CHEMCEPT LIMITED
Mathematical Modelling, Chemical Engineering Software and Engineering Consultancy
Consultancy
Crays Pond, Reading, England
Mathematical Modelling, Chemical Engineering Software and Engineering Consultancy
Software Products

a
allows for the velocity profile in the pipe (for turbulent flow, it is about
0.94).
G
is the mass flow through the pipe
v
is the molar specific volume;
v
_{
1
}
and
v
_{
2
}
are the specific volumes at inlet and outlet
M
is the mean molecular weight of the mixture
f
_{
m
}
is the mean friction factor
l
is the pipe length
d
is the pipe diameter

Equation (1) applies for gas, liquid and two-phase flows. It also applies for
isothermal and adiabatic flows.

For liquids, the first term is zero.

For liquids and vapours, friction factor varies very little with distance along
the pipe. However, for two-phase flows, friction factor depends strongly on
pressure.
The correct choice of mean is then essential for computing the
frictional pressure drop accurately.
To good approximation, the same mean
applies for single-phase flow and two-phase flow. The user has the choice of
Colebrook, Churchill, Darcy-Churchill and Spitzglass friction factor
correlations.
Figure S1 compares the computed friction factors for smooth
pipes. It also shows the friction factor computed by the Blasius Equation,
which is the most accurate for smooth pipes. It is seen that there is an 18%
spread between the estimates. This spread gives an estimate of the intrinsic
error in estimating turbulent pressure drops. Smooth pipe conditions are very
well defined. Rough pipes are less well defined, and we may expect larger
errors in estimated friction factors. Flaresign aims to solve pressure drop
equations to within 2% to 3%. We see that this precision is an order of
magnitude better than the intrinsic accuracy of the equations that we are
solving.

The relationship between
P
and
v
depends on the fluid, and for gas and two-phase flows, on the heat flux
conditions (isothermal, adiabatic or finite flux). Using the Chemcept
Pressure-Virial Equation of State, the integral term in Equation (1) reduces to:

dP/ v = PdP/ [F
_{
1
}
{T} + F
_{
2
}
{T}P] (2)

Equation (2) applies to both single phase and two-phase flow. For isothermal
flow, there is a simple analytical solution.

For adiabatic (or finite heat transfer) flow, temperature is related to
pressure by energy balance. Flaresign estimates temperatures at every node on
the network from which, for each pipe length, it approximates the relationship
between T and P. The equations can then be solved analytically. The
temperatures at each node are recomputed by energy balance and a super-linear
iterative scheme is employed to reach converged estimates of the temperatures.
The adiabatic option is not available in the current release.

The accuracy of the analytical solutions can be appreciated by the following
examples:

Single phase isothermal flow of ethylene gas at 298 K
Mass Flow Rate 100 kg/s
Discharge directly to atmosphere at 1.0 bar
Pipe Length = 10 km
Pipe Internal Diameter = 0.7 m
Pipe Roughness = 0.04572 mm
Colebrook Friction Factor Formula.

The 1-step analytical solution is compared to an accurate numerical solution in
Table 1

Table 1. Flow of Ethylene Gas in 10-km pipe.

Pressure in (bar) Pressure out (bar)
Numerical Solution 9.86708 1.00129
Analytical Solution 9.86669 1.00129

It is seen that the analytical solution differs from the accurately computed
solution by only 0.004%. This difference is negligible for a calculation that
probably has intrinsic errors of 10%. Furthermore, few numerical integrations
will be converged this accurately. Note that the pipe shows almost a
factor-of-ten pressure drop. Such a large pressure ratio in one pipe can occur
in emergency release networks, but is unusual. Over 45% of the pressure drop
is in the last 25% of the pipe length. Thus, the remaining 75% of the piping
could use a smaller diameter with negligible increase in overall pressure drop.
A smaller pipe is possible because, although the output to the large pipe is
80% choked, the inlet is only 8% choked. Furthermore, at 75% of the length,
the pipe is still only 15% choked. Thus, there is a large velocity margin
before choked conditions are approached. There would be cost benefits in
smaller pipe and an increased safety margin because of the higher pressures
than can be safely held in a smaller diameter pipe of a given gauge. For these
reasons, it is unlikely that practical applications will show a greater
proportional pressure drop in one length of pipe. The result confirms that the
analytical solution gives excellent accuracy as well as several orders of
magnitude decrease in computer run-time.

The approximation is less accurate for 2-phase flow because the two-phase
friction factor varies strongly with mixture pressure. Thus, the choice of
mean friction factor in equation (1) becomes more critical. Table 2 gives a
comparison of accurate and approximate 2-phase pressure drop computations. The
two-phase calculation is for the same conditions as for the single-phase
calculation except that the gas flow is reduced to 50 kg/s and a liquid water
flow of 50 kg/s is introduced.

Table 2. Flow of Ethylene water mixture in a 10-km pipe.

Pressure in (bar) Pressure out (bar)
Numerical Solution 13.99070 1.00018
Analytical Solution 13.93840 1.00018

(The difference between the outlet pressures and the single-phase outlet
pressures arises because we insert a hypothetical zero pressure drop KO drum.
The resulting reduced gas flow has a lower pressure drop as it is discharged to
atmosphere).

The resulting error in using the approximate analytical solution is 0.4%. This
error is small compared to the inherent error in two-phase flow friction-factor
correlations. These errors are typically 20%. Both the Dukler method and the
Lockhardt-Martinelli method show similar accuracy (or lack of it). We use a
modified Lockhardt-Martinelli method because it has the benefit of giving an
upper-bound estimate of the two-phase friction factor within the accuracy of
the single-phase friction factor methods employed. Thus, the user can be
assured that the pressure drop is not underestimated.

As for the single-phase example, the example probably illustrates extreme
conditions, not likely to be met in practice. Thus, over 50 % of the pressure
drop is in the last 25% of the length. At the exit of the pipe, the flow is
56% choked, but is only 7% choked at the inlet to the last 25%. For a given
mass flow rate, two-phase friction factors increase as the pressure drops
(roughly as the inverse square root of pressure). Thus, pressure drop is
forced into the low-pressure section of a pipe even more than is the case for
single-phase flow. There is then a stronger incentive to have larger diameter
pipes at the low-pressure end and smaller diameter pipes at the high-pressure
end. For shorter pipes, the analytical solution gives much closer agreement
with an accurate numerical integration. Thus, the discrepancy of 0.4% is
unlikely to be exceeded in any practical application.

2) Sonic and Choking Flow.

The maximum velocity that can be achieved in a uniform pipe is the choking
velocity. For the adiabatic flow of a single-phase gas, the choking velocity
is the sonic velocity. Flaresign gives solutions of equation�(1) that are less
than the choking velocity for all single- and two-phase flow regimes. However,
it is desirable to be able to compute the choking velocity to ensure that
networks are not designed to get close to the choking velocity in pipe flow.
Choking conditions can also arise in pipe fittings and at pipe expansions.
Thus, we need to be able to compute choking flow for any fluid.

For flow in pipes, Whalley shows that the liquid velocity can be treated as the
same as the gas velocity for inertial calculations. This assumption may be
incorrect in computing hold-up. However, the total kinetic energy of the
liquid is the same as if it were flowing at the same velocity as the liquid.
Changes in hold-up and radial flow distribution compensate through the
different flow regimes maintain the inertial identity. Sudden accelerations
and decelerations may occur in fittings, and cause the liquid velocity to
change relative to the gas. However, we assume that we can maintain the
identity in order to compute two-phase choking flow.

The basic thermodynamic equation for choking flow can be derived from equation
(1). For uniform flow across the pipe, it is:

u
_{
X
}
= v[-v
^{
2
}
(dP/dv)
_{
X
}
/M] (3)

In equation (3),
u
is velocity; the other symbols are as for equation (1). Subscript
"X"
specifies what is held constant. Thus, for isothermal flow
"X"
is temperature, for isentropic flow,
"X"
is entropy. The Chemcept 2-phase equation of state enables equation (3) to be
solved analytically for conditions of practical interest. A number of
competitive programs assume that the compressibility factor,
Z
, remains constant when the partial differential is calculated. Flaresign
employs a full equation of state so that
dZ/dv
is automatically computed. Thus, Flaresign provides a more accurate estimate
than some systems that employ more complex physical-property correlations.

Flaresign corrects equation (3) to allow for radial velocity profile.

3) Expansions and Contractions.

Flaresign is one of very few systems that has a theoretically consistent
treatment for the flow of compressible and two-phase fluids through fittings.

There are well-established correlations for liquid flow pressure-drop through
fittings. These same correlations are applied to gas flows with low
pressure-drops (when compressibility effects can be ignored). There have been
a limited number of publications showing that the correlations cannot be
applied when there is a gas flow with significant pressure drop. (See for
example, the early work by Benedict, Carlucci and Swetz).

We consider first sudden expansions. For an incompressible fluid, a simple
momentum balance gives an excellent estimation of pressure change.
Theoretically, the incompressible formula should not apply to a compressible
fluid. When pipe diameter increases, the fluid velocity falls and the pressure
increases. For a compressible fluid, the increase in pressure gives a decrease
in specific volume so that the velocity falls further. The further fall in
velocity gives rise to a further increase in pressure. It is possible to
consider these effects in performing a modified momentum balance across a
sudden pipe expansion. The resulting expression correlates the data of
Benedict and co-workers better than their original correlation. This modified
formula is employed in Flaresign. It has the benefit that, at low
pressure-drops, it automatically reduces to the incompressible flow formula.
We have access to no experimental data for high-pressure gases when
non-idealities become important. However, our simple theory accounts for the
data we can find. Accordingly, we have used the Chemcept-data equation of
state to extend the method to non-ideal gas flows. With slightly less
justification, we also apply the method to two-phase flows through pipe
expansions. A theoretical treatment indicates that the resulting error should
be small. The resulting expressions reduce to the incompressible formula for
100% liquid. Thus, we have the benefit of a single simple method that applies
equally to ideal and non-ideal gases, to liquids and to gas/liquid mixtures.

For contractions, we adopt the model that the fluid accelerates reversibly to a
vena-contracta from which it expands irreversibly to the new (reduced) pipe
diameter. The pressure gain in the expansion step is treated exactly as the
expander model. The pressure drop in the reversible acceleration is obtained
by solving the flow equations using the Chemcept-data equation of state.
Again, the resulting correlation reproduces the Benedict et al results better
than their own correlation. It has the benefit of automatically reducing to
the incompressible fluid model under limiting conditions.

As in all its other models, Flaresign corrects for turbulent radial velocity
profile.

4) Fittings with one inlet and one outlet.

Flaresign treats all such fittings as having the same size inlet pipe and outlet
pipe. Where there is a difference, an additional expander or contractor must
be added to the simulation. There are three ways of modelling these fittings.
First, they can be modelled using K-values (the "velocity head" method).
Secondly, they can be modelled using an "equivalent pipe length". Thirdly,
they can be modelled as a K-value plus a pipe length. This third method is the
most reliable if the parameters are known. The K-value method is better than
the equivalent pipe-length method. Flaresign allows all three options. Flaresign
also allows the option of automatically converting all K-values to equivalent
lengths or vice-versa.

For the K-value method, Flaresign uses a modification applicable to compressible
and to multi-phase fluids. Published K-values are derived from tests with
incompressible fluids. Theoretically, the K-value is derived from
relationships representing the acceleration and deceleration of the fluid
through the fitting. For example, excellent estimates of K-values for bends
can be derived by recognizing that the fluid is decelerated in its original
flow direction. Consequently, the pressure is higher in the direction-change
zone. The fluid is then accelerated in its new direction. Mathematically, the
pressure drop calculation is identical to the calculation for an expansion
followed by a contraction. Thus, as for expansions and contractions, we would
expect the pressure drop for compressible fluids to be given by a modified
formula. The point of highest pressure will be a point of lowest velocity and
highest density. At the low-pressure outlet, the density will be less, and
hence the velocity higher. The overall pressure-energy loss is then greater
than for an equivalent incompressible fluid. To allow for compressibility,
K-values are converted to equivalent area changes. In accordance with theory,
for incompressible fluids, Flaresign gives exactly the same pressure drop as for
the equivalent K-value. For compressible and two-phase fluids, a higher
pressure-drop is estimated.

Fittings such as valves operate by restricting the flow and then letting it
expand. Thus, the flow accelerates to the narrowest opening (or vena
contracta, where that is smaller) from which it then decelerates to the outlet.
For such fittings, the K-value is used to estimate an equivalent area, which,
in contrast to the bend area, is smaller than the pipe area. The fitting is
modelled as a reversible acceleration to the constriction followed by an
irreversible deceleration. For incompressible fluids, this model gives the
same pressure drop as the K-value model. However, for compressible fluids, it
gives a wider pressure excursion, and hence a larger pressure drop. The model
applies also to non-ideal gases and two-phase flows. Sonic or choking flow
frequently arises in fittings. Flaresign checks for choking/sonic flow and gives
a higher pressure-drop if choking conditions are reached. Flaresign also checks
for liquid cavitation in flow through fittings and gives a higher pressure-drop
if it occurs. (This correction is approximate in the current version because
gas solubility and vapour pressure is not accurately computed).

Note that the pressure drop in fittings is primarily an inertial effect from
alternate acceleration and deceleration. The equivalent pipe-length method
relies on friction factor being insensitive to pipe diameter and Reynolds
Number. None of the equivalent pipe-length methods is accurate, and none more
accurate than that used by Flaresign. The equivalent pipe-length employed in
Flaresign is satisfactory within � 40% for single-phase flows. When the
disposition of fittings is unknown in the early design stages, this accuracy is
satisfactory. However, two-phase friction factors can be more than an order of
magnitude greater than single-phase friction factors; they are also sensitive
to pressure. Thus, unless you know the pressure, flow rate and pipe diameter
in advance, it is difficult to assign an equivalent pipe length for 2-phase
flow through a fitting. Flaresign has a modified method for estimating
equivalent pipe-length that is normally within a factor of two, but can be a
factor of four in error. Such errors are normally acceptable when total
fitting pressure drop is a small fraction of the total. However, bear in mind
that, in a flare network, the total pipe-length equivalent to fittings can
amount to several hundred metres. On balance, it is better to use K-values
when they are available. Flaresign provides the most comprehensive range of
K-value based tools for handling non-ideal and two-phase flow through fittings.

The Spitzglass equation is now mainly of historical interest, but is still
included in some current regulations. It employs a smaller equivalent pipe
length appropriate to the smaller gas pipe sizes that were current when it was
developed almost a century ago. Flaresign allows the method to be used at any
pressure for any single-phase or two-phase flow. However, application should
be restricted only to low pressure gas pipelines.

5) Junctions.

For most junctions, the area of the pipes entering the junction exceeds that of
the single pipe leaving the junction. This situation is always the case if one
of the inlets (typically the straight-through inlet) has the same diameter as
the outlet. Flaresign divides the outlet area into two parts, one corresponding
to each inlet flow. The division is arranged such that both inlet streams come
to the same velocity as they are mixed in the outlet pipe. Normally, both
inlet streams are accelerated to reach a common outlet velocity. In such
cases, Flaresign treats the two inlet streams as both entering area reduction
fittings. The program also caters for the cases when the relevant areas
increase or remain the same (which can happen, for example, if one inlet has no
flow). The relevant inlet is then treated as an expansion. Equally, it may be
that one inlet stream accelerates and the other decelerates. The total fitting
pressure-drop comes from combining area-change and direction-change effects.
The user can apply a K-value and/or an equivalent length (to estimate
direction-change pressure-drop). Thus, Flaresign provides a junction modelling
facility that is based on sounder science than most of its competitors.

6) Knock-out drums.

Knock-out drums provide liquid/gas separation by a combination of settling and
inertial impaction resulting from sudden changes in gas flow direction or
centrifugal force. The balance of mechanisms depends on the design of the
particular separator. Flaresign provides a simple modelling capability which
assumes that gas and liquid are separated with 100% efficiency. Pressure drop
is computed based on changes in direction, which are modelled as equivalent to
expansions and contractions. The sequence of direction changes is represented
by a single equivalent area. A facility is provided to compute this equivalent
area from calibration data, namely one or more pressure drops for known flow
conditions.

7) Flares.

Flaresign computes the pressure drop through a flare tip assuming adiabatic
discharge of the gas to atmosphere. The adiabatic pressure drop (and
temperature drop) is computed from the equation of state also allowing for
change in Heat Capacity Ratio as the gas expands. Each flare tip is
characterized by an equivalent discharge area. A facility is provided to
compute this area from Manufacturer's calibration data (flow versus pressure
for a known gas mixture and known inlet temperature).

Flaresign provides protection against a failure in the unlikely circumstance of
choked flow from an open pipe. Sonic discharge from an open pipe is greater
than the choked flow in an isothermal pipe. If a user specifies isothermal
flow with discharge from an open pipe at a sufficiently high Mach Number, the
flow would be greater than the maximum possible flow in the pipe. It would
then be impossible to solve equation (1). Such an unlikely failure is
eliminated by limiting the flow in the pipe feeding a flare nozzle to its
choked flow. Such detailed attention to potential error conditions ensures
that Flaresign is resistant to run-time failure.

8) Design Options.

Flarenet provides the following design options:

1) Simulation of specified flow network with fixed outlet pressure and
adjustable inlet pressures.

2) Design to give a maximum specified fraction of choking flow at any point in
the network.

3) Design to give a maximum specified velocity at any point in the network.

4) Design to give a maximum specified velocity head at any point in the network

5) Design so that no specified inlet pressure is exceeded.

In each case, the pipes can be sized exactly, or can be selected from preset
standards. The standards can be from a specified schedule, or from users own
standards. For example, some companies use a heavier gauge on smaller pipes
(to provide physical strength), and some omit some of the sizes allowed by
standard gauges to reduce the pipe size inventory.

Flame Shape and Radiation Calculations.

Flaresign offers the following calculations:

1) Flame length and 3-dimensional axial trajectory.
2) Flame lift-off (distance between flare tip and start of combustion zone).
3) Flame diameter at flame base (adjacent to flare tip) and flame tip (remote
from flare tip).
4) Heat generated by combustion and stoichiometic air consumed.
5) Proportion of heat generated that is radiated.
6) The radiation intensity on a surface inclined to the flame.

The methods used are as follows:

1. Flame length and shape.

1) API RP 521-1968*
2) Brzustowski Method*
3) GKN Birwelco Method*
4) Kaldair Method*
5) Shell Method*

NOTE: * We use these titles to identify the methods. We do not guarantee that
we have correctly interpreted the publications on which our code is based.
None of the companies or individuals named have endorsed our implementations.
Some of the methods apply to proprietary flare tip designs, and may not apply
exactly to other designs. All methods have been abstracted from publications
in the public domain, and most have been modified by ourselves. We believe
that our implementations give a good preliminary-design basis but, particularly
where proprietary flare tip designs are used, tip manufacturers should be
consulted to fine-tune the calculations.

We summarize here some of the features of our implementation.

We employ an analytical integration of the API method.

The Brzustowski method, as published, shows a number of problems. First, it
gives infinite length flames in zero wind speeds. For wind speeds above about
1 m/s, it gives a shorter flame than the API flame. When the user selects the
Brzustowski option we compute both the Brzustowski and API flame lengths and
return the shorter of the two. Secondly, it includes a conditional expression
that, for an infinitesimal change in fuel flow rate, can give rise to a sudden
change in radiation intensity of over 25%. The correlations are modified so
that, at the conditional break, both expressions give the same answer, which is
the geometric mean of the two Brzustowski answers. For higher and lower
values, the modified correlations asymptotically approach the unmodified
Brzustowski values. Thirdly, the Brzustowski correlations are in terms of the
horizontal projection of the flame. All other flame shape correlations are in
terms of the axial distance along the flame. Consequently, the "mid-point" of
the Brzustowski flame is not the mid-point of the axis of the flame. The
distance along the axis varies with flare and wind conditions. This anomaly
makes it difficult to modify the Brzustowski flame in order to calculate line
and surface source radiation. For consistency with other flame models, we
compute all distances along the axis of the flame, rather than the horizontal
projection of the flame. The computed radiation intensity increases slightly
because the mid-point of the axis is nearer to the flare tip than is the point
corresponding to the mid-point of the horizontal projection.

The flame shape equations recommended by GKN Birwelco authors includes a term
for "uplift velocity". The velocity corresponds to the vertical convective
velocity at the flame tip. We compute that velocity from simple theory.
However, we believe that GKN Birwelco use an experimentally determined value
that is, in effect, an empirical correction to give a good match with observed
flame shapes. Their equations also include a factor "f" for which they give no
guidance. We calculate a theoretical value based on momentum balance.
Finally, they have a condition under which effective flare-tip
discharge-velocity is brought to zero. Their published equation gives an
illogical large velocity jump for a minuscule change in wind speed. We have
modified the condition so that our equation behaves similarly, but does not
exhibit the illogical jump. We integrate their equations numerically.

We have empirically correlated the data published by McMurray (of Kaldair), so
that flame length depends on heat release rate and Mach number. Thus,
"supersonic" flares are shorter than subsonic flares. We employ their
published analytical integration.

We have considerably modified the correlations published by Chamberlain
describing the model based on extensive Shell data. The modifications should
give minor differences under normal conditions, but behave more sensibly when
extrapolated to unusual conditions. Thus, Chamberlain employs an approximate
empirical relationship between mean molecular weight and stoichiometric air
requirement. We use the Chemcept-data database that gives exact stoichiometric
requirements for any mixture. For "supersonic" flares, Chamberlain presents
equations for computing the hypothetical conditions (temperature, density,
velocity, jet cross-sectional area etc) at which an expanding supersonic jet
from a sonic flare tip reaches atmospheric pressure. We use a set of equations
consistent with the equations we use in pipe network simulation. The result
should be little different, but (for us) ensures consistency, maximizes code
reuse and minimizes testing. Chamberlain presents equations that show the
effect on flame length when a flare is tilted into or out of a wind. However,
they show a dramatic difference in length when tilted into a wind of 0.00001
m/s and into a wind of �0.00001 m/s. We believe that the observed effect
results primarily from a balance between buoyancy and inertial forces.
Accordingly, we employ a modified formula that depends only on the orientation
to the horizontal. For near-vertical flares, it gives results almost identical
to the equations presented by Chamberlain. Chamberlain presents equations for
flame tilt as a function of wind speed and Richardson Number. The equations
are clearly wrong for tilt angles of more than a few degrees. Specifically,
they can show flames that tilt into the wind, horizontal flames that tilt
downwards, and flames that tilt by more than 360 degrees. Flaresign employs a
completely new set of equations. These equations are derived from a model in
which flame velocity is resolved into three components at right angles, namely
vertical, horizontal in the wind direction and horizontal normal to the wind
direction. The resulting equations give very similar results to the Shell
equations for tilts of less than 45 degrees. However, their asymptotic
behaviour is sensible. Specifically, the asymptotic behaviour in very high
winds is that the flame tilts in the direction of the wind and lies horizontal.
All the equations have been modified such that they are in consistent SI
units. Despite the large number of changes made, the extensive published
experimental data probably make this model the most reliable.

2. Flame Diameter.

We require the flame diameter as a function of axial distance in order to
compute the surface area of the flame. Chamberlain gives correlations for the
flame base diameter and the flame tip diameter. To good approximation, flame
diameter increases linearly with axial distance. The flame tip diameter
correlation provides an excellent fit of extensive observations of full-scale
flames. This correlation has been applied to all flame models. The
correlation at the flame base is more scattered, but is less important in
determining total flame surface area. The full published correlation has been
applied to the Shell model, but a simplified correlation has been applied to
the remaining four models.

3. Comparison of Flame Length and Shape Calculations.

The five models are compared for twelve sets of conditions covering a light gas
(methane), a heavy gas (butane) and both subsonic and supersonic flow. The
flare tip conditions are summarized in Table 3, above. For all tests, the
temperature of the gases entering the flare tip was taken to be 300 K. In
adiabatic expansion, the gas cools in its passage through the flare tip. For
supersonic flames, the gas cools further as the gases leaving the flare tip
expand further. The cooled gas temperatures are reported in Table 3. Tables
R1 to R4 summarize the flame length and orientation computations. They were
performed for wind speeds of 0 m/s, 2 m/s, 10 m/s and 50 m/s. These speeds
correspond to still air, 4.5 mph (light breeze), 22 mph (brisk wind), and 110
mph (extremely strong gale). For given conditions, the estimated lengths
differ by about a factor of four. The differences may result in part from
differences in flare tip design and in difficulty in defining flame length.
The latter difficulty probably accounts for differences of �20%. There is
also a difficulty relating to conduct of field trials. It is not possible to
control wind speeds in conducting full-scale trials. Thus, most trials are
conducted in the wind-speed range 2 m/s to 10 m/s. If we restrict the
comparison to wind speeds of 2 m/s to 10 m/s, we get closer agreement. If we
further restrict comparisons to results obtained from tests conducted by major
oil companies, we get still closer agreement. Thus, if we focus on Brzustowski
(trials conducted with Esso), Kaldair (trials conducted with BP), and
Chamberlain (trials conducted by Shell), agreement is within the accuracy with
which flame lengths can be estimated. The level of agreement with Butane is
surprising because no large-scale trials have been conducted with this gas.
For both gases, the agreement covers both flame length and the extent to which
flames are deflected in the wind.

It may be surprising that such a level of agreement is achieved between models
based on such different theory. Thus, Brustowski uses Lower Flammable Limit as
a prime parameter, Kaldair uses heat release, and Shell uses stoichiometric
oxygen requirement. In practice, for hydrocarbons, there is a strong
correlation between these parameters. Thus, any flame length correlation in
one parameter can be converted to a correlation in the other by substituting
the relationship between the relevant physico-chemical properties.

4. Proportion of Heat Released by Radiation.
Flaresign offers the following estimation options.

1) User supplied value of the fraction
(F)
of combustion heat that is radiated.

2) User-supplied judgement of flare type: 0.0 = low velocity flare tip, 1.0 =
high-velocity "smokeless" flare tip. For each flare tip type, the program
includes an empirical correlation of published tabulations of fraction versus
fuel-gas molecular weight. It then interpolates between these correlations.

3) A correlation derived from data published by Chamberlain. The Shell
correlation in terms of jet velocity is recast in terms of jet Mach Number. A
dimensionally consistent correlation for their "small flame" correction is
included.

4) A combination of methods (2) and (3) that replaces the "subjective
judgement" of method (2) with an interpolation derived from method (3).

5) A user-supplied value of "Surface Emissive Power" (SEP). SEP is the
radiation per unit flame-surface area. The Chamberlain paper suggests a value
of around 230 kW/m
^{
2
}

6) A default value of SEP that depends only on mean molecular weight of the
fuel.

7) A user-supplied value of flame surface emissivity. (Can provide upper bound
on
"F"
).

8) Program estimates emissivity from absorption strength of luminous gas and
flame geometry.

The only methods that have any theoretical justification are the
emissivity-based methods (options 7 and 8). With the exception of weather
conditions, this F�parameter is the most uncertain in computing incident
radiation from flares. The scatter within one set of measured results is
typically �50%. The uncertainty in predicting radiation intensities for
facilities not-yet-built, is higher. The range of F-parameter options within
Flaresign can inform engineering judgement and is adequate for preliminary
design. However, values clearly depend on detailed flare tip design, and flare
tip manufacturers must be consulted before finalizing designs. Furthermore,
the values also depend on the flame models and the radiation models. For
example, a surface radiation model distributes the energy radiated uniformly
over the flame surface. The flame gets wider as it rises (inverted cone
shape), so that most of the heat is radiated from the top part of the flame.
In contrast, line source models distribute the heat radiated uniformly along
the axis of the flame. Received radiation depends on the inverse square of the
distance to the radiation source. Thus, line-source models predict that most
of the radiation received at a point below the flame is from the lower half of
the flame. For the same source intensity, a line source model will give higher
received radiation levels. Empirically reported values of F are derived from
measured radiation intensities. Thus, for the same received radiation,
estimated values of F will be lower for line-source models than for point
source models. Similarly, values for point source models will probably be less
than for surface source models. It is for this reason that F values published
by Shell workers (who exclusively employ surface radiation models) are higher
than for earlier authors, who used point or line source models.

As computer-based methods of computing radiation replace pocket calculators and
spreadsheets, designers are increasingly moving from line and point source
models to the more realistic surface-source radiation models. It is prudent to
increase values of F obtained from older publications that assumed line or
point sources.

5. Radiation intensity at "target" locations.

Flaresign uses accepted accurate numerical integration techniques to compute the
intensity of radiation at any selected point, or set of points. Where there
are several flames (flares), the total radiation from all sources is
accumulated. The 3-dimensional orientation of the sources is fully taken into
account. Thus, the radiation from parts of the flame surface that are not
normal to the receiving target are reduced. A full discussion of the
characteristics is given in notes reporting test results.
.