Ramin Shamshiri ABE6981 HW_08

7/30/2019 Ramin Shamshiri ABE6981 HW_08

1/15

Ramin Shamshiri ABE 6986, HW #8 Due 10/29/08

Homework #8

Due 10/29/08

Kinetic coefficients for phosphorus transport in a packed bed reactor

Refernce: Overman, A.R.,Data points for Figure 7 of the article are given in Table 1 below.

v

(cm/min)

Ka

(min-1)

Kd

(min-1)

Kr

(min-1)

0.118 1.10 0.0314 0.00013

0.256 2.00 0.0790 0.00024

0.539 2.75 0.133 0.00035

0.900 3.00 0.155 0.00047

Table 1: Dependence of coefficients for adsorption (K a), desorption (Kd), and reaction (Kr) on flow velocity (v)

for phosphorus transport in a packed bed reactor.

1. Plot the data for the rate coefficients vs. velocity on linear graph paper.Answer:

Plot of coefficients for adsorption (Ka) vs. flow velocity (v), coefficients for desorption (Kd) vs. flow velocity (v)

and coefficients for reaction (Kr) vs. flow velocity (v) are shown in Figure 1, 2 and 3 respectively.

Figure 1: of data for adsorption coefficient (Ka) vs. flow velocity (v)


2/15


Figure 2: Plot of data for desorption coefficient (kd) vs. flow velocity (v)

Figure 3: Plot of data for reaction coefficient (kr) vs. flow velocity (v) (Y axis scale 10-4

)


3/15


2. Assume that each curve can be described by the hyperbolic equation =

+

which can be converted to the linear form

=

+

Perform linear regression to obtain values for parametersa andb for adsorption, desorption andreaction.

Answer:

v (cm/min) Ka (min-1 ) Kd(min

-1 ) Kr (min-1 )

()

()

()

0.118 1.10 0.0314 0.00013 0.107273 3.757962 907.692308

0.256 2.00 0.0790 0.00024 0.128000 3.240506 1066.66667

0.539 2.75 0.133 0.00035 0.196000 4.052632 1540.00000

0.900 3.00 0.155 0.00047 0.300000 5.806452 1914.89362

Linear model Poly:

f(v) = p1*v + p2p1 = 0.2502 (0.1788, 0.3215)

p2 = 0.06943 (0.03068, 0.1082)

=

=

+

= .+ .

a=3.9968 & b=0.27749

R-square: 0.9913 & r=0.9956

=

+ =

.

.+

Linear model Poly:

f(v) = p1*v + p2p1 = 2.91 (-1.294, 7.114)

p2 = 2.895 (0.6123, 5.179) =

=

+

= .+ .

a=0.3436 & b=0.9948

R-square: 0.816 & r=0.9033

=

+ =

.

.+

Linear model Poly:

f(v) = p1*v + p2p1 = 1320 (899.8, 1740)

p2 = 759.1 (530.9, 987.2)

=

= + = . +

a=0.0007575 & b=0.5750R-square: 0.9892 and r=0.9945

=

+ =.

.+


4/15


3. Draw the curves forKa,Kd, andKr vs. v on (#1).Answer:

Figure 4

Figure 5


5/15


Figure 6

Figure 7


6/15


Figure 8

Figure 9


7/15


4. For Table 1, calculate Ka/Kdfor each velocity.Answer:

v

(cm/min)

Ka

min-1

Kd

min-1

Ka / Kd

0.118 1.10 0.0314 35.03184

0.256 2.00 0.0790 25.31645

0.539 2.75 0.133 20.67669

0.900 3.00 0.155 19.35483

5. For Table 1, calculateK= Kr (Ka/Kd) for each velocity.Answer:

v

(cm/min)

Ka

(min

-1

)

Kd

(min

-1

)

Kr

(min

-1

)

Ka / Kd K= Kr (Ka/Kd)

(min

-1

)0.118 1.10 0.0314 0.00013 35.03184 0.004554

0.256 2.00 0.0790 0.00024 25.31645 0.006076

0.539 2.75 0.133 0.00035 20.67669 0.007237

0.900 3.00 0.155 0.00047 19.35483 0.009097


8/15


6. Plot results of #4 and #5 on linear graph paper.Answer

Figure 10

Figure 11


9/15


7. Use regression equations forKa andKd to calculate and draw the curve forKa/Kdvs. v.Answer:

Approach No.1

=

+ =

.

.+

=

+

=.

.+

=

..+ ..+

=.+ .

.+ .

Figure 12


10/15


Approach No.2

=

=

+

= .+ .

=

=

+

= .+ .

=.+ .

.+ .=

Figure 13


11/15


As expected, both approaches result same graph. This is shown in Figure below:

Figure 14


12/15


Calculating regression line corresponding to ka/kd vs. v:

v

(cm/min)

Ka

min-1

Kd

min-1Ka / Kd

0.118 1.10 0.0314 35.03184

0.256 2.00 0.0790 25.316450.539 2.75 0.133 20.67669

0.900 3.00 0.155 19.35483

Linear model Poly:

f(v) = p1*v + p2

p1 = -17.64 (-49.72, 14.44)

p2 = 33.09 (15.67, 50.51)

= .+ .

R-square: 0.7367 and r=0.8583

Figure 15


13/15


8. Use Eq. (2) to estimate parametersa andb for the overall k vs. v. Draw the curve on #6.Answer:

v

(cm/min)

K= Kr (Ka/Kd)

(min-1 )

()

0.118 0.004554 25.91049

0.256 0.006076 42.13334

0.539 0.007237 74.480010.900 0.009097 98.93621

Eq. (2):

=

+1

Linear model Poly:

f(v) = p1*v + p2p1 = 93.76 (55.35, 132.2)p2 = 17.87 (-2.995, 38.73)

=

=

+

= .+ .

a=0.01066 & b=0.1905R-square: 0.9733 and r=0.9865

=

+ =

.

.+

Figure 16


14/15


=

+ =

.

.+

Figure 17


15/15


9. How do these results compare to those of Figures 9 and 10 of the 1976 article?Answer:In Figure 9 of the 1976 article, dependence of adsorption coefficient on pore velocity shows that the values of R

decrease as velocity increases. This relationship is similar to the relationship between ka/kd vs. flow velocity (v)

shown in Figures 10, 12, 13 and 15. In Figure 10 of the 1976 article, dependence of kinetic rate constant ( k) onpore velocity shows values ofk increase as velocity increase. The relationship seems to have a curvature trend.

Similar relationship between coefficients for adsorption (Ka), desorption (Kd), and reaction (Kr) vs. flowvelocity (v) is shown in Figures 1,2 and 3. The curvature behavior for each of these relationships was calculatedand is summarized below (Eq. I, II and III). High correlation coefficients (r1=0.9956, r2=0.9033, r3=0.9945)

admits this result and indicates strong relationship. It was also shown that similar relationship exists between K

and flow velocity (Eq. IV below).

=

+ =

.

.+

Eq. I

=

+ =

.

.+

Eq. II

=

+ =.

.+

Eq. III

=

+ =

.

.+

Eq. IV

10.Discuss your results.Answer:Plots of (Ka), (Kd), and (Kr ) vs. v first showed positive (increasing) curvature trend between data. (Figure 1, 2

and 3). Assuming that this relationships can be illustrated and would follow hyperbolic equations, linearization

procedure was used to obtain equation constant parameters (a and b) for individual relationship between

adsorption, desorption and reaction (Kr) vs. flow velocity. Plot of calculated hyperbolic equations on thecoefficients ((Ka), (Kd), and (Kr )) vs. v data as well as the high values of correlation coefficient confirmed our

initial assumption.

Calculating values of ka/kd, a negative (decreasing) curvature trend was observed. Once again, hyperbolicequation was assumed and constant parameters were obtained. This time, results also confirmed our

assumption.

Lastly, values of K= kr (ka/kd) was calculated and plotted vs. v. A positive (increasing) curvature trend was

observed which was similar to the Figure 10 of article 1976. As a general result, it can be stated that the all ofthese coefficients (k) values has and very well follow hyperbolic relationship with the flow velocity in the given

data range (0, 1).

Date post:	04-Apr-2018
Category:	Documents
Upload:	raminshamshiri
View:	227 times
Download:	0 times

Ramin Shamshiri ABE6981 HW_08

Documents