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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