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Mathematical Model of Steam Injected into Steam/air Mixture
Determined for Temperature Control of Flexible Energy System
MARTIN PIES, STEPAN OZANA
VSB-Technical University of Ostrava
Department of Cybernetics and Biomedical Engineering
17. listopadu 15, 708 33 Ostrava
CZECH REPUBLIC
[email protected], [email protected]
Abstract: This paper deals with the description of the mathematical model resulting in adjustment of the properties
of technological gas used in turbine drive unit of the power plant. The paper gives a description of mathematical
model of thermal balance for technological gas which is mixed by injected saturated steam, resulting in air/steam
mixture. Model of this mixer is embedded into mathematical model of the power plant, serving as one of the
actuating term for regulation of this mixture in the power plant. Mixer model is implemented as S-function in
Simulink environment.
Key–Words: Mathematical models, Complex systems, Steam, Temperature control, Thermal equilibrium
1 Introduction
2 Mathematical model of the mixer
for power gas/steam
Vitkovice Power Engineering joint-stock company
counts among the big power plant boiler manufacturers. At present time Vitkovice develops power plant
boiler of the new type [5].
Mathematical model of the mixer for power
gas/steam, referred to as mixer M401 includes
algebraic equations describing mixing process of
steam and air/steam mixture coming from recuperation exchanger. Mathematical model supposes that
input temperature of air/steam mixture coming to
injection M401, is higher than boiling point of the
water at a given pressure. It means the mixture will
contain no condensed water.
The output of mixer M401 is power gas whose
composition is determined by concentration of dry air
wda and by concentration of the steam ws . The following table Tab. 1 describes the physical quantities
involved in mathematical model of mixer M401.
This paper comes out from the previous work [2]
that introduced a basic overview of technological solution of the model of power plant, so called Flexible
Energy System.
Considered power plant uses a mixture of superheated steam and dry air as the main heat-carrier media. The most turbines in current power plants are
driven by superheated steam. The reference [4] introduced water injection into a humid air. The other
way ho to regulate the temperature of mixture consisting of dry air/steam is injection of saturated steam into
this mixture. This injection also affects steam concentration in the resulting air/steam mixture. The output
temperature of such mixture can be determined by use
of existing thermodynamic tables for water and steam,
see [6].
symbol
h
M
p
Q
Heat-carrier media used in the power plant model
described in [2] comprises of mixture of air and steam.
So far there are no tables with thermodynamic properties for such mixture.
T
w
The goal of presented paper is to introduce a possible solution for determination of reset temperature
of the mixture that is cooled by the steam.
ISBN: 978-1-61804-099-2
description
unit
enthalpy
mass flow rate
pressure
heat added / drained per
second
temperature
concentration of mixture
components
[kJ/kg]
[kg/s]
[Pa]
[kJ/s]
[◦ C]
[kg/kg]
Table 1: Symbols occurring in the model
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Indexes
da, in
da, out
is
pg
pg, in
pg, out
s
s, in
s, out
correspond to dry air in incoming
power gas
correspond to dry air in outcoming
power gas
correspond to input steam injected
to power gas
correspond to power gas
correspond to incoming power gas
correspond to outcoming power gas
correspond to steam
correspond to steam in incoming
power gas
correspond to steam in outcoming
power gas
The assumption to be fulfilled is that both incoming media have the same pressure ppg , corresponding the power gas pressure at the mixer output. The
first step includes determination of water steam concentration ws,out and dry air concentration wda,out in
power gas. The amount of water vapor in the air/steam
mixture is given by ratio between concentration of the
steam in input mixture ws,in and quantity of air/steam
mixture Mpg,in , and by amount of injected steam Mis .
This injected steam changes ratio of concentration
ws,in to ws,out and cools down input air/steam mixture
at the same time. Ratios of particular concentrations
are illustrated in Fig. 1.
Total quantity of power gas is given by formula
(1).
Mpg,out = Mpg,in + Mis
(1)
Figure 1: Definition of power gas in the block representing mixer M401.
where rda and rs are specific gas constants of dry air
and water vapor.
Enthalpy of power gas hpg , created as a mixture
of incoming power gas and the steam is composed of
three enthalpy elements. The first one is the enthalpy
of dry air hda . This enthalpy can be determined by set
of the tables stated in [1] using this command:
d
Tpg,in
Concentration of steam coming to the mixer
M401, is given by formula (2).
Mis
wis =
Mpg,in + Mis
The second element is the enthalpy of water steam
hs contained in a humid air. This enthalpy can be determined by set of the tables stated in [6] using this
command:
(2)
Overall concentration of water vapor in the mixture is given by equation (3).
ws,out = (1 − wis )ws,in + wis
ps,in
(3)
Concentration of dry air in power gas wda,out is
then a supplement to one.
wda,out = 1 − ws,out
Tpg,in
wda,out · rda
ps = ppg · 1 −
wda,out · rda + ws,out · rs
pda = ppg − ps
ISBN: 978-1-61804-099-2
hs = xsteam(’h_pT’,psin,Tpgin)
partial pressure of the steam in a power gas
[bar] (conversion Pa → bar necessary)
temperature of incoming power gas [◦ C]
Partial pressure of the water steam in incoming
power gas ps,in expresses partial pressure of the water
steam in power gas before mixing the water and power
gas. This partial pressure is computed according formulas (6) and (7).
For wis = 0 according (3) we get
(4)
Partial pressure of the water steam and a dry air
in power gas are determined by (5).
"
hda = humde(d,Tpgin)
relative humidity level [kg/kg] (d = 0)
temperature of incoming power gas [◦ C]
#
ws,out = (1 − 0) · ws,in + 0 ⇒ ws,out = ws,in (6)
(5)
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3 Calculation of power gas temperature
Then
"
ps,in = ppg · 1 −
wda,in · rda
wda,in · rda + ws,in · rs
pda,in = ppg − ps,in
#
Temperature of power gas being cooled by injection
of steam can be determined from entire heat Qpg by
use of suitable iterative method. To find mixture output temperature Tpg bisection method has been chosen. Illustrative scheme of the algorithm to find the
temperature is shown in Fig. 2. while T1 represents
first media temperature and T2 the second media.
In case difference of T1 and T2 does not exceed
0.1 ◦ C, algorithm stops and returns minimum of the
interval hT1 ; T2 i.
Or in case the temperatures are different, algorithm determines interval whose half is denoted as
working temperature Tpg,temp of cooled power gas.
This temperature is then substituted into relations (15)
and (16) in order to get the working heat of cooled
power gas mixture (referred to as Qpg,temp in diagram). Next step involves computation of relative error according formula (14).
(7)
The third component of the mixture enthalpy is
the input steam enthalpy his being injected to the
power gas. This enthalpy can be determined by set
of the tables stated in [6] using this command:
pis
Tis
his = xsteam(’h_pT’,pis,Tis)
partial pressure of injected steam [bar]
temperature of injected steam [◦ C]
Partial pressure of the input steam pis means the
difference of the partial pressures of the water steam
before and after mixing. It can be expressed according
(8).
pis = ps − ps,in
(8)
Thus it is possible to say that particular enthalpies
are functions of the following quantity:
hda = f (Tpg,in ) , where d = 0
hs = f (Tpg,in , ps,in )
his = f (Tis , pis )
δQpg =
|Qpg − Qpg,temp |
· 100 [%]
Qpg,temp
(14)
(9)
Entire enthalpy of power gas after mixing humid
air with the water will be:
hpg = hda · wda,out + hs · (1 − wis ) · ws,in + his · wis
(10)
In case wis is zero, then pis is also zero and relation (10) turns into:
ws,out = (1 − wis ) · ws,in + wis ; wis = 0
ws,out = (1 − 0) · ws,in + 0; wda,out = 1 − wda,in
hpg = hda · wda,out + hs · ws,out
(11)
In case air/steam mixture at the input of mixer
M401 contains no water vapor, (ws,in = 0), then ps,in
is zero and equation (10) changes to:
Figure 2: Diagram of the method to find the resulting
temperature of power gas.
ws,out = (1 − wis ) · ws,in + wis ; ws,in = 0
ws,out = (1 − wis ) · 0 + wis ; wda,out = 1 − wis
hpg = hda · wda,out + his · wis
(12)
If this relative error does not exceed user-defined
maximum (parameter of S-funcion), then algorithm
managed to find power gas temperature equaled to
Tpg,temp . In case the error exceeds this threshold, algorithm computes difference Qpg − Qpg,temp . According the sign of this difference between the heat values
it computes a new power gas working temperature
Tpg,temp1 . This temperature Tpg,temp1 is again substituted into (15) and (16) and relative error is computed
Resulting heat of power gas is described by (13).
Qpg = hpg · Mpg
(13)
This resulting heat Qpg is one of input parameters
for iterative algorithm described in chapter 4.
ISBN: 978-1-61804-099-2
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4 Simulation results
once again according (14). This is the way how algorithm keeps working until it finds working temperature Tpg,temp , which will determine the heat Qpg,temp ,
which is approximately the same as Qpg . This temperature can be considered as the output power gas
temperature Tpg .
This algorithm performs computation of enthalpy
hpg,temp and working heat Qpg,temp of power gas
mixture Qpg,temp according (15) and (16). These
enthalpies are functions of working temperature
Tpg,temp and partial pressure of water steam ps contained in resulting power gas mixture.
hpg,temp = hda (Tpg,temp ) · wda +
+ hs (Tpg,temp , ps ) · wda
(15)
Qpg,temp = hpg,temp · Mpg
(16)
The survey described in [3] shows that it is highly efficient to implement the mathematical models into Sfunctions by use of C language. The above mentioned
mixer M301 is implemented into Simulink block as its
level-2 S-function in C syntax, see Fig. 3. Parameter
of S-function is the limit for maximal error δQpg,max ,
which is compared to the level computed by (14).
As mentioned above, this heat is compared to the
heat Qpg computed according (13).
During the run of the algorithm there might be
need of computation of the enthalpy for temperature
that returns water enthalpy since the tables respect
phase transformation between water and steam. This
causes a significant increasing of Qpg,temp and consequent increase of relative error computed according
(14). This indicates condensation of the steam – substituted temperature Tpg,temp was under the saturation
temperature of power gas. In this case the bisection
algorithm will most likely and repeatedly use temperature Tpg,temp in relation (15) oscillating around saturation temperature Tpg,sat of the water in power gas.
Algorithm finishes without finding the steam enthalpy
and return NaN value.
If NaN is returned, then the temperature of power
gas is determined from the Xsteam tables as follows:
ps
Figure 3: Mathematical model of mixer M401 implemented in Simulink.
Linearly increasing signal Mis (t) was brought to
the input of M401, representing the amount of injected steam. Operating point of the injection was set
up as follows: Tpg,in = 475 ◦ C, Mpg,in = 2.261 kg/s,
ws,in = 0.06 kg/kg, Tis = 320 ◦ C, ppg = 5.584 bar.
Maximum error during the computation of the heat of
power gas was predefined as δQpg,max = 0.01 %.
Tpgsat = xsteam(’Tsat_p’,ps)
partial steam pressure [bar]
There’s one more issue to be remarked: tables for
water and steam start working from the least value of
6.12 mbar. Such low pressure might cause troubles in
determination of power gas temperature in particular
cases when partial pressure pis or ps,in is too small.
For these purposes, these partial pressures are
tested before running the computation of temperature
Tpg . Low values of pis indicates no steam is injected
and temperature Tpg will be replaced by Tpg ≃ Tpg,in .
Low value of ps,in means that concentration of steam
in incoming power gas ws,in is too small or dry air is at
the injection input. If it is so, then the enthalpies hda
and his are used for calculation of temperature Tpg .
ISBN: 978-1-61804-099-2
Figure 4: Temperature Tpg at mixer output M401 as a
function of amount of injected water Mis .
Fig. 4 shows the resulting the dependence of temperature of power gas Tpg at mixer output the amount
of injected water Mis . Simulation results show exponential dependence of the output temperature of
air/steam mixture Tpg on amount of injected steam
Mis . The injector M401 as a part of the whole Flex217
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ible Energy System is working in the operating point
described above. The amount of injected steam works
out approximately about Mis = 0.335 kg/s.
Fig. 5 shows resulting concentration ratios of dry
air wda,out and steam ws,out in power gas mixture at
the mixer output block.
der construction and these measurements are planned
for the end of 2012.
Acknowledgements: The work was supported by the
grant No. FR-TI1/073 of the Czech Department of
Industry and Commerce and by project SP2012/111,
named “Data Acquisition and Processing from Large
Distributed Systems II” of Student Grant Agency
(VSB - Technical University of Ostrava).
References:
[1] Rice University, Department of Chemical and Biomolecular Engineering. CHBE
301 - Material and Energy Balances. [web
page] http://www.owlnet.rice.edu/
˜ceng301/index.html, 2000. [Accessed
on 20th May 2011].
[2] N EVRIVA P., V ILIMEC L.: Simulation of the
Power Plant Dynamics, In International Conference on Modeling, Simulation and Control
ICMSC 2010, Cairo, Egypt.
[3] O ZANA S., M ACHACEK Z.: Implementation of
the mathematical model of a generating block in
Matlab&Simulink using S-functions, In International Conference on Computer and Electrical
Engineering, ICCEE 2009, Dubai, ISBN 978076953925-6.
[4] P IES M., O ZANA S., M ACHACEK Z.: Mathematical Model of Water Injected into Steam/air
Mixture Determined for Temperature Control
of Flexible Energy System, In Proceedings of
the 10th WSEAS International Conference on
System Science and Simulation in Engineering,
ICOSSSE ’11, Penang, ISBN 978-1-61804-0411.
[5] VITKOVICE POWER ENGINEERING:
Power production process with gas turbine from
solid fuel and waste heat and the equipment for
the performing of this process, Inventors: Vilimec Ladislav, Starek Kamil, Czech Republic,
Patent EP08466028
[6] IAPWS IF-97. Thermodynamical properties of steam and water. [web page]
http://www.x-eng.com/, 2007. [Accessed on 10th June 2011].
Figure 5: Concentration of dry air and steam wda,out
and steam ws,out at the output of mixer M401 as a
function of injected steam amount Mis .
5 Conclusion
Presented paper gave a description of mathematical
model of steam injection into steam/air mixture. It
is model describing steady temperature of power gas
after blending of the both media. For simulation of the
mathematical mode of the power plant, the dynamics
of pressure changes during the injection is handled by
using first order system block located at ppg input of
the mixer M401.
Currently the model calculates the new output
temperature of the output air/steam mixture Tpg,out
each step of the simulation. Improvement which is
currently under construction is the use of discrete state
variable which will hold the last computed value of
power gas temperature Tpg at block’s output. Its use
will make it possible to speed up the computation of
temperature Tpg , since for small or no changes of input media parameters it will be possible to use this
value for the next simulation step (with a certain error).
Future work on this project will include comparison of simulated results with the real values measured
at real Flexible Energy System which is currently un-
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