completed report engg815

8/14/2019 Completed report ENGG815

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TERM PROJECT

Temperature Profile in the Demethanizer Column

Present to

Dr. Nader Mahinpey

By

Teerawat Sanpasertparnich 200250594

This term project is a part ofComputer-Aided Processes, ENGG 815

Faculty of Engineering

University of Regina, Saskatchewan

WINTER 2006


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TABLE OF CONTENTS

CHAPTER PAGE

I ABSTRACT 1

II INTRODUCTION 2

III LITERATURE REVIEW 3

IV DESCRIPTION OF THE DEMETHANIZING SYSTEM 4

V METHODOLOGY 5

VI NUMERICAL METHOD 6

VII RESULT AND DISCUSSION 7

VIII CONCLUSION 8

IX REFERENCE 9


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Teerawat Sanpasertparnich, [email protected] 1

I. ABSTRACT

The demethanizing process has been used widely in the industrials to separate methaneout of any other hydrocarbon components. Basically, two phases, vapor and liquid phases, of

feed stream, which previously are separated by flash drum, flow to an upper and a lower tray of

demethanizer column, respectively. The vapor phase which mostly composed of methane,nitrogen and some ethane are heated by the reboiler at the bottom of the demethanizer column.

Methane and nitrogen, which have the lowest boiling point, readily flow upward through the topof the column, meanwhile the ethane which has a higher boiling point as well as the rest ofhydrocarbon components at a lower tray flow downward and so far they are withdrawn out of the

bottom of demethanizer column. This paper is focused on the behavior of temperature profile

along the demethanizer column. The calculated boundary conditions at the steady state using

PRO/II 7.0 are demonstrated. Further, for temperature profile calculation, the transformationfrom partial differential equation to the algebraic difference equation based on the central

difference approximation is applied on this particular problem. As the results, the calculated

solutions are in a good agreement with the analytical solutions using PRO/II 7.0.

II. INTRODUCTION

The purpose of this term project is to employ the algebraic numerical method to solve thescientific problems instead of using calculus method to directly solve the partial differential

equation. In addition, the content of this course outlines is to bring the computer aided software,

for example MATLAB, FEMLAB and so on, to solve for the particular engineering problems(Mahinpey, 2006). To appropriately fit with the course objectives, this term project has been

employed the PRO/II 7.0 (SIMSCI), and MATLAB as the software aided tools to calculate the

boundary conditions at the steady state of the demethanizer column and so far compute the

temperature profile along the demethanizer column based on the partial differential equation butusing algebraic difference equation by an approach of central difference approximation which,

further, it is solved by an approach of the Gauss-Seidel method.

The advantage of this studying helps to predict the behavior of the temperature profile atthe steady state by the degree of temperature inside the demethanizer column. In addition, The

advantage of this study is to help reduce the capital cost of the temperature sensors.

III. LITERATURE REVIEW

The temperature sensors are inexpensive, more reliable, and commonly used in the

industrials as purposed by Kister, 1990. However, the temperature sensors located far away havethe potential to get significant errors due to noise and pressure variation which could interferewith the temperature as reported by Rademaker et al., 1975. These issues were handled by using

the temperature and pressure drop compensation as proposed by Luyben and Boyd, 1975 and Yu

and Luyben, 1984 and as well as figured out the appropriated locations of temperature sensors as

proposed by Luyben, 1971, Parnis, 1987, and Bozenhart, 1988.

On the other hands, the model simulations have helped the scientists and researchers to

minimize error from the measurement and help to approximately predict the temperature profilesalong the column instead of installation of temperature sensors in various locations in column

(Luyben, 2006). There are several researches studying about the temperature profile model such

as using the Newton-Raphson methods to predict the temperature in the distillation column by Nelson, 1971; Komatsu and Holland, 1977; Carra et al., 1979, using modified tri-diagonal

methods by Susuki et al., 1971; Izarraraz et al., 1980; Tierney and Riquelme, 1982; Xu and


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Chen, 1988, using inside-out algorithms by Venkataraman et al., 1990, using partial least square

regression by T. Mejdell et al., 1991 and so on.

IV DESCRIPTION OF THE DEMETHANIZING SYSTEM

Each component in the stream inlet has differences in temperature, pressure, composition,and/or phase state. As the system goes to the equilibrium, each component which has different

concentration in each zone is separated. This separation process is called distillation. Thisgeneral concept has been applied to use in practical such as for distillation column. It starts whilethe feed streams flow through the distillation column and are separated into vapor and liquid

phases. Because of different gravity between vapor and liquid, therefore the vapor flows upward

while the liquid flows downward inside a column. Liquid reaching the bottom of the column is

partly vaporized in a reboiler and put some vapor back to the column. The rest of the bottomliquid is withdrawn as bottoms. Some vapor which flows upward is cooled and condensed to

liquid by condenser. Some liquid is returned to the column as reflux and contact with the vapor

which continuous moves upward as the counter current flow (Warren et al., 1993; Smith et al.,1996).

For the demethanizing system which is one type of distillation processes as illustrated inFigure 1, firstly hydrocarbon stream which mainly is composed of methane, ethane, propane, iso-

butane, n-butane, iso-pentane, n-pentane, n-hexane, n-heptane and N2 flows through the flash

drum to separate stream into two phases, the vapor phase and liquid phase. The vapor phase willflow through the expander and then to an upper tray of the demethanizer column. Meanwhile, the

liquid phase will flow through the liquid valve and so far to a lower tray of column. The

demethanizer column in this study is no reflux which has no condenser unit at the top of the

column because of obviously different boiling points between methane and the rest ofhydrocarbon components. In the demethanizer column, the hydrocarbon liquid flowing down to

the bottom will be heated by reboiler and then move through the top of the column. Methane and

nitrogen which have the significant lowest boiling point will flow out of the top of the column,

but the rest of hydrocarbon stream such as ethane, and propane will flow back to the column andconsequently are withdrawn out of the bottom of the column by reboiler.

V. METHODOLOGY

The steps to compute the temperature profile have been demonstrated in Figure 2. Firstly

the process flow diagram and the operating conditions as shown in Figure 1 and Table 1 are

drawn and put into PRO/II 7.0 to calculate for the temperatures at the boundary conditions in thesteady state. This study focuses on two cases of process flows, the first case by feeding the liquidstream flowing out of liquid valve to the 3

rdtray of column and the second case by feeding the

liquid stream flowing out of liquid valve to the 5th

tray of column. The calculated boundary

conditions are put into the grids in Figure 3 which represent the dimension of demethanizer

column. Next, the partial differential equation is applied to solve for the unknown temperaturesin various locations inside the column. This employs algebraic difference approximation by an

approach of Gauss-Seidel method. The results are reported and so far they are graphically

plotted. Further, these calculated results will be compared with the analytical solutions usingPRO/II.


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Stream Name

TemperaturePressureFlowratePhase

LIQUID_OUT

26.833139.696484.586

Liquid

FPSIALB-MOL/HR

Stream Name


VAPOR_OUT

-170.589139.696

2195.533Vapor

FPSIALB-MOL/HR

For Case 2

For Case 1LIQUID_VALVE

FLASH_DRUM

EXPANDER

1

2

3

4

5

6

7

8

9

10

DEMETHANIZERStream Name


VAPOR

-84.000587.700

1818.585Vapor

FPSIALB-MOL/HR

Stream Name


LIQUID

-84.000587.700861.534

Liquid

FPSIALB-MOL/HR

Stream Name


EXP_TO_DIST

-174.277139.696

1818.585Mixed

FPSIALB-MOL/HR

Stream Name


VALVE_TO_DIS

-131.381139.696861.534

Mixed

FPSIALB-MOL/HR

Stream Name


VAPOR_OUT

-169.912139.696

2196.732Vapor

FPSIALB-MOL/HR

Stream Name


STREAM_IN

-84.000587.700

2680.119Mixed

FPSIALB-MOL/HR Stream Name

TemperaturePressureFlowrate

Phase

LIQUID_OUT

27.085139.696483.387

Liquid

FPSIALB-MOL/HR

For Case 1 For Case 2

For Case 1

For Case 2

Figure 1 A scheme of demethanizing system with its operating conditions and calculated results

for case 1 and case 2

Process Flow Diagram (PFD)

of De-methanizing SystemOutputs: Tij

Operating

conditions PRO/II 7.0 Graphical plot

Boundary Conditions at the

surface of the column End

PDE transformed to algebraic

difference approximation

Gauss-Seidel Method

Boundary Conditions at

the steady-state

Outputs: Tij

End

YesNoas

Figure 2 The engineering problem solving flowchart


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Table 1 A main characteristic of hydrocarbon stream component in this study

Component Composition

(% mole)

Boiling point

(oF)

Methane 73.05 -258.745

Ethane 7.68 -127.480

Propane 5.69 -43.784iso-butane 0.99 10.886

n-butane 2.44 31.096

iso-pentane 0.69 82.180

n-pentane 0.82 96.906

n-hexane 0.42 155.714

n-Heptane 0.31 209.172

N2 7.91 -320.440

VI. NUMERICAL METHOD

This particular problem is the two-dimensional, steady state, and differential balance.

Therefore, the partial differential equation has been employed to solve this problem. The general

form of two-dimensional partial differential equation is shown in Eq.(1).

0Dy

TC

yx

TB

x

TA

2

22

2

2

=+

+

+

(1)

In addition, this problem is based on boundary value problem which suits with the elliptic

partial differential equation as demonstrated in Table 2 (Basmadjan, 1999).

Table 2 The categories of second order PDEs: Elliptic, Parabolic and Hyperbolic PDEsCriterion Type of PDE Example Properties

B2-4AC < 0 Elliptic Laplace's equation Boundary value problem

B2-4AC = 0 Parabolic Fourier's equation Mixed boundary value problem

and initial value problem

B2-4AC > 0 Hyperbolic Wave equation Mixed boundary value problem

and initial value problem

or initial value problem

0y

u

x

u2

2

2

2

=

+

2

2

2

22

t

u

x

uc

=

t

uC

x

u2

2

=

By heat balance, the equation can be expressed as

0y

q

x

q=

+

(2)

Then, applying the Fouriers law of heat conduction as follow,

)i

T(Ckqi

= by Ckk = (3)

Substituting Eq.(3) into Eq.(2) results in

0y

T

x

T2

2

2

2

=

+

(4)

The Eq.(4) is called the Laplace equation. This equation is similar to the general type of

partial differential equation in Eq(1) by the constant A and C being equal to 1 but B being equal

to zero.


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Next, the differential equation form in Eq.(4) is transformed into the algebraic equation

based on the central derivative difference approximation. The reason to use the central differenceapproximation is because it has the residual error (O(h

2)) less than the forward and backward

difference approximations (O(h)) as demonstrated in Table 3 (Chapra et al, 1998).

Table 3 The type of 2nd

derivative difference approximation

Type of difference approximation

Forward

Backward

Central

)h(Oh

)x(f)x(f2)x(f)x(f

2

i1i2ii +

+=

++

)h(Oh

)x(f)x(f2)x(f)x(f

2

2i1iii +

+=

)h(Oh

)x(f)x(f2)x(f)x(f 2

2

1ii1ii +

+=

+

Case 1 Case 2

-131.381F

-174.277FTray 1 (Ti, j = -169.912F)i=1, 2; j=1, 5 except T2,1

Tray 2 (Ti, j = -152.956F)i=3, 4; j=1, 5

Tray 3 (Ti, j = -134.015F)i=5, 6; j=1, 5 except T6, 1

Tray 4 (Ti, j = -133.258F)i=7, 8; j=1,5

Tray 5 (Ti, j = -132.899F)i=9,10; j=1,5

Tray 6 (Ti, j = -130.833F)i=11, 12; j=1, 5

Tray 7 (Ti, j = -117.836F)i=13, 14; j=1, 5

Tray 8 (Ti, j = -71.699F)i=15, 16; j=1, 5

Tray 9 (Ti, j = -16.170F)i=17, 18; j=1, 5

Tray 10 (Ti, j = 27.085F)i=19 ,20; j= 1, 5

i = 1

i = 20

j = 1 j = 5

Tray 5 (Ti, j = -132.624F)i=9,10; j=1,5 except T1 0 ,1

i = 20

Tray 9 (Ti, j = -16.131F)i=17, 18; j=1, 5

Tray 10 (Ti, j = 26.834F)i=19 ,20; j= 1, 5

Tray 7 (Ti, j = -116.765F)i=13, 14; j=1, 5

Tray 8 (Ti, j = -71.069F)i=15, 16; j=1, 5

Tray 6 (Ti, j = -129.823F)i=11, 12; j=1, 5

-131.381F

-174.277F

j = 1

i = 1

Tray 3 (Ti, j = -148.612F)i=5, 6; j=1, 5

Tray 4 (Ti, j = -144.446F)i=7, 8; j=1,5

Tray 2 (Ti, j = -156.362F)i=3, 4; j=1, 5

Tray 1 (Ti, j = -170.590F)i=1, 2; j=1, 5 except T2,1

j = 554 unknown temperatures

Figure 3 The grids for case 1 and case 2 with their boundary conditions and 54 unknown

temperatures in each column

Thus,

0y

TT2T

x

TT2T

2

1j,ij,i1J,i

2

j,1ij,ij,1i=

++

+ ++(5)

By simplifying the equation by defining yx = , hence0T4TTTT j,i1j,i1j,ij,1ij,1i =+++ ++ (6)

Therefore, the Eq.(6) can be solved by rearrange into the matrix form by{ } { }bT]A[ =

Then, the transformation of boundary condition values is substituted into Eq.(6). The result has

been written as shown in Eq.(7).


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{ } { }1x54j,i1x54j,i

54x54

bT

4...00000

:...:::::

0...01-41-

0...1-01-41-

0...01-01-4

=

(7)

In order to solve for this simultaneous algebraic equations in Eq.(7), iterative method isintroduced. Basically, there are the Gauss-Seidel and Jacobi methods which generally are used

to solve for simultaneous algebraic equation. In this study, it used the Gauss-Seidel method to

solve. Then, the Eq.(7) can be simplified as follows.

1,154,154,133,122,111 a/)Ta...TaTab(T =

2,254,154,233,222,222 a/)Ta...TaTab(T = (8)

: : : : : : :

54,5454,154,5433,5422,545454 a/)Ta...TaTab(T =

By simplifying Eq.(8), the result is shown as follow.

{ } { } { }1ii T]C[dT = (9)

By, { }

=

54,5454

2,22

1,11

a/b

.

.

.

a/b

a/b

d ,and

=

0.../aa/aa/aa

....

....

....

/aa...../aa0/aa

/aa...../aa/aa0

]C[

54,5454,354,5454,254,5454,1

2,22,542,22,32,22,1

1,11,541,11,31,11,2

Here, the initial values are defined as zero (Ti = 0) then they are substituted into Eq.(9), until the

approximate error is satisfied with the defined criterion as shown in Eq.(10) (the specified errorin this study is defined as 0.001%). Then the calculated results will be graphically plotted.

s

i

1ii

i,a %100xT

TT

=

(10)

The following is the M-file code written in a basis of the Eqs.(8)-(10) for solving this particularproblem.-------------------------------------------------------------------------------------------------

function [x,T] = demettempprof(A,b,deType)

%specified error is defined to be equal to 0.001%.

%size of matrix A is limited at 54x54.

%the initial values of temperature start at Tij = 0 for i = 2..4; j = 2..19.

%Case is the choice either Case 1 or 2.

es = 0.001; ea = es + 1;

C = A;

for i = 1:54

C(i,i) = 0;

x(i) = 0;

end

x = x';

for i = 1:54

C(i,1:54) = C(i,1:54)/A(i,i);

end

for i = 1:54

d(i) = b(i)/A(i,i);

end

iter = 1;

while ea > es

xold = x;

for i = 1:54

x(i) = d(i) - C(i,:)*x;


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if x(i) ~= 0

ea(i) = abs((x(i)-xold(i))/x(i))*100;

end

end

iter = iter+1;

end

k = 0;

for i = 2:19

for j = 2:4

T(i,j) = x(1+k);

k=k+1;

end

end

for i = 1:20

for j = 1:5

xx(i,j) = j;

yy(20-i+1,j) = i;

end

end

%Set the boundary conditions resulting from PRO/II 7.0 (SIMSCI).

if deType == 1

for i = 0:1

for j = 0:1

T(1+i,1+4*j) = -169.912; T(3+i,1+4*j) = -152.956;

T(5+i,1+4*j) = -134.015; T(7+i,1+4*j) = -133.258;

T(9+i,1+4*j) = -132.899; T(11+i,1+4*j)= -130.833;

T(13+i,1+4*j)= -117.836; T(15+i,1+4*j)= -71.699;T(17+i,1+4*j)= -16.170; T(19+i,1+4*j)= 27.085;

end

end

T(1,2)= -169.912; T(1,3)= -169.912; T(1,4)=-169.912;

T(20,2)= 27.085; T(20,3)= 27.085; T(20,4)= 27.085;

T(2,1)= -174.277; T(6,1)= -131.381;

elseif deType == 2

for i = 0:1

for j = 0:1

T(1+i,1+4*j) = -170.590; T(3+i,1+4*j) = -156.362;

T(5+i,1+4*j) = -148.612; T(7+i,1+4*j) = -144.446;

T(9+i,1+4*j) = -132.624; T(11+i,1+4*j)= -129.823;

T(13+i,1+4*j)= -116.765; T(15+i,1+4*j)= -71.069;

T(17+i,1+4*j)= -16.131; T(19+i,1+4*j)= 26.834;

end

end

T(1,2)= -170.590; T(1,3)= -170.590; T(1,4)=-170.590;T(20,2)= 26.834; T(20,3)= 26.834; T(20,4)= 26.834;

T(2,1)= -174.277; T(6,1)= -131.381;

end

axes('position',[0 0 .25 1]);

pcolor(xx,yy,T);

shading interp;

hold on;

[ch,h] = contour(xx,yy,T,40);

clabel(ch,h,'fontsize',8,'label spacing',800);

colorbar;

light

-------------------------------------------------------------------------------------------------

VII. RESULT AND DISCUSSION

The calculated temperature results are based on the iterative method by an approach of

Gauss-Seidel method, until the approximate error (a) is less than or equal to the specified error(s) as shown in Eq.(9). Here the iterative steps will stop after the approximate error is less than

or equal to 0.001%. As the results, the calculated temperatures for case 1 and case 2 have been

demonstrated in Table 4 and 5, respectively.


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Then, the outputs and boundary conditions are set into the same matrix form in

dimension 20x5 as previously illustrated in Figure 3 for case 1 and case 2. So far by using thecontour plot for both cases, the temperature profiles are graphically shown in Figure 4.

Tray 1

Tray 2

Tray 3

Tray 4

Tray 5

Tray 6

Tray 7

Tray 8

Tray 9

Tray 10

Nodeposition(i)

Temperature(oF)

Node position (j)

Tray 1

Tray 2

Tray 3

Tray 4

Tray 5

Tray 6

Tray 7

Tray 8

Tray 9

Tray 10

Nodeposition(i)

Temperature(oF)

Node position (j)

Figure 4 The temperature profiles of demethanizer column for case 1 and case 2, respectively

Table 4 and Figure 7 summarize the comparisons between the calculated solutions andthe analytical solutions using PRO/II 7.0. As the results, the calculated solutions are in a good

agreement with the analytical solutions.

Table 6 The calculated solutions comparing with the analytical solutions using PRO/II 7.0

Analytical

solution

Calculated

results|t| (%)

Analytical

solution

Calculated

results|t| (%)

1 -169.912 -164.997 2.892% -170.590 -166.158 2.598%

2 -152.956 -150.394 1.675% -156.362 -155.419 0.603%

3 -134.015 -134.909 0.667% -148.612 -148.111 0.337%

4 -133.258 -133.290 0.024% -144.446 -142.367 1.439%

5 -132.899 -131.514 1.042% -132.624 -131.768 0.646%

6 -130.833 -126.008 3.688% -129.823 -125.282 3.498%

7 -117.836 -107.474 8.794% -116.765 -106.580 8.722%8 -71.699 -63.452 11.502% -71.069 -62.923 11.461%

9 -16.170 -12.724 21.313% -16.131 -12.690 21.330%

10 27.085 27.085 0.000% 26.834 26.834 0.000%

Avarege of true error 5.160% 5.063%

Case 1 (oF)

Tray

Position

Case 2 (oF)

Note the calculated results are based on the average temperature at the same row but different column.


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-200

-150

-100

-50

0

50

1 2 3 4 5 6 7 8 9 10

Tray number in the column

Temp

erature(oF)

CASE 1: Analytical solution based on PRO/II

CASE 1: PDE based on AE method

CASE 2: Analytical solution based on PRO/II

CASE 2: PDE based on AE method

Figure 5 The comparisons between calculated results and analytical solution using PRO/II 7.0

VIII. CONCLUSION

By means of partial differential equation (PDE) and algebraic difference approximation

by an approach of Gauss-Seidel method, the mathematical model provides the results that have agood agreement with the results obtained from analytical solutions using PRO/II 7.0. The results

obtained from the mathematical model have an approximate average true error of five percents.

IX. REFERENCES

1. Graduate course materials for ENGG 815, Dr. Nader Mahinpey, 2006.2. S. C. Chapra and R. P. Canale, Numerical methods for engineers with programming andsoftware application, 1998, 3

rdEdition, McGraw-Hill International Editions.

3. W. L. McCabe, J. C. Smith, P. Harriott, Unit operations of chemical engineering, 1993, 5 thEdition, McGraw-Hill International Editions.

4. J.M. Smith, H.C. Van Ness, H.M. Abbott, Introduction to chemical engineeringthermodynamics, 1999, 5

thed., McGraw-Hill International Editions.

5. PRO/II 7.0, SimSci-Esscor, http://www.simsci-esscor.com6. MATLAB, The MathWorks, http://www.mathworks.com7. W.L. Luyben, Evaluation of criteria for selecting temperature control trays in distillationcolumns, 16 (2006) 115134.


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completed report engg815

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