1.
Introduction
Steam
shows different behaviors in compressible flow from those of ideal gas because
of its inconsistent characteristics depending on the conditions. In
this concern, different approaches have been used in the analyses of compressible
flow in power plant engineering, depending on their conditions of steam,
i.e. superheated steam, saturated steam and saturated water.
This
category of analysis in power plant engineering includes cascade heater drain
analysis, safety valve vent stack analysis and steam blow-out analysis. There
are many reference papers for these subjects as listed in the References at
the end of this paper. Ref. No. 1 through 3 are for cascade
heater drain analysis; Ref. No. 4 through 6 are for safety valve vent stack
analysis; and Ref. No. 7 is for steam blow-out analysis.
The
reference papers use basically the equations for ideal gas with various experimental
coefficients and experimental equations in order to supplement the inadequacy
of ideal gas laws and equations for the compressible flow analysis of steam.
And they use different experimental equations and coefficients depending
on the conditions of steam. In case of Ref. NO 1, for example,
a pseudo-isentropic exponent is used to apply Fanno-line equation
to the two-phase flow of saturated water. Also they use several
fundamental laws additionally to check whether the analysis results are correct
or not.
Anyway,
the target of the analysis method presented by these papers is to provide results
with positive margins, and the system so designed should have conservative
margins in any case.
The
page presents the basic equation of flow dynamics for compressible flow analysis
and the equation of ideal gas flow analysis
2.
Basic Equations of Flow Dynamics
Basic
equations which applies to both ideal gas and steam are as follows.
-
Energy Equation (The Fist Law of Thermodynamics)
-
Continuity Equation
-
Equation of Motion
The
equations presented here are for one-dimensional, steady flow.
The
general energy equation in infinitesimal form is as below.
q
+ w = du + d(p * v) + d(V^2 / 2/ g) + d(h)
| where, | q | : Heat added per mass of flowing fluid |
| w | : Work added per mass of flowing fluid | |
| u | : Internal Energy | |
| p | : Static Pressure | |
| v | : Specific Volume | |
| V | : Fluid Velocity | |
| h | : Elevation Head |
Since
the process of nozzle and short length pipe can be assumed as a adiabatic
process with no external work, q and w become zero in the equation above.
According to the definition of enthalpy, du + d(pv) = dH, and in
case of gas and steam the elevation head(dh) can be neglected.
Therefore,
the energy equation to be applied to nozzle and short length pipe is as below.
d(H)
+ d(V^2 / 2 / g / J) = 0
| where, | H | : Enthalpy, kcal/kg |
| V | : Fluid velocity, m/sec | |
| g | : Gravity acceleration = 9.81 m/sec2 | |
| J | : Joule constant, 427 kg-m/kcal |
The
above equation means that the fluid velocity is generated by the consumption
of enthalpy and the fluid velocity is another form of energy the enthalpy transformed.
This equation is effective as long as the process is adiabatic
even though there is friction in the process.
The
enthalpy at zero velocity is called as total enthalpy which is the maximum value
of enthalpy in the process, and the total enthalpy is calculated as below.
Ht
= H + V^2 / 2 / g / J
| where, | Ht | : Total Enthalpy, kcal/kg |
If
the precess is adiabatic, then the total enthalpy value through the process
is constant.
The
continuity equation below is applied to all fluid flow process.
W
/ A = V / v
| where, | W | : Mass flow rate, kg/sec |
| v | : Specific volume, m3/kg | |
| V | : Fluid velocity, m/sec | |
| A | : Flow path area, m2 |
The
equation of motion is one of the energy equation which represents the energy
balance among pressure head, velocity head, elevation head and friction head.
The momentum equation says the same thing with the equation of motion,
but in force terms instead of head terms in the equation of motion.
If
we neglect the elevation head for gas and steam flow, the momentum equation
in force terms is described as below in infinitesimal forms.
A
* dP = T * 3.14 * D * dL + W * dV / g
| where, | A | : Flow path area |
| dP | : Pressure difference | |
| T | : Wall Shear Stress | |
| dL | : Duct length | |
| D | : Hydraulic diameter of duct | |
| g | : Gravity acceleration | |
| W | : Mass flow rate | |
| dV | : Velocity difference |
The
first term in the equation represents the force for flow by pressure difference,
the second represents the resistance against flow by friction, and the third
represents the momentum change by velocity due to specific volume change.
The
equation means that the force by pressure difference is used to overcome the
friction and to make momentum change.
In
the equation, the friction term written in wall shear stress can be replaced
with the term in the friction factor commercially used, i.e. T = f * V^2 / v
/ 8. Under the assumption that the duct flow path area is constant
along the duct length, when dividing the equation by area(A or 3.14 * D^2 /
4) and multiplying by specific volume(v), we can write the equation as below.
v
* dP = f * dL / D * V^2 / 2 / g + W / A * v * dV / g
| (?) | A | : Flow path area, m2 |
| dP | : Pressure difference, kg/m2 | |
| f | : Friction Factor | |
| dL | : Duct length, m | |
| D | : Hydraulic diameter of duct, m | |
| g | : Gravity acceleration = 9.81 m/sec2 | |
| dV | : Velocity difference, m/sec | |
| W | : Mass flow rate, kg/sec | |
| v | : Specific volume, m3/kg |
In
the compressible flow analysis, critical pressure should be calculated in order
to find out the possibility of choked flow, and in order to calculate the critical
pressure the calculation of sonic velocity is required.
For
compressible fluid, the sonic velocity is calculated as below.
Vc
= (dP / dRo * g)^0.5 @ isentropic infinitesimal pressure change
| where, | Vc | : Sonic Velocity, m/sec |
| dP | : Infinitesimal pressure difference, kg/m2 | |
| dRo | : Density change by dP in isentropic process, kg/m3 | |
| g | : Gravity acceleration, 9.81 m/sec2 |
Critical
pressure exists in compressible fluid flow, where the fluid velocity equals
the sonic velocity of the fluid.
Before the
downstream pressure reaches the critical pressure, the compressible mass
flow rate increases as much as the downstream pressure is decreased. However,
once the downstream pressure reaches or is less than the critical pressure,
the compressible mass flow rate does not increase even though the downstream
pressure is decreased further.
This
specific phenomenon is called as "choked flow", and the energy difference
between the choked exit and ambient conditions is dissipated by shock wave and/or
turbulence.
For
the case of compressible nozzle flow, the pressure, mass flow rate and velocity
have the relationship described below.
Type
of Flow
|
Pressure
|
Mass
Flow Rate
|
Velocity
|
Sub-critical
Flow
|
P2
= P3 > Pc
|
F
< Fc
|
V2
< Vc
|
Critical
Flow
|
P2
= P3 = Pc
|
F
= Fc
|
V2
= Vc
|
Choked
Flow
|
P2
= Pc > P3
|
F
= Fc
|
V2
= Vc
|
| where, | Pc | : Critical Pressure |
| Fc | : Critical Flow Rate | |
| Vc | : Critical Velocity = Sonic Velocity |
If
a convergent-divergent de Laval nozzle is used, then a supersonic flow can be
made. However, such a supersonic flow is normally not used in power
plant engineering.
The
first thing in analysis of compressible flow of nozzle and piping is to calculate
the critical pressure(Pc) and then to check whether the discharge pressure(P3)
is higher than the critical pressure. If P3 >= Pc, the nozzle
or piping exit pressure(P2) is selected as P3, and if P3 < Pc then P2 is
selected as Pc.
In
case of orifice, actually there is no critical pressure. The mass
flow rate increases as much as the discharge pressure is decreased till zero
absolute pressure. This is because, as the fluid velocity increases,
the friction generated at the edge of the orifice is absorbed into the flowing
fluid itself and increase the fluid temperature, the mass flow rate per unit
area decreases and more mass flow can pass the orifice. However,
the increase rate of mass flow rate is very slow when the discharge pressure
is less than approximately 50% of the inlet pressure which is the critical pressure
ratio of the nozzle of general gas.
4.1
Nozzle
4.1.1
Mass Flow Rate per Unit Area
The
mass flow rate per unit area of nozzle expanding isentropically (P * v^k = Const.), is
calculated by the following equation, which is derived from the energy equation,
continuity equation and Boyle-Charles equation (P * v = R * T).
W
/ A = ((2 * g * k) / (k - 1))^(0.5) * (P1 / v1)^(0.5) * (r^(2/k) - r^((k+1)/k))^(0.5)
| where, | r | : Pressure ratio = P2 / P1 |
| P1 | : Nozzle inlet pressure, kg/m2 abs. | |
| v1 | : Nozzle inlet specific volume, m3/kg | |
| W | : Mass flow rate, kg/sec | |
| A | : Nozzle throat area, m2 | |
| g | : Gravity acceleration = 9.81 m/sec2 |
4.1.2
Critical Pressure and Nozzle Throat Pressure
The
critical pressure of ideal gas expanding isentropically in nozzle, is calculated
by the following equation, which is derived from the equation above where the
mass flow rate per unit area has the maximum value.
Pc
= (2 / (k + 1))^(k / (k - 1)) * P1
| where, | Pc | : Critical pressure, kg/m2 abs. |
| k | : Specific heat ratio of ideal gas = cp/cv | |
| P1 | : Nozzle inlet total pressure at V1 = 0, kg/m2 abs. |
The
nozzle exit pressure(P2) is selected considering the discharge pressure(P3)
according to the method described in the Clause 3 above.
As
described above, there is no choked flow in orifice flow, and the mass flow
rate is calculated by the following equation, which is applicable for turbulent
flow.
Y
= 1 - 0.41 * (P1 - P2) / P1 / k
W
/ A = 0.598 * Y * (2 * g * (P1 - P2) / v1)^(0.5)
The
difference of pipe analysis from that of nozzle is the involvement of
friction and heat loss. The existence of friction means that the
process is not isentropic and the existence of heat loss means that the process
is not adiabatic.
In
pipe flow analysis, friction is always there. However, the heat
loss is not.. When analyzing long distance lines such as cross country gas pipe
lines, the heat loss consideration is must. However, when analyzing relatively
short pipe lines, the heat loss can be neglected. Almost all pipe
lines in power plant engineering is short enough to be analyzed without the
consideration of heat loss.
The
equation for ideal gas flowing through constant cross-sectional area duct with
friction and heat loss is called as Rayleigh Line equation, while the equation
for ideal gas flowing through constant area duct with friction but in adiabatic
process, is called as Fanno Line equation. Herein, the Fanno Line
equation is introduced for its applicability to power plant engineering.
The
assumption given to Fanno Line equation is as below.
Assumption
of Fanno Line Equation
|
Ideal
gas (constant specific heat)
|
Steady,
one-dimensional flow
|
Constant
friction factor over the length of duct
|
Adiabatic
flow (no heat transfer through wall)
|
Effective
conduit diameter D is four times hydraulic radius (cross-sectional
area divided by wetted perimeter)
|
Elevation
changes are unimportant compared with friction effects
|
No
work added to or extracted from the flow
|
The
Fanno Line equation is for the flow which has sonic velocity(Mach no. = 1) at
the duct(or pipe) exit plane, i.e. choked flow.
4.3.1
Choked Flow
As
depicted in the diagram below for choked flow, the pipe exit pressure(P2) is
same with the critical pressure(Pc) and the fluid velocity at pipe exit is same
with sonic velocity, i.e. Mach no. = 1. And, the discharge pressure
is equal or less than the critical pressure.
The
Fanno Line equation defines the relationship among pipe inlet Mach no.(M1),
flow resistance coefficient(K) and specific heat ratio of the ideal gas(k).
The Fanno Line equation written below is derived from energy equation,
continuity equation, equation of motion, Boyle-Charles equation (P
* v = R * T) and sonic velocity equation (Vc = (k * g * R * T)^(0.5)).
K
= 1 / k * (1 / M1 ^ 2 - 1) + (k + 1) / 2 / k * ln(M1 ^ 2 * (k + 1) / ((k - 1)
* M1 ^ 2 + 2))
| where, | K | : Flow resistance coefficient, K = f * L / D |
| k | : Specific heat ratio = cp/cv | |
| M1 | : Pipe inlet Mach no. |
The
equations for critical ratios of pressure, temperature and velocity are as blow,
which are shown in term of pipe inlet Mach no.(M1).
Pc
/ P1 = M1 * (((k - 1) * M1 ^ 2 + 2) / (k + 1))^(0.5)
Vc
/ V1 = 1 / M1 * (((k - 1) * M1 ^ 2 + 2) / (k + 1))^(0.5)
Tc
/ T1 = ((k - 1) * M1 ^ 2 + 2) / (k + 1)
| where, | Pc | : Critical pressure |
| Vc | : Critical velocity = Sonic velocity | |
| Tc | : Critical temperature |
4.3.2
Critical Pressure
When
the flow resistance coefficient(K) and specific heat ratio(k) are known, the
pipe inlet Mach no.(M1) can be calculated by the Fanno Line equation above.
With M1 calculated, the critical ratios can be calculated. However,
in order to get the conditions of pipe inlet and outlet from the critical ratios,
either of pipe inlet or outlet condition should be known.
Furthermore,
since the Fanno Line equation is based on choked flow, the critical pressure
at the pipe exit should be known in order to select the pipe exit pressure.
Therefore, the Fanno Line equation with the critical ratio equations
is not enough to analyze the compressible pipe flow of ideal gas.
In
case of compressible nozzle flow, since the flow conditions along the nozzle
are determined by isentropic process, knowing of the nozzle inlet condition
with discharge pressure is good enough for analysis. However, in
case of the adiabatic pipe flow with friction, the flow conditions is indeterminate
and vary depending on the mass flow rate per unit area(W/A) and pipe resistance
coefficient(K).
Therefore,
in order to analyze the adiabatic pipe flow with friction the mass flow rate
per unit area and pipe resistance coefficient should be known in addition to
pipe inlet condition and pipe discharge pressure.
Pipe
inlet condition here means the total condition at zero velocity. Choked
flow of adiabatic pipe with friction is depicted below.
Critical
pressure of adiabatic pipe flow with friction for ideal gas can be derived as
below.
The
critical temperature ratio equation above may be rewritten as below by substituting
T1 with T0 where the velocity equals zero and M0 = 0.
Tc
/ T0 = 2 / (k + 1)
Applying
the Boyle-Charles equation (P * v = R * T), the equation above can be written,
(Pc
* vc) / (P0 * v0) = 2 / (k + 1)
Pc
= 2 / (k + 1) * (P0 * v0) / vc (Eq. 4.3.2 - 1)
P0
and v0 are the known, while vc can be calculated from the continuity equation
(W / A = Vc / vc)and sonic velocity equation (Vc = (k * g * P * v)^(0.5)), wherein
the mass flow rate(W) and pipe area(A) are constant through the pipe length.
Two
equations can be reduced for vc as below by eliminating Vc.
vc
= k * g * Pc / (W / A)^2 (Eq. 4.3.2 - 2)
Substituting
vc with (Eq. 4.3.2-2), (Eq. 4.3.2-1) can be written,
Pc
= (W / A) * (2 * P0 * v0 / k / (k + 1) / g)^(0.5)
The
critical pressure calculation equation above is applicable to all adiabatic
pipes, whatever processes the flow go through in the pipe.
4.3.3
Sub-critical Flow
Sub-critical
pipe flow has the pipe exit pressure(P2) same with discharge pressure(P3) and
sub-sonic velocity at the pipe exit. The sub-critical pipe
flow can be analyzed using a imaginary pipe between P2 and Pc as depicted below.
The
pressure ratio of the imaginary pipe which is in choked flow, is as below by
Fanno Line equation.
Pc
/ P2 = M2 * (((k - 1) * M2 ^ 2 + 2) / (k + 1))^(0.5)
Since
the Pc and P2 are known, the above equation can be rearranged for M2 as below.
M2
= (((1 + (k - 1) * (k + 1) * (Pc/P2)^(2))^(0.5) - 1) / (k - 1))^(0.5)
With
M2 known, the resistance coefficient of the imaginary pipe(Ki) can be calculated
by Fanno Line equation as below.
Ki
= 1 / k * (1 / M2 ^ 2 - 1) + (k + 1) / 2 / k * ln(M2 ^ 2 * (k + 1) / ((k - 1)
* M2 ^ 2 + 2))
Since
total pipe including actual pipe(K) and imaginary pipe(Ki) is in choked flow
again, the pipe inlet condition can be calculated according to the method described
in Clause.4.3.1.
The
general sonic velocity equation described in Clause 2.4 may be rewritten in
isothermal process as below.
Vc
= (dP / dRo * g)^0.5 @ isentropic infinitesimal pressure change
=
(k * dP / dRo * g)^(0.5) @ isothermal infinitesimal pressure change
Integrating
the isothermal equation by using Boyle-Charles equation (P * v = R * T),
we can get the sonic velocity equation of ideal gas as below.
Vc
= (k * g * R * T)^(0.5) = (k * g * P * v)^(0.5)
| where, | Vc | : Sonic velocity, m/sec | ||||
| k | : Specific heat ratio, cp/cv | |||||
| R | :
Gas constant, R = R0 / M
|
|||||
| g | : Gravity acceleration, 9.81 m/sec2 | |||||
| T | : Gas absolute temperature, degree Kelvin | |||||
| P | : Gas absolute pressure, kg/m2 abs. | |||||
| v | : Gas specific volume, m3/kg |
When
fluid flows through increaser or reducer, there is flow resistance by vortex
in addition to that by friction due to wall shear stress.
When
fluid flows through a increaser having the cross-sectional area changed
suddenly, the pressure just behind of main stream exiting from the smaller cross-sectional
area is lower than that of main stream by vortex, which disturbs the free expansion
of main stream and acts as flow resistance.
When
analyzing the compressible flow through increaser and reducer, the flow resistance
by vortex should be considered, even though the cross-sectional area changes
suddenly and so there is no friction by wall shear stress.
Meanwhile,
even if there is no friction by wall shear stress and there is only flow resistance
by vortex, the flow process is not insentropic process, i.e. not reversible
adiabatic process, but a kind of polytropic process without external work
having the polytropic exponent normally higher than isentropic exponent(k). This
is true even though the flow process is adiabatic process. Reversible
adiabatic process has no increase in entropy. But adiabatic process
has the increase of entropy, even though there is no heat transfer through boundary.
Of course, the adiabatic process includes the reversible adiabatic
process.
In
order to understand the flow mechanism and K value calculation of increaser
and reducer, the theory of incompressible flow is presented first.
A.
Increaser
In
a increaser having cross-sectional area suddenly enlarged as depicted above,
Bernoulli equation between the location 1 and 2 can be written as below neglecting
the elevation head for gas. Bernoulli equation is the equation of
motion for incompressible flow. The flow resistance by vortex is
called as hf in the equation below.
P1
* v + Vel1^2 / 2 / g = P2 * v + Vel2^2 / 2 / g + hf
Rearranging
the equation for hf,
hf
= (Vel1^2 - Vel2^2) / 2 / g - (P2 - P1) * v (Eq.
4.5.1 - 1)
Equation
of momentum between location 1 and 2 is as below.
(P1
- P2) * A2 = W / g * (Vel2 - Vel1)
Using
continuity equation (W = A2 * V2 / v), the equation above may be arranged as
below.
(P2
- P1) * v = Vel2 * (Vel1 - Vel2) / g (Eq. 4.5.1
- 2)
Eliminating
pressure terms from (Eq. 4.5.1 - 1] and (Eq. 4.5.2 - 2) and rearranging for
hf, we may have,
hf
= (Vel1 - Vel2)^2 / 2 / g = (1 - A1 / A2)^2 * Vel1^2 / 2 / g (Eq.
4.5.1 - 3)
The
equation above for hf is theoretical equation. However, it was revealed
by experiments that the equation represents the actual value very well for the
increaser having the cross-sectional area enlarged suddenly.
Comparing
(Eq. 4.5.1 - 3) with Darcy equation, the value corresponding to the resistance
coefficient(K) is (1 - A1 / A2)^2 which is based on the velocity of
location 1. Conversion to the velocity of location 2 can be
done by dividing the value by (A1 / A2)^2.
In
case of the increaser having the cross-sectional area enlarged gradually, there
is the friction by wall shear stress in addition to the flow resistance due
to vortex, because the fluid flows in contact with the increaser wall. While
there are many books providing with the experimental equations for the
resistance coefficient of increaser, herein the experimental equations
provided in Ref. No. 8 is introduced.
If
? <= 45 o (based on Aup velocity)
K
= 2.6 * Sin(?/2) * (1 - Aup / Adown)^2
if
45 o < ? <= 180 o (based on Aup velocity)
K
= (1 - Aup / Adown)^2
We
can see in the equations above that only the flow resistance by vortex exists
if the increaser angle is larger than 45o and the value is same with the theoretical
value. If the increaser angle is smaller than 45o, the flow resistance
is increased due to additional friction by wall shear stress.
The
flow pattern of reducer having the cross-sectional area contracted suddenly
is as depicted below.
Due
to fluid inertia the cross-sectional area of flow path is contracted smaller
than the physical area of the smaller end of the reducer passing over the location,
and then enlarged to the physical area of downstream pipe. The flow
resistance by the contraction itself is so small to be negligible,
and the flow resistance of reducer mainly consists of the flow resistance by
enlargement as in increaser, plus the friction by wall shear stress in
case of the gradually-contracted reducer.
The
experimental equations for the resistance coefficient of reducer presented in
Ref. No. 8 is as below.
If
? <= 45 o (based on Adown velocity)
K
= 0.8 * Sin(?/2) * (1 - Adown / Aup)
if
45 o < ? <= 180 o (based on Adown velocity)
K
= 0.5 * Sqr(Sin(?/2)) * (1 - Adown / Aup)
Since
reducer has always both flow resistance by enlargement and the friction by wall
shear stress except the reducing angle of 180o, the equations above include
both also. Meanwhile, the flow resistance only by enlargement can
be calculated as below, which is the case of 180o reducing angle.
K
= 0.5 * (1 - Adown / Aup)
4.5.2
Analysis of Ideal Gas
In
order to apply the resistance coefficient for incompressible flow to compressible
flow, the specific volume change due to pressure change should be considered.
If
the flow process is isentropic process, then the analysis can be done straight
using several equations. However, since the process of increaser
and reducer is not isentropic, the analysis should be done by try-and-error
method using the basic equations.
The
ideal gas equations which can be used for irreversible adiabatic process are
as below.
1)
Boyle-Charles equation
P
* v = R * T
2)
Continuity equation
W
/ A = V / v
3)
Energy equation
H0
= H + V^2 / 2 / g
4)
Equation of Motion
v
* dP + (V1^2 - V2^2) / 2 / g - hf = 0
5)
Enthalpy equation
dH
= cp * dT
6)
Constant pressure specific heat equation
cp
= k * R / J / (k - 1)
6)
Entropy equation
dS
= cp * Ln(T2 / T1) - R / J * Ln(P2 / P1)
For
enthalpy and entropy calculation of ideal gas, the normal ambient condition(0
oC and 10332 kgf/m2 abs.) shall be used as reference condition.
In
the analysis, it is assumed that the downstream condition of increaser or reducer
is known by previous analysis of downstream pipe, and the upstream condition
of increaser or reducer shall be calculated using try-and-error method.
It
is noted that the upstream-end flow velocity of increaser is higher than that
of downstream-end, and so the upstream-end pressure of increaser is lower than
that of downstream-end. Meantime, the lower limit of the upstream-end
pressure of increaser is the critical pressure at the upstream-end.
Vice-versa,
the upstream-end pressure of reducer is higher than that of downstream-end,
and the higher limit is the inlet pressure of total pipe. The
upstream-end pressure of reducer can not be lowered below the critical
pressure in any case.
Hereinafter,
the upstream end of increaser and reducer will be called as Loc 1 and the downstream
end as Loc 2.
A.
Calculation Method of Increaser
Since
the lower limit of Loc 1 pressure is the critical pressure, the pressure of
Loc 1 is the value between Loc 2 pressure and the critical pressure. Loc
2 pressure is the higher limit and the critical pressure is the lower
limit.
If
Loc 2 pressure is equal or lower than the critical pressure, Loc 1 pressure
is the critical pressure and the other condition of Loc 1 is the critical
condition.
The
Loc 1 enthalpy is the value between the total enthalpy of the fluid(the higher
limit) and the insentropic expansion enthalpy at Loc 1 pressure(the lower
limit).
The
condition of Loc 1 is selected by try-and-error method searching the Loc 1 pressure
and enthalpy ranges described above, which meets the equation of motion.
B.
Calculation Method of Reducer
Calculation
method of reducer is same with that of increaser, except that the pressure range
to search is different.
The
key difference of steam analysis from that of ideal gas is the use of steam
table instead of Boyle-Charles equation, enthalpy and entropy equation, and
the use of try-and-error method to get steam conditions from steam table
for each iteration.
5.1.1
Input Data
The
input data required for the analysis of nozzle are as below.
| - | Stagnated condition of nozzle inlet where steam velocity equals zero. Two properties are required for defining the steam condition, e.g. pressure(P1) and enthalpy(H1) |
| - | Nozzle discharge pressure (P3) |
5.1.2
Critical Pressure, Nozzle Throat Pressure and Mass Flow Rate per Unit Area
Nozzle
analysis of steam is very simple because the process can be analyzed as isentropic
process.
First,
get from steam table the nozzle inlet entropy(S1).
Select
a nozzle throat pressure(P2) lower than the nozzle inlet pressure(P1), and get
steam enthalpy(H2) at nozzle throat from steam table using P2 and
S1. The condition is the isentropic expansion condition.
Then
calculate the sonic velocity at the nozzle throat condition selected, and then
the velocity energy by the sonic velocity.
If
the sonic velocity energy is different from the enthalpy difference of H1 -
H2, then select another nozzle throat pressure(P2) and try again.
If
the sonic velocity energy converges on the enthalpy difference of H1 - H2, then
calculate the mass flow rate per unit area at nozzle throat using the sonic
velocity and specific volume selected and finish the calculation.
Nozzle
analysis is rather simpler than pipe analysis because the process can be analyzed
using isentropic process, as described above. However, the adiabatic pipe
with friction is different because the process is polytripic of which process
is dependent on the given conditions of pipe. Therefore, the mass
flow rate per unit area and pipe resistance coefficient shoul be known for analyzing
the adiabatic pipe with friction.
There
are two kinds of compressible steam pipe analyses in power plant engineering.
The
First is to calculate pipe inlet and outlet conditions from the stagnated
pipe inlet condition, pipe cross-sectional area, mass flow rate and
pipe discharge pressure known. These kinds of analyses include the safety
valve vent stack analysis and cascade heater drain pipe analysis.
The
second is to calculate the maximum mass flow rate in addition to the pipe inlet
and outlet conditions from the stagnated pipe inlet condition, pipe cross-sectional
area and pipe discharge pressure given. This kind of analysis includes
the steam blow-out pipe analysis.
Actually
the second analysis is the iteration of the first analysis by try-and-error
method, in which the mass flow rate is searched for the pipe inlet pressure
converging on the stagnated pipe inlet pressure given.
In
this Clause, the method of the first analysis is described.
5.2.1
Input Data
The
input data required for the analysis of adiabatic pipe with friction are
as below.
| - | Stagnated condition of pipe inlet where steam velocity equals zero. Two properties are required for defining the steam condition, e.g. pressure(P0) and enthalpy(H0) |
| - | Pipe discharge pressure(P3) |
| - | Pipe cross-sectional area(A),(to be constant) |
| - | Mass flow rate(W) |
| - | Resistance coefficient of pipe K = f * L / D |
5.2.2
Critical Pressure(Pc)
The
critical pressure is calculated using the facts that the velocity at critical
condition is sonic velocity and that the sum of the static enthalpy and
the velocity energy at critical condition is same with the stagnated pipe inlet
enthalpy.
In
the calculation, it should be noted that the sonic velocity is calculated by
using insentropic infinitesimal pressure change, even if the pipe process
is not isentropic.
The
critical pressure must exist in the pressure range below the stagnated pipe
inlet pressure. If the critical pressure does not exists
below the stagnated pipe inlet pressure(P0), this means that the mass flow rate
of input data can not flow through the pipe given even at choked flow condition
and the pipe condition given by the input data does not exist.
For
each critical pressure selected for try-and-error, the maximum enthalpy is the
stagnated pipe inlet enthalpy(H0) and the minimum enthalpy is the enthalpy
expanded through isentropic process to the critical pressure selected.
Consequently,
the critical pressure is selected by try-and-error method by searching the pressure
range below the stagnated pipe inlet pressure and searching the maximum
and minimum enthalpies at each pressure described above, in which the mass flow
rate calculated by the sonic velocity and the specific volume is converging
on the mass flow rate of input data.
5.2.3
Pipe Exit Condition(Location 2)
The
pipe exit pressure(P2) is selected by comparing the critical pressure(P2)
and the pipe discharge pressure(P3). Selection method is described
in Clause 3 above.
The
maximum value of the pipe exit enthalpy(H2) is the stagnated pipe inlet enthalpy(H0)
and the minimum value is the isentropically expanded enthalpy to pressure P2
selected. The pipe exit enthalpy(H2) is selected by try-and-error
method searching between the maximum and minimum values, in which the mass flow
rate calculated by the velocity generated by the enthalpy difference converges
on the mass flow rate of input data. Other properties of steam at
pipe exit can be gotten from steam table using P2 and H2.
5.2.4
Pipe Inlet Condition(Location 1)
The
pipe inlet condition is calculated using the momentum equation. The
momentum equation for a pipe with friction means that the force by pressure
difference between the inlet and outlet equals the flow resistance fore by pipe
wall friction plus the momentum increase by velocity increase.
The
momentum equation in differential form is as below.
v
* dP = f * dL / D * V^2 / 2 / g + W / A * v * dV / g
| where, | A | : Pipe cross-sectional area, m2 |
| dP | : Pressure difference, kg/m2 | |
| f | : Friction factor | |
| dL | : Pipe length, m | |
| D | : Pipe diameter , m | |
| g | : Gravity acceleration = 9.81 m/sec2 | |
| dV | : Velocity difference, m/sec | |
| W | : Mass flow rate, kg/sec | |
| v | : Specific volume, m3/kg |
Rewriting
the friction term of the above equation using the square of the continuity equation
(V^2 = (W / A)^2 * v^2), we have,
v
* dP = f * dL / D * (W / A)^2 * v^2 / 2 / g + W / A * v * dV / g
Dividing
the equation by v^2, then we have
(1
/ v) * dP = f * dL / D * (W / A)^2 / 2 / g + W / A * (1 / v) * dV / g (Eq.
5.2.4 - 1)
Meanwhile,
the differential form of continuity equation is as below.
dV
= (W / A) * dv (W / A = constant) (Eq. 5.2.4 - 2)
Substituting
dV of (Eq. 5.2.4 - 1) by dV of (Eq. 5.2.4 - 2), integrating the equation (Eq.
5.2.4 - 1) from pipe inlet to pipe outlet, and then substituting f by K using
K = f * L / D, we have,
K
= { Integral(dP / v)(from P2 to P1) } / (W/A)^(2) * 2 * g - 2 * ln(v2/v1) (Eq.
5.2.4 - 3)
| where, | K | : Pipe resistance coefficient, K = f * L / D |
| dP | : Incremental pressure difference, kg/m2 | |
| P1 | : Pipe inlet pressure, kg/m2 abs. | |
| P2 | : Pipe exit pressure, kg/m2 abs. | |
| W | : Mass flow rate, kg/sec | |
| A | : Pipe cross-sectional area, m2 | |
| g | : Gravity acceleration = 9.81 m/sec2 | |
| v1 | : Pipe inlet specific volume, m3/kg | |
| v2 | : Pipe exit specific volume, m3/kg |
Fanno
Line equation for ideal gas is the equation that the pressure integral
term of the above equation has been solved using the Boyle-Charles equation.
However, steam is not simple because the relationship of pressure
and specific volume can not be expressed by a equation.
For
steam, the integral can be solved by summing up the reciprocal of specific volume
for incremental pressure changes with reasonable accuracy. For specific
volume, the algebraic average value of the incremental pressures is used.
The
pipe inlet pressure(P1) is selected as the pressure at which the resistance
coefficient calculated by (Eq. 5.2.4 - 3) equals the resistance coefficient
of input data. The summing-up starts from the pipe exit pressure(P2)
to the stagnated pipe inlet pressure(P0). Other steam properties
at pipe inlet are calculated as described for the pipe exit condition in Clause
5.2.3 above..
Meanwhile,
if the resistance coefficient which meets the value of input data can not be
found till summing up to P0, that means the pipe condition given by the input
data does not exist.
5.2.6
Sensitivities of Program Variables
As
described in Clause 1 Introduction, the methods described in Clause 5 have been
programmed and then the sensitivities of major variables have been investigated
and summarized as below.
| 1) |
The
sonic velocities calculated by either of 100 kg/m2 or 10 kg/m2 infinitesimal
pressure changes show no distinct difference, and 100 kg/m2 pressure
change has been selected for use in the program.
|
| 2) |
It
was found in the integration work of the momentum equation(Eq. 5.2.4
- 3) that the incremental pressure change by ratio basis is
appropriate rather than by algebraic addition. In the
program runs it was found also that the pressure increases by either
of 1% or 0.1% show no distinct difference, and 1% has been selected
for use in the program.
|
| 3) |
The
sonic velocity calculation was tried for sub-cooled water, but the
result was not effective because the specific volume change for
infinitesimal pressure change got from steam table is too small
to make the result meaningful. However, the
sonic velocity of flashing saturated water was found effective.
|
The
analysis method of steam flow through increaser and reducer is same with that
of ideal gas described in Clause 4.5 above, except that the steam table is used
instead of Boyle-Charles equation, enthalpy and entropy equation.
In
Ref. No. 7, it was described that the downstream pipe size should be less
than the upstream pipe size without any explanation. The truth is
that the method given in the paper is not appropriate for the increaser analysis.
It does not mean that compressible steam does not flow through the
increaser. Using the method described here, the compressible steam
flow analysis through increaser can be solved without any problem.
References
:
1.
Analytical Approach for Determination of Steam/Water Flow Capability in Power
Plant Drain Systems by G.S. Liao and J.K. Larson, ASME Publication 76-WA/Pwr-4
2.
Heater Drain Systems by A.L. Cahn, Bechtel Power Corp., presented for Feedwater
Heater Workshop of EPRI held in July, 1979.
3.
Flow of a Flashing Mixture of Water and Steam through Pipes by M.W. Benjamin
and J.G. Miller, Transactions of the ASME, Oct., 1942
4.
ASME B31.1-1992, Appendix II Nonmandatory Rules for The Design of Safety Valve
Installations
5.
Analysis of Power Plant Safety and Relief Valve Vent Stacks by G.S. Liao, Bechtel
Power Corp., Transactions of the ASME, 1974
6.
Crosby Pressure Relief Valves Engineering Handbook, Crosby Gage & Valve
Company, March 1986
7.
Cleaning of Main Steam Piping and Provisions for Hydrostatic Testing of Reheaters
(GEK - 27065D)", General Electric Co.
8.
Crane Technical Paper No. 410, Flow of Fluids, Crane Co., 1977
9.
Principles and Practice of Flow Meter Engineering by L. K. Spink, Foxboro
Source: http://www.engsoft.co.kr
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