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Novel predictive tools for design of radiant andconvective sections

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Novel predictive tools for design of radiant and convective sectionsof direct fired heaters

Alireza Bahadori *, Hari B. VuthaluruDepartment of Chemical Engineering, Curtin University of Technology, GPO Box U1987, Perth, WA 6845, Australia

a r t i c l e i n f o

Article history:Received 14 July 2009Received in revised form 25 November 2009Accepted 25 November 2009Available online 21 December 2009

Keywords:CorrelationDirect fired heaterHeat fluxRadiant sectionConvection section

a b s t r a c t

Direct fired heaters are used considerably in the energy related industries and petroleum industries forheating crude oil in the petroleum refining and petrochemical sectors. The aim of the current study isto formulate simple-to-use correlations to design the radiant and convective sections of direct fired heat-ers. The developed tools are easier than currently available models and involves a fewer number ofparameters, requiring less complicated and shorter computations. Firstly, a simple correlation is devel-oped to provide an accurate and rapid prediction of the absorbed heat in the radiant section of a firedheater, expressed as a fraction of the total net heat liberation, in terms of the average heat flux to thetubes, the arrangement of the tubes (circumferential), and the air to fuel mass ratio. Secondly, anothersimple correlation is developed to approximate external heat transfer coefficients for 75, 100, and150 mm nominal pipe size (NPS) steel pipes arranged in staggered rows and surrounded by combustiongases. Finally, a simple correlation is presented to predict the gross thermal efficiency as a function ofpercent excess air and stack gas temperature. This study shows that the proposed method has a goodagreement with the available reliable data in the literature. The average absolute deviations betweenreported data and the proposed correlations are found to be around 1.5% demonstrating the excellentperformance of proposed predictive tool. The proposed simple-to-use method can be of significant prac-tical value for the engineers and scientists to have a quick check on the design of radiant and convectivesections of direct fired heater. In particular, mechanical and process engineers would find the proposedapproach to be user-friendly involving no complex expressions with transparent and easy to understandcalculations.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Operation strategies of energy intensive plants strongly affectthe production cost [1,2]. The data gathered around the processare not sufficient to analyze the plant behavior. Mathematicalmodels supply valuable information on the behavior of the plantand can be used to search for optimal operating conditions[1,3,4–6]. Knowledge of combined convective and radiative energytransfer in participating media is crucial for the determination ofheat fluxes on the walls of systems in numerous engineering appli-cations. Examples include boilers of power generating equipment,fossil fuel-fired industrial furnaces for materials processing, high-temperature heat recovery equipment, combustors and rocketengines, hypersonic propulsion, entry and re-entry vehicle protec-tion, and numerous others [7]. Gas to wall heat transfer in suchsystems results from coupled convection and radiation processeswhich cannot, in general, be calculated separately [7].

Direct fired heaters are used considerably in the chemical pro-cessing industries and oil and gas industries for heating crude oilin the petroleum refining and petrochemical sectors. In a typicalpetroleum refinery, there may be 25–75 direct fired heaters withdifferent configurations, and each heater may have different typesof burners [8]. Proper care and attention to these heaters can pro-long run lengths and increase reliability and safe operation. For aradiant burner, both the radiant power density and radiant effi-ciency are of practical importance. They are determined not onlyby the design of the burner/emitters but also by the combustionoperating conditions [9]. Heaters are usually designed for uniformheat distribution. The average radiant heat flux specified is definedas the quotient of total heat absorbed by the radiant tubes dividedby the total outside circumferential tube area inside the firebox,including any fittings inside the firebox. The rows of convectiontubes exposed to direct radiation shall be considered as being inthe radiant section and the maximum radiant heat absorption rateshall apply to these tubes, irrespective of whether extended sur-face elements are used or not [10]. The maximum radiant heat fluxdensity is defined as the maximum heat rate to any portion of anyradiant tube. Direct fired heaters vary in size from 0.15 MW small

0306-2619/$ - see front matter � 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.apenergy.2009.11.028

* Corresponding author. Tel.: +61 8 9266 1782; fax: +61 8 9266 2681.E-mail address: alireza.bahadori@postgrad.curtin.edu.au (A. Bahadori).

Applied Energy 87 (2010) 2194–2202

Contents lists available at ScienceDirect

Applied Energy

journal homepage: www.elsevier .com/ locate/apenergy

package regeneration gas heaters to 300 MW steam hydrocarbonreformer heaters. In the gas processing industry, the usual rangeis 0.3–6 MW [11]. There are two basic configurations: cylindricaland cabin. The simplest design is of vertical-cylindrical configura-tion with only radiant tubes. The net thermal efficiency (NTE) isabout 60% and the stack gas temperature is 650 �C or more. Theburner in the floor fires upward [11]. A stainless steel baffle slowsthe exit flow of the hot gases and reradiates heat back to the toppart of the tubes. There is a short stack that usually has no damper.The design is low cost and suited for low cost fuel. Adding a con-vection section improves the NTE to about 80%. The radiant sectionmay be either cylindrical or cabin, and the coil configuration eitherhelical or serpentine. These heaters cost more than the all-radianttype but they use less fuel for any given duty [11]. Fig. 1a and bshows a simple design of a vertical-cylindrical direct fired heaterwith radiant tubes and convection section [11]. A fired heatercan be considered as an enclosure containing gaseous heat source,heat sink and a refractory; in which heat is generated by the sourceand is transferred to the sink [12]. In comparison to convection andconduction heat transfer modes, radiation transfer is the dominantheat transfer mechanism in fired heaters. Heat transfer in a furnacestrongly depends on system geometry, nature of surface and gases,and the relative position of sinks and sources in the systems [12].

In the previous work [12], a mathematical model based on mul-ti-zone method was developed for simulating performance ofindustrial furnaces. Emphasis of their work is on the use ofthree-dimensional zone method to deal with radiative heat trans-fer for the furnaces, boilers and other fired heaters [13]. Accordingto the literature [14,15], because of the difficulties of direct-ex-change area (DEA) calculations and solving the matrices involvedin calculation of total exchange areas [16,17] the zone methodhad limited application in three-dimensional complex geometries.

In zone method, the enclosure is subdivided into a finite num-ber of isothermal volumes and surface area zones [15]. Energy bal-ance and other governing equations are then applied to theradiative exchange between any two zones [14,19] by employingpre-calculated ‘‘exchange areas” [18]. In the zone method, ex-change areas (direct and total) that are more general form of viewfactors are used. Finally, a set of simultaneous non-linear equationsare numerically solved to find unknown temperatures and heatfluxes [12].

The radiant tubes are placed above the bridge wall so that theyare, in effect, double fired. The radiant section or firebox should:

� Obtain complete combustion of the fuel with a reasonableamount of excess air, i.e., 10–15%.

� Contain the flame and avoid impingement on the tubes.� Distribute the radiant heat flux.� Cool the combustion gases to 800–1000 �C to protect the con-

vection section.

The proportions of the firebox are the key to good performance[11].

The total heat liberation consists of the lower heating value ofthe fuel and the sensible heat in combustion air, recirculated fluegas, and fuel and atomizing steam, all heat contents referred to adatum of 15 �C.

Nomenclature

A coefficientB coefficientC coefficientD coefficientA area, m2

Eg gross thermal efficiency percentF absorbed fraction of total heat liberation in the radiant

section of a direct fired heaterG air to fuel mass ratio, kg/kgh heat transfer coefficient, W/(m2 �C)H enthalpy, kJ/kgHHV higher or gross heating value, kJ/(standard m3)LHV lower heating value, kJ/(standard m3)m mass velocity, kg/sQ the allowable heat flux to the tubes, W/m2

P nominal pipe size in mmr ratio of flue gases to heat release, kg/(MW h)X excess air percent, X

Subscriptsc convectivef filmg gaso outsidep piper radiantLM log mean base

Fig. 1. Direct fired heater, vertical-cylindrical, helical coil with convection section(a) and cross section of radiant coil (b)[11].

A. Bahadori, H.B. Vuthaluru / Applied Energy 87 (2010) 2194–2202 2195

Lower heating value (LHV) of the fuel is known, or by stoichi-ometry. In this study the proposed predictive tool is for direct firedheaters with one row of 200 mm NPS pipes, spaced two pipe nom-inal sizes (NPS). Other designs require correction factors whichneeds to be multiplied by air to fuel mass ratio prior to the appli-cation of this new proposed predictive tool [11].

Generally the flame length should be 60% of the firebox lengthand the clearance between the flame and tubes at least 0.5 m.For small cylindrical heaters, the tube circle should be equal tothe length of the firebox. For small cabin heaters, the width, height,and tube length should be equal. For large heaters the height of acylindrical heater is twice the tube circle, and for cabins a good ra-tio of width to height to length is 1:2:4 [9].

1.1. Convection section

The purpose of the convection section is to transfer as muchheat as possible from the combustion gases leaving the radiant sec-tion. As always there is the trade-off between capital cost, i.e., add-ing more tubes, and operating cost, i.e., improved thermalefficiency. The construction is similar to that for the radiant sec-tion, a steel plate shell with internal castable or ceramic fiber insu-lation. The tubes are staggered, and the space between the sidewalland the tube is filled with ‘‘corbels” to prevent the flue gases frombypassing the end tubes [11]. The first two rows of the convectionsection are called shock tubes and they ‘‘see” the firebox flame. Thefirst row receives the full radiant heat flux and also some convec-tive heat transfer. It has the highest heat transfer flux in the heaterand is always bare tubes. The second shock row receives about onethird of the radiant flux as well as convective heat transfer from theflue gas. It is also bare tubes. If long radius return bends are used,the third row will receive radiant heat and it too should be baretubes [11].

2. Developing simple equations

The required data to develop the first predictive tool include thereliable data [10,11] for various absorbed heat fraction in the radi-ant section of a fired heater (F), air to fuel mass ratio (G) and theaverage heat flux to the tubes (Q).

For first predictive tool, various absorbed heat fractions in theradiant section of a fired heater (F) are predicted rapidly as a func-tion of air to fuel mass ratio (G) and the average heat flux to thetubes (Q) by proposing simple equations. The following methodol-ogy [20–24] has been applied to develop first simple predictivetool:

1. Correlate the absorbed heat fraction in the radiant section of afired heater (F) as a function of air to fuel mass ratio (G) for aaverage heat flux to the tubes (Q).

2. Repeat step 1 for other values of average heat flux to the tubes(Q).

3. Correlate corresponding polynomial coefficients, which areobtained in previous steps versus average heat flux to the tubes(Q), so we have a = f(Q), b = f(Q), c = f(Q), d = f(Q) (see Eqs. (2)–(5)).

The derived equations are applied to calculate new coefficientsfor Eq. (1) to predict absorbed heat fraction in the radiant section ofa fired heater (F). Table 1 shows the tuned coefficients for Eqs. (2)–(5) according to the data [10,11].

So, Eq. (1) represents the proposed governing equation in whichfour coefficients are used to correlate the absorbed heat fraction inthe radiant section of a fired heater (F) as a function of ratio of air

to fuel mass ratio (G) for various average heat fluxes to the tubes(Q) where the relevant coefficients have been reported in Table 1.

In brief, Eq. (1) provides a reliable estimate of the absorbed heatin the radiant section of a fired heater as a fraction of the total netheat liberation, in terms of the average heat flux to the tubes andthe air to fuel mass ratio

F ¼ aþ bGþ cG2 þ dG3 ð1Þ

where

a ¼ A1 þ B1Q þ C1Q 2 þ D1Q 3 ð2Þb ¼ A2 þ B2Q þ C2Q 2 þ D2Q 3 ð3Þc ¼ A3 þ B3Q þ C3Q 2 þ D3Q 3 ð4Þd ¼ A4 þ B4Q þ C4Q 2 þ D4Q 3 ð5Þ

In the above equations, ‘F’ and ‘G’ are the absorbed heat fraction inthe radiant section of a fired heater and the air to fuel mass ratio,respectively. ‘Q’ is the average heat flux to the tubes. The tunedcoefficients in Eqs. (2)–(5) are given in Table 1.

Eq. (1) is for fired heaters with one row of 200 mm NPS pipes,spaced two pipe nominal sizes (NPS). Correction factor for otherdesigns, to be multiplied by ‘C’ from Eq. (6) prior to applying Eq.(1).

The coefficients in Eqs. (2)–(5) are correlated as a function ofallowable heat flux to the tubes (Q) in (W/m2). The tuned coeffi-cients used in these equations are given in Table 1. These tunedcoefficients help to cover reported data in the air to fuel mass ratio(G) variation from 5 to 40 kg/kg. In order to consider the effect ofpipe size on absorbed fraction of total heat liberation in the radiantsection of a direct fired heater the following coefficient (C) (Eq. (6))is proposed as a function of pipe nominal size in meter:

C ¼ aþ bP þ cP2 þ hP3 ð6Þ

Table 2 shows the coefficients for Eq. (6). Tables 3 and 4 showcorrection factors to correct air to fuel mass ratio (G) as a functionof pipe spacing and number of pipes rows and pipe diameters andcorrection factors for ratio of air to fuel (see Tables 5–10).

Table 1Tuned coefficients used in Eqs. (2)–(5) for radiant section.

Variable symbol Coefficients

A1 1.7718493787B1 �1.001917635 � 10�4

C1 3.7534689295 � 10�9

D1 �4.191035072 � 10�14

A2 �1.366921187 � 10�1

B2 1.531156947 � 10�5

C2 �5.963858747 � 10�10

D2 6.684546867 � 10�15

A3 6.519753696 � 10�3

B3 �8.1321392297 � 10�7

C3 3.085147199 � 10�11

D3 �3.435592722 � 10�16

A4 �1.108513055 � 10�4

B4 1.383758239 � 10�8

C4 �5.172400596 � 10�13

D4 5.748038181 � 10�18

Table 2Tuned coefficients used in Eq. (6).

Variable symbol Coefficient

a 1.09266468b �.995014836c �0.995014836h �4.17804154

2196 A. Bahadori, H.B. Vuthaluru / Applied Energy 87 (2010) 2194–2202

So, Eq. (7) represents the proposed governing equation in whichfour coefficients are used to correlate gross thermal efficiency per-cent (Eg) for a gas as a function of stack gas temperature (T) and ex-cess air percent (X). This second predictive tool (Eqs. (7)–(11))determines the gross thermal efficiency from the excess air and

stack gas temperature. Especially for insulated heaters or furnaces,the combustion efficiency is close to the gross thermal efficiency.The difference is the heat lost through the walls to thesurroundings.

Eg ¼ aþ bTþ c

T2 þd

T3 ð7Þ

where

a ¼ Aa þ BaX þ CaX2 þ DaX3 ð8Þb ¼ Ab þ BbX þ CbX2 þ DbX3 ð9Þc ¼ Ac þ BcX þ CcX2 þ DcX3 ð10Þd ¼ Ad þ BdX þ CdX2 þ DdX3 ð11Þ

So, Eq. (12) predicts approximate external heat transfer coefficientsfor 75, 100, and 150 mm nominal pipe size (NPS) steel pipes ar-ranged in staggered rows and surrounded by combustion gases asa function of mass velocity and gas temperature. This third predic-tive tool (Eqs. (12)–(16)) determines flue gas convection-coeffi-cients for flow across staggered banks of bare tubes.

ln ðhÞ ¼ aþ bmþ c

m2 þd

m3 ð12Þ

where

Table 3Factors to correct air to fuel mass ratio (G) as a function of pipe spacing and number ofpipe rows.

Pipe spacing 2 � NPS 3 � NPSRows of pipes Multiply ‘G’ by

1 1 0.92 1.34 1.14

Table 4Correction factors for ratio of air to fuel.

Nominal pipe size (mm) Multiply ‘‘fuel mass ratio (G)” by

50 1.0575 1.04

100 1.02150 1.01200 1250 0.998

Table 5Tuned coefficients used in Eqs. (8)–(11) for gross thermal efficiency percent.

Symbol Temperature less than 400 �C Temperature more than 400 �C

Aa 3.5867271007 1.80332283009Ba 5.97137044 � 10�4 �1.199075376 � 10�1

Ca �1.041459411 � 10�4 2.116704420 � 10�3

Da 2.706436001 � 10�7 �1.940342135 � 10�5

Ab 8.0707270036 � 102 4.6693314389 � 103

Bb �3.800851813 2.440655075 � 102

Cb 1.3006679819 � 10�1 �4.6657851398Db �3.538013277 � 10�4 4.444289104 � 10�2

Ac �2.5316310679 � 105 �3.063354486 � 106

Bc 2.1062599069 � 103 �1.690959337 � 105

Cc �5.1739316869 � 101 3.40089217929 � 103

Dc 1.46145027479 � 10�1 �3.3787223665 � 101

Ad 2.73393876175 � 107 7.128880427 � 108

Bd �3.078059945 � 105 3.935986933 � 107

Cd 6.622921581 � 103 �8.2133524884 � 105

Dd �1.927246931 � 101 8.5237594268 � 103

Table 6Tuned coefficients used in Eqs. (13)–(16) for prediction of convection coefficient.

Symbol Pipe diameter, 89 mm Pipe diameter, 114 mm Pipe diameter, 168 mm

A1 4.918737132 4.955004725 4.597974856B1 �9.08329572 � 102 �8.475657696 � 102 �6.62358437 � 102

C1 3.75751009 � 105 2.42256423 � 105 1.95364904 � 105

D1 �5.814371094 � 107 �2.731986387 � 107 �1.99840303 � 107

A2 �1.72287207 �2.626696297 �1.856362745B2 �3.87317314 � 101 4.065789429 � 102 �4.454201835 � 102

C2 �1.257490884 � 105 2.747044931 � 104 2.64223037 � 105

D2 3.864705344 � 107 �1.704080417 � 107 �5.249216789 � 107

A3 5.7447120608 � 10�1 1.367664112 9.5280937416 � 10�1

B3 1.84636037 � 102 �2.472637134 � 102 1.465756727 � 102

C3 3.703569982 � 104 �6.031842028 � 104 �7.406550924 � 104

D3 �2.146592072 � 107 1.9544314875 � 107 1.82637445 � 107

A4 �3.742830645 � 10�2 �2.426865939 � 10�1 �2.243710488 � 10�1

B4 �9.504240999 � 101 1.662288813 � 101 3.6490362918 � 101

C4 8.9541624502 � 103 3.181288441 � 104 �2.4517973416 � 104

D4 2.8005463579 � 106 �7.15701890 � 106 2.77569628819 � 106

Table 7Coefficients for Eq. (17).

Fuel type Coefficient Value

Natural gas a 1280Natural gas b 12Fuel oils a 1320Fuel oils b 12

Table 8Coefficients for Eq. (18).

Flue gas type Coefficient Value

Flue gas LHV a �30.7302785Flue gas LHV b 1.19090080Flue gas HHV a 205.192307Flue gas HHV b 1.19465034

A. Bahadori, H.B. Vuthaluru / Applied Energy 87 (2010) 2194–2202 2197

a ¼ A1 þB1

Tþ C1

T2 þD1

T3 ð13Þ

b ¼ A2 þB2

Tþ C2

T2 þD2

T3 ð14Þ

c ¼ A3 þB3

Tþ C3

T2 þD3

T3 ð15Þ

d ¼ A4 þB4

Tþ C4

T2 þD4

T3 ð16Þ

Eq. (17) calculates ratio of flue gases to heat release as a function ofexcess air percent. Eq. (18) predicts the enthalpy of the exit gasfrom radiant section as a function of temperature.

r ¼ aþ bX ð17ÞH ¼ aþ bT ð18Þ

2.1. Proposed methodology to design radiant and convection sections

After developing the relevant simple predictive tools, the fol-lowing step-by-step methodology is recommended to design radi-ant and convection sections in direct fired heaters.

1. Estimate the ratio of flue gases to heat release using new pro-posed Eq. (17).

2. Calculate density of flue gas and ratio of air to fuel, kg/kg.3. Determine fraction of total heat liberation absorbed in radiant

section from new proposed simple correlation in this work(using Eqs. (1)–(5)).

Table 9The accuracy of proposed predictive tool for predicting approximate external heattransfer coefficients.

Massvelocity,kg/m2 s

Temperature,�C

Nominalpipesize(NPS),mm

Reportedexternalheattransfercoefficient[11]

Calculatedexternalheattransfercoefficient

Averageabsolutedeviationpercent

0.5 100 75 11 10.904 0.92 100 75 26 26.571 2.23 100 75 33 33.051 0.20.5 300 75 12.6 12.496 0.82 300 75 29.2 29.626 1.53 300 75 37 36.836 0.440.5 600 75 15 14.912 5.92 600 75 34 34.164 0.53 600 75 43 42.248 1.80.5 100 100 10 9.9215 0.81 100 100 16 15.965 2.23 300 100 34.2 33.669 1.552 200 100 26 26.0108 0.043 600 100 39 38.551 1.2.5 400 150 10.2 10.09 1.12.4 100 150 23 22.748 1.12 200 150 21 21.601 2.93 300 150 28.82 28.221 2.11 600 150 17 16.564 2.60.5 600 150 11 10.884 1.06

Average absolute deviation percent 1.6%

Table 10The accuracy of proposed predictive tool for predicting gross thermal efficiencypercent of gas with HHV = 37.3 kJ/m3.

Excessairpercent

Temperature,�C

Reported grossthermal efficiencypercent [11]

Calculated grossthermalefficiencypercent

Absolutedeviationpercent

0 20 88 88.16 0.220 50 87.5 87.13 0.4240 100 85 85 060 200 79 79.055 0.0780 300 72 71.496 0.7

100 400 63 63.18 0.3150 20 87.8 87.68 0.14200 50 85 85.306 0.36

0 400 75 74.932 0.120 500 68 67.978 0.0340 600 59 59.187 0.3260 700 48.5 48.787 0.680 800 38 37.386 1.62

100 900 23 24.361 5.92150 400 57 57.434 0.76200 600 28.5 31.866 11.8

Average absolute deviation percent 1.46%

12 14 16 18 20 22 24 26 28 30 32

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

Ratio of Air to Fuel (kg/kg)

Fra

ctio

n of

Tot

al H

eat L

iber

atio

n th

at is

Q/A=18000DataQ/A=24000DataQ/A=30000DataQ/A=36000DataQ/A=42000DataQ/A=48000Data

Abs

orbe

d in

Rad

iant

Sec

tion

Fig. 2. Prediction of absorbed fraction of total heat liberation in the radiant section of a direct fired heater as a function of air to fuel mass ratio, kg/kg and the allowable heatflux to the tubes (W/m2) from Eqs. (1)–(5) and comparison with some typical data [11].

2198 A. Bahadori, H.B. Vuthaluru / Applied Energy 87 (2010) 2194–2202

4. Calculate heat transfer (rates) and radiant heat transfer area.5. Determine the heat content rate of the combustion gases leav-

ing radiant section.6. Calculate heat transfer coefficient by new proposed simple cor-

relation for convection section (Eqs. (12)–(16)).7. Predict the required surface area.8. Calculate the length of heat transfer surface.

3. Results

Fig. 2 illustrates the results of proposed predictive tool for pre-dicting the absorbed fraction of total heat liberation in the radiantsection of a direct fired heater as a function of air to fuel mass ratio,kg/kg and the allowable heat flux to the tubes (W/m2), comparingwith some typical data [10,11]. As can be seen, the results of thenew proposed predictive tool are accurate and acceptable. It alsoshows the emissivity of combustion gases decreases at higher airto fuel mass ratio, and increases for lower allowable heat flux tothe tubes.

Fig. 3 shows the results of the proposed Eq. (6) to calculate thecorrecting coefficient ‘C’ of the allowable heat flux to the tubes as afunction pipes nominal size in meter. Figs. 4 and 5 show the accu-racy of proposed predictive tool to estimate the percent gross ther-mal efficiency as a function of stack gas temperature and excess airpercent in comparison with the reported data [10,11]. Thesegraphs show excellent agreement between proposed predictivetool and reliable data in the literature. Figs. 6 and 7 show externalheat transfer coefficients for 75, 100 mm nominal pipe sizes (NPS)for steel pipes arranged in staggered rows and surrounded by com-bustion gases as a function of mass velocity and gas temperature.These graphs also demonstrate the excellent performance of theproposed predictive tool.

4. Case study

Given below is an example [11] to demonstrate the applicationof the proposed predictive tool showing the easiness of handlingthese predictive tools for design purposes.

0.05 0.1 0.15 0.2 0.25 0.30.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

Nominal Pipe Size, m

Coe

ffici

ent

Fig. 3. Results of the proposed Eq. (6) to calculate the correcting coefficient ‘C’ of the allowable heat flux to the tubes as a function pipes nominal size (Eq. (6)).

250 300 350 400 450 500 550 600 650 70050

55

60

65

70

75

80

85

90

Stack Gas Temperature, K

Gro

ss T

herm

al E

ffici

ency

, Per

cent

Excess Air=0 PercentDataExcess Air=20 PercentDataExcess Air=40 PercentDataExcess Air=60 PercentDataExcess Air=80 PercentDataExcess Air=100 PercentDataExcess Air=150 PercentDataExcess Air=200 PercentData

Fig. 4. Gross thermal efficiency percent as a function of stack gas temperature and excess air percent for temperature less than 400 �C.

A. Bahadori, H.B. Vuthaluru / Applied Energy 87 (2010) 2194–2202 2199

0.5 1 1.5 2 2.5 310

15

20

25

30

35

40

45

Mass Velocity

Con

vect

ion

Coe

ffici

ent

T=100°CDataT=200°CDataT=300°CDataT=400°CDataT=500°CDataT=600°CData

Fig. 6. Prediction of convection heat transfer coefficient Wm2 �C

� �as a function of mass velocity kg

m2 S

� �and temperature for 89 mm OD steel pipe.

600 700 800 900 1000 1100 12000

10

20

30

40

50

60

70

80

Stack Gas Temperature, K

Gro

ss T

herm

al E

ffici

ency

, Per

cent

Excesa Air=0 PercentDataExcesa Air=20 PercentDataExcesa Air=40 PercentDataExcesa Air=60 PercentDataExcesa Air=80 PercentDataExcesa Air=100 PercentDataExcesa Air=150 PercentData

Fig. 5. Percent gross thermal efficiency as a function of stack gas temperature and excess air percent for temperatures more than 673 K.

0.5 1 1.5 2 2.5 35

10

15

20

25

30

35

40

Mass Velocity

Con

vect

ion

Coe

ffici

ent

T=100°CDataT=200°CDataT=300°CDataT=400°CDataT=500°CDataT=600°CData

Fig. 7. Prediction of convection heat transfer coefficient Wm2 �C

� �as a function of mass velocity kg

m2 s

� �and temperature for 114 mm OD (outside diameter).

2200 A. Bahadori, H.B. Vuthaluru / Applied Energy 87 (2010) 2194–2202

Problem statement: Estimate the radiant tube area for a3000 kW regeneration gas heater. To avoid overheating the tubes,a radiant flux of 30,000 W/m2 is specified. The design calls for100 mm NPS Sch 80 tubes on a 2400 mm tube circle. The fuel is0.61 relative density gas with LHV of 37 260 kJ/m3. Use 20% excessair. Fuel gas and combustion air are supplied at 15 �C. The heaterNTE is 80%. The tubes are arranged in one row at 200 mm spacing.

Design the convection section for the above 3000 kW regenera-tion gas heater. The heat loss is assumed to be 2% of the heat re-lease. Use six 100 mm NPS Sch 80 tubes on 200 mm centre-to-centre spacing with 2400 mm effective length in each row. Aftertwo rows of bare shock tubes use finned pipe, 118 fins/m, 32 mmhigh, 2.7 mm thick. Assume pipe wall temperatures of 90–240 �Cacross the finned part of the convection section and average valuesof 250 and 260 �C for the two shock row.

4.1. Solution

r¼1520 ðkg flue gasÞ=ðMW hÞ¼417 ðkg flue gasÞ=ð106 kJÞðfrom Eq: ð17ÞÞ

r�LHV¼417�37;260¼15:537kg flue gasm3 fuel gas

Mass of 1 m3 fuel gas¼1�0:61�2923:68

¼0:747 kg

Mass of combustion air¼15:537�0:747¼14:79 kg

G¼14:790:747

¼19:8kg air

kg fuel

Correction factor for 100 mm tube is 1.02 (from Eq. (6)).

Corrected G ¼ 1:02� 19:8 ¼ 20 ðkg air=kg fuelÞa ¼ 1:012639040591 ðfrom Eq: ð2ÞÞb ¼ �0:033609556267 ðfrom Eq: ð3ÞÞc ¼ 6:13560449361� 10�4 ðfrom Eq: ð4ÞÞd ¼ �6:0428565753� 10�6 ðfrom Eq: ð5ÞÞF ¼ 0:53 ðfrom Eq: ð1ÞÞ

Q ¼ UHT� RNTE

¼ 3000� 0:530:8

¼ 1988 kW

Radiant heat transfer area ¼ Qr

I¼ 1988� 1000

30;000¼ 66:9

The surface area of 100 mm NPS pipe is 0.359 square meter permeter.

Total tube length ¼ 66:9=0:359 ¼ 186:4 m

There are 37 vertical tubes in a cylindrical heater with a2400 mm diameter tube circle when the tubes are 200 mm cen-tre-to-centre:

Tube length ¼ 186:4=37 ¼ 5:04 m

Then Qtotal is calculated:

Q total ¼ Duty=GTE ¼ 3000=0:80 ¼ 3750 kWr ¼ 1500 kg flue gas=ðMW hÞ

Flow rate of flue gases = 3.75 (1500) = 5625 kg/h.Assuming the setting loss of 2% or 75 kW occurs in the radiant

section, the heat content rate of the combustion gases leaving radi-ant section can be estimated to be:

Qradiant exit ¼ 3750� 1988� 75 ¼ 1687 kW

¼ 6073:2� 103 kJ=h

The enthalpy of the exit gas from radiant section:

H ¼ 6073:2� 103=5625¼ 1080 kJ=kg

Tg ¼ 918 �C ðflue gas-LHVÞ ðfrom Eq: ð18ÞÞArea of gas flow ¼ ðNumber of tubesÞðLÞðSpacing-DÞ

¼ ð6Þð2:4Þð0:2� 0:114Þ ¼ 1:24 m2

Gg ¼ 5625=ð1:24� 3600Þ ¼ 1:23 kg=ðs m2Þ

First shock row: assume the average gas temperature is 885 �Cand tube wall temperature is 260 �C.

Tgmean ¼260þ 885

2¼ 573 �C

ho ¼ 21:6W

m2 �C

� �

A = 0.359 m2 per linear meter and 14.4 is linear meter per row.

Atube ¼ 14:4ð0:359Þ ¼ 5:17 m2

Q C ¼ hoAðDTÞ ¼ ð21:6Þð5:17Þð885� 260Þ ¼ 69;795 W

Flux ¼ Q=A ¼ 30;000 W=m2

Q r ¼ ðQ=AÞA ¼ 30;000ð5:17Þ ¼ 155:1 kWQ c þ Q r ¼ 69:795þ 155:1 ¼ 224:9 kWQ exitgases ¼ ð1687� 224:9Þ ¼ 1462:1 kW

Hexitgases ¼ 5263:6� 103=5625 ¼ 935:7 kJ=kgTgexit ¼ 820 �C ðFlue gas-LHVÞ ðfrom Eq: ð18ÞÞ

Second shock row is analogous except that the radiant heat fluxis one third of that for the first row, i.e., 10,000 W/m2.

Q r ¼ 10;000� 5:17 ¼ 51:7 kW

With ho ¼ 21 W=ðm2 CÞQ c ¼ ð21Þð5:17Þð885� 260Þ ¼ 67 kWQ c þ Q r ¼ 67þ 51:7 ¼ 111 kWQ exitgases ¼ 1462:1� 119 ¼ 1343 kW

Hexitgases ¼ 4864� 103=5625 ¼ 864:7 kJ=kgTgexitgases ¼ 762 �C ðfrom Eq: ð18ÞÞ

Finned rows : the combustion gas mass velocity increases be-cause of the increased cross sectional area of finned pipe.

Acs ¼ 114=1000þ ð118ð2:7Þð178� 144ÞÞ=106 ¼ 0:134 m2=

ðLinear meterÞ

Gg ¼562

14:4ð0:2� 0:134Þð3600Þ ¼ 1:64 kg=ðs m2Þ

Q f ¼ 3000� 1988� 224:9� 111 ¼ 676:1 kWQ exit ¼ 3750� 3000� 75 ¼ 675 kW

Hgexit ¼2:43� 106

5625¼ 432 kJ=kg

TgðexitÞ ¼ 395 �C ðfrom Eq: ð18ÞÞ

Assuming that HHV is 10% more than LHV, the gross heater effi-ciency is calculated by Eqs. (7)–(11):

a ¼ 3:559176613 ðfrom Eq: ð8ÞÞb ¼ 7:8025197� 102 ðfrom Eq: ð9ÞÞc ¼ �2:3056447� 105 ðfrom Eq: ð10ÞÞd ¼ 2:3678256� 107 ðfrom Eq: ð11ÞÞEg ¼ 73:161% ðfrom Eq: ð7ÞÞ

Pipe and gas temperatures are:

A. Bahadori, H.B. Vuthaluru / Applied Energy 87 (2010) 2194–2202 2201

TP1 ¼ 90 �C; TP2 ¼ 240 �C; Tpav ¼ 165 �CTg1 ¼ 762 �C; Tg2 ¼ 395 �C; Tgav ¼ 578:5 �CDTLM ¼ ½ð762� 240Þ � ð395� 90Þ�=ðln ð522=305ÞÞ ¼ 405 �C

T film ¼165þ 578:5

2¼ 372 �C

ho ¼ 25W

m2 �CQ f ¼ hoAoDTLM

Ao ¼676:1� 103

25� 405¼ 66:78 m2

This is the theoretically required surface area. The fin efficiencyis 87% and the external surface area of the finned pipe is 3.676 m2/linear m.

Lpipe ¼66:78

0:87� 3:676¼ 20:88 m

5. Conclusions

In this work, simple-to-use predictive tools, which are simplerthan current available models involving a large number of param-eters, requiring more complicated and longer computations, is for-mulated to design radiant and convective sections of direct firedheaters. Over the past decades, several methods have been devel-oped to design radiant and convective sections of direct fired heat-ers However, in practice, these approaches are not easy to use,since they require a detailed understanding of complex mathemat-ical formulations. According to the authors’ knowledge, there is nosimple-to-use predictive tool in the literature to design radiant andconvective sections of direct fired heaters. In view of this status,our efforts have been directed at formulating simple-to-use predic-tive tool that can help engineers to design radiant and convectivesections of direct fired heaters. The predictive tool proposed inthe present work is simple and unique expression which is non-existent in the literature. In addition, the proposed predictive toolsare smooth and well-behaved (i.e. smooth and non-oscillatory)equations which should allow for more accurate predictions.

Acknowledgements

The lead author acknowledges the Australian Department ofEducation, Science and Training for Endeavour International Post-graduate Research Scholarship (EIPRS), the Office of Research &Development at Curtin University of Technology, Perth, WesternAustralia for providing Curtin University Postgraduate ResearchScholarship and the State Government of Western Australia forproviding top-up scholarship through Western Australian Energy

Research Alliance (WA:ERA). Useful comments from three anony-mous reviewers and the editor are also acknowledged which ledto improvements in the original version of the paper.

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