SOURCE MODELS

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SOURCE MODELS
Prepared by Associate Prof. Dr. Mohamad
Wijayanuddin Ali Chemical Engineering
Department Universiti Teknologi Malaysia
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Flow of Liquids Through Pipes
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Figure 5 Liquid flowing through pipe. The
frictional flow losses between the fluid and the
pipe wall result in a pressure drop across the
pipe length. Kinetic energy changes are
frequently negligible.
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Table 1 Roughness factor, e, for clean pipes.
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Figure 6 Plot of Fanning friction factor, f,
versus Reynolds number.
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Figure 7 Plot of 1/? ƒ, versus Re ? ƒ. This form
is convenient for certain types of problems. (see
Example 2.)
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Where the summation is over all of the valves,
unions, elbows, and so on within the piping
system. Table 2 provides selected values for the
equivalent lengths. Note that Table 2 includes
corrections for contractions and expansions in
the piping system. For many problems associated
with pipe flow the contribution due to the
kinetic energy term in the mechanical energy
balance is negligible and then check the validity
of the assumption at the completion of the
calculation. For problems involving laminar flow,
the solution is always direct. Turbulent flow
problems with an unknown pipe diameter, d, always
require a trial-and-error solution. Other types
of turbulent flow problems might be direct or
trial-and-error depending on the work and kinetic
energy terms.
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Table 2 Equivalent pipe lengths for various pipe
fittings (Turbulent flow only).
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Example 3
Water contaminated with small amounts of
hazardous waste is gravity drained out of a large
storage tank through a straight, commercial steel
pipe 100 mm in ID. The pipe is 100 m long with a
gate valve near the tank. The entire pipe
assembly is mostly horizontal. If the liquid
level in the tank is 5.8 above the pipe outlet,
and the pipe is accidentally severed 33 m from
the tank, compute the flow rate of material
escaping from the pipe.
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Solution
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Figure 8
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Start over from the mechanical energy balance,
but this time include the KE term Solving for
u The solution to this equation requires a
trial and error procedure since f is a function
of u. the procedure is. a. Guess a value for the
friction factor, f, b. Determine average
velocity, u, from above equation, c. Determine
Reynolds number, Re, d. Compute f from Colebrook
equation, Equation 25, and e. Iterate until
value of ƒ converges.
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Flow of Vapor Through Holes
For flowing liquids the kinetic energy changes
are frequently negligible and the physical
properties (particularly the density are
constant. For flowing gases and vapor these
assumptions are only valid for small pressure
changes (P1/P2 lt 2)and low velocities ( 0.3
speed of sound in gas). Energy contained within
the gas or vapor as a result of its pressure is
converted into the kinetic energy as the gas or
vapor escapes and expands through the hole. The
density, pressure and temperature change as the
gas or vapor exits through the leak. Gas and
vapor discharges are classified into throttling
and free expansion releases. For throttling
releases, the gas issues through a small crack
with large frictional losses very little of
energy inherent with the gas pressure is
converted to kinetic energy. For free expansion
reeases, most of the pressure energy is
converted to kinetic energy the assumption of
isentropic behavior is usually valid. Source
models for throttling releases require detailed
information on the physical structure of the
leak they will not be considered here. Free
expansion release source models require only the
diameter of the leak.
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Figure 9 A free expansion gas leak. The gas
expands isentropically through the hole. The gas
properties (P, T) and velocity change during the
expansion.
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Example 4
A 0.1 inch hole forms in a tank containing
nitrogen at 200 psig and 80F. determine the mass
flowrate through this leak.
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Solution
For the diatomic gas nitrogen, g 1.4.
Thus, An external pressure less than 113.4 psia
will result in choked flow through the leak.
Since the external pressure is atmospheric in
this case, choked flow is expected and Equation
40 applies. The area of the hole is
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Flow of Vapor Through Pipes
Vapor flow through pipes is modelled using two
special cases adiabatic or isothermal behavior.
The adiabatic case corresponds to rapid vapor
flow through an insulated pipe. The isothermal
case corresponds to flow through an uninsulated
pipe maintained at a constant temperature an
underwater pipeline is an excellent example. Real
vapor flows behave somewhere between the
adiabatic and isothermal cases. Unfortunately,
the real case is very difficult to model and no
generalized and useful equations are available.
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Adiabatic Flows
An adiabatic pipe containing a flowing vapor is
shown in Figure 11. For this particular case the
outlet velocity is less than the sonic velocity.
The flow is driven by a pressure gradient across
the pipe. This expansion leads to an increase in
velocity and an increase in the kinetic energy of
the gas. The kinetic energy is extracted from the
thermal energy of the gas a decrease in
temperature occurs. However, frictional forces
are present between the gas and the pipe wall.
These frictional forces increase the temperature
of the gas. Depending on the magnitude of the
kinetic and frictional energy terms either an
increase or decrease in the gas temperature is
possible.
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Figure 11 Adiabatic, non-choked flow of gas
through a pipe. The gas temperature might
increase or decrease, depending on the magnitude
of the frictional losses.
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For most problems the pipe length (L), inside
diameter (d), upstream temperature (T1) and
pressure (P1), and downstream pressure (P2) are
known. To compute the mass flux, G, the procedure
is as follows. 1. Determine pipe roughness, e
from Table 1. Compute e/d. 2. Determine the
Fanning friction factor, f, from Equation 27.
This assumes fully developed turbulent flow at
high Reynolds numbers. This assumption can be
checked later, but is normally
valid. 3. Determine T2 from Equation
51. 4. Compute the total mass flux, G, from
Equation 52. For long pipes, or for large
pressure differences across the pipe, he velocity
of the gas can approach the sonic velocity. This
case is shown in Figure 12. At the sonic velocity
the flow will be choked. The gas velocity will
remain at the sonic velocity, temperature, and
pressure for the remainder of the pipe. For
choked flow, Equations 46 through 50 are
simplified by setting Ma2 1.0. The results are
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Figure 12 Adiabatic, choked flow of gas through
a pipe. The maximum velocity reached is the sonic
velocity of the gas.
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For most problems involving choked, adiabatic
flows, the pipe length (L), inside diameter (d),
and upstream pressure (P1) and temperature (T1)
are known. To compute the mass flux, G, the
procedure is as follows. 1. Determine the Fanning
friction factor, f, using Equation 27. This
assumes fully developed turbulent flow at high
Reynolds number. This assumption can be
checked later, but is normally valid. 2. Determine
Ma1, from Equation 57. 3. Determine the mass
flux, Gchoked, from Equation 56. 4. Determine
Pchoked from Equation 54 to confirm operation
at choked conditions.