1. Introduction

Fired process heaters are furnaces in which a process fluid flowing inside tubes located in the furnace is heated by gases produced by the combustion of a liquid or gaseous fuel. Such heaters are widely used in oil refineries and petrochemical plants for heating purposes. They are of two basic types: vertical cylindrical or box-type heaters ^{[1, 2, 3]}. For the most part, tubular heaters consist of two main sections: a radiant section, variously called a combustion chamber or firebox, in which heat is transferred mainly by radiation, and a convection section followed by the stack. Roughly-speaking, about 45-55% of the total heat release in the furnace is transferred to the process fluid in the radiant section, leaving about 25-45% of the total heat release to be either transferred to the process fluid in the convection section or carried by the flue gases through the stack and is lost ^[4].

2. Heat Transfer Mechanisms in Fired Heaters

In the usual practice, the process fluid is first heated in the convection section preheat coil which is followed by further heating in the radiant section. In both sections heat is transferred by both mechanisms of heat transfer, viz. radiation and convection, where radiation is the dominant mode of heat transfer in the radiant section and convection predominates in the convection section as the average temperature in this section is much lower. The heat-absorbing surface in both sections is the outside wall of the tubes mounted inside the heater.

Total heat transfer to the process fluid can be estimated using the following equation:

(1)

The radiant heat transfer follows the relationship:

(2)

and convective heat transfer follows the relationship:

(3)

A number of key variables serve as a basis for the determination of heater performance. These include ^[5]:

1. Inlet and outlet process fluid temperatures.

2. Effective flue gas and flame temperatures.

3. Tube skin or tube wall temperature.

4. Heat transfer rates to the process fluid.

5. Flue gas composition.

6. Process fluid flow rate.

7. Fuel flow rate.

8. Process fluid pressure drop.

9. Pressure profile in the heater and stack.

Of these variables, the effective gas and flame temperatures are most important for a comprehensive heat transfer analysis of the heater radiant section. Furthermore, efficiency calculations require prior calculation of these two temperatures ^[6].

3. Effective Gas Temperature (T_g)

The effective gas temperature is the temperature controlling radiant transfer in the heater radiant section. For a "well mixed" radiant section this temperature is assumed to be equal to the bridgewall temperature, i.e. the exit temperature of the flue gases leaving the radiant section. For most applications, this is an acceptable assumption with the notable exception of high temperature heaters with tall narrow fireboxes and wall firing where the effective gas temperature may be 95 to 150°C higher than the bridgewall temperature. In this and other cases where the two temperatures differ widely and an adjustment may be necessary, the use of a more accurate gas temperature may have to be considered or the radiant section may have to be divided into zones for the energy balance calculations ^[7].

Furthermore, complete flue gas mixing in the firebox is normally assumed in most methods used for the estimation of the effective gas and other radiant section temperatures, including the widely-used Lobo-Evans method ^[8]. This ignores in effect the existence of longitudinal and transverse temperature gradients. Cross, on the other hand, evaluated the temperature gradients in fired heaters in order to predict the bridgewall temperature ^[9].

4. Derivation of Effective Gas Temperature Equation

There are two primary sources of heat input to the radiant section, the combustion heat of the fuel Q_rls, and the sensible heat of the combustion air Q_air, fuel Q_fuel and the fuel atomization fluid (for liquid fuel when applicable). Of this heat input some heat is absorbed in the radiant section by the radiant Q_R and shield tubes Q_shld, while the remaining heat is either lost through the casing Q_losses or carried by the flue gas to the convection section Q_{flue gases}. By setting up a heat balance equation for fuel gas the flue gas temperature can then be calculated as follows ^[10]:

(4)

Where:

(5)

(6)

(7)

(8)

Q_r is the radiant heat transfer

(9)

and Q _conv is the convective heat transfer in the radiant section.

(10)

(11)

(12)

(13)

Substitution of the appropriate terms in Equation. 4:

(14)

The average tube wall temperature is given by ^[11] :

(15)

The Newton-Raphson method ^[12] was used to solve the heat balance equation and determine the effective gas temperature, for which two Matlab programmes were written (Appendices 1 and 3).

5. Flame Temperature (t_f)

Flame temperature is the temperature attained by the combustion of a fuel. This temperature depends essentially on the calorific value of the fuel. A theoretical or ideal flame temperature may be calculated assuming complete combustion of the fuel and perfect mixing. But even when complete combustion is assumed, the actual flame temperature would always be lower than the theoretical temperature. There are several reasons for this, chiefly dissociation of the combustion products at higher temperatures and heat loss. Up to a flame temperature of about 1370^◦C, the burned mixture generally includes such ordinary gases as CO₂, N₂, SO₂, H₂O and residual O₂ (from excess air). At higher temperatures, however, CO₂ appreciably dissociates to CO and O₂; H₂O to O₂ and OH^-; O₂ to O^-2; H₂ to H⁺; N₂ to N^-3; and NO (produced by N₂ and O₂) to N^-3 and O^-2. These dissociation reactions absorb an enormous amount of energy, substantially lowering the flame temperature ^{[13, 14]}. Further lowering of the flame temperature is also caused by radiation and conduction to the walls of the combustion chamber. Highly turbulent flames usually suffer an appreciable heat loss.

Some work has been done on the calculation of flame temperature, including work by Stehlik and others who studied furnace combustion and drew furnace temperature and enthalpy profiles ^[4]. Vancini wrote a programme in assembly language for the calculation of the average flame temperature, taking into account dissociation at higher temperatures ^[13].

6. Derivation of Flame Temperature Equation

In this paper, a simple heat balance serves as the basis for calculating the flame temperature. The increase in enthalpy between the unburned and burned mixtures is assumed to be equal to the heat produced by the combustion. When the fuel is fired, the heat liberated raises the temperature of the combustion products from t₁ to t₂ so that the following relationship is satisfied:

(16)

Where:

Q_combustion = Heat of combustion of fuel.

W_i = Mass of a flue gas component.

Cp_i = Molar heat of a flue gas component.

t₁ and t₂ = Initial and final temperatures.

The use of Equation (16) allows the calculation of the flame temperature t by iteration using a programmable calculator. The variation of Cp_i with temperature can be approximated by a polynomial, having the obvious advantage of being integrated easily. Using a third-degree polynomial, Cp_i can be written as:

(17)

Where, a_i, b_i, c_i and d_i are constants dependent on the nature of the gas. Assuming t₁ to be negligible (= 0), Equation (16) thus becomes:

(18)

Integrating:

(19)

It is customary to call the parenthetic term in Equation (19) the mean molar heat:

(20)

By taking mean molar heats instead of true molar heats, the integration of Equation (16) may be dispensed with. The molar heats at constant pressure for air and flue gases are given in Table 1.

(21)

Equation (21) allows the calculation of the theoretical flame temperature, t₂, by iteration using a programmable calculator. In order to compensate for the factors that tend to lower the theoretical flame temperature, the heat of combustion is usually multiplied by an empirical coefficient. The values normally used for this coefficient are only estimates; this is why the temperature calculated with any method can only approximate actual values. For an accurate calculation of the actual flame temperature, account must be taken of heat losses through the casing by setting up heat balance equation for fuel gas as follows:

(22)

Where:

(23)

(24)

The Newton-Raphson method ^[11] was used to solve the heat balance equation and determine the actual flame temperature, for which two Matlab programmes were written (Appendices 2 and 3).

To illustrate the use of the programme, an example is worked out for an actual crude oil heater used in an atmospheric topping unit at the Homs Oil Refinery (Cabin 43-5-16/21 N). In this example, fuel gas is fired with 25% excess air. Ambient temperature = 15°C, exit gas temperature = 400°C. Table 2 gives the furnace characteristics for the radiant section and fuel and combustion air.

The effective gas temperature equation, derived using programme (1), has the following form:

(25)

The first derivative of the effective gas temperature equation is:

(26)

where B, C and D are constants dependent on the type of fuel, percentage of excess air, operating conditions and geometrical characteristics of the fired heater. These constants can then be estimated using Programme 1 as follows (Table 3):

(27)

(28)

These equations were solved by the Newton-Raphson method in programme (3) to give an effective gas temperature in the fire box equal to 1278K.

The flame temperature equation, derived using programme (2), has the following form:

(29)

The first derivative of the flame temperature equation is:

(30)

Where a, b, c, d and e are constants estimated dependent on the type of fuel and its gross calorific value, the percentage of excess air and the operating conditions of the fired heater. These constants can then be estimated using Programme 2 as follows (Table 4):

(31)

(32)

These equations were solved by the Newton-Raphson method using programme (3) to give the actual flame temperature of 2128K.

7. Conclusion

Using Matlab programming and the Newton-Raphson method, it was possible to calculate simply and accurately both the effective gas temperature in a fired heater and the actual flame temperature. The calculation was based on heat transfer analysis of the fired heater taking into account heat absorption and losses in the radiant section of the heater. This calculation can be an important tool, not only in the operation and daily running of fired heaters, but also and more importantly for their design.

Nomenclature

AHeat exchange surface area (m²).

A_cpCold plane area of tubes bank in radiation section (m²)

A_{cp shld}Cold plane area of shield tubes bank (m²).

A_tArea of tubes bank in Radiation section (m²).

C_{P air}Molar heat of combustion air (kJ/kmol.K).

C_PfuelSpecific heat of fuel (kJ/kg.deg).

C_{Pflue gas}Average specific heat of flue gases flowing to a bank of bare tubes (kJ/kg.K).

C_piMolar heat of a flue gas component (kJ/kmol.K).

D_i, D_oInside and outside diameters of tube (mm).

eTube thickness (e = R_o-R_i) (mm)

FExchange factor = 0.97

GCV Gross calorific value of fuel (kJ/h).

h_convFilm convection heat transfer coefficient (kJ/m².K.h).

L_tubeEffective tube length (m)

m_airFlow rate of combustion air (kg/h).

m_fuelFlow rate of fuel (kg/h).

m_{flue gas}Flow rate of flue gas (kg/h).

NCVNett calorific value of fuel (kJ/h).

n _RNumber of tubes in radiation section

N _{tube (shld)}Number of shield tubes.

Q_airSensible heat of combustion air (kJ/h).

Q_CTotal heat transfer (kJ/h).

Q _combustion Combustion heat of fuel (kJ/h)

Q_conv.Convective heat transfer in the radiant section(kJ/h).

Q_fuel Sensible heat of fuel (kJ/h).

Q_{flue gas}Sensible heat of gas leaving the radiant section(kJ/h)

Q_lossesAssumed radiation heat loss (kJ/h)

Q_RTotal heat transferred to radiant tubes (heat absorbed by radiant tubes) (kJ/h).

Q_rRadiant heat transfer (kJ/h).

Q_rlsHeat release by burners (kJ/h).

Q_shldRadiant heat to shield tubes (kJ/h).

R_i, R _oInside and outside radii of tube (mm).

S_i, S_oInside and outside heat surface areas of tube (m²).

S_tubeTube spacing (m).

t₁Temperature of fuel and air (°C)

t_fFlame temperature (°C)

T_fFlame temperature (K).

T_gEffective gas temperature in firebox (K).

T_inInlet process fluid temperature (K)

T_outOutlet process fluid temperature ( K)

T_wAverage tube-wall temperature (K).

W_i Mass of flue gas component (kmol/h).

Greek symbols

αRelative effectiveness factor of the tubes bank.

σStefan-Boltzman constant = 2.041×10^-7 kJ/h.m².K⁴.

References

[1]	P. Wuithier (ed), Raffinage et génie chimique, L'institut français du pétrole, Paris, 1972, Tome 1.
	In article
[2]	Nelson, W. L., Petroleum Refinery Engineering, 4th ed. McGraw-Hill, New York, 1958.
	In article
[3]	Al-Haj Ibrahim, H., Fired process heaters, Ch. 16, pp. 327-364, in: Matlab, Modelling, programming and simulations, Ed. E. P. Leite, Sciyo, 2010.
	In article
[4]	Stehlik, P., et al., Furnace integration into process justified by detailed calculation using a simple mathematical model, Chemical Engineering and Processing, 34, pp. 9-23, 1995.
	In article		CrossRef
[5]	Fired Heaters: A Guide to Performance Evaluation, Prepared by Equipment Testing Procedures Committee, AIChI Equipment Testing Procedure, 1989.
	In article
[6]	Al-Haj Ibrahim, H., Al-Qassimi, M.; Matlab program computes thermal efficiency of fired heater, Periodica Polytechnica, Chemical Engineering, Vol. 52, No. 2, pp. 61-69, 2008.
	In article
[7]	Effective gas temperature in firebox, Available at: www.firedheater.com.
	In article
[8]	Lobo, W. E, Evans, J. E.; Heat Transfer in Radiant Section of Petroleum Heaters, Trans. Am. Inst. Chem. Engrs. 35, pp. 748-778, 1939.
	In article
[9]	Cross, A., Evaluate Temperature Gradients in Fired Heaters, Chemical Engineering Progress, Vol. 98, No.6, pp. 42-46, 2002.
	In article
[10]	Heat Balance in the Radiant Section, Available at: www.firedheater.com.
	In article
[11]	Walas, S. M., Fired heaters, Chemical Process Equipment, Selection and Design, Butterworth-Heinmann, 1990.
	In article
[12]	Chapra Steven C., Applied Numerical Methods with MATLAB for Engineers and Scientists, 1st edition, McGraw-Hill Companies, Inc, 2005.
	In article
[13]	Vacini, C. A., Program calculates flame temperature, Chemical Engineering, pp.133-136, March 22, 1982.
	In article
[14]	Flame Temperature Calculator, Available at: www.firedheater.com.
	In article
[15]	Perry, R. H., Green, D. W.; Perry's Chemical Engineers' Handbook, 8th Edition, McGraw-Hill Publishing, 2008.
	In article

Appendix 1

% Program for determination of effective gas temperature.

% Qin=Qrls+Qair+Qfuel

% Input :

ti=input(' Inlet temperature of process fluid (C) =');

Tin=ti+273;

to=input(' Outlet temperature of process fluid (C)= ');

Tout=to+273;

ts=input('Stack Temperature (C)=');

mfuel=input(' Flow rate of fuel (kmol/h)=');

mair=input(' Flow rate of combustion air (kmol/h)=');

mflue=input(' Flow rate of flue gases (kmol/h)=');

N=input (' Number of tubes in radiation section=');

Nshld=input(' Number of shield tube=');

L=input (' Effective tube length(m)=');

Do=input(' External diameter of tube in convection section(m)=');

C=input(' Center-to-Center distance of tube spacing(m)=');

NCV=input('Net Calorific Value of fuel(kJ/kmol)=');

Cpfuel=input(' Molar heat of fuel(kJ/kmol.deg.)=');

% Consrtant :

% Stefan-Boltzman Constant(kJ/h.m^2.K^4)

Sigma=2.041*10^(-7);

F=0.97; % Exchange factor

alpha=0.835; % Relative effectiveness factor of the tubes bank

% Heat input to the radiant section

% Combustion heat of fuel

Qrls=mfuel*NCV;

% Q=mair*Cpair*(tair-tdatum)

tair=25;

tdatum=15;

% Molar heat of air

Cpair=33.915+1.214*10^(-3)*(tair+tdatum)/2;

% Sensible heat of air

Qair=mair*Cpair*(tair-15);

% Qfuel=mfuel*Cpfuel*(tfuel-tdatum)

tfuel=25;

% Sensible heat of fuel

Qfuel=mfuel*Cpfuel*(tfuel-tdatum);

Qin=Qrls+Qair+Qfuel;

% Heat taken out of radiant section

% Qout=QR+Qshld+Qlosses+Qflue

% Heat absorbed by radiant tubes

% QR=Qr+Qconv

% Radiant heat transfer

% Qr=sigma+(alpha*Acp)*F*(Tg^4-Tw^4)

% Tw = Average tube wall temperature in degrees Kelvin

% Tg = Effective gas temperature in degrees Kelvin

Tw=100+0.5*(Tin+Tout);

% Cold plane area of tube bank

Acp=N*C*L;

% Qconv=hconv*At*(Tg-Tw)

At=N*pi*Do*L; % Area of tubes in bank

hconv=30.66; % Film convective heat transfer coefficient; (kJ/h.m2.c)

% Radiant heat to shield tubes

% Qshld=Nshld*sigma*(alpha*Acp)shld*F*(Tg^4-Tw^4)

% alpha=1, for the shield tubes can be taken to be equal to one.

Acpshld=Nshld*C*L;

% Qshld=Nshld*sigma*(1*Acpshld)*F*(Tg^4-Tw^4)

% Heat losses through setting

Qlosses=0.05*Qrls;

% Qflue=mflue*Cpflue*(Tg-Tdatum)

Tdatum=tdatum+273;

% Molar heat of flue gases

Ts=ts+273;

Cpflue=29.98+3.1157*10^(-3)*(Ts+Tdatum)/2;

% Sensible heat of flue gases

% Qflue=mflue*Cpflue*(Tg-Tdatium)

% Output:

A=Qin-Qlosses;

B=A+Sigma*F*(alpha*Acp+Acpshld)*Tw^4+hconv*At*Tw+mflue*Cpflue*Tdatum;

C=Sigma*F*(alpha*Acp+Acpshld);

D=hconv*At+mflue*Cpflue;

syms Tg

% Equation of effective temperature.

y=C*Tg^4+D*Tg-B;

disp('Equation for effective gas temperature')

func=y

% Finding of first derivative of effective gas temperature equation.

disp('first derivative of effective gas temperature equation')

dfunc=diff(func)

Appendix 2

% Programme for determination of flame temperature equation.

% Input :

% Flow rate of fuel (kmol/h)

% Flow rate of flue gases (kmol/h)

% Molar Composition of flue gases: XCO2, XH2O, XN2,

% XO2 and XSO2

% Molar heats of flue gases (kJ/kmol.K)

% Percentage of heat losses

% Output:

% Flame temperature (K)

Mfuel=input('Flow rate of fuel(kmol/h)=:');

Mfluegas=input('Flow rate of flue gases(kmol/h)=:');

X=input('percentage of heat losses=:');

GCV=input('Gross calorific value of fuel (kJ/kmol)=:');

XC=input('Molar fraction of CO2=:');

XH=input('Molar fraction of H2O=:');

XN=input('Molar fraction of N2=:');

XO=input('Molar fraction of O2=:');

XS=input('Molar fraction of SO2=:');

td=15; % Datum temperature (C)

% Molar heats at constant pressure for flue gases

% CpCO2=43.2936+0.01147*T-818558.5*T^(-2)

% CpH2O=34.42+6.281*10^(-4)*T+5.611*10^(-6)*T^2

% CpN2=27.2155+4.187*10^(-3)*T

% CpO2=34.63+1.0802*10^(-3)*T-785900*T^(-2)

% CpSO2=32.24+0.0222*T-3.475*10^(-6)*T^(2)

% Heat evolved by fuel during combustion (kJ/h)

Q=GCV*Mfuel;

% Heat losses (kJ/h)

Qloss=X*Q;

Qt=Q-Qloss;

syms tf

CpCO2=43.2936+0.01147*tf-818558.5*tf^(-2);

CpH2O=34.42+6.281*10^(-4)*tf+5.611*10^(-6)*tf^2;

CpN2=27.2155+4.187*10^(-3)*tf;

CpO2=34.63+1.0802*10^(-3)*tf-785900*tf^(-2);

CpSO2=32.24+0.0222*tf-3.475*10^(-6)*tf^(2);

% Integration of mean molar heats

Cpm=int(XC*CpCO2+XH*CpH2O+XN*CpN2+XO*CpO2+XS*CpSO2);

H=Cpm-Qt/Mfluegas;

disp('Equation of actual flame temperature')

func=H

% Finding of first derivative of flame temperature equation.

disp('Finding of first derivative of flame temperature equation')

dfun=diff(func)

Appendix 3

% Solution of effective gas and flame temperature by

% Newton Raphson method

function root=newtraph(func,dfunc,xr,es,maxit)

% newtraph(func,dfunc,xguess,es,maxit):

% Uses Newton-Raphson method to find a function

% input:

% func=name of function

% dfunc=name of derivative of function

% xguess=initial guess

% es=(optional) stopping maximum allowable iterations

% output:

% root =real root

% if necessary, assign default values

if nargin<5, maxit=50; end % if maxit blank set to 50

if nargin<4, es=0.001; end % if es blank set to 0,001

% Newton-Raphson

iter=0;

while (1)

xrold=xr;

xr=xr-func(xr)/dfunc(xr);

iter=iter+1;

if xr~=0, ea=abs((xr-xrold)/xr)*100; end

if ea<=es|iter>=maxit, break, end

end

root=xr;

if root>1300

root=root+273;

fprintf('The actual flame temperature is %8.0f K\n',root)

else root=xr+273;

fprintf('The Effective gas temperature is%8.0f K\n',root)

end