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Journal of Engineering Volume 18 July 2012 Number 7

859

Calculation Of Emission From Fixed Roof Storage Tank

Kadhum Judi Hammud

University of Baghdad

Abstract:

In this paper we present study to estimate amount of emission from storage tank which have

fixed roof.

Fixed roof field storage tank of crude oil production significant sources of hydrocarbon

emissions; these emissions may vary as a result of flashing, working, and standing effects. Prediction

methods were needed to determine as accurately as possible the amount of hydrocarbon emissions and

to help assess the level of cost-effective emission controls.

Several predicting method for determine breathing and working losses from tank containing

stabilized cured oil and produse are available but were not applicable to exact working, breathing, and

flashing conditions experienced by a tank under study.

The objective of this study was to find best models that would consider all major variables, to

predict emissions from storage tanks which were experiencing working, breathing, and flashing

effects.

The result was get from mathematical method in fixed roof tank was powerful to study every

parameter effects on emission.

Keyword: Estimate, storage tank, fixed roof, floating roof, equation of state.

CALCULATION OF EMISSION FROM FIXED ROOF STORAGE TANK

Kadhum Judi Hammud

ج

860

1. Introduction:

It was proposed that one aggressive detailed

model be developed to predict hydrocarbon

emissions from fixed-roof storage tanks. The

model included the effects of liquid surface

working, crude feed flashing, and solar heating

(breathing) on hydrocarbon emissions from a

tank top vent into the atmosphere. In order to

do this proposed model simultaneously solved

the momentum, continuity, species, and energy

equations.

To predict hydrocarbon emissions from

fixed—roof storage tanks.

2.1-Momentum

The momentum equations can be

written in the following form for Cartesian (1)

coordinate;

Horizontal X Directional Momentum Equation

2x

2

2x

2z

2y

2

zV

yV

zxV

yxV

∂∂

+∂∂

=∂∂

∂+

∂∂

∂. (1)

Horizontal Y Directional Momentum Equation

2y

2

2y

2z

2x

2

zV

xV

zyV

xyV

∂

∂+

∂

∂=

∂∂∂

+∂∂

∂

(2)

Vertical Z Directional Momentum Equation

⎥⎦

⎤⎢⎣

⎡∂∂

+∂∂

+∂∂

µ=∂∂

ρ+⎟⎠⎞

⎜⎝⎛ ρ−ρ 2

z2

2z

2

2z

2z

z

__

zV

yV

xV

zVVg

(3)

Where

Vx gas velocity in horizontal

north/south direction, ft/hr

Vy gas velocity in horizontal east/west

direction, ft/hr

Vz gas velocity in vertical direction,

ft/hr

g acceleration of gravity, ft/hr2

ρ gas density lb/ft3

__

ρ average gas density on horizontal

phase lb/ft3

The above three gas phase momentum

equations were solved simultaneously together

with a horizontal planar adjustment check. This

check was the integration of vertical velocity

on any horizontal plane to assure mass balance

in the vertical direction.

The planar continuity equation1

∫ ∫= dxdyVI z ………………………………

……………………...….…... (4)

Where

I: Planar continuity integral, ft3/hr

Adjustments in the vertical gas velocity

using Equation (4) were minor, on the order of

1.0 to 10 percent1.

Figure (1) is a top view of a cylinder as

viewed with a Cartesian grid. The computer

program selects points on the grid closest to the

circular boundary to closely simulate a circular

figure. Grids points lying outside the circular

figure are not used, and thereby, represent an

approximate loss of 25 percent of the computer

stored velocity matrix; however, the increase in

computational speed by a factor of 5 more than

offsets this disadvantage in computer storage.


861

Figure (2) is a tank side view showing

the gas space velocity values at the walls,

dome, and liquid surface.

The numerical technique used to solve

equations 1, 2, and Essentially the technique

required central differencing of the special

derivatives that gave a stable balanced

approximation of the velocities. The

momentum equations were solved for velocity

using the Gauss-Seidel iterative method, and

the solution methodology was stable since the

equations possessed diagonal dominance.

The solution method was found to be

stable and was quickly attained by starting the

solution at the top dome plane and sequentially

moving down to the liquid surface. At any

plane, k, the vertical velocity, Vz, was solved

by first using Equation (3). The value of Vz on

the k+1 plane had to be assumed the same as

the value on the k plane in order to calculate Vz

on the k + 1 plane. This approximation was

made for the first pass only in order to get the

solution started. Subsequent calculations of Vz

used the previous pass value of Vz on the plane

to get updated values of Vz on the k plane.

Results showed that this technique is extremely

useful because only two passes down the tank

gas space were needed for a converged

solution. After Vz was calculated on a plane,

the mass balance integral, Equation (4), was

used to insure mass balance in the vertical

direction. If mass balance was not achieved,

the vertical gas velocities were adjusted. In all

cases considered, this adjustment was between

1.0 and 10.0 percent of the mass balance

integral, I. The horizontal velocities, Vx and Vy,

were then calculated using Equations (1) and

(2). After Vx, Vy, and Vz values had been

calculated on a plane, the next lower plane was

selected and the procedure repeated. At the

liquid surface plane, the calculations were

WEST

NO

EAS

SOU

Figure (1) Cartesian Coordinates Approximating a Cylindrical

Boundary (Top View)

Figure (2) Momentum Equation Boundary Conditions


Kadhum Judi Hammud

ج

862

repeated starting at the top dome and

sequentially moving down again to the liquid

surface. In the cases considered, the values of

the velocities remained within 1.0 percent of

their previous pass estimates.

2.2-Species

The partial differential equation which

describes the diffusion and bulk velocity

transport of a chemical species within the gas

space of a fixed—roof storage tank was given

by Brid et. al(1960) as(1)

zCV

yCV

xCV

tC

zC

yC

xCD i

zi

yi

xi

2i

2

2i

2

2i

2

im ∂∂

+∂∂

+∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

(5)

Where

Ci : gas phase concentration of species i, lb

moles/ft3

Dim : gas phase diffusivity of species i, ft2/hr

Equation (5) descretised using central

difference for the second order derivatives and

forward difference for the time and inertial

derivatives (5). These differences yielded

excellent numerical stability.

The boundary condition at the side walls

resulted from a zero mass flux and zero gas

velocity constraint at the wall is given as (6)1:

0y

Cx

xC

y ii =∂∂

+∂∂

(6)

By observing Figure (3) it can be seen

that Equation (6) gives ∂Ci/∂y = 0 at the left

and right hand boundaries and ∂Ci/∂x = 0 at the

top and bottom boundaries. At the boundary

positions located in between those mentioned,

Equation (6) shows that there is a relationship

between the concentration derivatives for all x

and y values which is taken into account in the

computer program.

At the top dome the concentration

gradient with respect to the vertical direction is

zero ( 0z

Ci =∂∂

). This condition is valid during

exhale but not during inhale conditions at the

vent ports which are located in the top dome. In

the vent ports during gas inhale, the flux of all

volatile species is considered to be zero,

giving1:

im

zi

z

i

DV

Cz

C=

∂∂

= )0( (7)

Emission of each species is calculated

by integrating the boundary concentration at

the vent with respect to time to give the total

amount of volatile species emitted, Ei.

dtVCE ventzventii ∫= )()( (8)

Where

iE : emissions flux of species i, lb moles/ft2

At the liquid surface the gas phase

concentration, Ci (gas sur), is assumed to be in

equilibrium with the liquid surface

concentration, Ci (liq sur). These gas phase and

liquid phase concentrations are related through

the vapor/liquid equilibrium constant, K;

liq

gasSurliqigasSuri C

CKCC )()_(

)(

××= (9)

Where

Ci (gas sur): gas concntration of species i at liquid

surface, lb moles/ft3

Ci (liq sur): liquid concentration of species i at

surface, lb moles/ft3


863

C(gas): total concentration in gas phase, lb moles/ft3 C liq: total liquid phase concentration, lb

moles/ft3

K :vapor/liquid equilibrium constant

The value of K for each hydrocarbon is

obtained from the Depriester charts (2) but

could also be obtained from Davis (3). The

value of K for a dissolved gas such as nitrogen,

hydrogen, hydrogen sulfide and carbon dioxide

were obtained from Edmister and Lee 1984 (4)

or could be estimated from experimental data,

in this study predicting K values was

depending on using equation of state (5).

The liquid phase composition at the liquid

surface, Ci (liq sur), was related to the bulk

liquid phase composition deep below the liquid

surface, Ci (liq blk), by specifying a boundary

layer film thickness, FILM, that a species

would have to diffuse through. This

assumption related the gas phase flux at the

liquid surface, Ni (at z=H), to the bulk liquid

composition which allowed the depletion of

volatile components at the liquid surface1.

DilFilmN

CC Hzatiblkliqisueliqi

×−= = ).(

)_()_( (10)

Where

Dil: liquid phase diffusivity of species i, ft2/hr

Film: thickness of liquid surface resistance

layers ft

Ni: molar flux of species i in gas phase, lb

moles/ft2 hr

This type of boundary condition was

needed in order to take into account the

mixedness of the liquid phase. A small value of

FILM such as 0.0001 ft would imply a well

mixed liquid in which Ci (liq sur) = Ci (liq blk),

whereas a large value of FILM would relate to

a stagnant liquid phase where Ci (liq sur) could

be much less than Ci (liq blk). A large value of

FILM would be on the order of 1.0 feet. The

variable, Dil, is the liquid phase diffusivity of a

species; the value of Dil in Equation (11) was

estimated by the Wilke correlation (1).

µ

×××−=

6.

)boil_liq(iCTMb8E4.7

Dil (11)

In Equation (11) Mb is the molecuar

weight of the mixture, T is the liquid

temperature, µ is the liquid viscosity and Ci (liq

boil) is the molar concentration of species “i”

as liquid at its normal boiling point. Other

0yCi =∂∂ 0

yCi=∂∂

0xCi =∂∂

0xCi =

0

yC

xxC

y ii =∂∂

+∂∂

Figure (3-3) Species Equation Boundary Condition Top View19


Kadhum Judi Hammud

ج

864

values of Dil could be supplied to the program

by the user.

The gas phase velocity in the vertical

direction at the liquid surface is represented

by1:

∑ == +=

)gas(

)Hz_at(i)Hz_at(z C

NdtdHV (12)

Equation (12) is composed of two

effects which establish the gas phase velocity

at the liquid surface; 1) the velocity of the

physically moving surface, dH/dt, and 2) the

velocity resulting from all the species trying to

evaporate into the gas phase. Together these

two effects establish the effective velocity of

the liquid surface.

Figure (4) is a tank side view showing the

composition boundary conditions at tank

internal surfaces.

2.Energy

The partial differential equation which

describes the conduction and bulk velocity

transport of energy (gas temperature) within

the gas space of a tank is given by (1)(6):

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

zTV

yTV

xTV

tTC

zT

yT

xTk zyxρ2

2

2

2

2

2

(13)

Where k gas phase thermal

conductivity, BTU/ft hr °F

T liquid temperature, °F

Equation (13) was discredited and

numerically solved in the same manner as the

species equation (Equation (5)). The necessary

boundary conditions needed to solve the

internal gas phase temperature Equation (13)

are shown in Figure (5). These boundary

conditions take into account conduction,

convection, and radiation energy and establish

internal tank surface temperatures on the top

dome, liquid surface, and side walls.

The tank top dome temperature is established

by Equation (14) which relates the solar flux to

the sky irradiation, external wind convection,

dome conduction, and dome temperature

change(6).

( )dtdodTLCpQQQEdoQ domeconvskysolar ρ+++=× (14)

Where

Qconv dome convection energy flux to ambient,

BTU/ft2hr

Qdome dome conduction energy flux, BTU/ft2hr

Qsky energy flux reradiated to sky, BTU/ft2hr

Qsolar incident solar radiation flux, BTU/ft2hr Figure (4)

Species Equation Boundary Conditions

Figu


865

Cp gas phase heat capacity, BTU/lb mole °F

The solar radiation, Qsolar, is a

function of many tank external variables.

Variables such as clear sky to hazy sky ratio,

day of the year, time of day, latitude, altitude,

and emissivity of the tank external surface,

Edo, are extremely significant in predicting the

net incident solar radiation flux on a tank

external surface. The instantaneous value of

Qsolar can be predicted from the relationships

found in Duffie and Beckman (1980) or could

be supplied to the computer program from field

data. The sky reradiation energy flux, Qsky,

can be predicted by:

( )44sky TskyTdoEdoQ −σ= (15)

Where

Qsky: energy flux reradiated to sky, BTU/ft2hr

Tdo: dome outside temperature, °F

σ: stephan — Botlzmann constant

Edo: emissivity of dome outside surface

In Equation (15) the outside dome temperature,

Tdo, will vary with time of day as a result of

solving Equation (14). The effective sky

temperature, Tsky, is a function of daily

ambient air temperature and can be computed

from equations as given in Duffie (1980).

The convective heat loss to the ambient

air is given by(6)

)( TairTdoUQconv −×= (16)

The convective energy loss flux is equal to the

heat transfer coefficient times the difference

between the outside dome temperature, Tdo,

and the ambient air temperature, Tair. The heat

transfer coefficient, U, is a function of tank size

and wind velocity as given in Duffie7 (1980).

The energy that conducts from the outside

dome surface to the inside dome surface, Qdome,

is given by:

( )

⎟⎠⎞

⎜⎝⎛ +

−=

KiLi

KdLd

TdiTdoQdome

(17)

Where Ki: vapor/liquid equilibrium constant for species. Li: dome insulation thickness ft. Equation (17) contains the dome metal thermal

conductivity, Kd, and the dome metal

thickness, Ld, along with thermal conductivity

and thickness of any insulation. Equation (17)

essentially relates the inside dome temperature,

Tdi, to the outside dome temperature, Tdo. The

equations expressing the wall external and

internal temperatures are given by

( )dtwodTLCpQQQEwoQ domeconvskysolar ρ+++=×

(14a)


Kadhum Judi Hammud

ج

866

( )44sky TskyTwoEwoQ −σ=

(15a)

)( TairTwoUQconu −×= (16a)

( )

⎟⎠⎞

⎜⎝⎛ +

−=

KiLi

KdLw

TwiTwoQdome (17a)

The radiation energy fluxes are

calculated simultaneously by Equation (18) in

order to determine the inside dome, wall, and

liquid surface temperatures.

The Equations (18), (19) and (20) were

obtained from Incropera and Dewitt (1981)(7).

( ) 13)31(12211 FJJFJJQrad −+−=

( ) 23)32(21122 FJJFJJQrad −+−= (18)

( ) 32)23(31133 FJJFJJQrad −+−=

Where

Qradl: radiation flux from dome, BTU/ft2hr

Qrad2: radiation flux from wall, BTU/ft2hr

Qrad3: radiation flux from liquid, BTU/ft2hr

F12: view factor dome to wall

F13: view factor dome to liquid

F21: view factor wall to dome

F23: view factor wall to liquid

F31: view factor liquid to dome

F32: view factor liquid to wall

J1: radiosity of dome, BTU/ft2hr

J2: radiosity of wall, BTU/ft2hr

J3: radiosity of liquid surface, BTU/ft2hr

Equation (18) relates the net radiation

leaving a surface to the radiosity, J, of the

surfaces and the view factors, F, of the

surfaces. The subscripts 1, 2, and 3 refer to the

dome, wall, and liquid surfaces respectively.

The view factors are determined from

the following relationships:

⎟⎠⎞

⎜⎝⎛ −××+×−= 1

RH

RH158.0

RH62.0113F

(19)

F31=F13

F12=1-F13

F32=F12

F21=F32×R/ (2H)

F23=F21

( ) ( )( ) ⎥

⎦

⎤⎢⎣

⎡+×−+

×+××−+×=

13F12F)1E1(1E3J13F2J12F1E11G1E1J

( ) ( )( ) ⎥

⎦

⎤⎢⎣

⎡+×−+

×+××−+×=

23F21F)2E1(2E3J23F1J21F2E12G2E2J

(20)

( ) ( )( ) ⎥

⎦

⎤⎢⎣

⎡+×−+

×+××−+×=

32F31F)3E1(3E2J32F1J31F3E13G3E3J

The values of

G1= σ×Tdi 4

G2= σ × Twi 4

G3 =σ × T14

Where

H: outage height, ft

R: tank radius, ft

El: emissivity of dome inside

E2: emissivity of wall inside

E3: emissivity of liquid surface

In Equation (1)7 H is the tank outage

and R is the tank radius. The values of the

radiosities are calculated by Equation (20)7.

An energy balance at the inside dome surface

then relates the Qdome to the radiation leaving


867

the dome and to the energy of conduction

leaving the dome and flowing inside the tank6;

dzdTgK

QQ g1raddome −= (21)

The second term in Equation (21) is

the conduction of energy into the gas space at

the dome. The equation for the wall energy is

the same as Equation (21) with the subscripts

changed from dome to wall and the

temperature gradient taken in the horizontal

plane.

The metal temperature distortion

involved where the top dome and wall intersect

was disregarded in the model for the following

reasons: Using a cooling fin analogy, 90

percent of the temperature distortion caused at

the intersection is recovered by the metals

within 4 inches of the intersection. Similarly,

the wall metal temperature distortion involved

where the liquid basin and the wall intersect

was also disregarded by using the same cooling

fin analogy. The liquid basin temperature was

assumed not to distort near the wall/basin

intersection because of the large size of the

basin.

The liquid surface energy balance

relates the net radiation flux with the

conduction fluxes into the gas and liquid

phases together with the energy needed by the

evaporating materials from the liquid surface:

∑ λ×+×

−×

−= ig

3rad Ndz

dTlKldz

dTgKQO

(22)

Where

O: Liquid Surface enrgy

λ: heat of vaporization BTU/lb mole

Kg: thermal conductivity of gas, BTU/ft hr °F

The following procedure is used by

the computer model to establish the solar

fluxes on the top dome and on each section of

the walls. The walls were divided into 42

vertical sections so that the direction of solar

energy with time of day could be taken into

account. The equations needed to establish the

solar insolation on the tank exterior surfaces .

1. The latitude angle of the tank location

is specified by the user, phi. Phi is

positive north of the equator.

2. The day of the year is specified by the

user, Nday.

3. The earth angle of declination is

calculated, del.

⎟⎟⎠

⎞⎜⎜⎝

⎛ +××=

365N284

360sin45.23del day

where

del: earth’s angle of declination, degrees

4. The sunset angle is calculated, Ws, by:

tan(phi)tan(del)-cos(Ws) ×=

5. The number of daylight hours is

calculated, N; 15

Ws2N ×=

6. The time of dawn and sunset are

calculated; 2N12tdawn −= ,

2N12tset +=

7. At a specific time of day set the solar

angle, W;


Kadhum Judi Hammud

ج

868

( )tdawntime15WsW −×+=

8. Calculate the angle of incidence or the

angle between the beam radiation on a

horizontal surface and the normal to

that surface, Θz,

cos(Θz) = sin(del)×sin(phi)+cos(del)

×cos(phi) ×cos(W)

9. Calculate the hourly extraterrestrial

radiation on a horizontal surface,

Io=414× {1+.033×cos(360×365

Nday ) ×

[sin(phi) ×sin(del)+cos(phi) ]×cos(del)

×cos(W)}

10. Calculate beam radiation, Tb;

⎟⎟⎠

⎞⎜⎜⎝

⎛Θ

−×+=

)zcos(kexp1aaoTb

where ao, al and k are functions of

altitude and day of year

11. Calculate diffuse radiation Td,

Td=0.271-0.2939×Tb

12. Specify the clearness ratio which is the

ratio of the clearness of the sky to that

of the sky on a perfectly clear day,

Iclear.

13. Calculate the diffuse fraction of total

radiation,

IdI = 1-0.1×Iclear if Iclear <0.48, IdI =

1.1+0.0396×Iclear-0.789×Iclear2 if 0.48 <

Iclear < 1.1, IdI = .2 if Iclear > 1.1

14. Calculate the total radiation on a

horizontal surface, Ihor,

Ihor = Iclear× (Tb+Td) ×Io

15. Calculate the diffuse radiation, Id, Id =

IdI×Ihor

16. Calculate total beam radiation, lb =

Ihor-Id

17. Select a portion of the wall and

calculate the angle from the normal of

the surface to the meridian, Gamma,

18. Calculate the angle of solar incidence

with the normal of the surface, Θ,

cos(Θ)=-

sin(del)×cos(phi)×cos(Gamma)+cos(del)×

sin(phi)×cos(Gamma)×cos(W)+ cos(del)

×sin(Gamma)×sin(W)

19. Calculate the total solar energy on that

particular vertical surface, Ivert,

Ivert = Id + Ib×cos(Θ)/cos(Θz)

By using this procedure, the solar energy

that is incident on the top dome and on 42

different positions on the side walls can be

estimated for a storage tank located at a

specific place and on a specific day of the year.

2.4-Flash Calculation

The mathematical model takes into account the

possible flashing of crude oil as high pressure

feed stocks are reduced to atmospheric

conditions. The crude is analyzed at a specified

temperature and pressure by the computer

program to determine the vapor and liquid

phase amounts and compositions. This analysis

allows the crude energy content to be

established so that when the crude pressure is

dropped to ambient, the resulting flashed vapor

and liquid mixture has the same energy content

as the specified high pressure crude. This

adiabatically flashed crude is assumed to be

separated into pure vapor and pure liquid

streams. These streams can then be routed to a


869

storage tank at the user’s discretion. For

example, the liquid stream is sent to a specific

location in the tank’s liquid basin while the

vapor stream may be routed to a specific top

dome location, or to the liquid basin area, or it

may simply bypass the tank altogether.

Specifically, the hydrocarbon species that

identify the crude composition are; methane,

ethylene, ethane, propylene, propane,

isobutane, n-butane, isopetane, n-pentane, n-

hexane, n-heptane, n-octane, n-nonane, and

decane. Materials that are higher in molecular

weight than n-decane will be lumped together

and designated as n-undecane plus; the 4

hydrocarbon soluble gases incorporated in the

program are carbon dioxide, hydrogen,

hydrogen sulfide, and nitrogen. The vapor

liquid equilibrium constants (K values) for the

hydrocarbon species were obtained from Peng

– Robinson equation of state.

Equilibrium ratios (K-values) are used

in the phase behavior techniques to predict

composition change in liquids and gas when

the phase diagram of particular hydrocarbon

fluid. The area enclosed by the bubble point

and dew point curves is the region of pressure

temperature combination at witch both gas and

liquid phases will exist. The curves within the

two phase region show the percent of the total

hydrocarbon volume, which is liquid for any

temperature and pressure.

In many references there are many procedure

have been developed to occupy solution of

flash calculation and to increase the accuracy

of the solution. Which in general depend on

iteration procedure .

The ratio of the vapor mole fraction to the

liquid mole fraction for a given component is

known as the equilibrium ratio, or alternatively

as the K-value, and is defined as

iii /xyK = (23)

Empirical correlations can be used to provide

an initial estimate of the equilibrium ratios. The

Wilson equation (24) (3) was used in this study

ri

rii

i P

)T1)(1ω(15.37exp

K⎥⎦

⎤⎢⎣

⎡−+

= (24)

Where

ciri T/TT = (25)

ciri P/PP = (26)

2.4.1-Material Balance

To predict mole percent of each

component for liquid and vapor in the tank in

this study martial balance principle was used.

Material balances are based on the fundamental

“law of conservation of mass” therefore the

system was assumed containing a total of one

mole of chemical species, and having an over

all composition represented by the set of mole

fractions {zi}. Let L be the moles of liquid,

with mole fractions {xi} and let V be the moles

of vapor, with mole fractions {yi}. The

material-balance equations are

L+V=1 (27)

zi= xiL+yiV ( i=1,2,3……,n) (28)


Kadhum Judi Hammud

ج

870

Choosing to eliminate L from these equations,

we get

zi=xi(1-V)+yiV (29)

As a matter of convenience, we write Raoult’s

law as

yi= Ki xi (30)

Where Ki is known as a “K-value”, given here

by

Ki= yi/xi (31)

The problem is to calculate for system of

known overall composition {zi} at given T and

P the fraction of the system that is vapor V and

the compositions of both the vapor phase {yi}

and liquid phase {xi}. This problem is known

to be determinate on the on the basis of

Duhem’s theorem, because tow independent

variables (T and P) are specified for a system

made up of fixed quantities of its constituent

species.

On the basic material balances and the

definition of a K- value we derived the

equation following eqution:

)1K(V1Kzy

i

iii −+= (i=1, 2… n) (32)

Since xi=yi/Ki an alternative equation is

)1K(V1zx

i

ii −+= (1, 2… n) (33)

Since the both sets of mole fractions must sum

to unity, ∑ xi= ∑ yi = 1. Thus, if we sum eq.

(32) over all species and subtract unit from this

sum, the difference Fy must be zero; that is ,

∑ =−−+

= 01)1K(V1

KzFi

iiy (34)

Similar treatment of Eq. (33) yields the

difference Fx, which must also be zero.

∑ =−−+

= 01)1K(V1

zFi

ix

(35)

Solution to a P, T-flash problem is

accomplished when a value of V is found that

makes either the function Fy or Fx equal to

zero. However, a more convenient function for

use in a general solution procedure is the

difference :FFxFy =−

0)1K(V1

)1K(zFi i

ii∑ =−+

−= (36)

The advantage of this function is apparent from

this derivative:

( )∑−+

−−=

i2

i

2ii

)1K(V1)1K(z

dVdF

(37)

Since dF/dV is always negative, the F

vs. V relation is monotonic, and this make

Newton’s method, a rapidly converging

iteration procedure, well suited to solution for

V. Newton’s method here gives

j

jj1j )dV/dF(

FVV −=+

(38)

Where j is the iteration index and Fj and

(dDf/dv) j are found by Eqs. (36) and (37). In

these equations the K-values may be computed

from

i

ii x

yK = (39)

2.4.2-The Fugacity

The chemical potential that determines

whether a substance will undergo a chemical


871

reaction or diffuse from one part of system to

another.(4)

For constant-composition fluid at

constant temperature chemical potential can be

calculated as

dP V dG = (40)

From equation of state we can replace v

by its value for ideal gas as shown

PRTV =

(41)

This for pure ideal gas

( ))Pln(dRTdPP

RTdG ×=⎟⎠⎞

⎜⎝⎛=

(42)

The chemical potential of real fluid can

be expressed by replacing pressure in equation

with propriety called fugacity f

( ))fln(dRTdG ×= (43)

Since equation (43) defines fugacity in

differential form, a reference value is required.

P)f(Lim0p

=→

(44)

Note that fugacity simply replaces

pressure in the ideal gas equation to form a real

gas equation. Fugacity has pressure unites.

Combination of equation (40) with (43)

gives :

dpRTV)f(lnd ⎟

⎠⎞

⎜⎝⎛= (45)

Equation (45) merely states that at low

pressures the fluid acts like an ideal fluid.

At equilibrium, the chemical potential

for the liquid must equal chemical potential for

the gas. For pure substance this means that at

any point along the vapor pressure line, the

chemical potential of the liquid must equal the

chemical potential of gas. Thus equation (43)

shows that the fugacity of liquid must equal the

fugacity of gas at equilibrium on the vapor

pressure curve. So gas liquid equilibrium can

be calculated under the condition that

lg ff = (46)

The following expression can be

derived form equation (45) under the constraint

of equation (44) (4)

∫∞

⎟⎠⎞

⎜⎝⎛ −+−−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ VdVP

VRT

RT1Zln1Z

pfln

(47)

For a pure substance, the ratio of

fugacity to pressure pf is called fugacity

coefficient.

The situation with regard to mixtures

somewhat more difficult to visualize. However,

equilibrium is attained when the chemical

potential of each component in the liquid

equals to chemical of that component in the

gas.

The chemical potential of the

component of mixture may be calculated as:

)f(lnRTddG ii = (48)

The reference value for fugacity in this

equation is

iii0pppyflim

i

==→

(49)

That is, as pressure approaches zero the

fluid approaches its ideal behavior and the

fugacity of component approaches the partial

pressure of that component.


Kadhum Judi Hammud

ج

872

The chemical potential for a component

of mixture at equilibrium must be the same in

both gas and liquid. Thus equation (49) shows

that at equilibrium the fugaci ties of a

component must be equal in both gas and

liquid. So, For all component i gas liquid

equilibrium can be calculated under the

condition

i li g ff = (50)

This is analogous to development of the

equations for ideal solutions, where the partial

pressure of the liquid was set equal to the

partial pressure of gas.

The value of fugacity for each

component is calculated with an equation of

state and the fugacity coefficient for each

component of a mixture is defined as the ratio

of fugacity to the partial pressure

pyf

i

ii =ϕ (51)

Fugacity coefficient may be calculated as (4)

∫∞

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−=ϕV

n,V,Tii ZlndV

np

VRT

RT1ln

i

(52)

Further, the ratio of the fugacity coefficient can

be used to calculate K- value

i

i

i

gi

i

Li

gi

Lii y

x

Pyf

Pxf

K ==ϕϕ

= (53)

2.4.Cubic Equation of state

The equation that relats pressure, molar

or specific volume, and temperature for any

homogenous fluid in equilibrium states called

equation of state.

For an accurate description of phase

behavior of fluids over wide ranges of

temperature and pressure an equation of state

more comprehensive than the virial equation is

required. Such an equation must be sufficiently

general to apply to liquids as well as to gases

and vapors. Yet it must not be so complex as to

present excessive numerical or analytical

difficulties in application.

Polynomial equations that are in cubic

in molar volume offer a compromise between

generality and simplicity that is suitable to

many purposes. Cubic equations are in fact the

simplest equations capable of representing both

liquid and vapor behavior. The first general

cubic equation of state was proposed by Van

Der Waals .

In this study, Peng-Robenson Equation

of state was applied to estimate phase behavior

since it was proved to be its one of the best

method as compared by authors and searchers

for volatile oil.

2.4.4-Peng – Robinson Equation of

State (PR EOS)

Peng and Robinson (1976) proposed the

following equation of state

( ) ( ) ( )bVbbVVa

bVRTP

mmmm −++α

−−

=

(54)


873

Where

[ ]25.0r )T1(m1 ++=α (55a)

22699.05423.13746.0m ω−ω+=

(55b)

For PR EOS, the constants a and b are

defined by the following equations:

c2c

2 P/TR45724.0a = (56a)

cc P/RT0778.0b = (56b)

In vapor-liquid calculation for mixture Peng

– Robinson suggested the same mixing rule of

Soave (1972).

( ) ( ) ( )∑∑= =

−αα=αM

1i

M

1jij

5.0jijiijm k1aaxxa

(57a)

∑=

=M

1iiim bxb

(57b)

The values of binary interaction coefficients

(Kij) that are used in Peng-Robinson (EoS) are

set zero in this work.

Compressibility factor is calculated from

the following equation:

( ) ( ) ( ) 0BBABZB2B3AZ1BZ 32223 =−−−−−+−+ (58)

Where A and B are calculated from

( )2RTPaA α

= (58a)

RTbPB = (58b)

The fugacity coefficient of component i in a

mixture is calculated from the following

equation:

( ) ( ) ( ) ⎟⎠⎞

⎜⎝⎛

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

ΨΨ

⎟⎠⎞

⎜⎝⎛−−−

−=Φ

BZBZ

bb

BABZ

bZb

m

ii

m

ii 414.0

414.2ln282843.2

ln1

ln

(59)

Where

( ) ( )∑=

−αα=ΨN

1jij

5.0jijiji k1aaX (59a)

( ) ( )∑∑= =

−αα=ΨN

1i

N

1jij

5.0jijiji k1aaxx (59b)

2.4.5-Heptane and heavier

component properties

Compositional analysis of petroleum

fluids are generally reported as series of pure

components form methane through hexane

with the remaining components grouped

together as heptane plus, (or hexane plus

sometimes). The heptane plus contained a

mixtures of hydrocarbon components heavier

than hexane which typically includes paraffin,

aromatics & naphthenes. Due to this the

properties of heptane plus (critical pressure,

critical temperature and acentric factor) will be

very effective especially with volumetric and

phase behavior calculations. Therefore using

equation of state will be difficult in

characterizing the heavy fractions of

hydrocarbon mixtures. However, several

correlations have been revised and extended for

several times. In this study to predict the

critical properties for the plus fraction for PR

EOS Riazi and Duubert correlation is used.

Riazi and Duubert correlation will depend on

boiling point and specific gravity. The equation

has the form:

( ) ( )Cc

bB

7Ta +γ=θ (60)

Critical pressure and critical temperature can

be calculated from the above equation. Where

(θ) is a physical property to be predicted, (a, b,


Kadhum Judi Hammud

ج

874

c) are correlation constants as shown in Table

(1).

The value of the boiling point estimated

from Whitson relationship as follows; 3

15427.015178.0

CB

7c

)M(T7

5579.4

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡γ=

++

(61)

Where TB is the boiling point in degree Rankin.

The correlation coefficient (R) of

equation (60) is 0.992 when using the equation

for estimating the critical pressure and R equals

0.9946 when using the equation for estimating

critical temperature.

Table (1) Correlation Constant for Riazi

and Dubert Equation and Modified Riazi

and Dubert Equation

Riazi and Dubert Correlation Constant

PC TC

A 12281*109 24.2787

B -2.3125 0.58848

C 2.3201 0.3596

The third property that must be found

is acentric factor. Edmister (4 ) suggested an

approximate correlation for find the acentric

factor that depend on three variables; critical

pressure, critical temperature and boiling point.

The boiling point has been calculated from

Whitson relation and unit for critical pressure

is in atmospheres. Edmister4 Correlation has

the form.

11T/T

Plog73

BC

C −⎥⎦

⎤⎢⎣

⎡−

=ω (62)

Conclusion 1- The first method which are gives

good results depending on the application of the program to the field data

2- Application results different from one place to another because of the weather factors including temperature, winds, sun rays and the length of the day time

3- Rate of the evaporation in fixed roof tanks is higher than the rate notice in floating roof tanks due to the additional weight applied on the liquid inside the tank in floating roof tank case.

References

[1] R Byron Bird, Waren E. Stewart,

Edwin N. light foot, transports

phenomena John Wiley 1960.

[2] DePriester, C.L. chem. Eng.

Progr. Symposium ser. 7:149

(1953).

[3] Perry, R.H. and Chilton, C.W.

"Chemical Engineering

Handbook" 5th ed. McGraw Hill

company, New York, 197

[4] Edmister w.c and Lee B. I.

"Applied hydrocarbon


875

thermodynamic" 2th ed. Gulf

publishing Co. Houston 1984.

[5] Madi N Al-Dulaimy M.Sc.

dissertation university of Baghdad

Jan 2001 Comprehensive

computer model from PVT

analysis.

[6] Holman J.P heat transfer 8th

edition MCG 1996.

[7] Incropera, F.P., D.P. Dewitt,

fundamentals of Heat Transfer,

John Wiley, 1981.

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