION ENGINEERING
5.1. Determination of Kinetic Parameters of the Saponification Reaction in a PFR
5.2. Determination of Kinetic Parameters of the Saponification Reaction in a CSTR
5.3. Experimental and Numerical Determination of Kinetic Parameters of the
Saponification Reaction in a Series of CSTRs
5.4. Determination of Optimal Agitation Conditions in a Stirred Tank
5.5. Determination of Residence Time Distribution in a Packed Bed
5.6. Modeling of Water Level in a Tank under Steady State and Dynamic Conditions
2
5.1. DETERMINATION OF KINETIC PARAMETERS OF THE SAPONIFICATION
REACTION IN A PFR
Keywords: Tubular reactor, plug flow reactor, saponification, integral method, differential
method.
Before the experiment: See your TA before the experiment. Read the booklet carefully. Be aware
of the safety issues. Make sure to do all solution preparation calculations before you come to lab.
Ethyl Acetate Mol wt.: 88.10, Density: 0.898 g/cm3 (liquid)
Sodium Hydroxide Mol wt.: 40.00 (pellet)
5.1.1. Aim
To study the saponification of ethyl acetate in a plug-flow reactor and to determine the kinetic
parameters using integral and differential methods.
5.1.2. Theory
5.1.2.1. Plug Flow Reactor
In a tubular reactor, the feed enters at one end of a cylindrical tube and the product stream leaves at
the other end. The long tube and the lack of stirring prevent complete mixing of the fluid in the
tube. Hence the properties of the flowing stream will vary from one point to another, namely in both
radial and axial directions.
In the ideal tubular reactor, which is called the “plug flow” reactor, specific assumptions are made
about the extent of mixing:
1.
no mixing in the axial direction, i.e., the direction of flow
2.
complete mixing in the radial direction
3.
uniform velocity profile across the radius.
The absence of longitudinal mixing is what defines this type of reactor [1]. The validity of the
assumptions will depend on the geometry of the reactor and the flow conditions. Deviations, which
are frequent but not always important, are of two kinds:
3
1.
mixing in longitudinal direction due to vortices and turbulence
2.
incomplete mixing in radial direction in laminar flow conditions
For a time element t and a volume element V at steady state, the mass balance for species ‘i’ is
given by Eq. 5.1.1
vA CA │v t- vA CA│v+Δv t - rAVt = 0
(5.1.1)
where
vA : volumetric flow rate of reactant A to the reactor, L/s
CA : concentration of reactant A, mol/L
rA : rate of disappearance of reactant A, mol/Ls
Dividing Eq. 5.1.1 by V and t results in Eq. 5.1.2
𝑣𝐴 𝐶𝐴 |𝑉 − 𝑣𝐴 𝐶𝐴 |𝑉+∆𝑉
r𝐴
=
∆𝑉
𝑣𝐴
(5.1.2)
and taking limit as V  0 gives Eq. 5.1.3.
dC𝐴
r𝐴
= −
dV
𝑣𝐴
(5.1.3)
Eq. 5.1.4 is the relationship between concentration and size of reactor for the plug flow reactor.
Here rate is a variable, but varies with longitudinal position (volume in the reactor, rather than with
time).
dV
dC𝐴
= −
𝑣𝐴
r𝐴
(5.1.4)
The conversion, X, is defined by Eq. 5.1.5
X =
(initial concentration – final concentration)
(initial concentration)
(5.1.5)
and Eq. 5.1.4 can be rewritten as Eq. 5.1.6.
dV
dX
= −
𝑣𝐴 C𝐴0
r𝐴
Equation 5.1.6 can be used with differential method to obtain the reaction parameters [2].
(5.1.6)
4
If the equation 5.1.4 is integrated with following boundary conditions,
At the entrance:
V=0
CA = CA0 (inlet concentration of reactant A)
At the exit:
V = VR (total reactor volume)
CA = CA (exit conversion)
V𝑅
∫
0
C𝐴
dV
dC𝐴
= −∫
𝑣𝐴
C𝐴𝑂 r𝐴
(5.1.7)
With the elementary reaction assumption, reaction rate equation can be simplified as in Eq. 5.1.8.
where C𝐵 is concentration of reactant B and 𝜃𝐵 is the ratio of inlet concentrations of reactant B and
A (C𝐵0 /C𝐴0 )
r𝐴 = 𝑘C𝐴 C𝐵
(5.1.8)
C𝐴 = C𝐴0 (1 − 𝑋); C𝐵 = C𝐴0 (𝜃𝐵 − 𝑋)
Combining equations 5.1.7 and 5.1.8, the following expression (Eq. 5.1.9) is obtained.
V𝑅
∫
0
C𝐴
dV
dC𝐴
= −∫
Q𝐴
C𝐴𝑂 kC𝐴 C𝐵
(5.1.9)
It can be written in terms of conversion as Eq. 5.1.10
V𝑅
∫
0
C𝐴
dV
dC𝐴
= −∫
2
Q𝐴
C𝐴𝑂 kC𝐴0 (1 − 𝑋)(𝜃𝐵 − 𝑋)
(5.1.10)
This expression can be solved using the integral method to determine the reaction parameters [3].
5.1.2.2. Saponification Reaction
Sodium Hydroxide + Ethyl Acetate → Sodium Acetate + Ethanol
The reaction is 2nd order elementary within the range of 0-0.1 M concentration and 20-40 °C.
5
5.1.2.3. The Conductivity and Concentration Relations
The conductivity of the reaction mixture changes with conversion and therefore the extent of the
reaction can be monitored by recording the conductivity with respect to time. A calibration curve is
needed to relate conductivity data to concentration values.
5.1.3. Experimental Setup
The apparatus used in this experiment is shown in Figures 5.1.1 and 5.1.2. The plug flow reactor is
0.4 L.
Figure 5.1.1. Chemical reactor service unit.
Figure 5.1.2. Tubular reactor.
6
5.1.4. Procedure
1. Prepare 100 ml of 0.05 M NaOH solution in a 1000 ml beaker, record conductivity data.
2. Add 100 ml distilled water to the beaker to dilute the NaOH solution, make sure it is perfectly
mixed, and record conductivity data. Repeat this step six more times to prepare a calibration
curve.
3. Prepare 5 L of 0.05 M ethyl acetate and 0.05 M NaOH solutions. Pour these solutions into the
feed tanks.
4. Adjust a constant flow rate by setting the pump speeds of both reactants.
5. Record the conductivity when steady state is reached.
6. Repeat the same procedure for different flow rates.
Safety Issues: In this experiment, sodium hydroxide (NaOH) and ethyl acetate (EtAc) will be used
as reactants. NaOH is poisonous and corrosive. It may be fatal if swallowed and harmful if inhaled.
It causes burns to any area of contact and reacts with water, acids and other materials. EtAC is a
flammable liquid and vapor. It causes eye irritation. Breathing it may cause drowsiness and
dizziness. It may also cause respiratory tract irritation. Prolonged or repeated contact causes
defatting of the skin with irritation, dryness, and cracking. During the experiment, beware of risks
when handling with reactants. Make sure all the tanks are emptied, and all the electronic devices are
unplugged at the end of the experiment. In case of eye or skin contact, remove any contact lenses or
contaminated clothing or shoes. Immediately flush eyes or skin with plenty of cold water for at least
15 minutes. Cover the irritated skin with an emollient. In case of inhalation, remove to fresh air. If
breathing is difficult get medical attention immediately. In case of ingestion, do not induce
vomiting. Never give anything by mouth to an unconscious person. For further information, check
MSDS of sodium hydroxide and ethyl acetate [4, 5]. The solutions used in this experiment are dilute
in terms of the chemicals, therefore no need to a special treatment.
5.1.5. Report Objectives
1. Evaluate the data using the Integral Method. Calculate conversion.
2. Evaluate the rate constant using the Differential Method.
3. Compare the results of the two methods.
7
References
1. Perry, R.H. and D. Green, Perry’s Chemical Engineers’ Handbook, 8th edition, Mc Graw-Hill,
New York, 2008.
2. Levenspiel, O., Chemical Reaction Engineering, 3rd edition, Cambridge University, 1999.
3. Fogler, H. S., Elements of Chemical Reaction Engineering, 4th edition, Prentice-Hall Inc., 2006.
4. Sodium Hydroxide MSDS, http://www.sciencelab.com/msds.php?msdsId=9924998.
5. Ethyl Acetate MSDS, http://www.sciencelab.com/msds.php?msdsId=9927165.
8
5.2. DETERMINATION OF KINETIC PARAMETERS OF THE SAPONIFICATION
REACTION IN A CSTR
Keywords: Continuous stirred tank reactors, saponification, mathematical modeling, differential
equations.
Before the experiment: See your TA. Read the booklet carefully. Be aware of the safety
precautions. Make sure to do all solution preparation calculations before you come to lab.
Ethyl Acetate Mol wt.: 88.10, Density: 0.898 g/cm3 (liquid)
Sodium Hydroxide Mol wt.: 40.00 (pellet)
5.2.1. Aim
To study the dynamics of a CSTR during different stages of its continuous operation by using the
saponification of ethyl acetate reaction.
5.2.2. Theory
5.2.2.1. Continuous-Stirred Tank Reactors
Continuous-stirred tank reactors (Fig. 5.2.1) are used very commonly in industrial processes. For
this type of reactor, mixing is complete, so that the temperature and the composition of the reaction
mixture are uniform in all parts of the vessel and are the same as those in the exit stream [1].
Figure 5.2.1. Continuous stirred tank reactor.
Three stages of the continuous operation of a CSTR can be modeled.
9
1.
From beginning to overflow
2.
From overflow to steady state
3.
Steady state operation
The first and second stages are transient and they produce differential equations. The third stage is
represented by a steady state model which contains algebraic equations.
Stage One
This stage is semibatch. There is no output because the reactor contents do not yet reach the
overflow level. With assuming that the saponification reaction of ethyl acetate with sodium
hydroxide is second order overall, a material balance on either NaOH or ethyl acetate (both
reactants are at the same concentration and flow rate) gives Eq. 5.2.1 or 5.2.2:
rate of accumulation = rate of input - rate of consumption
𝑑(𝑉𝐶)
𝑑𝑡
= 𝑣𝐶0 − 𝑉𝑘𝐶 2
(5.2.1)
or
𝑉
𝑑𝐶
𝑑𝑡
+𝐶
𝑑𝑉
𝑑𝑡
= 𝑣𝐶0 − 𝑉𝑘𝐶 2
(5.2.2)
where
C = concentration
(M)
C0 = initial concentration
(M)
𝑣 = volumetric flow rate
(L/min)
k = reaction rate constant
t = time
V = volume of reactor
(min)
(L)
But ‘V’ is a function of time, and since the system is of constant density and flow rate, a total mass
balance gives:
𝑑𝑉
𝑑𝑡
=𝑣
since at t  0, V  0. Eq. 5.2.2 gives Eq. 5.2.3
or
𝑉 = 𝑣𝑡
10
𝑑𝐶
𝑑𝑡
=
𝐶0
𝑡
𝐶
− − 𝑘𝐶 2
(5.2.3)
𝑡
Eq. 5.2.3 is subject to C  C 0 at t  0 .
Stage Two
The second stage is continuous but not yet steady. The concentration is changing with time but the
volume of the reactants is constant. A material balance takes the form of equations 5.2.4 and 5.2.5
rate of accumulation = rate of input - rate of output - rate of consumption
𝑉
𝑑𝐶
𝑑𝑡
= 𝑣𝐶0 − 𝑣𝐶 − 𝑉𝑘𝐶 2
(5.2.4)
and therefore,
𝑑𝐶
𝑑𝑡
=
𝐶0
𝑡
𝐶
− − 𝑘𝐶 2
(5.2.5)
𝑡
where
𝑇 =𝑡−𝜏
(min)
𝑡 = 𝑉/𝑣
(min)
At steady state, C  C s , which is a particular solution to Eq. 5.2.5.
Stage Three
This is the easiest stage to model. A material balance results in equations 5.2.6 and 5.2.7
rate of input = rate of output + rate of consumption
𝑣𝐶0 = 𝑣𝐶𝑠 + 𝑉𝑘𝐶𝑠 2
(5.2.6)
𝑘𝜏𝐶𝑠 2 + 𝐶𝑠 − 𝐶0 = 0
(5.2.7)
The calculation of the specific rate constant k can be carried out by the Eq. 5.2.8,
𝑘=
(𝑣𝐴+ 𝑣𝐵 ) (𝐶𝐴0 −𝐶𝐴 )
𝑉
𝐶𝐴
2
L / mol.s
(5.2.8)
11
For a reactant A in a reactor operating at steady state, the volume (V) may be assumed constant and
the steady state concentration of NaOH in the reactor (CA) may be used to calculate the specific rate
constant (k) [2, 3].
In this experiment, the kinetic parameters of the saponification reaction will be calculated based on
the conductivity data collected during the experiment performed in a CSTR, and the results will be
compared with the literature. Using a computational tool, NaOH concentration on stream and the
rate constant will be determined and compared with the experimental data.
5.2.2.2. The Conductivity and Concentration Relations
The conductivity of the reaction mixture changes with conversion and therefore the extent of the
reaction can be monitored by recording the conductivity with respect to time. A calibration curve is
needed to relate conductivity data to concentration values.
5.2.3. Experimental Setup
The apparatus used in this experiment is shown in Figures 5.2.2 and 5.2.3. The reactor volume in
this experiment is 1.6 L.
CSTR (see Fig. 5.2.3.)
Figure 5.2.2. Chemical reactor service unit.
12
Figure 5.2.3. Continuous stirred tank reactor.
5.2.4. Procedure
1. Prepare 100 ml of 0.05 M NaOH solution in a 1000 ml beaker, record conductivity data.
2. Add 100 ml distilled water to the beaker to dilute the NaOH solution, make sure it is perfectly
mixed, and record conductivity data. Repeat this step six more times to prepare a calibration
curve.
3. Make up 5 liter batches of 0.05 M sodium hydroxide and 0.05 M ethyl acetate.
4. Remove the lids of the reagent vessels and carefully fill the reagents. Refit the lids.
5. Set the pump speeds of both reactants to give 50 ml/min flow rate.
6. Set the agitator speed controller to 7.0.
7. Switch on the feed pumps and agitator motor. Start the stopwatch.
8. Collect conductivity data each minute for 45 minutes.
Safety Issues: In this experiment, sodium hydroxide (NaOH) and ethyl acetate (EtAc) will be used
as reactants. NaOH is poisonous and corrosive. It may be fatal if swallowed and harmful if inhaled.
It causes burns to any area of contact and reacts with water, acids and other materials. EtAC is a
flammable liquid and vapor. It causes eye irritation. Breathing it may cause drowsiness and
dizziness. It may also cause respiratory tract irritation. Prolonged or repeated contact causes
defatting of the skin with irritation, dryness, and cracking. During the experiment, beware of risks
when handling with reactants. Make sure all the tanks are emptied, and all the electronic devices are
unplugged at the end of the experiment. In case of eye or skin contact, remove any contact lenses or
contaminated clothing or shoes. Immediately flush eyes or skin with plenty of cold water for at least
15 minutes. Cover the irritated skin with an emollient. In case of inhalation, remove to fresh air. If
breathing is difficult get medical attention immediately. In case of ingestion, do not induce
vomiting. Never give anything by mouth to an unconscious person. For further information, check
13
MSDS of sodium hydroxide and ethyl acetate [4, 5]. The solutions used in this experiment are dilute
in terms of the chemicals, therefore no need to a special treatment.
5.2.5. Report Objectives
You may assume that the saponification reaction of ethyl acetate with sodium hydroxide is second
order overall.
1. Prepare a calibration chart for NaOH concentration and conductivity.
2. Prepare a spreadsheet to calculate sodium hydroxide concentration, sodium acetate concentration
and the degree of conversion of sodium hydroxide and sodium acetate for each of the
conductivity data taken over the period of the experiment.
3. Draw sodium hydroxide concentration versus time, sodium acetate concentration versus time,
and the degree of conversion of sodium hydroxide versus time graphs.
4. Calculate experimental specific rate constant (k) from material balance and compare it with the
one obtained from Arrhenius equation using the data found from literature.
5. Draw sodium hydroxide concentration versus time using MATLAB starting from differential
equation obtained from material balance, estimate steady state concentration of sodium
hydroxide and compare it with the experimental one.
6. Derive all of the equations you use in the theory part.
References
1. Fogler, H. S., Elements of Chemical Reaction Engineering, 4th edition, Prentice-Hall Inc., 2006.
2. Levenspiel, O., Chemical Reaction Engineering, 3rd edition, Cambridge University, 1999.
3. Perry, R.H., and D. Green, Perry’s Chemical Engineers’ Handbook, 7th edition, Mc Graw-Hill,
New York, 1997.
4. Sodium Hydroxide MSDS, http://www.sciencelab.com/msds.php?msdsId=9924998.
5. Ethyl Acetate MSDS, http://www.sciencelab.com/msds.php?msdsId=9927165.
14
5.3.
EXPERIMENTAL
AND
NUMERICAL
DETERMINATION
OF
KINETIC
PARAMETERS OF A REACTION IN A SERIES OF CSTRS
Keywords: CSTR, CSTR in series, saponification, conversion.
Before the experiment: Make sure to do all solution preparation calculations before you come to
lab. For further information about the chemicals, look over [1, 2].
Ethyl Acetate Mol wt.: 88.10, Density: 0.898 g/cm3 (liquid)
Sodium Hydroxide Mol wt.: 40.00 (pellet)
5.3.1. Aim
To observe transient changes in concentrations of three CSTRs in series using of experimental
measurements and computational methods and to determine the rate constant of the saponification
reaction under steady state conditions.
5.3.2. Theory
In continuously operated chemical reactors, the reactants are pumped at constant rate into the
reaction vessel and the chemical reaction takes place as the reaction mixture flows. The reaction
products are continuously discharged to the subsequent separation and purification stages. The
extent of the required separation and purification process depends on the efficiency of the reactor,
and so the selection of the correct type of the reactor for a given duty is most important since the
economics of the whole process could hinge on this choice. Normally, the efficiency of a chemical
reactor is measured by its ability to convert the reactants into the desired products with the
exclusion of unwanted by-products; this is measured by yield. However, additional factors such as
safety, ease of control and stability of the process must also be considered. Many of these factors
depend on the size and shape of the reactor. The size of reactor for a purposed feed rate depends on
the reaction kinetics of the materials undergoing chemical change and on the flow conditions in the
reactor. On the other hand, the flow conditions are determined by the cross sectional area of the
path through which the reaction mixture flows; i.e. the shape of the reactor. Thus, it is apparent that
size and shape are interrelated factors, which must be taken together when considering continuous
reactors [3].
15
5.3.2.1. Continuously Stirred Tank Reactors (CSTRs)
CSTRs are usually cylindrical tanks with stirring provided by agitators mounted on a shaft inserted
through the vessel lid. In addition, the tank is fitted with auxiliary equipment necessary to maintain
the desired reaction temperature and pressure conditions [3].
The well-stirred tank reactor is used mostly for liquid phase reactions. In normal operation, a steady
continuous feed of reactants is pumped into the vessel and since there is usually negligible density
change on reaction, an equal volume of the reactor contents is displaced through an overflow pipe
situated near the top of the vessel [3].
F0
C0
F1
C1
C1
VR
Figure 5.3.1. Single continuous stirred tank reactor.
For a single stirred tank reactor (Figure 5.3.1), it is assumed that the design of the agitator provides
complete mixing of the vessel contents to achieve uniform temperature and composition distribution
throughout the vessel. This premise of complete mixing then implies that the reactor outlet stream is
identical in temperature and composition to the bulk reactor contents [4].
For an effective reactor volume ‘V’, molar input and output ‘F’, and moles N the mole balance for
component A is given by Eq. 5.3.1:
𝐹𝐴0 − 𝐹𝐴1 + 𝑟𝐴 𝑉𝑅 =
𝑑𝑁𝐴
𝑑𝑡
(5.3.1)
For an irreversible reaction, the reaction rate term defined as Eq. 5.3.2 [3]:
𝑟𝐴 = 𝑘𝐶𝐴1 𝑛
(5.3.2)
The reaction rate 𝑟𝐴 is constant within the reactor and time when steady state operation has been
established. The right hand side of the Equation 5.3.1 vanishes under steady state operation
therefore the equation becomes algebraic.
16
The conversion of the reactant can be defined as Eq. 5.3.3:
𝑥=
𝐹𝐴0 − 𝐹𝐴1
𝐹𝐴0
(5.3.3)
Combining equations 5.3.1 and 5.3.3 under steady-state conditions, the design equation of a single
CSTR is obtained as Eq. 5.3.4 [4].
𝑉𝑅 = 𝐹𝐴0
𝑥
−𝑟𝐴
(5.3.4)
Single Tanks with Simple Reactions
For a first order reaction under steady state conditions, dividing Eq. 5.3.1 by volumetric flow rate
yields 𝑄 (𝑣𝑜𝑙𝑢𝑚𝑒⁄𝑡𝑖𝑚𝑒) as in Eq. 5.3.5 [3]:
𝐶𝐴0 = 𝐶𝐴1 (1 +
𝑘𝑉𝑅
)
𝑄
(5.3.5)
𝑉𝑅 ⁄𝑄 is the space time and will be given the symbol ‘’. Hence, Eq. 5.3.6 forms as
𝐶𝐴1
1
=
𝐶𝐴0 (1 + 𝑘𝜏)
(5.3.6)
Since the input and the outlet volumetric flow rates are equal, the conversion ‘𝑥’ can also be written
as Eq. 5.3.7:
𝑥=
𝐶𝐴0 − 𝐶𝐴1
1
=1−
(1 + 𝑘𝜏)
𝐶𝐴0
(5.3.7)
and as is usual with first order reactions the result is independent of the feed concentration.
For a second order reaction starting from the overall mole balance around each tank individually at
steady state conditions, Equation 10.3.1 under equimolar feed becomes as Eq. 5.3.8:
𝐶𝐴0 = 𝐶𝐴1 + 𝑘𝜏𝐶𝐴1 2
(5.3.8)
and the positive root of this quadratic equation gives Eq. 5.3.9:
𝐶𝐴1 =
(−1 + √1 + 4𝑘𝜏𝐶𝐴0 )
2𝑘𝜏
(5.3.9)
17
hence the conversion equation becomes as Eq. 5.3.10,
𝑥 =1+
−1 + √1 + 4𝑘𝜏𝐶𝐴0
2𝑘𝜏𝐶𝐴0
(5.3.10)
Transient Behavior
Under unsteady state conditions the mass balance turns out to be an ordinary differential equation
that can be solved numerically in order to simulate the theoretical behavior of concentration and
conversion with respect to time. For a second order reaction this equation is derived from the Eq.
5.3.1 by substituting the expressions of concentration, space time, and conversion as Eq. 5.3.11 [4].
𝑑𝑥
𝑥
= 𝑘𝐶𝐴0 (1 − 𝑥)2 −
𝑑𝑡
𝜏
(5.3.11)
where
𝑥 = conversion of reactant A
𝐶𝐴0 = Initial concentration of A
𝑘 = Reaction rate constant
𝜏 = Space time (V/Q)
Multiple Tank Cascade with Simple Reactions (CSTRs in Series)
A major shortcoming of a single stirred tank is that all of the reaction takes place at the low final
reactant concentration and hence requires an unduly large reactor hold-up. If a number of smaller
well-stirred reactors are arranged in series, only the last one will have a reaction rate governed by
the final reactant concentration and all of the others will have higher rates. Hence, for a given duty
the total reactor hold-up will be less than for a single tank. This saving in reactor volume increases
as the required fractional conversion increases and also as the number of installed tanks increases.
In fact, all of the desirable features of the CSTR may be retained while the low hold-up
characteristics of a plug flow tubular reactor are approached, e.g. five to ten tanks in a cascade are
likely to give high conversion values at low hold-up times similar to a pug-flow reactor. It is a
matter of economics to balance the cost of the number of tanks against their reduced size [3].
There are other operational advantages in carrying out reactions in series of stirred tanks. For
example if one vessel in the cascade has to be put out of commission for any reason, it may be by-
18
passed and production continued at a slightly reduced rate whereas failure of a single CSTR would
entail complete loss in production [3].
5.3.2.2. Saponification Reaction
NaOH
+
CH3COOC2H5

CH3COONa
+
C2H3OH
Sodium Hydroxide
+
Ethyl Acetate

Sodium Acetate
+
Ethyl Alcohol
(A)
(B)
(C)
(D)
The reaction can be considered first order with respect to sodium hydroxide and ethyl acetate i.e.
second order overall, within the limits of concentration (0-0.1M) and temperature (20-40°C) studied
[5].
To calculate the specific rate constant k, the overall mass balance may be written as [4]:
Rate of change within the reactor = Input – Output + Accumulation
i.e. for NaOH in a reactor operating at steady state the volume may be assumed constant. The steady
state concentration of NaOH in reactor (CA) may be used to calculate the specific rate constant (k)
as in Eq. 5.3.12 and Eq. 5.3.13:
𝑘=
𝑘=
𝑄 (𝐶𝐴0 − 𝐶𝐴 )
𝑉
𝐶𝐴2
(𝑄𝐴 + 𝑄𝐵 ) (𝐶𝐴0 − 𝐶𝐴 )
𝑉
𝐶𝐴2
(5.3.12)
(5.3.13)
5.3.2.3. Conductivity and Concentration Relations
The conductivity of the reaction mixture changes with conversion and therefore the extent of the
reaction can be monitored by recording the conductivity with respect to time. Conductivity data
obtained throughout the experiment has to be converted into concentration data of the system. A
calibration chart showing the relation between the conductivity of the fluid and the concentration of
the ionized material can be obtained by measuring known concentrations of the material's
conductivity.
19
The reaction carried out in a Continuous Stirred Tank Reactor or series CSTRs eventually reaches
steady state when a certain amount of conversion of the starting reagents has taken place. The
steady state conditions will depend on concentration of reagents, flow rate, volume of reactor and
temperature of reaction.
In this experiment, the reaction rate constant for the saponification reaction will be calculated based
on the conductivity data collected during the experiment carried on three CSTR in series and
compared with the literature value. In addition, the final conversion will calculated numerically
using suitable computational tools.
5.3.3. Experimental Setup
The experimental setup used in this experiment is shown in Figure 5.3.2.
Figure 5.3.2. Chemical reactor service unit.
5.3.4. Procedure
1. Prepare a calibration curve for the conductivity vs. concentration of NaOH (for the solutions
with the concentrations 0.005M, 0.01M, 0.02M, 0.03M, 0.04M, 0.05M).
2. Prepare 3.0 liter batches of 0.05M sodium hydroxide and 0.05M ethyl acetate.
3. Remove the lids of the reagent tanks and carefully, fill with the reagents. Refit the lids.
4. Set the pump speed control to give 50 ml/min flow rate.
5. Start agitators.
6. Switch on both feed pumps and agitator motor, and start recording conductivity.
7. Collect conductivity data for 30 minutes in 30 seconds intervals.
Safety Issues: In this experiment, sodium hydroxide (NaOH) will be used as reactant. NaOH is
poisonous and corrosive. It may be fatal if swallowed and harmful if inhaled. It causes burns to any
20
area of contact and reacts with water, acids and other materials Prolonged or repeated contact
causes defatting of the skin with irritation, dryness, and cracking. During the experiment, beware of
risks when handling with reactant. Make sure all the tanks are emptied, and all the electronic
devices are unplugged at the end of the experiment. In case of eye or skin contact, remove any
contact lenses or contaminated clothing or shoes. Immediately flush eyes or skin with plenty of cold
water for at least 15 minutes. Cover the irritated skin with an emollient. In case of inhalation,
remove to fresh air. If breathing is difficult get medical attention immediately. In case of ingestion,
do not induce vomiting. Never give anything by mouth to an unconscious person. For further
information, check MSDS of sodium hydroxide and ethyl acetate [6]. The solutions used in this
experiment are dilute in terms of the chemicals, therefore no need to a special treatment. After the
experiment make sure all the tanks are emptied, and all the electronic devices are unplugged.
5.3.5. Report Objectives
1. Find the reaction rate constants for each CSTR starting from the design equation of a CSTR.
2. Find the rate constant of the reaction from the literature and calculate the error.
3. Find the NaOH conversion corresponding to each measurement.
4. Draw conversion vs time graphs for each reactor during the whole experiment (Draw on the
same graph).
5. Solve the differential equations numerically to find the conversions using MATLAB or
POLYMATH software (Use literature value for the reaction rate constant).
6. Compare the conversion values obtained from numerical solution with experimental
observations, showing experimental and numerical conversions on a single graph.
References
1. http://akkimyaas.com/urunler/aspx.
2. http://www.merckmillipore.com/pharmaceutical-ingredients/
3. Cooper, A. R. and G. V. Jeffreys, Chemical Kinetics and Reactor Design, Oliver & Boyd,
Norwich, 1971.
4. Fogler, H. G., The Elements of Chemical Reaction Engineering, 2nd edition, Prentice Hall, New
York, 1992.
5. Perry, R. H. and D. Green, Perry’s Chemical Engineers’ Handbook, 6th edition, Mc Graw-Hill,
New York, 1988.
6. Sodium Hydroxide MSDS, http://www.sciencelab.com/msds.php?msdsId=9924998.
21
5.4. DETERMINATION OF OPTIMAL AGITATION CONDITIONS IN A STIRRED TANK
Keywords : Mixing, baffle, propeller, impeller, mixing efficiency, impeller Reynolds number.
Before the experiment: Read the booklet carefully. Be aware of the safety issues. See your TAs.
5.4.1. Aim
To observe various flow patterns with respect to different sizes of impellers and baffles and to
investigate the effect of size and types of agitators on mixing efficiency and power consumption.
5.4.2. Theory
Agitation means forcing a fluid mechanically such that it flows with any type of flow pattern inside
a vessel. The operation of agitation, which includes mixing as a special case, is now well
established as an important component in a wide variety of chemical processes. Mixing can be
defined as putting two or more separate phases together and allowing them to be randomly
distributed with respect to one another. Specifically, agitators are applied to three general classes of
problems:
(i)
To produce static or dynamic uniformity in multicomponent multiphase systems,
(ii)
To faciliate mass or energy transfer between the parts of a system not in equilibrium,
(iii)
To promote phase change in a multicomponent systems with or without a change in
composition.
Mixing in tanks is an important area when one considers the number of processes, which are
accomplished in tanks. Essentially, any physical or transport process can occur during mixing in
tanks. Qualitative and quantitative observations, experimental data, and flow regime identifications
are needed and should be emphasized in experimental pilot studies in mixing [1].
Fluid mechanics and geometry are key points to understand mixing. The fluid mechanics transports
the material about the tank, whereas the geometry determines the fluid mechanics. In fact, the
geometry is so important that the processes can be considered geometry specific. Liquid-liquid
dispersion depends on the geometry of the impeller, blending, the relative size of the tank to the
impeller, and power draw.
22
Mixing efficiency in a stirred tank is affected by various number of parameters such as presence of
baffles, impeller speed, impeller type, clearance, tank geometry, solubility of substance, eccentricity
of the impeller.
A vortex is produced owing to centrifugal force acting on the rotating liquid. If vortex reaches the
impeller severe air entrainment occurs. The depth and the shape of the vortex depend on impeller
and vessel dimensions as well as on rotational speed [2].
Baffles are flat vertical strips set radially along the tank wall. Baffles prevent vortex formation. In
baffled tanks, a better concentration distribution throughout the tank and therefore improvement in
the mixing efficiency is achieved. The larger the width of the baffles, the better is the mixing. In the
unbaffled vessel with the impeller rotating in the center, centrifugal force acting on the fluid raises
the fluid level at the wall and lowers the level at the shaft [2, 3].
Flow patterns can be changed according to the type of impellers, and fall into three categories:
axial, radial and tangential [2]:
Radial Flow: Radial Flow discharge is parallel to the impeller radius toward to the vessel wall. If a
radial impeller is not positioned close to the surface or the tank bottom, the flow will split into two
streams upon impinging on the tank wall. Each flow loop will continue along the wall and then
return to impeller.
Axial Flow: Axial flow discharge coincides with the axis of impeller shaft, so when the impeller
operates in a down pumping mode, the flow impinges on the bottom of the tank and spreads out in
all directions toward the wall. The flow rises along the walls up the liquid surface and is pulled back
to the impeller. Since axial flow impeller produce only one loop, fluids mix faster and blend time is
reduced compared to radial flow impellers. The fluid does not take sharp turns near impellers and
because of this, power consumption is less than that of radial flow impellers at the same speed and
same diameter.
23
Figure 5.4.1. Axial and radial flow impellers showing time-averaged flow patterns
in a stirred tank [2].
Tangential Flow: A tangential flow pattern is naturally induced by swirling or vortexing flow that
fluids assume in an unbaffled tank. Tangential flow patterns offer very little mixing because the
velocity gradients are very small.
Consider a tank where a newtonian fluid of density , and viscosity  is agitated by an impeller of
diameter D and rotating at rotational speed N. Let the tank diameter be T, the impeller width W and
the liquid depth H. The power requirement of the impeller P represents the rate of energy
dissipation within the liquid and depends on the following variables of Eq. 5.4.1:
P = P (,, N, g, D, T, W, H,)
(5.4.1)
It is not possible to obtain the functional relationship in Eq. 5.4.1 because of the complex geometry
of the vessel, impeller and other inserts such as heating coils. Using dimensional analysis, the
number of variables describing the problem can be minimized and Eq. 5.4.1 reduces to Eq. 5.4.2
[3]:
𝑃
𝜌𝑁3 𝐷5
= 𝑓(
𝜌𝑁𝐷2 𝑁2 𝐷 𝑇 𝑊 𝐻
𝜇
,
𝑔
, ,
, , 𝑒𝑡𝑐)
(5.4.2)
𝐷 𝐷 𝐷
where
P
N 3 D 5
is the Power number, Po;
number, Fr.
ND 2

is the Reynolds number, Re;
N2D
g
is the Froud
24
Power consumption is related to fluid density, fluid viscosity, rotational speed and impeller
diameter. Power is found by Eq. 5.4.3
P = 






Here “P” is the power, “𝝎” is angular speed (rad/s) and “𝝉” is torque. Torque generated during the
mixing process can be calculated by Eq. 5.4.4
 = F.r
(5.4.4)
where “F” is force and “r” is distance of the torque arm [3].
In this experiment, the different types of impellers are used to investigate the effects of size and
types of agitators on mixing efficiency and power consumption. Also, the effect of baffle on mixing
efficiency is studied.
5.4.3. Experimental Setup
Figure 5.4.2. A stirred tank apparatus.
25
5.4.4. Procedure
Part A
1. Fill the mixing apparatus with distilled water up to a depth of 0.3 m.
2. Attach flat paddle impeller to the end of the shaft and release the torque arm clamp so that the
dynamometer arm moves freely.
3. Set the dynamometer to neutral position and adjust the tension spring.
4. Adjust the length of the cord so that the indicator aligns with the mark on the datum line in the
neutral position.
5. Increase the speed of the impeller in small increments.
6. Observe the behavior of water to note whether there is vortex formation or not.
7. Record the speed, vortex height and force on the balance at each speed.
8. Repeat the experiment for each flat impeller (large-wide, large-narrow, small-narrow), propeller
and turbine with and without baffles.
9. Change the depth of agitator (large-narrow with and without baffles) within the tank and record
the speed, vortex height and force on the balance at each speed.
Part B
1. Attach the small narrow impeller and put the baffles into the tank.
2. Inset the conductometer, insert into the tank. Set the speed of the impeller to 200 rpm.
3. When the conductivity has a fixed value, add 25 grams of NaCl into the tank.
4. Record the time required for the conductivity to reach the steady state using a stopwatch.
5. Repeat the experiment with propeller by adding an additional 25 grams of NaCl.
Safety Issues: Pay attention to injury which is through misuse of motor, from electric shock, from
handling large or heavy components or from rotating components and to damage to clothing. Be
sure of that all valves are closed, the tank is emptied and all the electronic devices are unplugged at
the end of the experiment.
26
5.4.5. Report Objectives
1. Discuss the flow patterns observed with different agitators (impeller, propeller and turbine) of
different sizes with and without baffles. In which cases vortex and/or air entrainment occur?
2. Calculate the power consumption for each speed.
3. How do the size and types of agitators affect mixing efficiency and power consumption?
References
1. Geankoplis, C. J., Transport Processes and Separation Process Principles, 4nd edition, Allyn and
Bacon Inc., 2003.
2. Perry, R. H. and D. Green, Perry’s Chemical Engineers’ Handbook, 8th edition, McGraw-Hill,
2007.
3. Harnby, N., M. F. Edwards and A. W. Nienow, Mixing in the Process Industries, 2nd edition,
Butterworth-Heinemann, Oxford, 1992.
27
5.5. DETERMINATION OF RESIDENCE TIME DISTRIBUTION IN A PACKED BED
Keywords: Packed bed, residence time, step input, pulse input.
Before the experiment: Read the booklet carefully. Be aware of the safety precautions.
5.5.1. Aim
To determine the residence time distribution function and the mean residence time for two columns
with different types of packings using step and pulse input experiments.
5.5.2. Theory
The duration the smallest part of the fluid, e.g. a molecule, has spent in the reactor is called its
residence time. The residence time consists of the time elapsed since the molecule entered the
reactor (its age), and the remaining time it will spend in the reactor (its residual life time). It is
important to note that micromixing can occur only between molecules that have the same residual
lifetime. In a packed-bed reactor as shown in Figure 5.5.1, the fluid may flow through the reactor
non-uniformly because some pathways provide little resistance while others create more resistance
to flow [1].
Figure 5.5.1. Packed-bed reactor [1].
The residence time distribution indicates the deviations the from ideal reactor behavior. There are
three kinds of deviations from ideal tubular-flow reactor. Those are (1) channeling of the fluid
through the packing and the presence of stagnant fluid pockets, (2) diffusion in the longitudinal
direction and (3) the presence of the velocity and the temperature gradient in the radial direction.
The residence-time distribution (RTD) mainly is used to characterize and model nonideal reactors
in chemical engineering. The RTD is determined experimentally by using an inert chemical,
molecule, or atom, which is called a tracer, into the reactor at some time t=0 and then measuring the
tracer concentration in the effluent stream as a function of time (C(t)). The RTD function (E(t))
28
describes in a quantitative manner the amount of time different fluid elements spend in the reactor
[1, 2].
The cumulative RTD function (F(t)) can be defined as the fraction of effluent that has been in
reactor for less than time t, and it is expressed as:
𝑡
∫ 𝐸(𝑡)𝑑𝑡 = 𝐹(𝑡)
0
A plot of F(t) vs. time has the characteristics shown in Figure 5.5.2.
1
F(t)
0
Time, t
Figure 5.5.2. Cumulative RTD curve [2].
Variations in density, such as those due to temperature and pressure gradients, can affect the
residence time and are superimposed on effects due to velocity variations and micromixing. It has to
be supposed that the density of each element of fluid remains constant as it passes through the
reactor. Under these conditions the mean residence time, averaged for all elements of fluid, is
theoretically given by Eq. 5.5.1:
𝑡𝑚 =
𝑉
(5.5.1)
𝑄
where Q is the volumetric flow rate and the reactor volume, V, is equal to the volume occupied by
the maize molecules, which are molecules in the reactor completely at time t=0. For constant
density of fluids, Q is the same for the feed as for the effluent stream.
By defining of F(t), dF(t) is the volume fraction of the effluent stream that has a residence time
between t and t+F(t). Hence the mean residence time is also given by Eq. 5.5.2 [1]:
1
𝑡𝑚 =
∫0 𝑡𝑑𝐹(𝑡)
1
∫0 𝑑𝐹(𝑡)
1
= ∫0 𝑡𝑑𝐹(𝑡)
(5.5.2)
29
The residence time distribution can be obtained by response-type experiments, where a known
amount of tracer is introduced to the system and its effect on the concentration of the effluent is
measured. The three common types of response-type experiments include pulse, step, and
sinusoidal inputs (Figure 5.5.3). Step and pulse input experiments are figured out in this experiment.
For step input experiment, F(t) is defined as the ratio of concentration of tracer and maximum
concentration. For pulse input, it is the ratio of the area under the curve from t=0 to each selected
time and the whole area under the curve [2].
Pulse input
Step input
Figure 5.5.3. Step and Pulse inputs.
Besides the residence time distribution and the mean residence time, the concentrations can also be
used to obtain the effective diffusivity. In tubes, especially in packed beds, formation of eddies
disturbs the flow pattern. The eddy effect is represented by a dispersion coefficient in Fick’s
diffusion law. A plug flow reactor has a dispersion coefficient of 0 and a CSTR of ∞. For a packed
bed, the dispersion model is as follows [3],
𝐷𝐿 =
𝑢𝐿
4𝜋(𝑠𝑙𝑜𝑝𝑒)2
where
u: velocity of fluid
(m/s)
L: tube length
(m)
slope: slope at t/tmean =1 in C vs t/tmean graph
In this experiment, water and methyl orange, used as a tracer, will be flowed into two columns
which are packed with small and large particles. For two packed beds, the residence time
distribution function and the mean residence time will be obtained by using step and pulse input
experiments.
30
5.5.3. Experimental Setup
The experimental setup used in this experiment is shown in Figure 5.5.4.
Figure 5.5.4. Schematic diagram of a packed column.
1.) Large Packed Column
8.) Water Inlet Valve
2.) Small Packed Column
9.) Three-way Stopcock
3.) Burette
10.) Three-way Stopcock
4.) Rotameter for Water
11.) Three-way Stopcock
5.) Rotameter for Tracer
12.) Tracer Inlet Valve
6.) Constant-Head Device for Water
13.) Water Reservoir Valve
7.) Constant-Head Device for Tracer
14.) Tracer Reservoir Valve
5.5.4. Procedure
5.5.4.1. Column Specifications
Packed-bed
Diameter (cm)
Length (cm)
Void (cm3)
With large particle
1.545
133.5
185
With small particle
1.555
135.5
166
31
5.5.4.2. Pulse Input Experiment
1. Adjust the water flow rate to 120 ml/min using the flow meter (No. 4) and valve (No. 8).
2. Adjust the three-way stopcock (No. 9) of (No. 11) so the water goes into only one column.
3. While the water is flowing through the column at a constant flow rate, inject a known amount of
tracer solution into the column by means of a hypodermic syringe at t = 0.
4. At t = 10 sec. start to collect samples from the outlet of the column at definite time intervals (10
seconds) until the tracer disappears in the effluent.
5. Measure the % transmittance of the samples by spectrophotometer and convert them to
concentration units using the calibration curve that will be given by TA.
6. Perform the same procedure for other column, which contains packing material of different size.
5.5.4.3. Step Input Experiment
1. Adjust the three-way stopcock (No. 9 or No. 11) so that the direction of the tracer is open.
2. Adjust the tracer flow rate with flow meter (No. 5) and the valve (No. 12). Do not allow the
tracer to go in the column, allow that it flows into third column by using valve (No. 10).
3. Then turn the stopcock in the direction that closes the tracer flow and permits water flow (do not
turn off valve (No. 12)).
4. Adjust the water flow rate to the same value as the tracer flow rate.
5. After steady state is reached, turn the stopcock (No. 9 or 11) to let the flow of the tracer into the
column. This moment is taken to be zero time (t = 0).
6. After 10 seconds, start to collect samples until the color of the effluent stream does not change.
7. Perform the same procedure for other column.
Safety Issues: In this experiment, methyl orange will be used. It is toxic if swallowed. It
may cause eye and skin irritation. It may be harmful and cause respiratory tract irritation if inhaled.
During the experiment, beware of risks when handling with methyl orange. Be sure of that all
valves are closed and all the electronic devices are unplugged at the end of the experiment. In case
of eye or skin contact, immediately flush eyes or skin with plenty of water for at least 15 minutes
while removing contaminated parts. Get medical aid immediately. In case of ingestion, wash mouth
out with water and get medical aid immediately. In case of inhalation, remove from exposure and
move to fresh air immediately. If not breathing, give artificial respiration and give oxygen. For
further information about the methyl orange, look over [4]. The solutions used in this experiment
are dilute in terms of methyl orange, therefore no need to a special treatment.
32
5.5.5. Report Objectives
1. Calculate the void fraction of each column.
2. Calculate concentrations by using the calibration curve.
3. Plot concentration vs. time graphs in each case.
4. Transform concentration curve to cumulative RTD function curve (F(t)) in each case.
5. Calculate the mean residence time (tm) in each case theoretically and graphically.
6. Calculate percent error.
7. For the step-input experiment cases, estimate the effective diffusivity, (DL) by using the
Dispersion Model.
8. Discuss the effect of the different flow rates on residence time.
References
1. Fogler, H. S., Elements of Chemical Reaction Engineering, 4th edition, Prentice Hall Inc., USA,
2006.
2. Levenspiel, O., Chemical Reaction Engineering, 3rd edition, Cambridge University, 1999.
3. Perry, R.H. and D. Green, Perry’s Chemical Engineers’ Handbook, 7th edition, Mc Graw-Hill,
New York, 1997.
4. Methyl orange MSDS, http://www.sciencelab.com/msds.php?msdsId=9927361.
33
5.6. MODELING OF WATER LEVEL IN A TANK UNDER STEADY STATE AND
DYNAMIC CONDITIONS
Keywords: Mathematical model, differential equation, linear vs. nonlinear, system, water tank.
Before the experiment: Read the booklet carefully. Be aware of the safety precautions.
5.6.1. Aim
To propose linear and nonlinear models to describe the dynamic behavior of water level in a tank
and then simulate these two models to compare with the experimental data and in with each other.
5.6.2. Theory
5.6.2.1. Modeling
Modeling is the art of creating mathematical descriptions of, e.g. physical, chemical or
electrotechnical phenomena which appear in reality. These descriptions have to be relatively simple,
yet accurate enough to serve the purpose of the modeler [1].
It is important to realize that many different models exist, all describing some different parts of the
same reality. It depends on the point of view and intention of the modeler which part of that reality
is described. In studying a chemical process, the modeler may be interested in the chemical
reactions, in the physical working conditions, in the mechanical construction of the reactor, in the
possible environmental impact of the reactor, in the financial return on investments and in the
dynamic behavior of the process [2]. All these different views will yield different models.
The essence of the art of modeling is to select only those characteristics that are necessary and
sufficient to describe the process accurately enough to suit the objects of the modeler. The
simplified models allow the modeler to grasp the essentials from a turbulent and sometimes chaotic
world [1]. So, modeling is grasping the central issue from reality and translating it into an abstract
language such as a mathematical model. An important decision in deriving models is the selection
of the system boundary. The isolated part of the process will be called the system and all the parts of
reality not belonging to the system are attributed to the environment of the system [1].
34
5.6.2.2. Differential Equations
It is often the case that the mathematical models used for describing the systems involve an
equation in which a function and its derivatives play important roles. Such equations are called
differential equations displayed in Eq. 5.6.1, Eq. 5.6.2 and Eq. 5.6.3 [3].
dx
 k1 x  k 2
dt
(5.6.1)
dy
 F ( x) x  y
dx
(5.6.2)
dx
 Ax 2
dt
(5.6.3)
When an equation involves one or more derivatives with respect to a particular variable, that
variable is called an independent variable. A variable is called dependent if a derivative of that
variable occurs in the equation [3]. In Equation 5.6.1, x is the dependent and t is the independent
variable. When t is the independent variable, the following notation in Eq. 5.6.4 may be used:
x  Ax 2
where x 
(5.6.4)
dx
.
dt
The order of a differential equation is the order of the highest ordered derivative appearing in the
equation, whereas the degree of a differential equation is the power to which the highest derivative
is raised when the equation has been rationalized. For instance (Eq. 5.6.5),
y  2cy 3  y  0
(5.6.5)
is an equation of order two (or a second order equation) and first degree. Differential equation can
be classified as linear and nonlinear. A linear differential equation is one for which the following
two properties hold [3]:
1. If x(t) is a solution, then cx(t) is also a solution, where c is a constant.
2. If x1(t) and x2(t) is solutions, then x1 + x2 is a solution.
Equations 5.6.1 and 5.6.3 are linear, whereas Equations 5.6.2 and 5.6.5 are nonlinear differential
equations.
35
5.6.2.3. Mass Balance around a Tank
There are several ways of obtaining a model and deduction based on the fundamental physical laws
is one of them. Using the law of conservation of mass, mass of a liquid in a tank where a single
stream of the liquid is entering and another single stream is leaving can be modeled as Eq. 5.6.6 and
Eq. 5.6.7 [4]:
Input – Output = Accumulation
For a short time period of t,
Input
= F  int
Output
= F  outt
Accumulation = (inV ) - (outV )
t  t
t
where F is the inlet flow rate and F is the outlet flow rate,
V is the volume that the liquid occupies in the tank at a certain time.
 is the density of the fluid (subscripts in and out denote the inlet and outlet flows),
F  int – F  outt = (V ) - (V )
t  t
(5.6.6)
t
Dividing all terms by t and letting t 0:
F  in - F  out =
d ( V )
dt
(5.6.7)
If the fluid is assumed to be incompressible,  is constant, therefore all density terms cancel out.
Furthermore, V = A  h (assuming the tank is a rectangular prism) and A (base area) is constant,
leading to Eq. 5.6.8:
F – F = A
dh
dt
(5.6.8)
where h is the liquid level in the tank.
The outlet flow rate, F may be maintained by the help of a pump to keep it constant. However, if
no pump is used, outlet flow rate depends on parameters such as the water height, gravity, the
friction factor of the outlet pipe. To account for the non-time dependent factors (like gravity and
36
friction) an empirical constant R may be used, so the outlet flow rate may be modeled as following
[5]:
a. A linear model (Eq. 5.6.9) where F’  h
F–
h
dh
=A
dt
R
(5.6.9)
b. A nonlinear model (Eq. 5.6.10) where F’  h0.5
F–
dh
h 0 .5
=A
dt
R
(5.6.10)
5.6.3. Experimental Setup
Figure 5.6.1. Process control system with water tank.
1. Centrifugal pump
6. Water tank
2. Manual control valve
7. Overflow
3. Variable area flow meter
8. Motorized flow control valve
4. Diffuser
9. Drain valve
5. Sealing stopper
5.6.4. Procedure
1. Adjust the drain valve to open position.
2. Check the motorized valve by rotating the knob on the positioner clockwise.
3. Adjust the flow rate from 1 L/min to maximum flow rate with 0.1 L/min increase and record the
steady state heights reached at these flow rates (verify that initial water level height does not
affect the steady state value).
37
4. Starting from different initial water level heights (away from the corresponding steady state
values) and at different constant flow rates between 1 L/min to max flow rate, record the
dynamic behavior of the water level until it reaches the maximum water level.
5. Do not forget to measure the length and width of the water tank.
* Be careful to water level in drain tank. Add water when it is necessary.
Safety Issues: Pay attention to injury through misuse of pump and motor. Be sure of that the flow
meter and the system are closed at the end of the experiment.
5.6.5. Report Objectives
1. Plot flow rate vs. height data for steady-state and unsteady-state conditions.
2. From the steady state height values, find R (resistance) values for linear and nonlinear models
using linear regression. Discuss whether a difference in R values for different flow rates is
plausible. Which model seems to be a better choice?
3. Using these R values, simulate these two models for the flow rates you performed in the
experiment using numerical integration (simple Euler, Runge-Kutta 4th order, or by any other
numerical integration method) on FORTRAN, C or by MATLAB/SIMULINK.
4. Show the simulation results and experimental data for the same flow rates on the same graph.
Discuss which model explains to the experimental behavior better. Which parts of the dynamic
response are better modeled? Why do you think it is so? How can you improve the models? Give
recommendations.
References
1. P. P. J. van den Bosch and A. C. van der Klauw., Modeling, Identification and Simulation of
Dynamical System, CRC Press, Boca Raton, FL, 1994.
2. Luyben, W. L., Process Modeling, Simulation and Control for Chemical Engineers, 2nd edition,
McGraw-Hill, 1990.
3. Rugh, W. J., Linear System Theory, 1st edition, Prentice-Hall, 1993.
4. R.G. Rice and D. D. Do, Applied Mathematics and Modeling for Chemical Engineers, John
Wiley & Sons, 1994.
5. Seborg, D. E., T. F. Edgar and D. A. Mellichamp, Process Dynamics and Control, 1st edition,
John Wiley and Sons, USA, 1989.