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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.
Journal of Engineering Volume 18 July 2012 Number 7
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
CALCULATION OF EMISSION FROM FIXED ROOF STORAGE TANK
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
Journal of Engineering Volume 18 July 2012 Number 7
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
CALCULATION OF EMISSION FROM FIXED ROOF STORAGE TANK
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
Journal of Engineering Volume 18 July 2012 Number 7
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)
CALCULATION OF EMISSION FROM FIXED ROOF STORAGE TANK
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
Journal of Engineering Volume 18 July 2012 Number 7
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;
CALCULATION OF EMISSION FROM FIXED ROOF STORAGE TANK
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
Journal of Engineering Volume 18 July 2012 Number 7
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)
CALCULATION OF EMISSION FROM FIXED ROOF STORAGE TANK
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
Journal of Engineering Volume 18 July 2012 Number 7
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.
CALCULATION OF EMISSION FROM FIXED ROOF STORAGE TANK
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)
Journal of Engineering Volume 18 July 2012 Number 7
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,
CALCULATION OF EMISSION FROM FIXED ROOF STORAGE TANK
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
Journal of Engineering Volume 18 July 2012 Number 7
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.