INTERNATIONAL JOURNAL OF ENERGY RESEARCH

Numerical method for determining the allowable
medium temperature during the heating operation of a
thick-walled boiler element in a supercritical steam
power plant

Piotr Duda and Dariusz Rzasa

,y

Department of Thermal Power Engineering, Faculty of Mechanical Engineering, Cracow University of Technology, Al. Jana Paw"a II

37, 31-864 Cracow, Poland

SUMMARY

The new generation of steam power plants operates at pressures higher than the critical pressure and at very
high temperatures. They are called supercritical power plants and their thermal efficiency is improved by
increasing their operating pressure and temperature. Such a demanding working environment causes high
stresses in the construction, especially during the heating and cooling operations. Additionally, the cyclic
character of loading during operations causes material fatigue, known as low-cyclic fatigue. This phenomenon
may lead to the formation of fractures. Steam boiler manufacturers make efforts to design pressure
elements to meet these high requirements. They make recommendations for conducting start up and shut
down operations in order to keep the stresses in the construction elements within acceptable limits and obey the
safety regulations. Thus, it is important to find optimum parameters that can ensure proper heating and cooling
processes (Struct. Multidiscip. Optim. 2010; 40:529–535, Proceedings of the Congress on Thermal Stresses, 2007;
437–440).

Paper (Proceedings of the Congress on Thermal Stresses, 2007; 437–440) presents the method for determining

the optimum medium temperature, which ensures that the sum of the thermal stresses and stresses caused
by pressure at selected points do not exceed the allowable stresses. The presented optimum medium tem-
perature consists of the initial medium temperature step and later increases in the optimum rate of temperature
change. The extended version of the paper (Proceedings of the Congress on Thermal Stresses, 2007; 437–440) was
published in 2010 (Int. J. Energy Res. 2010; 34:20–35). Another paper (Proceedings of the 8th International
Congress on Thermal Stresses

, 2009; 2:399–402) presents the numerical optimization procedure, based on the

Levenberg–Marquardt algorithm that allows the optimum medium temperature to be established. This procedure
is based on the assumption that the thermal stresses in the entire construction elements do not exceed the allowable
stresses.

The aim of this paper is to present the method, which makes it possible to find the optimum parameters, so that

the total stresses during the start-up processes are kept at an acceptable level. The maximum absolute stresses,
caused by non-uniform temperature distribution and by pressure, are monitored not only at selected points but
also in the whole construction element.

The described method is of great practical significance and can be applied directly in the industry. It can be

utilized in supercritical as well as subcritical power plants. The method proposed can greatly enhance the
performance of the power units by reducing the duration of all the transient operations and extending their
longevity. The presented heating operation based on the optimum parameters is compared with the German boiler
regulation-Technische Regeln fu¨r Dampfkessel 301 (TRD) (Technische Regeln fu¨r Dampfkessel, 1986; 98–138).
Copyright r 2011 John Wiley & Sons, Ltd.

KEY WORDS

total stresses; steam boilers; heat transfer; heating optimization

Correspondence

*Dariusz Rzasa, Department of Thermal Power Engineering, Faculty of Mechanical Engineering, Cracow University of Technology, Al.

Jana Paw"a II 37, 31-864 Cracow, Poland.

y

E-mail: [e-mail]

Received 24 April 2010; Revised 20 October 2010; Accepted 27 December 2010

Copyright r 2011 John Wiley & Sons, Ltd.

Int. J. Energy Res. 2012; 36:

–

Published online 28 February 2011 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/er.182

703 709

5

703

1. INTRODUCTION

The start-up and shut-down processes of the power
block devices cause high stresses in the construction
elements. These processes should be carried out in such a
manner that the total stresses that originated from the
internal pressure and thermal loading do not exceed the
allowable limit. The optimum operating parameters, i.e.
the initial fluid temperature and rates of temperature
change, have a great impact on the stress level in the
entire construction element. These parameters can be
evaluated using the German boiler code—Technische
Regeln fu¨r Dampfkessel 301 (TRD) [1]. The TRD 301
procedure is based on the quasi-steady, one-dimensional
temperature distribution in the whole component.
However, the heating and cooling processes are transient
operations, and the quasi-steady state does not occur.
Furthermore, the complicated geometry of the compo-
nent causes difficulties in assessing the regions with the
highest stresses. Thus, the operations conducted, using
parameters estimated according to the TRD 301 code,
may account for the state, where the allowable stresses
are exceeded. Paper [2] shows the numerical method
based on the golden search method, which could be used
to find the optimum medium temperature so that the
maximum absolute thermal stresses in the whole
construction element would not exceed the allowable
stresses. Another papers [3,4] describe methods for
determining optimum medium temperature history,
based on the assumption that total stresses at selected
points do not exceed permitted stresses. The aim of this
paper is to present the method, which makes it possible
to find the optimum parameters, so that the total stresses
at any point of construction element during the start-up
processes are kept at an acceptable level. For that
purpose, the Levenberg–Marquardt (LM) method is
used [5,6]. The heating operation is presented for an
outlet header, mounted in a supercritical power block.

The outlet header is one of the most heavily loaded

elements of the power block. This component is
mounted in power units of 460 MW. The geometry is
presented in Figure 1.

The outlet header is designed for pressure p

w

5

29 MPa

and steam temperature T

w

5

5591C. The demanding

working environment cause high stresses in the elements.
The extreme working conditions of the outlet header
require special alloy steel to be employed in the con-
struction. The material must withstand the high working
temperature and pressure while operating. It is important
that the material properties are retained within the wide
temperature range, especially at the yield strength of R

e

.

The ferritic alloy steel, X10CrMoVNb9-1 (P91), is

widely used in the construction of power plants and
other sectors involving temperatures higher than
5001C. This steel has a ferritic structure and high yield
strength

R

e

5

334.5 MPa

at

the

temperature

T 5

5001C. The thermal and mechanical properties [7]

are presented in Figures 2 and 3.

In order to find the total maximum stress variation

during the heating operation, a three-dimensional stress
analysis, based on the finite element method (FEM), was
conducted. Since the outlet header geometry is symme-
trical, 1/2 of the construction element was modelled and
analysed. The construction element was divided into fi-
nite elements, as depicted in Figure 4.

The type of element used in the analysis is an eight-

node brick element. This kind of element guarantees
high quality results in a relatively short period of time.
The heat transfer coefficient on the heated surface was

Figure 1. Geometry of the outlet header in millimeter.

Figure 2. Mechanical properties of X10CrMoVNb9-1 (P91)

steel.

Method for determining the allowable medium temperature

P. Duda and D. Rzasa

704

Int. J. Energy Res. 2012; 36:703–709

2011 John Wiley & Sons, Ltd.

DOI: 10.1002/er

r

assumed to be a 5 2000 W m

2

K

1

. This value was

chosen for this construction element based on research
done [8,9]. The outer surface is perfectly isolated, so
there is no heat exchange between the component and
the surrounding environment.

2. MATHEMATICAL FORMULATION
OF THE OPTIMIZATION METHOD

During the heating process, fluid enters the inner space
of a construction element. It has an initial temperature
T

f

1

and then rises with a constant rate of temperature

change v

T

1

until the nominal working medium pressure

is reached. Next, the temperature of the medium
changes stepwise to a value T

f

2

and rises with a

constant rate of change of temperature v

T

2

. Due to the

high internal pressure and temperature gradients
across the structural element, high stress concentration
areas occur on the inner surface of the outlet header.
The objective is to choose the optimum parameters, i.e.
the optimum rates of temperature change v

T

1

, v

T

2

and

the optimum fluid temperature steps T

f

1

, T

f

2

, in such a

way that the heating process is conducted in the
shortest time and the allowable stresses s

a

are kept at

an acceptable level. The optimum parameters T

f

1

, v

T

1

and T

f

2

, v

T

2

are found when the following equation is

satisfied:

s

a

s

max

ðT

f

1

; v

T

1

; T

f

2

; v

T

2

; t

i

Þ ffi 0;

i ¼

1;

. . . ; m; ð1Þ

where s

max

denotes the highest absolute value of the

component stresses, and m denotes the number of time
points during the heating process. In other words,
minimizing the sum

SðxÞ ¼

X

m

i¼

1

½s

a

s

max

ðT

f

1

; v

T

1

; T

f

2

; v

T

2

; t

i

Þ

2

ffi 0;

i ¼

1;

. . . ; m

ð2Þ

allows the optimum parameters to be established.

Due to the nonlinear character of the considered

problem, the most appropriate method must be cho-
sen. For that purpose, the LM optimization algorithm
is used, which is an iterative technique that finds the
minimum of a function that is expressed as the sum of
the squares of the nonlinear functions. The updating of
the parameters at every kth iteration step is performed
based on the following rule

x

ðk11Þ

¼ x

ðkÞ

1

d

ðkÞ

:

ð3Þ

Figure 3. Thermal properties of X10CrMoVNb9-1 (P91) steel.

Figure 4. The outlet header divided into finite elements.

Method for determining the allowable medium temperature

P. Duda and D. Rzasa

705

Int. J. Energy Res. 2012; 36:703–709

2011 John Wiley & Sons, Ltd.

DOI: 10.1002/er

r

The unknown parameters of x are denoted as

x

1

¼ T

f

1

, x

2

¼ v

T

1

, x

3

¼ T

f

2

, x

4

¼ v

T

2

, where

d

ðkÞ

¼ ðH

ðkÞ

1

ldiagH

ðkÞ

Þ

1

ðJ

ðkÞ

Þ

T

½s

a

s

max

ðx

ðkÞ

Þ;

k ¼

0; 1;

. . .

ð4Þ

The H and J are called the Hessian and Jacobian

matrices, respectively. They can be expressed in the
following way

J

ðkÞ

¼

@s

max

ðxÞ

@x

T

x¼x

ðkÞ

¼

@s

1

@x

1

. . .

@s

1

@x

n

. . .

. . .

. . .

. . .

. . .

. . .

@s

m

@x

1

. . .

@s

m

@x

n

2
6

6

4

3
7

7

5

x¼x

ðkÞ

ð5Þ

and

H

ðkÞ

¼

@

2

s

max

ðxÞ

ð@x

T

Þ

2

x¼x

ðkÞ

¼

@

2

s

1

@x

1

@x

1

. . .

@

2

s

1

@x

1

@x

n

. . .

. . .

. . .

. . .

. . .

. . .

@

2

s

m

@x

n

@x

1

. . .

@

2

s

m

@x

n

@x

n

2
6

6

4

3
7

7

5

x¼x

ðkÞ

ð6Þ

The solution for the optimum parameters is

obtained if the assumed convergence criterion

x

ðk11Þ
i

x

ðkÞ
i

pE;

i ¼

1;

. . . ; n

ð7Þ

is fulfilled.

3. HEATING OPERATION BASED
ON THE GERMAN REGULATION
TRD 301

The German boiler code TRD 301 regulates the
heating and cooling processes of the power blocks.
The TRD 301 procedure allows the allowable rates of
temperature change in the fluid and the permitted
stresses during start-up and shut-down operations to
be estimated.

The start-up operating parameters calculated for

the

outlet

header

are:

v

T

1

¼ 7:1 K min

1

and

v

T

2

¼ 18:85 K min

1

, where v

T

1

and v

T

2

are the change

rates in the temperature at the beginning and the end of
each process, respectively. The highest permitted stress
value during the heating operation is s

a

5

126.7 MPa.

The temperature and pressure change during the heat-
ing operation is shown in Figure 5.

Consider a process where the working medium at a

temperature of 201C floods the inner space of the
outlet

header

that

has

a

uniform

temperature

T

0

5

201C. Next, the medium temperature changes

with the calculated rates of temperature change v

T

1

and

v

T

2

until working parameters of p

w

5

29 MPa and

T

w

5

5591C are reached. During this process, high

compressive stresses on the inner surface were
observed. They come from the temperature gradient
in the element wall and the internal pressure. The

recorded maximum compressive stress history during
that process is presented in Figure 6.

The maximum compressive stresses act in the z-di-

rection and reach their highest value at time t 5 3100 s.
Therefore, these are treated as the critical stresses for
the construction. The temperature distribution for time
t 5

3100 s is plotted in Figure 7.

The temperature gradient through the wall measured

at points P

1

and P

2

is over 601C. Figure 8 shows the

maximum compressive stresses during the start-up
process.

The maximum compressive stresses are located on

the inner surface near the opening edges and their
value reaches 222.4 MPa. The outer surface is sub-
jected to tensile stresses with a peak of 182.6 MPa. It
can be seen from the performed numerical analysis that
conducting the heating operation for the outlet header

Figure 5. Temperature and pressure history during the heating

operation based on TRD regulation.

Figure 6. Maximum compressive stress history during the start

up process according to TRD 301 regulations.

Method for determining the allowable medium temperature

P. Duda and D. Rzasa

706

Int. J. Energy Res. 2012; 36:703–709

2011 John Wiley & Sons, Ltd.

DOI: 10.1002/er

r

with parameters defined by the German boiler regu-
lations causes high stresses in the structure, much
higher than allowed.

4. APPLICATION OF THE PROPOSED
METHOD

In the previous chapter, the heating operation of the
outlet header, based on the German boiler regulation
TRD 301, was presented. It was shown that, during the
start-up process, the allowable stresses s

a

5

126.7

MPa are exceeded (Figure 6). In paper [10], the
numerical procedure based on the LM method was
presented, which allows the optimum medium tem-
perature history to be found, so that the maximum
thermal stresses in the whole construction element
would not exceed the allowable stresses. This section
presents a numerical method that is used to optimize
the heating operation in such a manner that the
maximum total stresses remain within the acceptable
limits and the operation is conducted in the shortest
time. For that purpose, four working parameters, T

f

1

,

v

T

1

, T

f

2

, v

T

2

, using the LM algorithm, were found,

where T

f

1

, T

f

2

are the step changes in temperature and

v

T

1

, v

T

2

denote the temperature change rates.

Consider a hot fluid that has an initial temperature

T

f

1

5

64.51C, flooding the inner space of the outlet

header at temperature T

0

5

201C. Subsequently, the

fluid is heated-up, with a constant rate of temperature
change of v

T

1

¼ 8:1 K min

1

. During this process, the

working medium pressure p

f

changes as a function of

temperature until time t

1

, when the working pressure

p

w

5

28.9 MPa is reached (Figure 9).

First, a finite element stress analysis for two known

optimum parameters, T

f

1

and v

T

1

, was performed. It

was proved that, during the heating operation, high
compressive thermal stresses are created on the inner
surface due to the temperature gradients in the wall.
However, the increasing working medium pressure
causes stresses that have the opposite sign to thermal
stresses. Thus, the stresses coming from internal
pressure reduce the magnitude of the thermal stresses.
As a result, the maximum compressive stresses
decrease with the increasing pressure, as shown in
Figure 10.

Since the working medium pressure does not depend

on the temperature after time t

1

and there is some

margin between the actual and permitted stress values,
the heating process can be conducted with a higher
temperature change rate. Therefore, two additional
parameters, T

f

2

5

4381C and v

T

2

¼ 11:4 K min

1

, were

introduced, where T

f

2

is a second step change of

temperature and v

T

2

denotes a second rate of

temperature change. The temperature and pressure
history for the four optimum parameters is presented
in Figure 11.

Figure 9. Temperature and pressure history for the two

optimum parameters.

Figure 7. Temperature distribution on the inner surface in 1C at

time t 5 3100 s.

Figure 8. Maximum compressive stress s

z

distribution in MPa

for t 5 3100 s.

Method for determining the allowable medium temperature

P. Duda and D. Rzasa

707

Int. J. Energy Res. 2012; 36:703–709

2011 John Wiley & Sons, Ltd.

DOI: 10.1002/er

r

Figure 12 presents the stress history during the start-

up process conducted with the four optimum para-
meters.

The component is heated up by the working medium

from inside, and the temperature distribution for time
t 5

2800 s is depicted in Figure 13.

The highest compressive stresses were recorded in

the z-direction at time t 5 2800 s. The stress con-
centration areas are located at the opening edges on
the inner surface, as plotted in Figure 14.

Maximum tensile stresses are located on the outer

surface with the highest value of 149.6 MPa.

5. CONCLUSIONS

The heating operation and subsequent stress analysis of
the outlet header were presented. It was proved that,

Figure 12. Maximum compressive stress history during the

start-up process according to the optimization method.

Figure 13. Temperature distribution on the inner surface in 1C

at time t 5 2800 s.

Figure 10. Maximum compressive stress history during heat-

ing operation based on two optimum parameters.

Figure 11. Temperature and pressure history for the four

optimum parameters.

Figure 14. Maximum compressive stress s

z

distribution on the

inner surface in MPa for t 5 2800 s.

Method for determining the allowable medium temperature

P. Duda and D. Rzasa

708

Int. J. Energy Res. 2012; 36:703–709

2011 John Wiley & Sons, Ltd.

DOI: 10.1002/er

r

during the heating operation, processed according to the
German boiler regulation TRD 301, high compressive
stresses are formed in the outlet header construction.
These stresses exceed the stress limit s

a

as specified by the

TRD 301 regulations by about 75%. Next, a new
developed numerical method based on the LM algorithm
was presented. The new method allows the optimum
working parameters for the heating operation to be
estimated based on the total stress limitation. It was shown
that the maximum compressive stresses were not exceeded.

The optimum working parameters can guarantee the

extended longevity of the components, which, ipso facto,
makes it possible for them to operate in a cost-effective
way. The new method is of great practical significance
and can be implemented in the industry, wherever the
heating and cooling processes take place. The techniques
used to estimate the optimum parameters are fast and
does not require a high financial outlay. The safety
regulations for power block devices are obeyed by con-
forming with the TRD 301 regulations.

NOMENCLATURE

H, J,

d

5

matrices

x

5

vector of searched parameters

k

5

thermal conductivity (W m

1

K

1

)

p

w

5

working pressure of medium (Mpa)

R

e

5

yield strength (Mpa)

S

5

sum of squares

t

5

time (s)

T

f

0

5

initial temperature of medium (1C)

T

w

5

working temperature of steam (1C)

v

T

5

rate of medium temperature change
(K min

1

)

Greek symbols

e

5

convergence criterion

s

a

5

allowable stresses (MPa)

s

x

, s

y

, s

z

5

total stresses in respective direction
(MPa)

ACKNOWLEDGEMENTS

Part of this work was conducted within the confines of
cooperation between TU¨V NORD EnSys Hannover

GmbH & Co. KG and Cracow University of
Technology.

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Method for determining the allowable medium temperature

P. Duda and D. Rzasa

709

Int. J. Energy Res. 2012; 36:703–709

2011 John Wiley & Sons, Ltd.

DOI: 10.1002/er

r