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ORIGINAL
Heat transfer simulation in a helically coiled tube steam generator
Bazargan Hassanzadeh • Ali Keshavarz •
Masood Ebrahimi
Received: 18 December 2012 / Accepted: 26 July 2013
� Springer-Verlag Berlin Heidelberg 2013
Abstract A symmetric helically coiled tube steam gen-
erator that operates by methane has been simulated ana-
lytically and numerically. In the analytical method, the
furnace has been divided into five zones. The numerical
method computes the total heat absorbed in the furnace,
while the existing analytical methods compute only the
radiation heat transfer. In addition, according to the
numerical results, a correlation is proposed for the Nusselt
number in the furnace.
List of symbols
a Absorption coefficient (m-1)
CCHP Combine cooling, heating and power
CHP Combine heating and power
cp Heat specific capacity (kJ kg-1 K-1)
do Outside diameter of pipe (m)
HCTSG Helically coiled tube steam generator
LHVf Low heating value of fuel (kJ kg-1)
_m Rate of mass flow (kg s-1)
Nu Nusselt number
Pr Prandtl number
Qcon Convection heat transfer (W)
Qr radiation heat transfer (W)
Re Reynolds number
T Temperature (K)
Greek symbols
ew Wall emissivity
q Density (kg m-3)
r Stefan–Boltzmann constant
Subscripts
a Air
e Exit gas
f Fuel
g Gas
o Combustion-side
p Product
r Radiation
ref Reference
W Wall
1 Introduction
Design and analysis of steam-generators because of their
vast applications in industry is of great importance. Due to
the high flame temperature of the combustion products in
the gas-fired furnaces, the thermal radiation is the most
important. However, the convection heat transfer in the
furnace is not negligible and in order to have precise results
in the simulations it must be considered. In addition, the
gas side convective coefficient has the greatest impact on
the overall heat transfer coefficient [1, 2]; therefore, it is a
key parameter in the design of steam generators.
It is revealed that there are a few investigations on the
external heat transfer coefficients for helically coiled tubes.
Rahul [3] determined the gas-side heat transfer coefficient
for coiled tube surfaces in a cross-flow of air. The length of
the test section was 1.5 m and the velocity of air ranged
from 1 to 8 m/s. He developed a correlation based on the
B. Hassanzadeh � A. Keshavarz (&) � M. Ebrahimi
Mechanical Engineering Faculty, K. N. Toosi University
of Technology, Vanak Sq. Molla Sadra St., Tehran, Iran
e-mail: keshavarz@kntu.ac.ir
B. Hassanzadeh
e-mail: bh.alfa@yahoo.com
M. Ebrahimi
e-mail: ebrahimi_masood@yahoo.com
123
Heat Mass Transfer
DOI 10.1007/s00231-013-1215-y
range of Reynolds numbers and pitch to tube diameter
ratios used in the experiment. Salimpour [4] selected three
heat exchangers with different coil pitches as test section
for both parallel and counter flow configurations. He per-
formed 75 test runs, which the tube-side and shell-side heat
transfer coefficients were calculated. He proposed Empir-
ical correlations for the shell-side and tube-side. Ghorbani
[5] reported an experimental investigation of the mixed
convection heat transfer in a coil-in-shell heat exchanger
for different Reynolds and dimensionless coil pitch. The
experiments were performed for both laminar and turbulent
flow inside coil. Effects of coil pitch and tube diameters on
shell-side heat transfer coefficient of the heat exchanger
were studied. Zhao and Wang [6] investigated the flow and
heat transfer characteristics of synthesis gas in membrane
helical-coil heat exchanger under different operating
pressures, inlet velocities and pitches numerically.
According to the literature, the most of the researches
concern about the heat exchangers but the present study
investigate the convection heat transfer coefficient of the
gas side for a helically coiled tube steam generator
(HCTSG).
In the simulation of HCTSG a numerical simulation is
presented to compute the total heat flux from the com-
bustion process to the heating surfaces of the coils. In
addition, an analytical method is used to estimate the
radiation heat transfer in the furnace region. After com-
putation of the heat flux, the Nu number and convection
coefficient are presented as a function of Re number for the
gas-side, for the both sides of the coils. By using the cor-
relation proposed for the Nu number, an analytical model is
now available to compute the total heat transfer of HCTSG
by using the zone method.
2 Geometry of the HCTSG
The geometry of the HCTSG with coiled tubes is
depicted in Fig. 1. While the combustion takes place in
the furnace, the radiation-surface absorbs heat and this
phenomenon cools the combustion products to some
extent. The exiting combustion gases are channeled from
the furnace to the region where their energy can be
transferred to the other side of the coils. This helps to
use the total heating surface of the coils. The coil pitch
is equal to the external diameter of the tube. The
geometry of the HCTSG is designed to occupy as less
space as possible. Therefore, the proposed HCTSG is
suitable for applications such as combine heating and
power (CHP) and combine cooling, heating and power
(CCHP) units to be used in the residential buildings,
institutions and hospitals to provide steam, cooling,
heating and power simultaneously.
If the coil pitch is greater than the external diameter of
the tube, the convection side gases will penetrate to furnace
side and disturbs the combustion process. This penetration
makes the analytical simulation more complicated and
almost impossible.
Since the convection heat transfer of the combustion
gases is proportional to the coil pitch, therefore as the coil
pitch decreases the convection heat transfer of gases will
decrease accordingly [3] and as a result the heating surface
area will increase. Table 1 presents the characteristics of
the boiler geometry, and input air–fuel conditions.
3 Numerical simulation
In the present paper, the pre-mixed combustion is used for
the combustion simulation. In the fossil fuel furnaces,
different methods can be used to simulate the thermal
radiation.
Optical thickness (aL) is a good criterion for selecting
the suitable radiation model. L is the beam length and a is
the gas absorption coefficient. The DO and DTRM are
appropriate for the thin optical thickness aL \1. For a
cylindrical furnace, the beam length is equal to the cylinder
diameter. According to the Fig. 1, the cylinder diameter is
larger than the coil diameter, therefore L \1. Furthermore
Fig. 1 Geometry of the HCTSG
Table 1 Boiler geometry, input conditions and fluid properties
Fuel and air inlet zone, R (mm) 120–170
Pipe diameter, do (mm) 42
HCTSG length (mm) 1,800
HCTSG diameter, Db (mm) 1,000
Coil diameter, Dc (mm) 700
Coil length (mm) 1,600
Pitch of coil (mm) 42
Mass flow rate of reactants (kg/s) 0.5–0.9
Inlet temperature (K) 303
Fuel mass fraction (CH4) 0.0504
Air components mass fraction
O2 0.2213
N2 0.7283
LHVfuel (MJ/kg) 50
Heat Mass Transfer
123
the gas absorption coefficient is also smaller than 1,
therefore aL \1 and as a result, the only radiation models
that can be used for the HCTSG are DO and DTRM.
However, the DTRM is very time consuming and does
not consider the dispersion effect in the calculation of the
radiation heat transfer; therefore, this model is not used in
this study. The DO model solves a wide range of optical
thicknesses from the surface-to-surface radiation to the
radiation in the combustion problems.
For the reason of shortening, the governing equations
and the models used in the simulation of the combustion
and radiation are listed in Table 2.
3.1 Computational details
In the numerical simulation according to high velocity of
inlet air and fuel to inside of micro-boiler, the flow is
assumed to be turbulent and k–e is used for turbulence
modeling. For the simulation of combustion process the
premixed Species Transport model is used and DO model
is utilized for calculating the radiation heat transfer.
The heating surfaces are treated as a gray heat sink of
emissivity 0.8 and assumed to be completely water-cooled
at a temperature of 400 K [8]. Absorption and scattering
coefficients used in the radiation model are taken as 0.5 and
0.01 m-1 [9] (Fig. 2).
The geometry of the model and mesh used in the sim-
ulation are shown in Fig. 3. Also the results independency
on the number of elements of the model is investigated and
presented in Figs. 5 and 6.
4 Analytical simulation of the furnace of HCTSG
The main portion of radiation in the gaseous and liquid fuel
flames belongs to the radiation from the tri-atomic gases
such as CO2, H2O, SO2 and soot, but in the solid fuel flame,
ash and solid particles convey a large amount of radiation.
In the gaseous fuel flame, the radiation from the mono
atomic gases and soot in the temperature about 2,000 K is
negligible, hence it is not considered in the HCTSG pre-
sented in this study [1].
Table 2 Governing equations of the numerical simulation [9, 11–14]
Equation name Equations Descriptions, assumptions and conditions
Conservation oot
qYið Þ þ r � qv~Yið Þ ¼ �r � j~i þ Ri þ Si q, Yi, j~i and Ri are the density, mass fraction, diffusion flux and net rate
of production in chemical reaction of the ith species respectively, v~ is
the gas velocity and Si is the rate of creation by addition from the
dispersed phase
Mass diffusion in
turbulent flowsj~i ¼ � qDi;m þ lt
Sct
� �rYi
Di, m is the diffusion coefficient for the ith species in the mixture, lt is
the turbulent viscosity and Sct is the turbulent Schmidt number
Reaction rate �R ¼ CACRqek
min mf ;ma
S;
mp
1þ S
� �
CR ¼ 23:6leqk2
� �1=4
�R is the reaction rate, CR is the reaction rate constant,e and k are the
dissipation of energy and the turbulence kinetic energy respectively.
mf, ma and mp are mass fraction of fuel, air and product respectively, l
is the viscosity and S = 17.189 is stoichiometric value. CA = 1 is an
empirical constant. The eddy dissipation model calculates the heat
released in the combustion. The combustion gas is taken as a mixture
of oxygen, nitrogen, carbon dioxide, water vapor and fuel gas. The gas
temperature is derived from the enthalpy equation where the specific
heat is calculated as the weighted sum of the individual specific heat
of the mixture components. The gas density is evaluated from the
ideal gas equation of state. All computations are conducted by using a
finite volume discretization scheme. Figure (2) illustrates the process
of radiative heat transfer in every element
Radiation transfer r � I r~; s~ð Þs~ð Þ þ aþ rsð ÞI r~; s~ð Þ
¼ an2 rT4
pþ
rs
4p
Z4p
0
I r~; s~0
� �U s~; s~
0� �
dX0
I is the radiation intensity, r~ and s~ are the position vector and direction
vector respectively n is the refractive index, rs and s~0
are the
scattering coefficient and scattering direction vector respectively. U is
the phase function and X0
is the solid angle. Effect of ash and solid
particles are neglected
Radiation heat flux on
the wallqin ¼
RIins~� ndX Iin is the intensity income to the wall
The net radiation heat flux
from the surface
equation
qout ¼ 1� eð Þqin þ n2ewrT4w
Tw is the wall temperature and ew is the wall emissivity
Heat Mass Transfer
123
As it can be seen in the Fig. 4, to calculate the radiation heat
transfer, the furnace has been divided into 5 separate and equal
zones. The governing equations including the radiation, con-
vection and the temperature of the exiting gas are solved for
each region independently. The radiation heat transfer between
gas and the heating surface, and the exit temperature for each
zone are calculated. The exit temperature from the fifth zone is
the exit gas temperature of the furnace.
4.1 Governing equations
It is assumed that the furnace and flame are two flat plates
with infinite surfaces. Therefore, the total radiation heat
transfer between gas and furnace is given as below [1, 2]:
Qr ¼ Arasr T4g � T4
w
� �ð1Þ
In which as is a combination of gas and pipe absorption
coefficient and is given as follow [1]:
as ¼agaw
1� ð1� agÞð1� awÞð2Þ
where, aw can be determined according to the pipe surface
temperature [8]. Tg is the average temperature of the flame
and the exit gas in each zone. For the first zone, the inlet gas
temperature is the same as the flame temperature, hence [7]:
Tfl � Tref ¼_mf LHVf þ _macpaðTa � Tref Þ
_mgcpg
ð3Þ
In which, _mf , _ma and _mgare the rate of mass flow of fuel,
inlet air and combustion products respectively, Ta and
Tref = 298 K are the inlet air temperature and reference
temperature correspondingly. cpa and cpg are the specific
heat capacities of air and combustion products respectively.
4.2 Radiation mean beam length
The HCTSG geometry is complicated therefore; the radia-
tion from different directions travels different distances until
they reach the heating surface. To simplify the radiation
calculations an average thickness of the radiation gas is used
and it is called the mean beam length (L). This parameter is
determined by the following equation [1, 2, 7, 8, 10].
L ¼ 3:6V
Ar
ð4Þ
In which, V (m3) and Ar (m2) are the gas volume and
enclosure surface respectively.
4.3 Emission from the gaseous flames
The flame may be luminous or non-luminous. The flame of
tri-atomic gases is non-luminous. Soot makes the flame more
luminous therefore, near the flame of the heavy oils, it looks
brighter and the brightness decreases along the flame since
the soot concentration decreases along the flame, this cause
the flame to look non-luminous at the furnace exit. The
emission from the gaseous flame is determined as follow [1]:
ag ¼ 1� ekyrpL ð5Þ
In which, r is the summation of molar concentration of
CO2 and H2O. p is the total pressure of the gas components,
which is assumed to be 1 atm, and ky is the absorption
coefficient of three atomic gases. Hence [1]:
ky ¼7:8þ 16rH2o
3:16ffiffiffiffiffiffiffirpLp � 1
� �1� 0:37Tg
1,000
� �ð6Þ
4.4 Exit gas temperature from each zone
According to the Fig. 4, from the first law of thermody-
namics for first zone it can be written that [2]:
_mf LHVf þ _macpaTa ¼ Qr1 þ Qcon1 þ _mtcpe1Te1 ð7Þ
Then:
Te1 ¼_mf LHVf þ _macpaTa
� �� ½Qr1 þ Qcon1�
_mtcpe1
ð8Þ
Fig. 2 Process of radiation heat transfer in each element
Fig. 3 The elements and mesh
generated for numerical
simulation of HCTSG
Heat Mass Transfer
123
In which Te1 is the exit gas temperature from the first
zone and cpe1 is the specific heat capacity of the gases at
the exit of the first zone. Accordingly for the second to the
nth zone the exit gas temperature would be:
Ten¼ _mtcpen�1
Ten�1� ½Qrn
þ Qconn�
_mtcpen
ð9Þ
Since Qr is temperature dependant, therefore,
computation of Te needs a trial and error procedure. The
exit temperature of the fifth zone is the gas exit temperature
from the furnace. Computation of Te is of great importance
because the exhaust of furnace transfers heat to the back of
the coil via the convective mechanism.
In the analytical simulation, to calculate the radiation
heat transfer and exit gas temperature from the furnace, it is
necessary to simulate the convection heat transfer as well.
Since there is no data for calculation of the convection
coefficient, the results of the numerical simulation is used
to calculate the convection heat transfer in each zone.
4.5 Convection coefficient
The heat resistance of the working fluid and the tube wall
with respect to the hot combustion gases is negligible.
Therefore, in computation of the overall heat transfer
coefficient only the hot combustion gas is considered [2].
The Nu for hot gas side in a straight tube is calculated as
below [1, 2]
Nu ¼ 0:023Re0:8Pr0:33 ð10Þ
For a helical coiled tube the Nu can be written as below
[3]:
Nu ¼ aRebPrc p
do
� �d
ð11Þ
In which p is the coil pitch, and a, b, c, and d are
unknown constants. Since Prandtl is independent from the
tube geometry, by comparing Eq. 10–11 the following
equation can be written.
Nu � Pr�0:33 ¼ a0Reb
0 P
do
� �c0
ð12Þ
In this study, the P and do are equal, hence:
Nu � Pr�0:33 ¼ a0Reb
0ð13Þ
where the magnitudes of a0
and b0
are determined through
curve fitting according to the numerical simulation.
The convection coefficient (h) in the inside and outside
of the coils, which is calculated from the convection heat
transfer in these regions, is related to the Nu by the fol-
lowing equations:
Nu ¼ h � Dh
kð14Þ
In which k is the gas conduction coefficient, and Dh is
the hydraulic diameter and can be determined from the coil
diameter (Dc) and boiler diameter (Db) for the convection
heat transfer in the inside and outside of the coil as below:
Dh; inside ¼ DC; Dh; outside ¼ Db � DC ð15Þ
5 Results
In the numerical simulation, the pre-mixed model is used
and the DO model is utilized for the radiation heat transfer.
External surface of the coil tubes is opaque with emissivity
coefficient of 0.8 [1, 8]. It is assumed that the flow inside
the tube is a two-phase flow therefore the constant wall
temperature for the internal surface of the tube is consid-
ered. Since the heat flux absorbed along the tube by the
external surface is different, therefore the temperature of
external surface of tube is not constant. However due to the
high conductivity and thinness of the wall tube, the tube
resistance is ignored and a constant-temperature condition
of 400 K for the external wall surface is considered.
To make sure about the independency of the results from
the number of elements in the numerical model some
analyses are done and presented in Figs. 5 and 6. Figure 5
Fig. 4 Sub-division of the
analytical solution domain into
zones
Heat Mass Transfer
123
shows the variation of maximum of gas temperature
gradient versus element numbers for inlet flow rate of
0.5 kg/s. Maximum of temperature gradient occurs in the
contact area between the hot gases and coil surface. Due to
this high gradient in this region, the number of elements
should be increased in the regions in the vicinity of the
tubes. According to Fig. 5 it can be seen that for the ele-
ment numbers more than 70,000 the variation of maximum
of gas temperature gradient is negligible. Also Fig. 6 shows
that for the element numbers of more than 80,000 the
variation of the total heat transfer, and heat transfer of the
external side of coil is not considerable.
Figure 7 shows the variation of the radiation heat
absorbed by the coil in the furnace with respect to the inlet
mass flow rate. According to the results as the mass flow
rate increases the Qr increases as well. This is due to the
higher temperature and absorption coefficient of the com-
bustion gases. In Fig. 7, the results of numerical simulation
and analytical solution are compared. The results show that
the numerical simulation is well agreed with the analytical
results. Figure 8 presents the effect of inlet mass flow rate
on the exit gas temperature (Te) from the furnace. It shows
that by increasing the inlet mass flow rate, the Te increases
as well. According to the limitations about the geometrical
size of the HCTSG it is very hard to decrease the gas
temperature therefore special attention must be given to the
material selection in the HCTSG to avoid overheating and
melting in the boiler. In Fig. 7 and 8 the simulation results
are compared with the analytical solution, the results are in
good agreement.
However, attention must be paid that to avoid over-
heating of tubes, the maximum temperature of exit gas
from the furnace should not exceed 1,200 �C [1, 2, 8]. This
temperature is calculated about 1,400 �C (Fig. 8) which is
not practical. Therefore to avoid overheating, mass flow
rate of more than 0.6 kg/s is not recommended.
There are two important potentials of error in the ana-
lytical results; the Tg in the Eq. 1 is assumed as the average
of input and output temperature from each zone, while each
particle has its particular temperature. The absorption
coefficient of tri-atomic gases is a function of temperature,
and partial pressure that are different in different locations
of the HCTSG.
The constant temperature contours in the HCTSG for the
0.5, 0.7 and 0.9 kg/s of inlet mass flow rate are presented in
55000 60000 65000 70000 75000 80000 85000 90000 95000
0.5
1
1.5
2
2.5
3
3.5
Faces
Max
imum
Tem
pera
ture
Gra
dien
t
Fig. 5 Variation of maximum of gas temperature gradient versus
element numbers
54000 60000 66000 72000 78000 84000 90000 96000440000
450000
460000
470000
480000
490000
500000
Faces
Q(K
W)
54000 60000 66000 72000 78000 84000 90000
304000
312000
320000
328000
336000
344000
Faces
Q(K
W)
Fig. 6 Variation of total heat transfer (right hand side), and heat transfer of external side of coil (left hand side) versus element numbers
Fig. 7 Radiation heat absorption by the radiation-heating surface
Heat Mass Transfer
123
Fig. 9. It can be seen that the maximum temperature occurs
near the burner.
Figure 10 presents the HCTSG centerline temperature in
different inlet mass flow rates. Consistent with the results,
the maximum temperature occurs near the burner where the
combustion takes place, and along the centerline, the
temperature decreases due to the heat absorption by the
heating surface via the convection and the radiation
mechanisms.
According to the Fig. 11 in the region near the burner, a
small bump is observed that is probably due to the
combustion process that the gas temperature reaches its
maximum in this region. Additionally, the overheating and
melting phenomena are more possible to occur at the end of
coil where the heat flux absorption becomes maximum.
The convection coefficient in the outside and inside of
the coils is calculated and presented in Figs. 12 and 13
respectively. The results show that the effective Re number
in the outside of coil is larger than that in the inside.
Therefore, the convective coefficient in the outside of the
coil is larger than that in the inside. For the HCTSG under
study in this paper p/do = 1, therefore the Nu number
according to the Eq. 13 is just a function of Re number.
The curve fitting for the numerical simulation results pre-
sented in Figs. 14 and 15 proposes the following mean Nu
numbers in the furnace-side and outside of the coils in the
HCTSG.
Nuoutside ¼ 0:752Pr0:33
Re0:555
37; 000 \ Re\ 65; 000 Pr ¼ 0:52 p=do ¼ 1ð16Þ
Fig. 8 Furnace exit gas temperature
Fig. 9 Constant temperature contours in HCTSG for different inlet
mass flow rate
Fig. 10 The variation of temperature along the furnace centerline
Fig. 11 The total heat flux absorbed by the coil tube surface along
the height of coil tube for different inlet mass flow rate
Heat Mass Transfer
123
Nuinside ¼ 0:0295Pr0:33
Re0:811
98; 300\Re\177; 000 Pr ¼ 0:55 p=do ¼ 1ð17Þ
According to the numerical results, the convection heat
transfer in the furnace side is about 25–30 % of the total
heat transfer in this region; therefore, in the calculation of
the total heating surface it plays an important role.
According to the Eqs. 16 and 17, whereas the mean gas
temperature in the inside and outside of coils is different
but the Pr number is remained constant approximately.
Therefore, it can be concluded that the changes of Pr for
the ideal gases is negligible. Also, the mean Re number in
the inside of the coil is larger than that in the outside. This
is due to the higher mean velocity of the flame, which is
assumed as the gas velocity in the furnace side.
According to the results, Eq. 16 can be used for calcu-
lation of convection heat transfer in the furnace in the
analytical zone method. In addition the Eq. 17 can be used
for designing helically coiled tube heat exchangers.
6 Conclusion
In the present study, a HCTSG has been modeled numer-
ically and analytically. The numerical simulation includes
the combustion, radiation and convection heat transfer
modeling. The fuel to be combusted in the HCTSG is CH4
and the pre-mixed model is used for the combustion
modeling. The radiation modeling is done by the use of DO
model. In the analytical simulation, the HCTSG is divided
into five zones. In each zone, the radiation heat absorbed by
the heating surface and the exit gas temperature are cal-
culated. The results of numerical and analytical simulation
are compared with each other. The results are in good
agreement and the difference is acceptable.
Finally, according to the importance of convection
coefficient in calculation of the heating surface area, an
equation resulting from the numerical simulation is pre-
sented for the gas-side Nusselt number.
100000 120000 140000 160000 180000
16
18
20
22
24
26
28
30
Re
h fs(
W/m
2 K)
Fig. 12 Convection coefficient with respect to the Re number in the
inside of coil
36000 40000 44000 48000 52000 56000 60000 6400028
30
32
34
36
38
40
42
44
Re
h cs(
W/m
2 K)
Fig. 13 Convection coefficient with respect to the Re number in the
outside of coil
35000 40000 45000 50000 55000 60000 65000
242
264
286
308
330
352
Re
Nu.
Pr-0
.33
Fig. 14 Nu number with respect to the Re number in the outside of
coil
100000 120000 140000 160000 180000
315
360
405
450
495
540
Re
Nu.
Pr-0
.33
Fig. 15 Nu number with respect to the Re number in the inside of
coil
Heat Mass Transfer
123
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Heat Mass Transfer
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