simulation models of steam drums based on the heat transfer

Applied Mathematical Sciences, Vol. 4, 2010, no. 74, 3687 - 3712

Simulation Models of Steam Drums

Based on the Heat Transfer Equations

Stefano Bracco

University of GenoaDIMSET (Department of Machinery, Energy Systems and Transportation)

Via Montallegro 1, 16145 Genoa, [email protected]

Abstract

The aim of this paper is the description of two simulation modelsthat have been realized in the Matlab/Simulink environment in order tostudy the heat transfer in the metal and insulation parts of steam drumsbelonging to heat recovery steam generators located in combined cyclepower plants. Both simulators derive from a finite differences modelbased on the discretization of the Fourier’s heat conduction equation butare characterized by a different Simulink structure; their main outputis the temperature and the thermo-mechanical stress distribution insidethe metal of a steam drum. Owing to their short runtimes and flexibility,the two simulators can be linked with the real-time dynamic simulationmodel of the combined cycle power plant and can be used to evaluatethe plant life consumption due to transient operating conditions whichdetermine high thermo-mechanical stresses in metal components. Thenthe paper reports on some simulation results concerning the models val-idation phase and the analysis of typical transient operating conditionsof a combined cycle power plant.

Keywords: Dynamic simulation, finite differences, heat transfer, insula-tion, metal, steam drum

1 Introduction

The paper is focused on the simulation of the heat transfer in steam drumsof combined cycle power plants, in steady-state and dynamic operating con-ditions [3] [5]. It’s important to say that the present analysis is part of alarge study related to the dynamic simulation of energy systems, in particularcombined cycle power plants, operating in the Italian deregulated electricity

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market based on a competitiveness mechanism, which rewards power genera-tion plants characterized by high flexibility, availability and reliability [3] [4].In this context, it’s necessary to remark that, as shown in [3], combined cyclepower plants, because of their high flexibility, are more and more subjectedto frequent startups and shutdowns, in order to maximize revenues by fit-ting their production schedules to the demand curve taking into account theelectricity price variability during the day. As a consequence of these tran-sient operating conditions, the most critical components of power plants aresubjected to large thermal stresses which determine considerable life consump-tions. So each power plant operator needs to adopt dynamic simulators whichcan predict the combined cycle power plant behavior in different steady-stateand transient operating conditions [4], in order to optimize the plant man-agement strategies without excessively compromising the plant useful life andtechnical performances. Nowadays many power plants are equipped with real-time simulation models which can predict both the plant behavior and its lifeconsumption, during transient operating conditions; these simulation modelscalculate thermal and mechanical stresses in most critical components in orderto investigate dangerous phenomena such as creep and low-cycle fatigue [1] [3].It’s important to remember that, in order to calculate the thermal stresses, it isnecessary to know, as input for the simulation models, the temperature distri-bution inside the metal parts of the plant components; this input derives froma thermal analysis, based on the Fourier’s heat conduction equation, whichis usually solved by means of a finite differences model or a finite elementssimulation tool [2] [6] [8] [10].The present paper describes two mathematical and simulation models, calledSIM CC and SIM CC m, that have been developed to evaluate the thermaland mechanical stress distribution inside the metal part of a steam drum in-stalled into a heat recovery steam generator of a combined cycle power plant.In particular, the Fourier’s heat conduction equation has been applied in orderto calculate the temperature values inside both the metal and the insulation ofthe drum: the paper reports the analytical solution of the Fourier’s equation,in steady-state and transient conditions. Then a finite differences method hasbeen adopted in order to implement the heat transfer mathematical model inthe Simulink environment, as reported also in [3] and [5]. As described inthe present paper, the two simulation models, both based on the discretizedFourier’s equation, differ in their structure in the Simulink environment: infact, SIM CC has a static blocks structure while SIM CC m is more flexiblebecause it has been developed writing the discretized Fourier’s equation invectorial form.The paper describes the simulation models validation phase which has beendone referring to literature studies; then the simulation results have been com-pared with the ones of a finite element analysis that has been done using the

Simulation models of steam drums 3689

software Ansys [3].Some simulation results are reported in order to show the effects, in terms ofthermo-mechanical stresses, of typical transient operating conditions, such asa night load decrease, on the steam drum of a modern combined cycle powerplant; this analysis permits to investigate how mechanical and thermal stressesinteract to generate the global stress inside the metal parts of the drum. Thesimulators can also be used to study the effects of different metal and insula-tion materials on the temperature distribution and to determine the insulationbest thickness which permits to have no high temperature values on the drumexternal surface, obeying safety standards [3].

2 The steam drum physical model

The steam drum is one of the most important components in a heat recoverysteam generator of a combined cycle power plant, that is a thermoelectricunit which produces electricity or, in cogenerative configuration, electricityand heat [3] [4] [5]. Figure 1 shows a simplified scheme of a combined cyclepower plant, which consists of a bottoming steam power plant downstream ofa topping gas turbine unit.

Figure 1: The simplified scheme of a combined cycle power plant

The main components of a combined cycle power plant are: the gas tur-bine, the Heat Recovery Steam Generator (HRSG), the steam turbine, thecondenser, the electrical generators and the transformers. The gas turbineunit consists of an air compressor, a combustion chamber usually fed withnatural gas or oil, a turbine and an electrical generator. The HRSG is a heatexchanger between the exhaust gases, leaving the gas turbine, and the wa-ter/steam circulating into finned tubes; a HRSG is characterized by a horizon-

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tal or a vertical configuration, depending on the flow direction of the exhaustgases. Most of HRGSs, installed in modern combined cycle power plants, areat three pressure levels (low, intermediate and high pressure) plus steam reheat[3] [4]. In the simplified scheme reported in Fig.1, the one pressure level HRSGis characterized by a horizontal gas flow natural circulation type configuration;within the HRSG it is possible to distinguish the following main components:the economizer, the evaporator and the superheater [5]. On the top of theHRSG, particularly externally, the steam drum is located. The steam drumcontains water and steam in saturation conditions: the water, leaving the econ-omizer, flows through the downcomer and then vaporizes in the risers, whichare heated by the exhaust gases; the produced saturated steam, leaving thetop of the steam drum, goes to the superheater. Then the superheated steamexpands into the turbine and becomes again saturated liquid passing throughthe condenser; nowadays, water-cooled or air-cooled condensers are installedin combined cycle power plants.It’s important to remark that in the developed mathematical model, a sim-plified configuration of the steam drum has been adopted: it consists of twocoaxial cylinders, the inner for the metal (characterized by an internal radiusequal to rint) and the outer for the insulation (r∗= internal radius, rext= ex-ternal radius). This simplified configuration does not consider the end platesand the downcomer and risers connections, that would call for a finite elementanalysis; as a consequence, the calculated thermo-mechanical stresses are thosewhich occur in a drum cross section far from the end plates. It has been neces-sary to adopt this configuration in order to develop quick simulators suitableto be coupled with the real-time dynamic simulation model of the combinedcycle power plant.The developed model considers: the heat convection between the inner cylin-der and the saturated steam, the heat conduction inside both the metal andthe insulation, the heat convection between the outer cylinder and the ambientair [2] [3] [10].

3 The steam drum mathematical model

A mathematical model has been developed in order to calculate the temper-ature and the thermo-mechanical stress distribution inside the metal and theinsulation of a high pressure steam drum; this model is very flexible and it isalso suitable to simulate the heat transfer through the intermediate or low pres-sure steam drums installed in a three pressure levels HRSG [3]. As mentionedbefore, this paper is focused on the description of the mathematical modelcreated to determine the temperature distribution inside both the metal andthe insulation of the drum; the paper does not report the equations used tocalculate the thermal and mechanical stresses, which are accurately described


in [3].

3.1 The heat conduction mathematical model

A mathematical model has been developed in order to evaluate the heat con-duction inside the metal and the insulation of the drum [2] [3] [5] [6] [10];applying the energy balance equation, in cylindrical coordinates (r, θ, z), to aninfinitesimal solid part of the drum it derives:

∑m

dφinm +∑

n

dφintn =∑

j

dφoutj + cρdV∂T

∂t(1)

where c and ρ are respectively the specific heat and the density of the metalor the insulation which the steam drum consists of, and dV = rdrdθdz is theinfinitesimal volume. The term containing the temperature partial derivativewith respect to time in Eq. 1 corresponds to the partial derivative with re-spect to time of the infinitesimal solid part internal energy. The inner thermalflows dφintn, reported in Eq. 1, are null due to the absence of internal heatsources while the heat conduction flows dφinm and dφoutj depend on the ma-terial thermal conductivity, the temperature gradient and the heat exchangearea perpendicular to the heat exchange direction. In particular,

∑m dφinm is

the sum of the inlet thermal flows:

∑m

dφinm = dφr + dφθ + dφz (2)

where:

dφr = −k · dAr · ∂T

∂r= −k · ∂T

∂r· rdθdz (3)

dφθ = −k

r· dAθ · ∂T

∂θ= −k

r· ∂T

∂θ· drdz (4)

dφz = −k · dAz · ∂T

∂z= −k · ∂T

∂z· rdθdr. (5)

while∑

j dφoutj is the sum of the outlet thermal flows:

∑j

dφoutj = dφr+dr + dφθ+dθ + dφz+dz (6)

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where:

dφr+dr = −k · dAr+dr · ∂T

∂r

∣∣∣∣r+dr

= −k ·(

∂T

∂r+

∂2T

∂r2dr

)· (r + dr) dθdz (7)

dφθ+dθ = −k

r· dAθ+dθ · ∂T

∂θ

∣∣∣∣θ+dθ

= −k

r·(

∂T

∂θ+

∂2T

∂θ2dθ

)· drdz (8)

dφz+dz = −k · dAz+dz · ∂T

∂z

∣∣∣∣z+dz

= −k ·(

∂T

∂z+

∂2T

∂z2dz

)· rdrdθ. (9)

Having calculated the outlet thermal flows dφoutj by means of 2nd orderTaylor polynomial expansion, omitting infinitesimal quantities of higher order,Eq. 1 becomes ([2], [3], [6], [8], [10]):

∂2T

∂r2+

1

r

∂T

∂r+

1

r2

∂2T

∂θ2+

∂2T

∂z2=

1

a

∂T

∂t→ ∇T =

1

a

∂T

∂t(10)

where a is the material thermal diffusivity:

a =k

cρ(11)

k being the thermal conductivity. As known, the thermal diffusivity is a phys-ical property connected with the speed of propagation of a temperature vari-ation inside the material [10]. The present mathematical model of the steamdrum considers two homogeneous materials, the metal and the insulation; sincethe metal is characterized by a thermal diffusivity higher than that of the in-sulation, the heat propagation is faster inside the metal.Considering the cylindrical symmetry of the steam drum, the temperature hasbeen assumed only as a function of both the radial coordinate r and the time;so, the Fourier’s equation becomes:

∂2T

∂r2+

1

r

∂T

∂r=

1

a

∂T

∂t. (12)

This partial differential equation has been implemented in the Simulinkenvironment to study the heat transfer through the metal and the insulationof the steam drum.


3.2 The analytical solution of the steam drum’s equa-tion in steady-state conditions

The Eq. 12 analytical solution has been found for both steady-state and tran-sient conditions [2] [3] [5].

Figure 2: The temperature distribution in steady-state conditions

In steady-state conditions, Eq. 12 is Laplace’s equation:

∂2T

∂r2+

1

r

∂T

∂r= 0. (13)

Considering only the metal part of the drum, the solution of the Laplace’sequation is:

T = T (r) = Tint − Tint − T ∗

ln(

r∗rint

) ln

(r

rint

)(14)

where Tint and T ∗ are respectively the temperatures at the inner and outer sur-face of the metal cylinder. Figure 2 shows the drum temperature distributionin steady-state conditions considering the saturated steam temperature higherthan the ambient air temperature; because of the metal high thermal diffusiv-ity, the temperature difference through the metal is significantly smaller thanthe one through the insulation. The thermal flow along the radial direction isequal to:

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φr CC =Tint − Text

12πkmetL

ln(

r∗rint

)+ 1

2πkinsLln(

rext

r∗) =

Tint − Text

Rtmet + Rtins

(15)

where Rtmet and Rtinsare the metal and insulation thermal resistances while

L is the drum length.

3.3 The analytical solution of the steam drum’s equa-tion in transient conditions

In transient operating conditions, the Fourier’s equation analytical solutioncan be found by the technique of separation of variables, as proposed in [8]and [9]. The adopted procedure assumes a product solution ([3], [5]):

T (r, t) = F (r) · G(t) (16)

which, substituted in the governing differential equation (Eq. 12), gives:

G · F ′′ +1

r· G · F ′ =

1

a· F · G′. (17)

By separating variables, the Eq. 17 becomes:

F ′′

F+

1

r

F ′

F=

1

a

G′

G. (18)

As a consequence, it is clear that both sides of Eq. 18 must equal a separa-tion constant −λ2, where the minus sign justifies the fact that the temperatureis a limited function; so the two following equations derive:

F ′′

F+

1

r

F ′

F= −λ2 (19)

1

a

G′

G= −λ2. (20)

The general solution of Eq. 20 is:

G = G(t) = c1e−aλ2t (21)


while the Eq. 19 is the Bessel’s equation of zero-order whose solution is:

F = F (r) = A1J0(λr) + B1Y0(λr) (22)

where J0 and Y0 are respectively the Bessel functions of zero-order of the firstand second kind [9]. As a consequence:

T (r, t) = e−aλ2t · [AJ0(λr) + BY0(λr)] . (23)

is a solution of Eq. 12. The general solution of Eq. 12 can be obtained viaa series expression of solutions like that in Eq. 23. In order to determine theA, B and λ values it is necessary to know both initial conditions, which areimposed at the initial instant ([2], [8], [10]):

T (r, 0) = f(r) (24)

and boundary conditions, which are imposed at the boundary of the drum [2][8] [10]; in particular, for the metal part they are:

T (rint, t) = Tint (t) = Tint, T (r∗, t) = T ∗ (t) = T ∗. (25)

As a consequence the solution of the Fourier’s equation is the sum of twoterms, the first being null in steady-state conditions [3]:

T (r, t) =

∞∑q=1

(∫ r∗

rintrf(r)T0 (λqr) dr∫ r∗

rintr [T0 (λqr)]

2 dr· e−aλ2

qt · T0 (λqr)

)+

+

⎛⎝Tint − Tint − T ∗

ln(

r∗rint

) · ln(

r

rint

)⎞⎠ = T1(r, t) + T2(r, t) (26)

where T0 (λqr) = [Y0 (λqrint) J0 (λqr) − J0 (λqrint) Y0 (λqr)].

3.4 The numerical solution of the Fourier’s equation

A finite differences model has been developed in order to implement the Fourierequation in the Matlab/Simulink environment, as shown in [3] and [5]. In thisfinite differences model both the metal and the insulation cylinder, which thedrum consists of, have been discretized in N coaxial cylinders: each cylindersurface is a node to which a temperature value corresponds. The Fourier’s heat

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conduction equation numerical solution has been found by substituting thefollowing finite differences, backward and forward, for the partial derivativesreported in Eq. 12:

∂T

∂r

∣∣∣∣i

=T (ri + Δr) − T (ri)

Δr=

T ti+1 − T t

i

Δr(27)

∂2T

∂r2

∣∣∣∣i

=∂

dr

(∂T

dr

) ∣∣∣∣i

=

∂T (r+Δr)∂r

∣∣∣∣i+1

− ∂T (r)∂r

∣∣∣∣i

Δr= ... =

T ti+1 − 2T t

i + T ti−1

(Δr)2(28)

∂T

∂t

∣∣∣∣i

= Ti =T t+Δt

i − T ti

Δt(29)

where Δr and Δt are respectively the space and the time discretization stepswhile i indicates the ith layer of the drum metal or insulation part.It’s important to remark that the developed discretization model is bidimen-sional, because it assumes that the temperature varies along the radial coor-dinate and the time, and explicit in time. The ith node temperature at the(t+1) instant is a function of the (i−1)th, ith and (i+1)th nodes temperaturescalculated at the t instant:

T t+Δti = f

(T t

i , Tti−1, T

ti+1, ri, a, Δt,Δr

). (30)

The resultant discretized Fourier’s equation is:

T t+Δti − T t

i

Δt=

a

(Δr)2

[T t

i+1

(1 +

Δr

ri

)− T t

i

(2 +

Δr

ri

)+ T t

i−1

](31)

and the stability of its solution depends on the following numerical criterion([3], [8]):

Δt ≤ 1

2

(Δr)2

a. (32)

4 The steam drum simulation models

The finite differences model, previously described, has been implemented in theMatlab/Simulink environment. In particular two different simulators, called


SIM CC and SIM CC m, have been realized in order to evaluate the tem-perature distribution and the thermo-mechanical stresses in the metal andinsulation parts of a steam drum [3] [5].The two simulation models have a different structure but both are based onthe discretized Fourier’s equation. The main inputs of the two simulators are:

• the steam pressure;

• the ambient temperature and pressure;

• the geometrical parameters of the steam drum: internal radius, metalthickness, insulation thickness, length;

• the metal and the insulation physical properties: thermal conductivity,specific heat, density, thermal diffusivity, Young’s modulus of elasticity,coefficient of thermal expansion.

On the other hand, the main outputs are:

• the temperature values inside the metal and the insulation;

• the thermo-mechanical stresses in the metal.

Figure 3 shows the Simulink mask which is the same main window for boththe two simulators.

Figure 3: The main window of the two simulation models

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4.1 The first simulation model

SIM CC, the first simulator, is characterized by a static structure based onmany Simulink subsystems, each corresponding to one of the metal or insu-lation cylindrical layers which the drum has been divided into. During thediscretization process, the number of metal and insulation nodes has been in-creased until the temperature values, calculated by the model, changed nomore [3] [5]. For a high pressure steam drum, the model gives acceptable re-sults considering the metal and the insulation divided into 18 (Nmet) and 10(Nins) cylindrical layers respectively, as shown by Fig. 4. This layers numberis suitable for any other steam drum installed in a HRSG of a combined cyclepower plant, since high pressure steam drums are the ones characterized bythe highest metal thicknesses [3].

r NODES

Figure 4: The drum discretization into coaxial cylinders

The geometrical dimensions which delimit the metal and the insulation ofthe steam drum are:

r0 = rint, r18 = r∗, r10ins = rext, (33)

the metal temperatures are T0, T1, ..., T18 while the insulation temperatures areT1ins, T2ins, ..., T10ins.The discretized Fourier’s equation used to calculate the temperatures of themetal layers is:

Timet =amet

(Δrmet)2

[T t

i+1met

(1 +

Δrmet

rimet

)− T t

imet

(2 +

Δrmet

rimet

)+ T t

i−1met

](34)

where:


Δrmet =r18 − r0

18(35)

while the insulation temperatures are calculated by:

Tiins=

ains

(Δrins)2

[T t

i+1ins

(1 +

Δrins

riins

)− T t

iins

(2 +

Δrins

riins

)+ T t

i−1ins

](36)

where:

Δrins =r10ins − r18

10. (37)

Figure 5: The simulator subsystem for the metal temperatures calculation

Figure 5 shows an extract of the Simulink subsystem which calculates thetemperature of the metal layers.It is necessary to say something about the calculation of the temperatures atthe boundary surfaces which are: the temperature of the internal surface incontact with the steam (T0), the temperature of the interface between the metaland the insulation (Tstar) and the temperature of the insulation external layerin contact with the ambient air (T10ins). In order to calculate T0 a finite volumesapproach has been used [3]. In fact, T0 is considered as the temperature ofa very thin metal layer, called Imet, in contact with both the steam at Tsteam

temperature and the metal layer characterized by the T1 temperature. T0 iscalculated by the integration of the following first order differential equation:

φin Imet − φout Imet = CImet ·∂T0

∂t(38)

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which is the energy balance equation applied to the Imet layer; CImet is the Imet

layer thermal capacity and the inlet thermal flow φin Imet is calculated by:

φin Imet = hsteamSint (Tsteam − T0) (39)

where hsteam is the convection coefficient which takes into account the heatexchange between the steam, in saturated conditions, and the metal internalsurface Sint of the drum. On the other hand, the outlet thermal flow φout Imet

is equal to:

φout Imet =2πkmetL

ln(

r1

r0

) · (T0 − T1) . (40)

A similar mathematical approach has been used to calculate T10ins [3]. Infact:

φin10ins− φout10ins

= C10ins · ∂T10ins

∂t(41)

where C10ins is the thermal capacity of the insulation 10th layer and the inletφin10ins

and outlet φout10insthermal flows are equal to:

φin10ins=

2πkinsL

ln(

r10ins

r9ins

) · (T9ins − T10ins) (42)

φout10ins= hambSext (T10ins − Tamb) (43)

where hamb is the convection coefficient which takes into account the heatexchange between the drum external surface Sext and the air at ambient tem-perature. In order to evaluate the temperature Tstar of the metal/insulationinterface a very thin fictitious layer has been modeled [3]; the energy balanceequation applied to this layer gives:

φinstar − φoutstar = Cstar · ∂Tstar

∂t(44)

where:

φinstar =2πkmetL

ln(

rstar

r18

) · (T18 − Tstar) (45)


φoutstar =2πkinsL

ln(

rstar+thstar

rstar

) · (Tstar − T1ins) . (46)

It has been necessary to add this fictitious layer in order to have, in steady-state conditions, the metal layers almost at the same temperature, as it is inaccordance with the heat transfer physical phenomenon [3].

4.2 The second simulation model

In order to have a more flexible simulator a matricial model, called SIM CC m,has been developed writing the discretized Fourier’s equation, related to eachmetal or insulation coaxial cylinder, in vectorial form [3]. This second modelis more flexible than the first because its Simulink structure permits to au-tomatically increase or decrease the number of metal and insulation layers,depending on the drum thickness, setting the desired value into the main datafile.In this subsection the metal layers model, implemented in this second simula-tor, is analyzed. This model considers the following metal layers:

• layer 0: it is the internal surface of the steam drum at the T0 temperaturecalculated by Eq. 38;

• layers from 1 to Nmet: these are the metal layers automatically generatedby the SIM CC m model;

• layer Nmet +1: it is the external surface of the metal in contact with thefirst layer of the insulation.

In order to describe the structure of this second model it is useful to writethe discretized Fourier’s equations system for the metal layers in the followingform:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

T1 = amet

(Δrmet)2·[T0 − T1

(2 + Δrmet

r1

)+ T2

(1 + Δrmet

r1

)]

T2 = amet

(Δrmet)2·[T1 − T2

(2 + Δrmet

r2

)+ T3

(1 + Δrmet

r2

)]...

Ti = amet

(Δrmet)2·[Ti−1 − Ti

(2 + Δrmet

ri

)+ Ti+1

(1 + Δrmet

ri

)]...

TNmet = amet

(Δrmet)2·[TNmet−1 − TNmet

(2 + Δrmet

rNmet

)+ TNmet+1

(1 + Δrmet

rNmet

)]

(47)

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where Δrmet is equal to:

Δrmet =rNmet+1 − r0

Nmet + 1. (48)

The differential equations system reported by Eq. 47 can be written invectorial form:

ˆT =amet

(Δrmet)2 · S (49)

where ˆT is the (Nmet × 1) vector of the partial derivatives of the layers tem-peratures with respect to the time, while S is the (Nmet × 1) vector equalto:

S = B × Y + A × T . (50)

Figure 6: The Fourier’s discretized equations system in vectorial form

Figure 6 shows the subsystem which models the Eq. 49 in the SIM CC msimulator [3].

B is a (Nmet × 2) matrix:

B =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 00 0...

...0 0...

...

0(1 + Δrmet

rNmet

)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(51)


while Y is the vector of the 0 and Nmet + 1 layers temperatures:

Y =

[T0

TNmet+1

]. (52)

A is a (Nmet × Nmet) matrix:

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

A(1, 1) A(1, 2) 0 . . . . . . 01 A(2, 2) A(2, 3) 0 . . . 0...

......

......

...0 0 1 A(i, i) A(i, i + 1) 0...

......

......

...0 . . . . . . 0 1 A(Nmet, Nmet)

⎞⎟⎟⎟⎟⎟⎟⎟⎠(53)

while T is the vector of the 1...Nmet layers temperatures.In order to define the A and B matrices it has been necessary to write the

following routine in the Initialization window of the ”Metal internal layers”subsystem Mask Editor [3]:

Δrmet =rNmet+1 − r0

Nmet + 1

K =amet

(Δrmet)2

A = zeros (Nmet, Nmet)

B = zeros (Nmet, 2)

for i = 1 : Nmet

A (i, i) = −(

2 +Δrmet

r0 + i · Δrmet

)end

for j = 1 : Nmet − 1

A (j, j + 1) = 1 +Δrmet

r0 + j · Δrmet

A (j + 1, j) = 1

end

B (1, 1) = 1

B (Nmet, 2) = 1 +Δrmet

rNmet

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4.3 The simulation models validation phase

The SIM CC and SIM CC m simulation models have been compared in steady-state conditions with analytical results [7] and modeling the same heat transferphenomenon by means of the Ansys software, using a finite element analysis[3]. Figure 7 reports the temperature distribution graphs [5], derived fromthe SIM CC model, for the metal and the insulation layers of a steam drumcharacterized by the following geometrical and thermal properties:

• Length=3.657 m, Inner Diameter=1.672 m, Metal Thickness=79.38 mm,Insulation Thickness=15 mm;

• Tsteam=286.85 ◦C, Tamb=11.85 ◦C;

• amet=1.7e -5 m2/s, ains=1.2e -6 m2/s.

M ETAL

INSULATION

M ETAL / INSULATION INTERFACE

20.16 °C

286.485°C

286.84 °C

286.84

286.48

286.40

286.50

286.60

286.70

286.80

286.90

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

T [°C]

M etal layers

258.22

20.160

30

60

90

120

150

180

210

240

270

1 2 3 4 5 6 7 8 9 10

T [°C]

Insulation layers

Figure 7: The steady-state temperature distribution

In steady-state conditions the temperature difference inside the metal layerscan be neglected, as well as the thermal stress, because of the metal higherthermal diffusivity in comparison with that of the insulation; as shown inFig. 7, the same results have been obtained by the Ansys model.


0 200 400 600 800 1000 1200250

300

350

400

450

500

550

600

Time [s]

T [K

]

Figure 8: The steam temperature variation due to a cold start

In transient operating conditions the steam drum simulation models have beenvalidated taking into account some literature results and with reference tosteam drums, installed in modern combined cycle power plants, characterizedby different geometrical dimensions and materials. The validation which refersto the study reported by Kim et al. in [7] is here described [3] [5].

SIM _CC

Paper

0−18 layers

Figure 9: The temperature distribution in the metal of the steam drum duringa cold start

As shown by Fig. 8, the analyzed transient consists of an increase in thesteam temperature due to a cold start of the combined cycle power plant. Asa consequence of this transient condition, the temperature of the metal layersincreases till a new steady-state condition is attained; Fig. 9 shows that theSIM CC results are in good agreement with those reported by Kim et al. in[7].

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0 200 400 600 800 1000 1200250

300

350

400

450

500

550

600

Time [s]

T [K

]

0−18 metal layers

1−10 insulation layers

Figure 10: The temperature distribution in the metal and in the insulation ofthe steam drum during a cold start

Figure 11: Maximum values of the thermal stress

In Fig. 10 the insulation layers temperatures are shown together with themetal temperatures: it is obvious noting that in steady-state conditions thetemperature drop is entirely concentrated in the insulation, as described in3.2. When a new steady-state condition is reached, the external surface ofthe drum is at a temperature of about 61.60 ◦C (334.75 K) which is too high,considering the maximum acceptable values reported by manufacturers; so thisindicates that it is necessary to increase the insulation thickness in comparisonwith the value of 15 mm considered by the present analysis [3].

Due to the thermal gradient inside the metal, very high thermal stressesoriginate; as shown by Fig. 11 and Fig. 12, both the SIM CC model and thereference paper indicate that the Von Mises equivalent thermal stress reachesthe maximum values in the first metal layer, the one in contact with the steam


SIM _CCPaper

Figure 12: The thermal stress distribution in the metal of the steam drumduring a cold start

0 400 800 1200 1600 2000 2400 2800 3200 3600575

576

577

578

579

580

581

582

583

584

585

Time [s]

T [K

]

Figure 13: The steam temperature decrease

[3] [5] [7].

4.4 A typical transient operating condition

Here is reported a typical transient operating condition which consists in aload decrease, during the night of a working day, for a combined cycle powerplant with a three pressure levels HRSG and a high pressure steam drumcharacterized by the following geometrical and thermal properties [3]:

• Length=12.9 m, Inner Diameter=2.184 m, Metal Thickness=90 mm,Insulation Thickness=150 mm;

• amet=1.427e -5 m2/s, ains=1.183e -6 m2/s.

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0 400 800 1200 1600 2000 2400 2800 3200 3600574

575

576

577

578

579

580

581

582

583

584

585

586

Time [s]

T [K

]

0−18 metal layersT steam

Figure 14: The metal temperatures decrease

0 100 200 300 400 500 600 700 800 900 1000583

583.2

583.4

583.6

583.8

584

584.2

584.4

584.6

584.8

585

Time [s]

T [K

]

0−18 metal layers

0−18 metal layers

T steam

Figure 15: The metal temperatures at the transient start

As shown by Fig. 13, during the analyzed transient the temperature of thesteam, in the high pressure steam drum, decreases from 311.70 ◦C (584.85 K)to 301.86 ◦C (575.01 K).As a consequence, the metal layers temperatures decrease too as shown bythe red curves plotted on Fig. 14. In the range from 0 to 750 s the metallayers temperatures are lower than the steam temperature, then they decreasefollowing the steam temperature trend with a time delay depending on themetal thermic inertia; at the end of the analyzed transient, the metal reachesa new steady-state condition characterized by temperature values lower thanthe steam temperature.Figure 14 shows also that in the range from 800 s to 1750 s the external metallayer is that characterized by the highest temperature values while it is the


1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 1950 2000 2050 2100574.4

574.6

574.8

575

575.2

575.4

575.6

575.8

576

Time [s]

T [K

]T steam

0−18 metal layers

0−18 metal layers

Figure 16: The time when the metal layers become colder than the steam again

0 400 800 1200 1600 2000 2400 2800 3200 36000

2

4

6

8

10

12

14

16

18

Time [s]

Von

Mis

es T

herm

al S

tres

s [M

Pa]

0123456789101112131415161718

Figure 17: The Von Mises thermal stress distribution in the metal layers

coldest at the transient start, as shown by Fig. 15; after 1750 s this externallayer becomes the coldest layer again, as shown by Fig. 16.Figure 17 reports the Von Mises equivalent thermal stress for the metal layers:it is obvious noting that the highest values occur in the range from 800 s to1000 s when the temperature difference between the internal and the externalmetal layers assumes the maximum values.During the transient the steam pressure decreases too, in particular from 101bar to 88 bar ; as a consequence, the Von Mises equivalent mechanical stressdecreases as shown by Fig. 18.It’s important to remark that the simulator evaluates the Von Mises thermo-mechanical stress which is calculated, for each metal layer, as the sum of theVon Mises equivalent thermal stress and the Von Mises equivalent mechanical

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0 400 800 1200 1600 2000 2400 2800 3200 360085

90

95

100

105

110

115

120

Time [s]

Von

Mis

es M

echa

nica

l Str

ess

[MP

a]

0−18 metal layers

Figure 18: The Von Mises mechanical stress in the metal layers

0 400 800 1200 1600 2000 2400 2800 3200 360080

85

90

95

100

105

110

115

120

125

Time [s]

Von

Mis

es T

herm

o−M

echa

nica

l Str

ess

[MP

a]

0123456789101112131415161718

0−18 metal layers

Figure 19: The Von Mises thermo-mechanical stress in the metal layers

stress [3]; Fig. 19 shows that in steady-state conditions, at the beginning andat the end of the analyzed transient, the Von Mises thermo-mechanical stresscoincides with the Von Mises equivalent mechanical stress because the VonMises equivalent thermal stress is negligible. On the other hand, in the rangefrom 700 s to 1600 s it’s necessary to consider also the thermal stress and sothe Von Mises thermo-mechanical stress differs from the Von Mises equivalentmechanical stress.


Table 1: NomenclatureSymbol Description Units

A heat exchange area [m2]a thermal diffusivity [m2/s]C thermal capacity [J/K]c specific heat [J/kgK]k thermal conductivity [W/mK]L length [m]R thermal resistance [K/W]r radius [m]T temperature [K]t time [s]V volume [m3]ρ density [kg/m3]φ thermal flow [W]

Subscript Description

ext externalin inletins insulationint internalmet metalout outlet

5 Conclusions

The present paper describes the study that has been done in order to simulatethe heat transfer through the metal and the insulation of steam drums, in-stalled in modern combined cycle power plants. In particular, two simulationmodels have been developed in the Matlab/Simulink environment: the firstsimulator is characterized by a static Simulink structure while the second ismore flexible, due to its vectorial structure.The main innovation introduced by the developed simulation models is thatthe steam drum temperature and thermo-mechanical stress calculation, due tothe adopted simplified geometry, has been done by means of a mathematicalapproach different from that used by a finite element analysis, and the Fourier’sequation has been discretized considering both the metal and the insulationof the steam drum. Both simulators have short runtimes and an user-friendlyinterface; then they are very flexible because they can be used to calculatethe temperature and stress distribution inside steam drums characterized bydifferent geometrical dimensions and materials.It’s important to remark that, due to their reduced runtime, both simulatorscan be coupled with the whole plant real-time simulation model, used as ob-server of the actual behaviour of the power plant. Moreover the simulatorsoutput values, above all thermo-mechanical stresses, can be the input of a lifeconsumption evaluation real-time model for combined cycle power plants.

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References

[1] D. Annaratone, Pressure Vessel Design, Springler-Verlag Berlin Heidel-berg, 2007.

[2] M. Bohn and F. Kreith, Principles of Heat Transfer, Thomson LearningCollege, 2000.

[3] S. Bracco, Dynamic simulation and useful life consumption evaluation ofcombined cycle power plants in the deregulated electricity market, Ph.D.Thesis, University of Genoa, Genoa, 2008.

[4] S. Bracco, G. Crosa and A. Trucco, Dynamic simulator of a combined cyclepower plant: focus on the heat recovery steam generator, In Proceedingsof Ecos 2007, vol.I (2007), 189 - 196.

[5] S. Bracco, Steady-state and dynamic heat transfer in steam drums ofcombined cycle power plants: mathematical and simulation model, InProceedings of ICNPAA Conference, 2008.

[6] D.Q. Kern, Process Heat Transfer, McGraw-Hill Inc., 1990.

[7] T.S. Kim, D.K. Lee and S.T. Ro, Analysis of thermal stress evolutionin the steam drum during start-up of a heat recovery steam generator,Applied Thermal Engineering, vol.20 (2000), 977 - 992.

[8] C.C. Lin and L.A. Segel, Mathematics Applied to Deterministic Problemsin the Natural Sciences, SIAM - Society for Industrial and Applied Math-ematics, Philadelphia, 1995.

[9] M.R. Spiegel, Fourier Analysis, Etas Libri, 1990.

[10] Y. Yener and S. Kakac, Heat Conduction, Taylor and Francis Group, 2008.

Received: June, 2008

simulation models of steam drums based on the heat transfer

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